Neural networks for tracking of unknown SISO

ISA Transactions 59 (2015) 363–374

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ISA Transactions journal homepage: www.elsevier.com/locate/isatrans

Neural networks for tracking of unknown SISO discrete-time nonlinear dynamic systems Muhammad Saleheen Aftab 1, Muhammad Shafiq n Department of Electrical & Computer Engineering, Sultan Qaboos University, Muscat, Oman

art ic l e i nf o

a b s t r a c t

Article history: Received 10 November 2014 Received in revised form 3 September 2015 Accepted 6 September 2015 Available online 9 October 2015

This article presents a Lyapunov function based neural network tracking (LNT) strategy for single-input, single-output (SISO) discrete-time nonlinear dynamic systems. The proposed LNT architecture is composed of two feedforward neural networks operating as controller and estimator. A Lyapunov function based back propagation learning algorithm is used for online adjustment of the controller and estimator parameters. The controller and estimator error convergence and closed-loop system stability analysis is performed by Lyapunov stability theory. Moreover, two simulation examples and one real-time experiment are investigated as case studies. The achieved results successfully validate the controller performance. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Lyapunov function Neural tracking Decentralized control Direct adaptive inverse control Indirect adaptive Inverse control Stable adaptive tracking

1. Introduction Adaptive tracking based on inverse dynamics control employs a controller whose dynamic behavior is exactly the inverse of that of the plant [1]. Therefore, a successful synthesis of inverse controller fundamentally depends upon the precise knowledge about the physical system at hand. However, most real world systems are harnessed with unknown nonlinearities, parametric and structural uncertainties and effects of various disturbances [2–4]. Consequently, development of an effective inverse controller, inevitably becomes a challenging task. This is where artificial neural networks (ANN) have gained immense popularity in past few decades. An artificial neural network is capable of approximating nonlinear functions through suitable learning mechanism. As a result, the application of neural networks in system identification and control has emerged as an active research topic [5–10]. Artificial neural networks have been effectively used as tracking controllers for unknown linear and nonlinear dynamic plants [11–13]. Numerous successful implementations have been reported in the literature that include applications in robotics, industrial and process control [14–21]. It has been shown that the neural networks can efficiently approximate inverse dynamics without requiring detailed knowledge of the plant [21,1]. However, the closed-loop stability and convergence of these systems are shown by using certainty n

Corresponding author. Tel.: þ 968 24142662. E-mail addresses: [email protected] (M.S. Aftab), mshafi[email protected] (M. Shafiq). 1 Tel.: þ968 24143775. http://dx.doi.org/10.1016/j.isatra.2015.09.003 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.

equivalence principle, computer simulation and experimental results [21]. In [22], an adaptive inverse control algorithm using Adaline neural network has been proposed. The tracking performance of this controller has been studied experimentally on an electro-hydraulic servo system. A multiple model approach is adopted in [23], in which a linear robust controller is used to insure stability, whereas a neural network controller handles the nonlinearities and uncertainties. In the multiple model technique, whenever the neural controller parameters start diverging, the linear controller takes over [23]. A combination of conventional nonlinear control techniques and the universal approximation capabilities of artificial neural networks has been proposed. For instance, in [24], Radial basis function (RBF) neural network is combined with sliding mode and backstepping control schemes and the resulting powerful algorithms have been tested on various mechanical systems. These schemes are costly and complex in the computation. A combination of higher-order neural networks and U-model based controllers is proposed to achieve adaptive tracking using computationally efficient techniques [25,26]. These algorithms are based on indirect adaptive control strategy, which is a consequence of the certainty equivalence principle. Although the performance of these adaptive tracking algorithms has been shown promising using computer simulations and experimental results but the convergence and stability of the complete closed-loop system using these algorithms is not proved. These techniques use conventional gradient descent backpropagation learning algorithm for network training [27–29]. In the gradient based learning techniques, the weights of the neural network are updated in the direction of the negative gradient of the error cost function to achieve the global minimum. However, the computations involved in the

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M.S. Aftab, M. Shafiq / ISA Transactions 59 (2015) 363–374

determination of the gradient and the vulnerability to frequent trapping at local minima make these algorithms considerably slow [30,28]. To solve this problem, variants of the original backpropagation algorithm have been proposed that include modified learning rates and use of momentum factors [31,32]. But in many cases, these modifications significantly increase the computational complexities [28,29]. This paper describes a gradient-free approach for neural network training. The proposed LNT control architecture uses two artificial neural networks with a typical single-hidden-layer structure. The weights update mechanism of the neural network controller and an estimator is based on the Lyapunov functions defined by the system's error dynamics. The analysis performed with the Lyapunov stability theory guarantees bounded error convergence and closed-loop stability. Two simulation examples are included to study the controller performance. The performance of the proposed algorithm has been compared with the conventional neuro-adaptive controller using computer simulations. The first case study demonstrates the application of LNT controller for position tracking of a single link robotic manipulator. The second example explains the effectiveness of the proposed controller under the effect of severe measurement noise acting on the plant output. In addition, a real time tracking experiment is successfully carried out on TQ MA3000 robotic manipulator in a decentralized control approach. The LNT control theory developed in this study is effectively validated through the analysis of the tracking response obtained in simulation and experiment case studies. The rest of this paper is organized as follows: the problem formulation is described in Section 2. The Lypunov function based neural network is discussed in Section 3. The adaptive tracking controller structure, its error convergence and stability analysis are described in Section 4. Section 5 is devoted to the comparative analysis and discussion of the computer simulation results. The methodology and results of the real-time tracking experiment are evaluated in Section 6. The paper is concluded in Section 7.

If (1) is not BIBO stable, then it can be stabilized with appropriate feedback mechanism. Assumption 2. : The system (1) is invertible i.e. there exists a dynamic function such that, 1 uðkÞ ¼ f ψ ðkÞ; rðk þ 1Þ Remark 2.1. : The proposed adaptive inverse control is applicable to stable (or stabilized) systems.

3. Lyapunov function based artificial neural networks Consider a typical single-hidden-layer (SHL) artificial neural network architecture, as shown in Fig. 1. This neural network is composed of m neurons in the input layer, n neurons in the hidden layer and one neuron in the output layer. The input neurons transmit the excitation input signals xN ðkÞ; xN ðk 1Þ; U U U xN ðk m þ 1Þ to the hidden neurons in a certain weighted proportion through parameters ð1Þ ð1Þ ð1Þ where wð1Þ ðkÞ wð1Þ ðkÞ ; w12 ðkÞ; U U U wji ðkÞ U U U wnm ðkÞ; 11

ji

represents the weight assigned to the connection between ith input and jth hidden neurons. Each hidden neuron performs a nonlinear activation function φ1 ð U Þ; φ2 ð U Þ; ::::::; φn ð UÞ on the cumulative weighted inputs. The outputs of the hidden neurons are then weighted ð2Þ ð1Þ ð2Þ through wð2Þ ðkÞ ; w12 ðkÞ; U U U w1j ðkÞ U U U w1n ðkÞ and sup11

plied to the output neuron for the summation and production of the neural network output uN ðkÞ: Note that wð2Þ 1j ðkÞ is the weight of the connection from jth hidden neuron to the output neuron. In order to minimize the error eN ðkÞ between the actual output uN ðkÞ and the desired response dN ðkÞ; the parameters of the neural network wð1Þ ji ðkÞ

and wð2Þ 1j ðkÞ are adjusted adaptively with the learning mechanism δN ðkÞ ¼ hN ðeN ðkÞ; dN ðkÞÞ using a Lyapunov function based error backpropagation learning algorithm. The output of the neural network is given by:

2. Problem formulation Consider a single-input single-output (SISO) discrete-time nonlinear system whose dynamic behavior is described by the following generalized difference Eq. (1): yðk þ 1Þ ¼ f ðψ ðkÞ; uðkÞÞ

Assumption 1. The system (1) is bounded-input bounded-output (BIBO) stable i.e. yðkÞ o 1 8 uðkÞ o 1

ð1Þ

with h iT ψ ðkÞ ¼ YðkÞ UðkÞ h i YðkÞ ¼ yðkÞ yðk 1Þ U U U U yðk lY þ1Þ h i UðkÞ ¼ uðk 1Þ uðk 2Þ U U U U yðk lU þ 1Þ where uðkÞ A ℜ and yðkÞ A ℜ are the system's input and output, respectively at the time instant k. f ð U Þ : ℜðlY þ lU Þ -ℜ is a smooth nonlinear function. The function f ð U Þ is assumed to have constant unknown parameters with unknown nonlinearities. The integers lY and lU are unknown and represent the order of YðkÞ and UðkÞ respectively. It is assumed that lU r lY to assure causality. The objective of this paper is to design a direct adaptive inverse controller for system (1), such that the tracking error eðkÞ between the system output yðkÞ and the reference trajectory rðkÞ asymptotically converges to a small value ε0 , in the neighborhood of zero, and a precise tracking performance is achieved, i.e. lim ‖yðkÞ rðkÞ‖-ε0 . k-1

Following two assumptions are necessary to guarantee stable tracking and the existence of system's inverse dynamics:

uN ðkÞ ¼

n X j¼1

wð2Þ 1j ðkÞSj ðkÞ

ð2Þ

where Sj ðkÞ is the output of the jth hidden neuron as given in (3): ! m X wð1Þ ðkÞx ðk iþ 1Þ ð3Þ Sj ðkÞ ¼ φj N ji i¼1

αðÞ

αðÞ

e where φj ðÞ ¼ tanhðαðÞÞ ¼ eeαðÞ and α is a positive constant. The þ e αðÞ error eN ðkÞ between the network output uN ðkÞ and the network desired response dN ðkÞ is given in (4):

eN ðkÞ ¼ uN ðkÞ dN ðkÞ

ð4Þ

By properly selecting a candidate Lyapunov function V N ðkÞ based on the network error dynamics and defining an appropriate weight update mechanism, the neural network stability and error convergence conditions can be established. These issues are thoroughly discussed in subsequent Theorem 1 and Lemma 1. ð2Þ Theorem 1. : If the weights wð1Þ ji ðkÞ and w1j ðkÞ of the neural network in Fig. 1 are updated with: ð2Þ w1j ðkÞ ¼

δN ðkÞ nSj ðk 1Þ

ð5Þ 0

1 wjið1Þ ðkÞ ¼ mxN ðk i þ1Þ

φ

1@ j

1

δN ðkÞ A

nwð2Þ 1j ðkÞ

ð6Þ


¼

n X j¼1

¼

365

0 wð2Þ 1j ðkÞ

φj @φ

0pffiffiffiffi 1@ j

11

βeN ðk 1Þ þ dN ðkÞAA nwð2Þ 1j ðkÞ

dN ðkÞ

qffiffiffiffi βeN ðk 1Þ þ dN ðkÞ dN ðkÞ eN ðkÞ ¼ ↓

qffiffiffiffi

βeN ðk 1Þ

ð8Þ

Since, 0 o β o1 the network error eN ðkÞ asymptotically converges to 0 as the time k increases.

Fig. 1. Typical single-hidden-layer artificial neural network architecture with ‘m’ input and ‘n’ hidden neurons.

pffiffiffiffi where δN ðkÞ ¼ hN ðeN ðkÞ; dN ðkÞÞ ¼ βeN ðk 1Þ þdN ðkÞ and 0 o β o 1 is the network learning parameter, then the stability of the neural network is guaranteed with the Lyapunov Stability Theory.

Proof. Let V N ðkÞ ¼ eN 2 ðkÞ be the candidate Lyapunov function of the neural network's error dynamics. It can be seen that V N ðkÞ ¼ 0 when eN ðkÞ ¼ 0 only and V N ðkÞ 4 0 for all other values of eN ðkÞ. Now the difference ΔV N ðkÞ can be written as:

ΔV N ðkÞ ¼ V N ðkÞ V N ðk 1Þ 2

¼ ðuN ðkÞ dN ðkÞÞ eN 2 ðk 1Þ 0 12 ! n m X X ð2Þ ð1Þ @ w1j ðkÞφj wji ðkÞxN ðk i þ 1Þ dN ðkÞA eN 2 ðk 1Þ ¼ i¼1

Putting for 0

ΔV N ðkÞ ¼ @

n X

j¼1

wð1Þ ji ðkÞ

from (6), we obtain: m X

wð2Þ 1j ðkÞφj

i¼1

1 φ 1 mxN ðk i þ 1Þ j

! 0pffiffiffiffi 1 βeN ðk 1Þ þ dN ðkÞA ðk i þ 1Þ x dN ðkÞÞ2 N nwð2Þ 1j ðkÞ

@

eN 2 ðk 1Þ 0 12 0 0pffiffiffiffi 11 n X @φ 1 @ βeN ðk 1Þ þ dN ðkÞAA dN ðkÞA ¼@ wð2Þ ðkÞ φ j j 1j nwð2Þ j¼1 1j ðkÞ eN 2 ðk 1Þ qffiffiffiffi 2 βeN ðk 1Þ þ dN ðkÞ dN ðkÞ eN 2 ðk 1Þ ¼ ΔV N ðkÞ ¼ ↓ ð1 βÞeN 2 ðk 1Þ

ð7Þ

Since, 0 o β o 1 therefore ΔV N ðkÞ o0: Hence, the stability of the neural network is guaranteed. Lemma 1. : Using the weight update mechanism described in Theorem 1, the following discrete-time error dynamics can be proven: qffiffiffiffi ð8Þ eN ðkÞ ¼ βeN ðk 1Þ

eN ðkÞ ¼ uN ðkÞ dN ðkÞ ¼

j¼1

¼

n X j¼1

An adaptive tracking control scheme can be implemented by using the Lyapunov function neural network in series with a nonlinear discrete-time SISO dynamic system, as shown in Fig. 2. The objective of this controller is to synthesize a control input that compels the plant output yðkÞ to track a desired reference smooth trajectory rðkÞ: The weights of the neural network controller are updated by that the error eN ðkÞ ¼ eðkÞ ¼ rðkÞ yðkÞ: At this point, it is essential to understand the difference between the two signals dN ðkÞ and rðkÞ; dN ðkÞ is the desired output of the neural network controller whereas rðkÞ is the desired output of the plant. Therefore, trajectory tracking cannot be achieved with dN ðkÞ ¼ rðkÞ: Instead, a mechanism is required to estimate an appropriate dN ðkÞ based on the actual output uN ðkÞ of the neural network controller so that yðkÞ is driven close to rðkÞ: In this context, we propose the Lyapunov function Neural network Tracking (LNT) control architecture, as depicted in Fig. 3. This scheme employs the Lypunov function neural network estimator (NNE) to generate the desired trajectory dN ðkÞ ¼ uE ðkÞ for the controller (NNC) to compute the control input uC ðkÞ ¼ uðkÞ: It must be noted that the NNE estimates the desired control effort by minimizing the error eE ðkÞ: On the other hand, the NNC learns the plant's inverse dynamics by minimizing the error eC ðkÞ; which is equal to the closed-loop error eðkÞ ¼ rðkÞ yðkÞ: It will be shown in the following sub-sections that the unique LNT control structure of Fig. 3 yields a direct relationship between the errors eE ðkÞ and eC ðkÞ that guarantees stable and asymptotic output tracking using the Lyapunov stability theory. 4.1. Error convergence analysis

Proof. Using (4): n X

Remark 3.2. : The excitation input signal xN ðkÞ has an effect on the convergence behavior of the tracking error. Eq. (6) clearly shows an inverse relationship between wð1Þ ji ðkÞ and xN ðkÞ: Although it is not possible to analytically find the exact numerical value, but it has been learnt through simulations that 0:1 o xN ðkÞ o1 usually gives better performance. The detailed discussion of this behavior is given in Section 5, Example 1.

4. Proposed adaptive tracking controller

¼ eN 2 ðkÞ eN 2 ðk 1Þ

j¼1

Remark 3.1. : The learning rate β has significant influence on the stability and convergence of the neural network output. As with the conventional gradient descent backpropagation algorithm [1,24], its smaller values (close to 0) slow down the error convergence; whereas larger values (close to 1) may result in an oscillatory convergence behavior.

ð2Þ w1j ðkÞ

φj 0

φ@

ð2Þ w1j ðkÞ

j

m X i¼1

! wð1Þ ji ðkÞxN ðk i þ1Þ

dN ðkÞ

0pffiffiffiffi 1 1 1 @ β eN ðk 1Þ þ dN ðkÞA φ mxN ðk iþ 1Þ j nwð2Þ ðkÞ i¼1 m X

xN ðk i þ 1ÞÞ dN ðkÞ

1j

Convergence of the error eN ðkÞ has been proved in Lemma 1. This means the neural networks weight learning is directional. In this section, a steady-state relationship between eC ðkÞ and eE ðkÞ is determined. Using the network equations of Section 3, we can write for the NNE as follows: ! n m X X ð1Þ wð2Þ ðkÞ φ w ðkÞx ðk i þ 1Þ uE ðkÞ ¼ E j E1j E ji j¼1

i¼1

366


Fig. 2. Adaptive inverse controller using Lyapunov function neural network.

Fig. 3. Proposed Lyapunov function neural networks based adaptive tracking controller (LNT).

¼

n X j¼1

0 @ wð2Þ E1j ðkÞφj

0pffiffiffiffiffiffi 1 1 1 β e ðk 1Þ þ uC ðkÞA xE ðk i þ 1ÞA φj 1 @ E E ð2Þ mxE ðk i þ 1Þ nw ðkÞ i¼1 m X

By canceling dynamics of the parameter learning process, following relationship is obtained: qffiffiffiffiffiffi ð9Þ uE ðkÞ ¼ β E eE ðk 1Þ þ uC ðkÞ qffiffiffiffiffiffi uE ðkÞ uC ðkÞ ¼ β E eE ðk 1Þ eE ðkÞ ¼

qffiffiffiffiffiffi βE eE ðk 1Þ

Similarly, for NNC: qffiffiffiffiffiffi uC ðkÞ ¼ βC eC ðk 1Þ þ uE ðkÞ uC ðkÞ uE ðkÞ ¼

ð10Þ

ð11Þ

Proof. Consider a discrete-time Lyapunov function defined by the error dynamics of NNE and NNC as follows: ð14Þ

Using (13) in (14):

ð12Þ

Comparing (10) and (12): qffiffiffiffiffiffi qffiffiffiffiffiffi βC eC ðk 1Þ ¼ βE eE ðk 1Þ

C

β V ðkÞ ¼ 1 þ E eE 2 ðkÞ βC

ð13Þ

where γ ¼ ββE C From (11) and (13), it can be concluded that by keeping βC 4 0 and 0 o β E o 1, asymptotic error convergence of NNE and NNC is guaranteed. However, the corresponding plant must satisfy Assumptions 1 and 2 expressed in Section 2.

ð15Þ

Now, the difference ΔVðkÞ ¼ VðkÞ Vðk 1Þ can be written as: β β ΔVðkÞ ¼ 1 þ E eE 2 ðkÞ 1 þ E eE 2 ðk 1Þ

βC βC βE 2 eE ðkÞ eE 2 ðk 1Þ ¼ 1þ βC

sffiffiffiffiffiffi

qffiffiffiffi

Theorem 2. : The LNT control architecture in Fig. 3 guarantees the closed-loop stability if β C 4 0 and 0 o β E o 1, provided Assumptions 1 and 2 are satisfied.

VðkÞ ¼ eE 2 ðkÞ þ ββE eE 2 ðkÞ

βC eC ðk 1Þ

βE e ðkÞ ¼ γ eE ðkÞ βC E

The closed-loop stability of the proposed control system can be guaranteed with the Lyapunov stability theory. Theorem 2 below establishes the necessary conditions for stability:

VðkÞ ¼ e2E ðkÞ þ e2C ðkÞ

qffiffiffiffiffiffi

qffiffiffiffiffiffi eE ðkÞ ¼ βC eC ðk 1Þ

eC ðkÞ ¼

4.2. Stability analysis

E1j

β 2 ¼ 1 þ E ðuE ðkÞ uC ðkÞÞ eE 2 ðk 1Þ

βC

But, from (9): uE ðkÞ ¼

qffiffiffiffiffiffi

βE eE ðk 1Þ þ uC ðkÞ


Fig. 4. Tracking performance with (a) the proposed LNT controller, and (b) the controller built with Eqs. (9) and (11).

367

Fig. 5. Effect of activation function on (a) the neural network weights, and (b) the closed-loop system output.

Therefore,

qffiffiffiffiffiffi βE βE eE ðk 1Þ þ uC ðkÞ uC ðkÞ βC β ¼ 1 þ E βE eE 2 ðk 1Þ eE 2 ðk 1Þ βC β ΔV ðkÞ ¼ 1 þ E 1 βE eE 2 ðk 1Þ βC

ΔV ðkÞ ¼ 1 þ

!

2 eE 2 ðk 1Þ

ð16Þ

By keeping βC 4 0 and 0 o β E o 1, ΔVðkÞ o 0 is assured. Therefore, the closed-loop system stability is guaranteed, provided the dynamic system fulfills Assumptions 1 and 2. Remark 4.1. : The preceding convergence and stability analysis describe the significance of the learning rates β C and β E : However, an analytical method to determine the optimal numerical values of these parameters is not known, but it has been observed in the numerical simulations that acceptable learning and error convergence behaviors are achieved by choosing t βC ¼ SYS ð17Þ ySYS where t SYS is the rise (or fall) time and ySYS is the DC gain obtained from the open-loop step response of the dynamic system (1). Furthermore, it has been found that selecting βE closer to 1 results in comparatively better tracking performance. Remark 4.2. : Eqs. (9) and (11) should not be misinterpreted as the replacement of (2). Since these equations have been derived by canceling the dynamics of the learning process, therefore they only represent the behavior of the neural network when the parameter learning becomes directional. To elaborate this point, a tracking control example of a 4th order linear system is considered here. The system dynamics are: GðsÞ ¼

s3 þ 10s2 þ 10s þ 5 s4 þ 15s3 þ 25s2 þ5s þ 2

ð18Þ

Note that system (18) fulfills Assumptions 1 and 2 described in Section 2. Further, we select β C ¼ 0:65 and β E ¼ 0:985: Fig. 4 compares the performances of the two controllers and it is evident that the proposed LNT controller in Fig. 4(a) gives a smooth output in the transient region as compared to the linear adaptive filter in Fig. 4(b) that has highly oscillatory response. Moreover, the

Fig. 6. Single link robotic arm response curves, (a) with open loop, (b) with feedback loop for stability.

tracking behavior of the two controllers becomes similar in the steady-state region, in accordance with the mathematical analysis performed in sub-Section 4.1. Additionally, we have also studied the effects of activation function on the network weights and the output of the closed-loop system. It has been observed that different values of α not only result in varying convergence behavior of the weights but also cause different output tracking patterns in the transient region. The corresponding results have been displayed in Fig. 5. These results clearly suggest that the output of the proposed Lyapunov function neural network has significant dependency on the activation function and weight parameters.

368


5. Simulation examples In this section, two simulation case studies are presented to evaluate the performance of the LNT controller. For comparative analysis, tracking performance achieved with the conventional gradient descent neural network (GDNN) controller has also been included. The following LNT control parameters are chosen for these case studies: m ¼ n ¼ 4 and xðkÞ ¼ 0:5: Example 1. : Consider a single link robotic arm [33], given by: y€ðtÞ þ λ1 ẏðtÞ þ λ2 sin ðyðtÞÞ ¼ λ3 uðtÞ

ð19Þ

where yðtÞ is the arm position (rad) and uðtÞ is the control torque (N m). The parameters λ1 , λ2 and λ3 depend upon the mass and length of the arm. For this simulation, the selected values are λ1 ¼ λ2 ¼ λ3 ¼ 1 [27]. Since the system dynamics are described in continuous time, uðtÞ and yðtÞ are discretized to uðkÞ and yðkÞ with

Fig. 7. Output vs. input curve showing same sign of input and output throughout the simulation time.

sampling time T ¼ 1 ms, for consistency with system definition in (1). The arm position is required to track a smooth sinusoidal trajectory: 2π k 2π k 2:5 sin ð20Þ rðkÞ ¼ 0:5 sin 20 100 In order to validate if the given dynamic system satisfies Assumptions 1 and 2, open-loop test has been performed and the results are shown in Fig. 6. Specifically, Fig. 6(a) portrays that the arm position remains stable in the vicinity of ‘0’, however, it destabilizes near ‘pi’ i.e. in the vicinity of ‘3’. The presence of sin ðyðtÞÞ in (19) explains this open-loop unstable behavior. Therefore, the system is stabilized by introducing output feedback with feedback gain K 4 1. Now, the new input to the system is unew ðkÞ ¼ λ3 ðuðkÞ KyðkÞÞ. Fig. 6(b) displays the post stabilization output response, and an stable output in both regions. In Fig. 7, the arm position is plotted against the input control torque. It is evident that the signs of uðkÞ and yðkÞ remain same throughout the simulation and consequently make the system invertible in the range of operation. Therefore, both Assumptions 1 and 2 have been satisfied and the LNT controller can now be implemented. The NNC learning rate is calculated using (17). In Fig. 6(b), the magnified region shows that t SYS 0:7 s and ySYS 0:25 rad. So β C ¼ 0:7 ¼ 2:8 is obtained. Moreover, we select β ¼ 0:985 and randomly E 0:25 initialize the neural network weights within [ 1, 1]. For the GDNN controller, η ¼ 0:001 is selected and the weights are randomly initialized. The tracking performance is displayed in Fig. 8. As shown in Fig. 8(a) and (b), a superior tracking in terms of stability and smoothness is achieved with the proposed LNT control scheme. The conventional GDNN controller gives undesirable oscillations of large amplitude initially due to its slow learning ability. Moreover, a few small oscillations are observed near the zero-crossing points. While a fast, smooth and oscillation-free tracking behavior is seen in the response of the LNT

Fig. 8. Performance comparison, (a) tracking with LNT controller, (b) tracking with GDNN controller, (c) LNT tracking error, and (d) GDNN tracking error.


Fig. 9. Control effort by (a) LNT controller and (b) GDNN controller.

Fig. 10. (a) NNC learning error, (b) NNE learning error, and (c) relationship between the errors of NNC and NNE.

Fig. 11. Effect of x(k) on controller performance.

Fig. 12. Open loop system response.

369

370


Fig. 13. Output vs. Input curve showing same sign of input and output throughout the simulation time.

controller. The closed-loop errors shown in Fig. 8(c) and (d) highlight the better control performance of the LNT controller. The steady-state tracking error of the LNT controller is about 10 times smaller than the GDNN controller, ignoring the high amplitude oscillatory period between 0 and 20 s. A similar situation is depicted in Fig. 9(a), where the LNT controller produces a smooth and bounded control torque. In contrast, the control signal of the GDNN controller is accompanied with unwanted oscillations, as shown in Fig. 9(b). It is also observable in Fig. 9(a) that the estimated and actual control torques produced by the NNE and NNC, respectively, are almost identical that is complemented by a small NNE learning error displayed in Fig. 10(b). Moreover, its learning pattern is similar to that of the NNC learning error plotted in Fig. 10(a). In fact, the mathematical relationship between p the two errors is effectively ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi validated in Fig. 10(c). Note that γ ¼ 0:985=2:8 ¼ 0:593.

Fig. 14. Performance comparison, (a) tracking with LNT controller, (b) tracking with GDNN controller, (c) LNT tracking error, and (d) GDNN tracking error.

Fig. 15. Control effort by (a) LNT controller and (b) GDNN controller.


371

Fig. 16. (a) NNC learning error, (b) NNE learning error, and (c) relationship between the errors of NNC and NNE.

Table 1 Specifications of TQ MA3000 robotic manipulator. Angular position

θWR ðtÞ

Definition

Waist Shoulder roll angle pitch angle

Type of axis Maximum rotation Maximum slew rate End effector

Major 7 1351 22:51/s

θSP ðtÞ

θEP ðtÞ

θRP ðtÞ

θRR ðtÞ

Elbow pitch angle

Wrist pitch angle Minor

Wrist roll angle

451/s

901/s

Pneumatic type; can carry up to 2 kg with 0.75 m dead-lift from the waist.

Example 2. : Consider the nonlinear discrete time system [34]:

ρðk þ 1Þ ¼ sin ρðkÞ þ uðkÞ 5 þ cos ρðkÞuðkÞ ð21aÞ yðkÞ ¼ ρðkÞ þ ζ ðkÞ Fig. 17. Sketch of TQ MA3000 5-DOF robotic manipulator.

Fig. 11 demonstrates the effect of changing xðkÞ on the behavior of the convergence of tracking error, measured as the ratio σ ¼ jeðkÞj1 jrðkÞj1 : Where j U j1 denotes the infinity norm of a signal. A small value of σ means a better performance of the controller. In Fig. 11, large values of σ are the consequence of xðkÞ o 0:1, which is categorized as a poor tracking performance. In contrast, small variations are observed in σ for xðkÞ 4 1: This phenomenon causes slow convergence of the tracking error. The range 0:1 o xðkÞ o 1 gives a comparatively better tracking performance in the sense of smooth learning and fast oscillation-free tracking.

ð21bÞ

where uðkÞ is the input, yðkÞ is the output and ζ ðkÞ is the random measurement noise. The amplitude of this random noise is bounded, ζ ðkÞ r 0:1. The output yðkÞ is required to track a smooth sinusoidal reference signal rðkÞ given in (22): 2π k 2π k þ 0:5 sin ð22Þ rðkÞ ¼ 1:5 sin 100 20 The open loop response is plotted in Fig. 12 when a unit step input is applied. Since, bounded input produces a bounded output, the system is BIBO stable. Moreover, Fig. 13 portrays that there is no sign change between uðkÞ and yðkÞ throughout the simulation. As a result, the dynamic system (21) is invertible within the range of operation.

372


Fig. 18. Decentralized LNT controller for major joints position tracking of MA3000 robotic manipulator.

The magnified portion in Fig. 12 depicts that t SY S 0:0006 s and ySYS 3:65: With these values, β C ¼ 0:0006 3:65 ¼ 0:00016 is obtained. Further, we select βE ¼ 0:75 and randomly initialize the neural network weights. The learning rate of the GDNN controller is chosen η ¼ 0:0001: The tracking results in the presence of sensor noise are shown in Fig. 14(a) and (b). It is evident that both controllers are affected by the noise, yet the tracking achieved with the LNT controller is remarkable, see Fig. 14(a). The LNT controller learns the plant's inverse dynamics quickly. On the other hand, Fig. 14(b) illustrates slow learning of the GDNN controller, with initial oscillations and large steady-state error. The error plots of Fig. 14(c) and (d) suggest that the GDNN controller exhibits very large tracking error compared to the LNT controller. At this point, it should be noted that usually, the noisy output is passed through a low pass filter before using it for the feedback or identification. However, in this example, we have not used the filtered output for feedback in order to study the effect of noise on the smoothness of the control signal. Fig. 15(a) and (b) demonstrate minimal effect of noise on the control signals that are bounded and fairly smooth. However, the control effort of the GDNN controller is not good enough to achieve desired tracking, as compared to the LNT controller. Fig. 15(a) also demonstrates that the difference between the outputs of the NNC and NNE is negligible, since uC ðkÞ closely follows uE ðkÞ: The learning errors of NCC and NNE are depicted in Fig. 16 (a) and (b) respectively. It is evident that both NNC and NNE have similar learning behaviors. Moreover, Fig. 16(c) effectively validates the p mathematical ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirelationship between eC ðkÞ and eE ðkÞ: Note that γ ¼ 0:75=0:00016 ¼ 68:5. The tracking control achieved in this example portrays the efficient performance of the proposed controller in the presence of the measurement noise.

6. Real time tracking experiment In this section, we consider a real-time application of the proposed LNT controller. A decentralized scheme for position tracking

of the major joints of TQ MA3000 robotic manipulator has been implemented. 6.1. TQ MA3000 robotic arm TQ MA3000 is a 5-DOF industrial robotic system, developed by TecQuipment Limited. As shown in Fig. 17, this robot has three major and two minor axes of rotation. The angular positions associated with these axes and their corresponding characteristics are given in Table 1. The DC motors connected at the joints of this robotic system are manufactured by Bodine Electric Company. These motors are characterized by low speed and high torque throughput, with very high gear-ratio. Brush type analog PWM servo amplifiers (Model 30A8C manufactured by Advanced Motion Controls) are used to drive the joint DC motors. In addition to driving the system, the servo amplifier protects the DC motors against short-circuits, over-heating, over-current and over-voltage problems. The joint positions of the robotic arm are measured with conductive plastic potentiometers. 6.2. PC interface and calibration In this experiment, the major axes joint positions i.e. θWR ðtÞ, θSP ðtÞ and θEP ðtÞ are controlled by implementing a decentralized tracking scheme. For real-time computation, a standard IBM PC with Intel Pentium IV processor and 3 GB RAM is used. The controller is programmed in Simulink real-time target environment and data acquisition is performed through Humusoft MF624 PCI interface card. The operating voltage range for input and output channels of the data acquisition card is limited between 7 10 V. For calibration, the output voltages and the corresponding joint positions for various input voltages are recorded and a linear voltage–angle relationship is established. The waist position θWR ðtÞ varies at 40°/V, the shoulder position θSP ðtÞ varies at 20°/V and the elbow position θEP ðtÞ varies at 30°/V. These calibration values are used to convert the system response from volts to degrees.


373

Fig. 19. Position tracking, control effort and closed-loop error of (a)–(c) elbow joint, (d)–(f) shoulder joint, and (g)–(i) waist joint.

6.3. Decentralized control architecture Multiple degree-of-freedom robotic manipulators have highly complex and coupled nonlinear dynamics. Here we present a decentralized control scheme based on the proposed LNT controller for position tracking. In this scheme, three separate LNT controllers are employed to control the elbow, shoulder and waist joints, as depicted in Fig. 18. These LNT controllers learn the inverse dynamics of the robotic system and produce an appropriate control command uEP ðkÞ, uSP ðkÞ and uWR ðkÞ such that the joint position tracking errors eEP ðkÞ, eSP ðkÞ and eWR ðkÞ are minimized. Since the joint positions of the robotic manipulator θEP ðkÞ, θSP ðkÞ and θWR ðkÞ are only locally stable, the system is stabilized via feedback gains K EP , K SP and K WR . The discrete time (DT) closed loop inputs to the robotic system ΔEP ðkÞ ¼ uEP ðkÞ θEP ðkÞ, ΔSP ðkÞ ¼

uSP ðkÞ θSP ðkÞ and ΔWR ðkÞ ¼ uWR ðkÞ θWR ðkÞ are converted into continuous time (CT) signals ΔEP ðtÞ, ΔSP ðtÞ and ΔWR ðtÞ by digitalto-analog converter on the data acquisition card. The robotic system produces position outputs as CT signals θEP ðtÞ, θSP ðtÞ and θWR ðtÞ which are converted into DT signals θEP ðkÞ, θSP ðkÞ and θWR ðkÞ by analog-to-digital converter for feedback. 6.4. Experiment results For experiments, following values of learning parameters are selected: for elbow LNT controller, β C ¼ 0:75 and β E ¼ 0:35; for shoulder LNT controller, βC ¼ 0:3 and β E ¼ 0:1; and for waist LNT controller βC ¼ 0:225 and β E ¼ 0:1. Moreover, for feedback stabilization, K EP ¼ K SP ¼ K WR ¼ 10 and for controller excitation inputs, xEP ¼ xSP ¼ xWR ¼ 0:25 have been selected. The desired signals for

374

joints position tracking are given in (23): π k r EP ðkÞ ¼ 60 sin 250 2π k r SP ðkÞ ¼ 30sin 50 þ 1801 π k r WR ðkÞ ¼ 80 sin 250


ð23Þ

Fig. 19 shows the experiment results for tracking of the major joint positions of the TQ MA3000 robotic manipulator. From Fig. 19 (a), (d) and (g), it is evident that smooth and stable tracking is achieved as the actual position closely follows the desired position profiles. Initially, a few ocsillations are observed but the response improves as the time progresses and the controllers learn the inverse dynamics. Moreover, the potential effects of joint couplings have been effectively neutralized by the proposed decentralized control scheme. As shown in Fig. (b), (e) and (h), the LNT controllers produce stable and bounded control effort. Fig. 19(c), (f) and (i) validates experimentally the relationship between eC ðkÞ and eE ðkÞ established in (13). Note that γ EP ¼ 0:683, γ SP ¼ 0:577 and γ WR ¼ 0:444, for the chosen values of the LNT learning parameters. It can be inferred that the closed loop error eC ðkÞ is indeed equivalent to the amplified estimator error eE ðkÞ.

7. Conclusions In this article, we have presented an adaptive inverse tracking controller for unknown SISO dynamic systems. The proposed scheme utilizes two Lyapunov function neural networks operating as controller and estimator. It has been proved mathematically that there exists a direct relationship between the controller error and the estimator error. This interesting result is used to establish the closed-loop system stability, based on the Lyapunov stability theory. The tracking behaviors of two nonlinear plants are studied using the LNT controller with computer simulations. The comparative study of these results with that of the GDNN is discussed. Further, a real-time tracking experiment is conducted on TQ MA3000 5-DOF robotic system. The analysis of these simulation and experiment studies effectively validates the theoretical implications of the proposed LNT control scheme. Contributions of the paper

1. Lyapunov function based neural network is developed for the adaptive tracking of Nonlinear plants. 2. The detailed convergence analysis of the algorithm is discussed in this paper. 3. The theoretical findings are verified using computer simulations. 4. The algorithm is successfully implemented on MA 3000 robotic arm for the position tracking of the end effector. Acknowledgments The authors acknowledge the support of Sultan Qaboos University, Oman for this research. Appreciations are due to the editor-in-chief and the anonymous reviewers for their valuable comments and suggestions to enhance the quality of this article.

References [1] Widrow B, Walach E. Adaptive inverse control: a signal processing approach, reissue edition. New Jersey: John Wiley & Sons; 2008. [2] Fateh MM, Khorashadizadeh S. Robust control of electrically driven robots by adaptive fuzzy estimation of uncertainty. Nonlinear Dyn 2012;69(3):1465–77.

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Modeling and adaptive control for supercapacitor in automotive applications based on artificial neural networks. Electr Power Syst Res 2014;106(0):134–41. http://dx.doi.org/10.1016/j. epsr.2013.08.016. [16] Horng JH. Neural adaptive tracking control of a DC motor. Inf. Sci. 1999;118(1– 4):1–13. http://dx.doi.org/10.1016/S0020-0255(99)00045-6. [17] Kittisupakorn P, Thitiyasook P, Hussain MA, Daosud W. Neural network based model predictive control for a steel pickling process. J. Process Control 2009;19 (4):579–90. http://dx.doi.org/10.1016/j.jprocont.2008.09.003. [18] Mohareri O, Dhaouadi R, Rad AB. Indirect adaptive tracking control of a nonholonomic mobile robot via neural networks. Neurocomputing 2012;88 (0):54–66. http://dx.doi.org/10.1016/j.neucom.2011.06.035. [19] Peng J, Dubay R. Identification and adaptive neural network control of a DC motor system with dead-zone characteristics. ISA Trans 2011;50(4):588–98. [20] Alvarez E, Riverol C, Navaza JM. Control of chemical processes using neural networks: implementation in a plant for xylose production. ISA Trans 1999;38 (4):375–82. [21] Cabrera JBD, Narendra KS. Issues in the application of neural networks for tracking based on inverse control. IEEE Trans Autom Control 1999;44 (11):2007–27. http://dx.doi.org/10.1109/9.802910. [22] Yao J, Wang X, Hu S, Fu W. Adaline neural network-based adaptive inverse control for an electro-hydraulic servo system. J Vib Control 2011;17(13):2007– 14. [23] Chen L, Narendra KS. Nonlinear adaptive control using neural networks and multiple models. Automatica 2001;37(8):1245–55. [24] Liu J. Radial basis function (RBF) neural network control for mechanical systems, 10. New York: AMC. New York, Springer; 2013. [25] Shafiq M, Butt NR. Utilizing higher-order neural networks in U-model based controllers for stable nonlinear plants. Int J Control Autom Syst 2011;9 (3):489–96. [26] Zhu Q, Wang Y, Zhao D, Li S, Billings SA. Review of rational (total) nonlinear dynamic system modelling, identification, and control. Int J Syst Sci 2013;46 (12):1–12. http://dx.doi.org/10.1080/00207721.2013.849774. [27] Rumelhart DE, McClelland JL, Group PDPR. Parallel distributed processing: explorations in the microstructure of cognition, 1 and 2. Cambridge, MA: MIT Press; 1986. [28] Hykin S. Neural networks: a comprehensive foundation. New Jersey: PrinticeHall, Inc.; 1999. [29] Gupta M, Jin L, Homma N. Static and dynamic neural networks: from fundamentals to advanced theory. New Jersey: John Wiley & Sons; 2004. [30] Du K-L, Swami MNS. Neural networks and statistical learning. London: Springer; 2014. [31] Huang CY, Chen LH, Chen YL, Chang FM. Evaluating the process of a genetic algorithm to improve the back-propagation network: a Monte Carlo study. Expert Syst Appl 2009;36:1459–65. [32] Örkcü HH, Bal H. Comparing performances of backpropagation and genetic algorithms in the data classification. 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International Journal of Control, Automation, and Systems (2011) 9(3):489-496 DOI 10.1007/s12555-011-0308-y

http://www.springer.com/12555

Utilizing Higher-Order Neural Networks in U-model Based Controllers for Stable Nonlinear Plants Muhammed Shafiq and Naveed R. Butt Abstract: The use of intelligent control schemes in nonlinear model based control (NMBC) has gained widespread popularity. Neural networks, in particular, have been used extensively to model the dynamics of nonlinear plants. However, in most cases, these models do not lend themselves to easy maneuvering for controller design. Therefore, a common need is being felt to develop intelligent control strategies that lead to computationally simple control laws. To address this issue, we recently proposed a U-model based controller utilizing nonlinear adaptive filters. The present work extends that concept further to include higher-order neural networks (HONN) for better approximation. The main feature of the proposed structure is its ability to capture higher-order nonlinear properties of the input pattern space while allowing the synthesis of a simple control law. The effectiveness of the proposed scheme is demonstrated through application to various nonlinear models and a comparison with the Backstepping controller is presented. Keywords: Adaptive tracking, higher order neural networks, IMC, U-model.

1. INTRODUCTION The primary objective of controller design is to influence the behavior of systems. A typical control objective is tracking which involves forcing system states to follow prescribed trajectories. Some of the main difficulties to be overcome by the designer are the nonlinear plant dynamics and the uncertainties caused by differences between actual and assumed dynamic models. This has led to an increased interest in developing intelligent control systems. Neural networks (NNs) form an important part of intelligent control and have the ability to model a large variety of nonlinear dynamic plants. Today, NNs are being widely used in a number of nonlinear control applications including model predictive control (MPC), internal model control (IMC) and adaptive control. Detailed discussions on the variety of available NNs and their application to control problems can be found in [1] and [2]. The principal topologies of neural networks that are currently most relevant for applications in power electronics have been reviewed including the detailed description of their properties in [3]. Robust smooth-trajectory control of nonlinear servo systems based on neural networks has been discussed in [4]. Among a vast variety of available NNs, higher-order neural networks (HONNs) are particulary suited for capturing the higher-order nonlinear properties of the __________ Manuscript received May 7, 2008; revised July 12, 2010; accepted January 29, 2011. Recommended by Editor Hyun Seok Yang. We acknowledge support of Sulatan Qaboos University, Muscat, Oman for this research work. Muhammed Shafiq is with the ECE, SQU, Muscat, Oman (email: [email protected]). Naveed R. Butt is with the Dept. of Electrical Engineering, Karlstad University, SE-651 88 Karlstad, Sweden (e-mail: naveed. [email protected]). © ICROS, KIEE and Springer 2011

input pattern space (see [1,5-10]). A novel onlinemodeling scheme for the switched reluctance motor using a B-spline neural network is developed in [11]. Sensorless control of single switch-based switched reluctance motor drive using neural network is discussed in [12]. These have been proved to have good computational, storage, pattern recognition, and learning properties and are realizable in hardware [9]. It must be noted that even though HONNs have excellent approximation capabilities, yet they fulfil only one criterion for a good model-based control scheme. Another equally important criterion for the utility of any modelling framework is its suitability for control law design. For instance, NARMAX (nonlinear autoregressive moving average with exogenous inputs) the most commonly used nonlinear modelling structure has the ability to represent a broad range of nonlinear systems [13]. However because of its overly complex structure, it does not lend itself to easy manoeuvring for controller design [14]. This shortcoming present in nonlinear model-based control strategies (including those utilizing NN models) has been clearly pointed out by a number of researchers (such as [15,16]). There is, therefore, an increasing interest in developing intelligent control strategies that lead to simplistic control laws. The present study attempts to address this issue by combining the approximation power of HONNs with the control-oriented nature of the U-model to achieve adaptive tracking of stable nonlinear dynamic plants. The present study attempts to address this issue by combining the approximation power of HONNs with the controloriented nature of the U-model to achieve adaptive tracking of stable nonlinear dynamic plants. The recently introduced U-model [15] has a more general appeal as compared to the polynomial NARMAX model [17] and the Hammerstein model. Based on the U-model a pole

Muhammed Shafiq and Naveed R. Butt

490

placement controller [15] and a new IMC (Internal Model Control) structure for dynamic nonlinear plants with known parameters [18] have recently been proposed. We previously proposed a U-model based controller utilizing nonlinear adaptive filters [19] and in this paper we extend that work to include HONNs for improved approximation. The main advantages of the proposed approach are its generality and simplistic control law. The proposed structure is depicted in the block diagram Fig. 1, where the dynamic nonlinear plant is modelled as a HONN whose U-model equivalence is used to synthesize a simple law for control inputs to the plant (this is discussed in detail in Section 3). The rest of the paper has been organized as follows. Section 2 presents a brief introduction to HONNs, the IMC, and the U-Model. An adaptive IMC structure based on HONNs (with U-model equivalence) is proposed in Section 3. The effectiveness of the proposed scheme is verified through simulations. These results are discussed in Section 4. Comparison with adaptive back stepping method is given in Section 5. The paper concludes in Section 6. 2. PRELIMINARIES

Fig. 2. A Higher-order neural networks.

2.1. Higher-order neural networks - HONNs Higher-order Neural Networks (HONNs) are particulary suited for capturing the higher-order nonlinear properties of the input pattern space. Higher-Order Neural Units (HONUs) are the basic building block of the HONNs as shown in Fig. 2. For a typical HONN the output is given by yM = φ ( z ),

(1)

n

n

i1

i1 ,i2

z = ω0 + ∑ωi1 xi1 + ∑ωi1i2 xi1 xi2 + " +

n

∑

i1 ,",ٛiN

be considered as a network with n inputs and a single output. This structure can be treated as a two-layered neural network and is capable of dealing with the problems of functional approximation and pattern recognition (see [1] for details). To accomplish an approximation task for given inputoutput data x(t), yM(t), the learning algorithm for the HONN can be easily developed on the basis of the gradient descent method. Let the error function be formulated as

(2)

ωi1" iN xi1 " xiN ,

where x = [ x1 , x2 ," xn ]T is the vector of neural inputs, yM is an output, and φ (⋅) is an activation function. It is possible to select φ (⋅) as a linear function [1]. If the order of HONN is high enough, then, as is known from Stone-Weiestrass theorem, (1) and (2) may

E (t ) =

1 [ y(t ) − yM (t )]2 , 2

(3)

where y(t) is the desired output and yM(t) is the output of the neural network at time t. Minimization of the error function by a standard steepest-descent algorithm yields the following set of learning equations.

ω0new = ω0old + η ( y − yM )φ( z ), ω new = ω old + η ( y − y )φ( z )v i1" i j

i1"i j

M

i1"i j

(4)

,

(5)

dφ ( z ) and vi1" i j denote the higherwhere φ( z ) = dz order correlation terms of n-dimensional input as: vi1 = xi1 ,

(6)

vi1i2 = xi1 xi2 ,

(7)

# vi1i2 "iN = xi1 xi2 " xiN 1 ≤ i1 , i2 ," , iN ≤ n.

Fig. 1. Proposed structure block diagram.

(8)

2.2. Internal model control: IMC IMC is one of the most popular control strategies used

Utilizing Higher-Order Neural Networks in U-model Based Controllers for Stable Nonlinear Plants

491

case of IMC presented in Fig. 3, the control signal u (t −1) can be obtained easily by using Newton-Raphson algorithm recursively, K

∑ α j (t )uij (t − 1) − U (t ) ui +1 (t − 1) = ui (t − 1) −

Fig. 3. IMC block diagram. in industrial process control. Its main features are its simple structure, fine disturbance rejection capabilities and robustness [20-25]. IMC can be used for both linear and nonlinear systems [26] and is especially suitable for the design and implementation of open-loop stable systems. Many industrial processes happen to be intrinsically open-loop stable. Fig. 3 depicts the basic structure of the IMC. Here the plant is represented by fP (·), the model by fM (·) and the controller by fC (·). The symbols r(t), u(t) and d(t) represent the reference input, the control signal and the additive noise respectively. A detailed analysis of the properties of IMC has been given by [27]. 2.3. The U-model The control-oriented U-model plays a central role in the adaptive scheme proposed in this paper. Following is the development of the U-model based on [15] (see also, [18,28]). Consider single-input single-output (SISO) nonlinear dynamic plant with an NARMAX representation of the form: ٛ y (t ) = f [ y (t − 1)," , y (t − n), u (t − 1)," u (1 − n), e(t )," , e(t − n)],

(9)

where y(t) and u(t) are the output and input signals of the plant respectively at discrete time instant t, n is the order of the plant, f (·) is a nonlinear function and e(t) represents the error due to measurement noise, model mismatch, uncertain dynamics and plant variations. The Umodel is obtained by expanding the nonlinear function f (·) of the above equation as a polynomial with respect to u (t −1) as follows: K

j =0

, ⎡K ⎤ j d ⎢ ∑ α j (t )ui (t − 1) ⎥ / du (t − 1) ⎣⎢ j =0 ⎦⎥ (12) where the subscript i is the iteration index. In (11), U(t) is an input signal to the inverse system (12). The control input u (t −1) is synthesized based on the signal U(t) using (12). The control input u (t −1) synthesized by (12) is responsible to force the plant output yP(t) to track the arbitrary continuous signal r(t). Further, U(t) = r(t) − ε(t) as shown in Figs. 1 and 3. Thus the problem of finding the inverse of the plant has been effectively reduced to an iterative root-solving procedure.

3. THE HONN CONTROLLER

We now discuss the proposed HONN control structure in details. The scheme assumes a stable nonlinear dynamic plant whose functional parameters or the functional structure need not be known. If the plant happens to be unstable then it is first stabilized using simple feedback. 3.1. Control input synthesis based on identified model We propose to identify the plant online using the HONN depicted in Fig. 2 and given by (1) and (2), as discussed in Section 2.1. If the order of HONN is high enough (1) and (2) may be considered as a network with n inputs and a single output. This structure can be treated as a two-layered Neural Network and is capable of dealing with the problems of functional approximation and pattern recognition. The input vector x can be composed of combinations of previous inputs and outputs and the exact format can be selected to best suit the plant at hand. Consider φ ( z ) = z and x = [u (t − 1), u (t − 2), " , u (t − n)]T, then (2) can be rewritten in the U-model format (10) by selecting the αj (t) parameters as N

y (t ) = ∑α j (t )u (t − 1) + e(t ), j

(10)

α0 = ω0 + ∑ωi1 u(t − i1 )ٛ i1 = 2

j =0

N

where K is the highest degree of the model input u (t −1), αj (t) is a function of past inputs and outputs u (t −2), " , u (t − n), y (t −1), " , y (t − n) and errors e (t − 1), " , e (t − n). To apply linear control system design methodologies to the nonlinear model a further transformation is applied as follows: y (t ) = U (t ),

(11)

N

The expression (11) is defined as the U-model. For the

N

+∑∑

N

∑ ωi i i u(t − i1 )u(t − i2 )u(t − i3 ) + ",

i1 = 2i2 =i1 i3 =i2

α1 = ω1 + +

K

j =0

(13)

i1 = 2i2 =i1

where U ( t ) = Φ[u (t − 1)] + e(t ) = ∑α j (t )u j (t − 1) + e(t ).

N

+ ∑ ∑ωi1i2 u(t − i1 )u (t − i2 )

123

N

∑ω1i u (t − i2 )

i2 = 2

N

2

N

∑ ∑ ω1i i u (t − i2 )u (t − i3 ) + ",

i2 = 2i3 =i2

23

(14)

#

α N = ω1"1.

(15)


492

Now, the HONN based identified plant can be written in the format of U-model as N

yM (t ) = ∑α j (t )u j (t − 1).

(16)

j =0

∑ α j (t )uij (t − 1) − U (t ) j =0

. N  d  ∑ α j (t )uij (t − 1)  / du (t − 1)  j =0  (17)

3.2. Control input synthesis for a third order HONN For instance, if a third-order HONN is selected (N = 3) with three inputs ( x = [ x1 , x2 , x3 ]T ), the NN model can be written as yM = ω0 + ω1 x1 + ω2 x2 + ω3 x3 + ω11 x12 + ω12 x1 x2 + ω13 x1 x3 + ω22 x22 + ω23 x2 x3 + ω113 x12 x3 + ω122 x1 x22 + ω123 x1 x2 x3

(18)

+ ω133 x1 x32 + ω222 x23 + ω223 x22 x3 + ω233 x2 x32 + ω333 x33 .

We note that the number of inputs n and the network order N are user parameters and that the present development can be readily extended for various selections of these parameters. The weights can be updated online using update (4) and (5). The main idea is to utilize the power of HONNs to capture the higher-order nonlinear properties of the input pattern space (the input to the HONN in this case consists of past inputs and outputs of the nonlinear plant). To facilitate the synthesis of a simplistic control law, the NN model of (2) is converted into its equivalent Umodel. The equivalent U-model will clearly vary according to the selection of the HONN order and according to the definition of the input vector x. For instance, if (18) is selected as the model in Fig. 2 and the neural input vector is select as x = [ x1 , x2 , x3 ]T = [u (t − 1), u (t − 2), u (t − 3)]T ,

where u(t) represents the input to the plant at discrete time t, then the equivalent U-model for (18) can be written as yM = α 0 (t ) + α1 (t )u (t − 1) + α 2 (t )u 2 (t − 1) + α 3 (t )u 3 (t − 1),

α1 (t ) = ω0 + ω12u (t − 2) + ω13u(t − 3) + ω122u 2 (t − 2) α 2 (t ) = ω1 + ω112u (t − 2) + ω113u (t − 3), α 3 (t ) = ω111

N

+ ω33 x32 + ω111 x13 + ω112 x12 x2

+ω233u (t − 2)u 2 (t − 3) + ω333u 3 (t − 3),

+ ω123u (t − 2)u(t − 3) + ω133u 2 (t − 3),

Therefore, the control input can be synthesized by

ui +1 (t − 1) = ui (t − 1) −

+ω223u 2 (t − 2)u (t − 3)

3.3. System operation There are two problems that may occur while using the Newton-Raphson root solving algorithm. First, the denominator in (17) may tend to zero near the solution. This happens at the local\global extremes of the function. This problem can be avoided by introducing ui +1 (t − 1) = N

+ ω22u 2 (t − 2) + ω23u (t − 2)u (t − 3) 2

3

+ ω33u (t − 3) + ω 222u (t − 2)

(20)

∑ α j (t )uij (t − 1) − U (t ) j =0

→ 0. N  j d  ∑ α j (t )ui (t − 1)  / du (t − 1)  j =0  δ is a small number in the direction of the solution. Second, the polynomial formed by (16) may not have a real solution. Selecting the order of HONN odd, always assure a real solution of the U-model. To solve these problems an improved computation for the NewtonRaphson algorithm has been suggested by [29]. Furthermore, stability of the Newton-Raphson algorithm is improved by introducing the following modification, ui (t − 1) + δ , when

ui +1 (t − 1) = ui (t − 1)ٛ N

−µ

α 0 (t ) = ω0 + ω2u (t − 2) + ω3u (t − 3)

(22) (23)

with this structure, the control inputs u (t −1) to the unknown nonlinear plant of Fig. 1 can be easily obtained using the Newton-Raphson polynomial root-solving algorithm given by (12). It must be noted that the proposed scheme leads to a very simple and general control law. This approach is therefore expected to prove extremely useful in the area of nonlinear control. As shown in Fig. 1, the output of the controller u(t) is fed to both the unknown plant and the HONN model. The mismatch error ε(t) input to the filter is the difference between the output y(t) and the output of the HONN model yM (t). The network parameters are updated using (4) and (5) such that the error ε(t) is minimized. A copy of the HONN parameters (which are also the parameters of the equivalent U-model) is fed to the controller online and the controller calculates the inverse of the unknown plant using the Newton-Raphson method based on the Umodel. If the plant to be controlled is unstable then it is first stabilized using standard feedback control algorithms.

(19)

where

(21)

∑ α j (t )uij (t − 1) − U (t ) j =0

  d  ∑ α j (t )uij (t − 1)  / du (t − 1)  j =0  N

,

(24)

where 0 < µ < 1. Small values of the parameter µ introduce robustness in sense of convergence on the cost of


slow convergence rate. With these adjustments, the proposed scheme presents a quite general framework for adaptive tracking of nonlinear dynamic plants. 3.4. Closed-loop stability Let us consider the IMC structure given in Fig. 3. fM (·) is HONN model of the plant fP (·). Using Fig. 3, the control input can be written directly as u (t ) = fC (r (t ) − ε (t )) = fC (U (t )).

(25)

If fC (·)is a time delay inverse of the fM (·) then fM ( fC (·)) = q−L where L is a positive integer and q−1 represents a back shift operator. Consider there is a mismatch between fM (·) and fP (·), then we assume f P ( fC (⋅)) = (1 + Δ)q − L . Now, using Fig. 3, we may write y (t ) = (1 + Δ )q − LU (t ) + d (t ),

The plant is identified using HONN shown in Fig. 2. This identified model is converted to the polynomial format given by (19). The control input u(t) is synthesized by U-model control law (17). An arbitrary reference input is selected and a sampling period of 10ms is used. For the update equations, a step size of 0.3 is used and all weights are initialized randomly between 0 and 1. The simulation results are depicted in figures 4 to 6. Fig. 4 shows that HONN identifies the plant quickly. The tracking performance is depicted in Fig. 5. Fig. 6 shows that the control signal is bonded and smooth. The results suggest that the proposed control law synthesizes a smooth control signal to accomplish the tracking.

(26)

−L

yM (t ) = q U (t ),

(27)

U (t ) = r (t ) − ε (t ),

(28)

−L

ε (t ) = Δq U (t ) + d (t ).

(29)

Using (28) and (29), we obtain U (t ) =

r (t ) (1 + Δq

−L

−

)

d (t ) (1 + Δq − L )

(30)

.

Putting the value of U(t) in (26) and rearranging the terms, we obtain y (t ) =

1+ Δ (1 + Δq

−L

)

r (t − L) −

1 − q−L (1 + Δq − L )

d (t ).

Fig. 4. Hammerstein model: identification error.

(31)

If Δ < 1 then the system represented by (31) will be stable. It is important to note that DC gain of the transfer 1+ Δ function is one. This assures the stability (1 + Δq − L ) and tracking of the reference signals. Further, the distur1 − q−L bance transfer function assures the decoup(1 + Δq − L ) ling of step disturbances.

Fig. 5. Hammerstein model: tracking.

4. APPLICATION TO NONLINEAR PLANTS To demonstrate the application of the proposed scheme, simulations were carried out for the adaptive tracking of the Hammerstein model and a DC motor model. All programs were run using the Simulink platform. This section presents the results of these simulations. 4.1. Adaptive tracking of Hammerstein model For the purpose of simulations the following Hammerstein model was used to serve as the unknown plant y (t ) = 0.5 y (t − 1) + x(t − 1) + 0.1x(t − 2), 2

2

x(t ) = 1 + u (t ) − u (t ) + 0.2u (t ).

(32) (33)

493

Fig. 6. Hammerstein model: control signal.


494

4.2. Adaptive tracking of DC motor speed The discrete time model [30] used for the simulations of DC motor speed tracking is given by:

ωr (n + 1) = K1ωr (n) + K 2ωr (n − 1) + K3 [ sign{ωr (n)}] ωr2 (n) + K 4 [ sign{ωr (n − 1)}] ωr2 (n − 1)

(34)

+ K5 va (n),

where va(n) is the time-varying motor terminal voltage and ωr (n) is the motor speed. K1, K2, K3, K4 and K5 are constants expressed in terms of motor parameters and are

given as: K1 =

2( La J + Ra J + La D − Ra DTs + K 2Ts ) , La J + 2 Ra J + 2 La D

K2 = −

La J , La J + 2 Ra J + 2 La D

K3 =

2ξ ( La + RaTs ) , La J + 2 Ra J + 2 La D

K4 =

2ξ La , La J + 2 Ra J + 2 La D

K5 =

2 KTs , La J + 2 Ra J + 2 La D

where Ts = sampling period in seconds Ra = armature resistance in ohms D = viscous constant in N.m.s K = torque constant in N.m/A ξ = load torque constant in N.m.s2 La = armature inductance in H J = rotor inertia in kg.m2. The motor model was identified as HONN, then HONN was converted to the polynomial format (19). The resulting plots are given in Figs. 7, 8 and 9. Fig. 7. Plant-model output mismatch.

5. COMPARISON WITH NONLINEAR ADAPTIVE BACKSTEPPING CONTROLLER The proposed intelligent controller is also compared with the most ”systematic” design approach available in nonlinear adaptive control, namely: backstepping [31]. The idea of backstepping is briefly explained with the help of a simple example. Consider the nonlinear plant y = u + ϕ ( y )T θ ,

(35)

where φ(·) and θ represent the nonlinearity vector and the parameter vector, respectively. The nonlinear adaptive backstepping controller for output tracking can be given as (with yd as the desired output trajectory) Fig. 8. DC motor speed tracking.

u = c( yd − y ) + ϕ ( yd − y )T θˆ

(36)

with parameter estimator

θˆ = Γϕ ( yd − y )( yd − y ).

(37)

Here c is a positive constant while Γ represents a positive definite symmetric matrix referred to as the “adaptation gain”. Note that although this controller adaptively estimates the parameter vector θ, it does require the exact knowledge of the nonlinearity function φ(·). On the other hand, the proposed intelligent controller performs the adaptive tracking of such a nonlinear plant without any knowledge of the nonlinearity function or the parameter vector. The nonlinear plant used for the comparison is Fig. 9. Magnified view of tracking.

y = u − 0.1log ( y + 0.1 ) − 0.1 y 2 .

(38)


[1] [2]

[3]

[4] Fig. 10. Tracking: backstepping controller.

[5]

[6]

[7] [8]

Fig. 11. Proposed controller. The system was run in SIMULINK with a sampling time of 1 ms. The desired trajectory was set as a slowly time varying arbitrary signal. The results are shown in Figs. 10 and 11. The mean square error (MSE) obtained by the backstepping controller was 0.145 while that achieved by the proposed intelligent controller was 2.13. The MSE in the proposed controller is higher due to the mismatch in the learning phase. Once the plant is learned the tracking error of the proposed controller is lesser than that of the backstepping controller. It is clear that the proposed scheme is capable of tracking the plant without requiring the knowledge of the nonlinearity at the cost of a slightly higher mean square error.

[9] [10] [11]

[12]

[13]

6. CONCLUSIONS A novel technique, based on the IMC structure is presented for the adaptive tracking of stable nonlinear dynamic plants. The proposed scheme combines the approximation power of HONNs with the control oriented nature of the U-model to form a technique that is general and leads to a simplistic control law. The proposed scheme was applied to the adaptive tracking of the nonlinear Hammerstein model output and DC motor speed. A comparison with the Backstepping controller was also presented.

[14]

[15]

[16]

495

REFERENCES M. M. Gupta, L. Jin, and N. Homma, Static And Dynamic Neural Networks, John Wiley and Sons, Inc., Hoboken, NJ, 2003. M. Norgaard, O. Ravn, N. K. Poulsen, and L. K. Hansen, Neural Networks for Modelling and Contol of Dynamic Systems, A Practitioner’s Handbook, Springer-Verlag, London, 2000. B. K. Bose, “Neural network applications in power electronics and motor drivesan introduction and perspective,” IEEE Trans. on Industrial Electronics, vol. 54, pp. 14-33, 2007. X. L. Wei, J. Wang, and Z. X. Yang, “Robust smooth-trajectory control of nonlinear servo systems based on neural networks,” IEEE Trans. on Industrial Electronics, vol. 54, pp. 208-217, 2007. A. Y. Alanis, E. N. Sanchez, and A. G. Loukianov, “Discrete-time adaptive backstepping nonlinear control via high-order neural networks,” IEEE Trans. on Neural Networks, vol. 18, pp. 1185-1195, 2007. D. E. Rumelhart and J. L. McClelland, Parallel Distributed Processing: Explorations in the Microstructure of Cognition: Foundations, vol. 1, MIT Press, Cambridge, MA, 1986. C. L. Giles and T. Maxwell, “Learning invariance and generalization in higher-order networks,” Applied Optics, vol. 26, pp. 4972-4978, 1987. X. Xu, E. Oja, and C. Y. Suen, “Modified hebbian learning for curve and surface fitting,” Neural Networks, vol. 5, no. 3, pp. 441-457, 1992. J. G. Taylor and S. Commbes, “Learning higherorder correlations,” Neural Networks, vol. 6, no. 3, pp. 423-428, 1993. N. Homma and M. M. Gupta, “A general secondorder neural unit,” Bull. Coll. Med. Sci., Tohoku Univ., vol. 11, no. 1, pp. 1-6, 2002. Z. Lin, D. S. Reay, B. W. Williams, and X. He, “Online modeling for switched reluctance motors using B-spline neural networks,” IEEE Trans. on Industrial Electronics, vol. 54, pp. 3317-3322, 2007. C. A. Hudson, N. S. Lobo, and R. Krishnan, “Sensorless control of single switch-based switched reluctance motor drive using neural network,” IEEE Trans. on Industrial Electronics, vol. 55, pp. 321329, 2008. I. J. Leontarities and S. A. Billings, “Input-output parametric models for nonlinear systems. part I: Deterministic nonlinear systems; part II: Stochastic nonlinear systems,” Int. J. of Control, vol. 41, no. 2, pp. 303-344, 1985. L. Piroddi and W. Spinelli, “An identification algorithm for polynomial NARX models based on simulation error minimization,” Int. J. of Control, vol. 76, no. 17, pp. 1767-1781, 2003. Q. M. Zhu and L. Z. Guo, “A pole placement controller for nonlinear dynamic plants,” J. Systems and Control Engineering, vol. 216, no. 1, pp. 467476, 2002. R. K. Pearson, “Selecting nonlinear model struc-

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tures for computer control,” Journal of Process Control, vol. 13, pp. 1-26, 2003. K. R. Sales and S. A. Billings, “Self-tuning control of nonlinear ARMAX models,” Int. J. of Control, vol. 51, no. 4, pp. 753-769, January 1990. M. Shafiq and M. Haseebuddin, “Internal model control for nonlinear dynamic plants using UModel,” Proc. of the 12th Mediterranean Conference on Control and Automation, Turkey, June 2004. N. R. Butt and M. Shafiq, “Adaptive tracking of non-linear dynamic plants, using the u-model,” IMechE Journal of Systems and Control Engineering, Proc. IMechE, Part I., vol. 220, no. 6, pp. 473-487, 2006. A. Datta and J. Ochoa, “Adaptive internal model control: Design and stability analysis,” Automatica, vol. 32, no. 2, pp. 261-266, 1996. C. E. Garcia and M. Morari, “Internal model control.1.a unifying review and some new results,” Ind. Eng. Chem. Process Des. Dev, vol. 21, pp. 308-323, 1982. S. C. Patwardhan and K. P. Madhavan, “Nonlinear internal model control using quadratic perturbation models,” Computers and Chemical Engineering, vol. 22, pp. 587-601, 1998. Q. G. Wang, B. Qiang, and Y. Zhang, “Partial internal model control.” IEEE Trans. on Industrial Electronics, vol. 48, no. 5, pp. 976-982, 2001. W. F. Xie and A. B. Rad, “Fuzzy adaptive internal model control.” IEEE Trans. on Industrial Electronics, vol. 47, no. 1, pp. 193-202, 2000. M. Shafiq and S. H. Riyaz, “Internal model control structure using adaptive inverse control strategy,” Proc. of the 4th Int. Conf. on Control and Automation, p. 59, 2003. C. G. Economou, M. Morari, and B. O. Palsson, “Internal model control. 5. extension to nonlinear systems,” Ind. Eng. Chem. Process Des. Dev., vol. 25, no. 1, pp. 403-411, 1986. M. Morari and E. Zafiriou, Robust Process Control. Prentice Hall, NJ, 1989. N. R. Butt, M. Shafiq, and T. Khan, “An adaptive root-solving controller for tracking of nonlinear dynamic plants,” Proc. of International Conference on

Industrial Electronics and Control Applications, Ecuador, vol. 1, pp. 144-149, November 2005. [29] Q. M. Zhu, K. Warwick, and J. L. Douce, “Adaptive general predictive controller for nonlinear systems,” IEE Proc. Control Theory Applic., vol. 138, no. 1, pp. 33-40, 1991. [30] A. Rubaai and R. Kotaru, “Online identification and control of a dc motor using learning adaptation of neural networks,” IEEE Trans. on Industry Applications, vol. 36, no. 3, pp. 935-942, May/June 2000. [31] M. Krstic, I. Kanellakopoulos, and P. Kokotovic, Nonlinear and Adaptive Control Design, John Wiley & Sons, New York, 1995. Muhammad Shafiq received his BE degree in Electronic Engineering from NED University, Karachi, Pakistan in 1989. He completed his PhD studies in information and computer engineering at Chiba University, Chiba, Japan in 1997. He worked as technical manager at Thalus, Sauadai Arabai. He served King Fahd University of Petroleum and Minerals as faculty member. He worked as a professor under higher education commission at GIKI and IIUI, Pakistan. Presently, he is working as faculty member in ECE, SQU, Oman.He worked with Pakistan Engineering Council as a member of accreditation teams formed for accreditating several electrical, electronic and mechatronics engineering programs in Pakistan. He was member of national curriculum development committee for communication systems under HEC, Pakistan. He has supervised serveral MS and PhD thesis. He is among the authors of more than eight journal and conference papers. His research interests include applied nonlinear control, adaptive control, and soft computing techniques applied to control systems. Naveed R. Butt received his B.S. degree in Lasers and Optoelectronics from the GIK Institute, Pakistan, in 2002. He received his Master’s degree in Systems Engineering (Automation and Control) in 2006 from KFUPM, Saudi Arabia. He is currently working towards a Ph.D. degree with Lund University, Sweden. He was previously a Research Assistant with KFUPM in 2004. His research interests include: statistical signal processing, system identification, and intelligent systems.

Electr Eng (2010) 91:405–415 DOI 10.1007/s00202-010-0149-3

ORIGINAL PAPER

U-model based learning feedforward control of MIMO nonlinear systems S. Saad Azhar Ali · Fouad M. Al-Sunni · Muhammad Shafiq · Jamil M. Bakhashwain

Received: 11 August 2008 / Accepted: 12 February 2010 / Published online: 8 March 2010 © Springer-Verlag 2010

Abstract In this paper, a learning feedforward controller (LFFC) using the U-model is proposed for a better tracking control of multivariable nonlinear systems over a finite time interval. The multivariable system is modelled using the U-model and the LFFC is established using Newton–Raphson method. U-model significantly simplifies the online synthesis of the feedforward control law. The proposed technique is verified on 2-link robot manipulator in real-time. The performance of the proposed U-model based LFFC is compared with a number of schemes under varying load conditions. Keywords MIMO nonlinear systems · LFFC · U-model · RBFNN · Robotics

S. S. A. Ali (B) Faculty of Engineering, Sciences and Technology, Iqra University, Karachi, Pakistan e-mail: [email protected] F. M. Al-Sunni Systems Engineering Department, King Fahd University of Petroleum and Minerals, 31261 Dhahran, Saudi Arabia e-mail: [email protected] M. Shafiq Department of Electrical and Computer Engineering, Sultan Qaboos University, Muscat, Oman e-mail: [email protected] J. M. Bakhashwain Electrical Engineering Department, King Fahd University of Petroleum and Minerals, 31261 Dhahran, Saudi Arabia e-mail: [email protected]

1 Introduction Feedback control is often applied to improve the dynamical behavior of electromechanical systems, attempting to achieve both high performance and robust stability. This usually results in a trade off between the two properties. A feedback controller that has a high performance often does not feature a robust stability. Small variations in process parameters may destabilize the system. This problem can be overcome by creating separate means for obtaining high performance and robust stability using an additional learning feedforward controller PI , Fig. 1 [1]. By adding a (learning) feedforward component to the feedback controller PC , an extra degree of freedom in the design of the controller is created. The feedforward part is intended to generate steering signals that make the output of the process Y (t) follow the reference R(t). This control concept is known as Learning feedforward Control (LFFC) [1,2]. LFFC can be regarded as a variant of ‘Iterative Learning Control’ (ILC) [3,4]. A learning feedforward controller (LFFC) using the Bartlet window function is proposed in [5] for a better tracking control of linear system over a finite time interval. LFFC is applied as a feedforward controller to the existing feedback controller. In [1], the robustness and easy design of a PD-feedback controller was combined with a learning control strategy for improved tracking performance. The learning feedforward controller was designed using the B-spline networks (BSN). A parsimonious (reduced dimensionality) LFFC has been applied to a linear motor motion system in [6]. The BSN-based LFFC is applied to linear motion control in [7]. YangQuan et al. [8] presents a frequency-domain analysis and design approach for LFFC using a dilated B-spline network. In [9], a support vector machine is proposed as the learning mechanism.

123

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Electr Eng (2010) 91:405–415

PI R(t)

-

E(t)

PC

2 Modelling framework and problem formulation

Uff(t-1)

+

U(t-1)

P

Y(t)

Ufb(t-1)

Fig. 1 Learning feedforward control scheme

In order to design an accurate feedforward controller, the process dynamics need to be known. The feedforward controller has to compensate for the process dynamics and disturbances. These requirements call for plant inverse. For a complex nonlinear or non-minimum phase plant, inverse finding poses theoretical and implementation difficulties. Several model based inversion techniques such as based on NARMAX model [10] or Hammerstein model [11] lead to complex nonlinear algebraic equations. However, the U-model proposed by [12] is a control oriented model which comes up with an explicit controller design methodology on approximate inverse concept. U-model has a more general appeal as compared to the polynomial NARMAX model and the Hammerstein model. U-model is polynomial in the control input U (t − 1) and the parameters of this polynomial are function of U (t − 2), . . . , U (t − m) and Y (t − 1), . . . , Y (t − n), where U (t) and Y (t) represent the plant inputs and outputs, respectively. In numerical analysis, the inverse of a polynomial can easily be estimated using any numerical technique such as the Newton–Raphson method. U-model has been used in the internal model control scheme for single and multivariable nonlinear systems [13,14]. Adaptive tracking of multivariable bilinear systems is also achieved using a simplified U-model based IMC scheme [15]. In this paper, we propose a novel LFFC design approach. MIMO nonlinear dynamic plant is identified online using the radial basis functions neural networks (RBFNN) in U-model structure. Then the inverse of the model is obtained by Newton–Raphson root solving technique. This inverse is embedded in the nonlinear IMC scheme. This approach gives same design procedure for both invertible and non-invertible nonlinear systems. The rest of the paper is organized as follows. The problem is stated in Sect. 2. The U-model structure is briefed in Sect. 3 along with the necessary background. Section 4 presents the proposed MIMO U-model LFFC scheme and the Newton– Raphson based controller. The stability analysis is also presented in Sect. 3. Simulation results are presented in Sect. 4 while the real-time experiment details and the results are presented in Sect. 5. Finally, the contributions are concluded in Sect. 6.

123

Most systems encountered in the real world are non-linear to some extent and in many practical applications non-linear models may be required to achieve an acceptable prediction accuracy. The NARMAX model provides a unified representation for a wide class of discrete-time non-linear systems. In a NARMAX description, the system is modelled in terms of a non-linear functional expansion of lagged inputs, outputs and prediction errors. Consider the NARMAX representation of a MIMO nonlinear plant as [16], Y (t) = F (Y (t − 1), . . . , Y (t − n), U (t − 1), . . . , U (t − m), E(t − 1), . . . , E(t − l)) ,

(1)

where Y (t) and U (t) are the output and input signals of the plant respectively at a discrete time instant t, E(t) represents the prediction errors compensating for measurement noise, model mismatch, uncertain dynamics and plant variation. F() is the non-linear functional of the inputs, outputs and errors. It can be observed that the NARMAX representation depends on n + m + l terms with equal number of parameters to be found out to establish an optimal model. Moreover, the controller design based on inversion of the NARMAX model involves complicated equations leading to cumbersome calculations. Similarly, an adaptive controller based on NARMAX representation of the system requires several update equations for all the parameters involved in the NARMAX model. The shortcoming of increasingly complex NARMAX model can be reduced by reducing the number of parameters. This lays the foundation for U-model. The basic idea behind U-model is to emphasize the role of the control signal, i.e. u(t − 1). The model is structured on a polynomial in u(t − 1). The effect of the rest of the terms are impounded in a single separate term. 2.1 The U-model structure Now we can remodel the NARMAX representation into a U-model structure. Consider a single-input single-output plant in MARMAX representation given by, y(t) = f (y(t − 1), . . . , y(t − n), u(t − 1), . . . , u(t − m), e(t − 1), . . . , e(t − l)),

(2)

The U-model is obtained by expanding the non-linear function of the above equation as a polynomial with respect to current control signal u(t − 1) as follows: ym (t) =

M

α j (t)u j (t − 1) + e(t),

(3)

j=0

where M is the degree of model input u(t − 1), α j (t) is a function of past inputs and outputs u(t − 2), . . . , u(t − m),

Electr Eng (2010) 91:405–415

407

y(t − 1), . . . , y(t − n) and disturbance errors e(t), . . . , e(t − l). It is important to bring up that U-model can be used to model the class of systems that can be represented by NARMAX representation. It is also well known that NARMAX model can represent a large class of nonlinear systems [17]. Once a system is represented by NARMAX representation, we can easily remodel the NARMAX representation into U-model. Note that plant delay has been assumed to be one. Without loosing generality the proposed control procedure is applicable for arbitrary known plant delay as well. Hence, the plant delay need to be known. 2.2 Problem statement The problem can now be stated as: given a MIMO nonlinear system with NARMAX representation as in Eq. 1, the objective is to synthesize the control input U (t − 1) such that Y (t) tracks an arbitrary piece-wise continuous trajectory R(t). Further, it is considered that the plant parameters are unknown and slowly time varying.

3 Proposed U-model based LFFC The proposed U-model based LFFC for multivariable nonlinear system is shown in Fig. 2. The multivariable nonlinear plant is modelled in U-model format online and the inverse is the established using the Newton–Raphson Method. R(t) is the arbitrary reference trajectory. U (t − 1) is the control input, Y (t) and Ym (t) are plant and model outputs. 3.1 MIMO U-model

1

2

Ym (t) = A0 (t)+ A1 (t) U (t − 1)+ A2 (t) U (t − 1)+. . . , M

j

output vectors. U is the vector with jth power of the control inputs u i (t − 1) as, j

j

j

U (t − 1) = [u 1 (t − 1) u 2 (t − 1) . . . u pj (t − 1)]T ,

j

A j (t) U (t − 1) = F (U (t − 1)) .

(4)

(5)

and A j (t) are matrices instead of scalars α j . For a system with unknown parameters, the matrices A j are estimated online using gradient descent adaptive algorithm. In this work, normalized least mean squares (nLMS) principle [18] is adopted for the update of the parameters. However, the matrix A0 is modelled using radial basis functions neural networks (RBFNN). The reason for incorporating the RBFNN is to assist the nonlinear modelling. Neural networks are capable of learning models of backlash, saturation, and deadzone. These nonlinearities may be present in cascade with the actual nonlinear system, particularly in actuators and sensors. The effect of these unknown nonlinearities can be compensated using the neural networks such as multilayered feedforward neural networks (MFNN)and RBFNN. The A0 (t) is estimated using the MIMO RBFFN output as, A0 (t) = Y (t) = W(t),

(6)

where W and (t) are the weights and the basis function, respectively (for details, see [19]). The weights of the RBFNN and the rest of the parameters A j are estimated online, and updated using the normalized least mean square (nLMS) principle. The weight update equations for the weights W and the A j are W(t + 1) = W(t) + µ(t)Err (t)(t)T ,

The SISO U-model is extended for multivariable systems as,

or Ym (t) =

The model output Ym (t) is a function of the current control signal U (t − 1), where U (t − 1) and Ym (t) are the input and

(7)

j

A j (t + 1) = A j (t) + µ(t)Err (t) U (t − 1)T ,

(8)

where µ(t) is the nLMS learning rate. The Stability and convergence analysis of the proposed MIMO U-model using the small gain theorem is presented in [20].

j=0

3.2 The Newton–Rapshon based controller P

I U-model Inverse

R(t)

-

E(t)

PC Feedback Controller

Uff(t-1)

+ Ufb(t-1)

U(t-1)

Y(t)

P Plant

The feedforward controller is established if the controllermodel cascade results in unity gain. This implies that if the controller is excited by the signal Ym (t), the control signal U (t − 1) will force the model and so the plant to track Ym (t). Therefore, reference tracking can be achieved by setting R(t) = Ym (t) = F (U (t − 1))

Pm

Ym(t)

U-model

-

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Fig. 2 U-model based learning feedforward control scheme

F (U (t − 1)) = R(t).

(9) (10)

Equation 10 is system of multivariable nonlinear equations. This system of equations can be solved by any recursive nonlinear equations solver, such as the Newton–Raphson method [21]. Starting from an initial approximate solution,

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for instance Uk (t −1), a better solution Uk+1 (t −1) is sought with the correction vector H = [h 1 . . . h n ] such that,

3.4 Stability

Uk+1 (t − 1) = Uk (t − 1) + H,

In LFFC structure, the plant is stabilized using a feedback controller. If a stable feedback controller stabilizes the plant, then the over all closed-loop system will remain stable if the control signal generated by the feedforward controller is bounded [23]. In the U-model based adaptive LFFC structure the controller has two parts. (a) The estimated U-model and (b) inverse of U-model. To ensure the stability of the controller, it is necessary to show that the estimated U-model converge to the true stable (or stabilized) plant. The convergence of the Umodel relies on the selection of a suitable learning rate that guarantees robust and faster convergence speeds in the presence of noise and load variations. Therefore, it is necessary to analyze the conditions on the learning rate in the adaptation algorithm that guarantees overall stable behavior. In the following subsection, the adaptation algorithm is discussed and is associated to a feedback structure as presented in time domain analysis of adaptive algorithms in [22]. The stability of the feedback structure will in return guarantee the overall stability of the U-model.

(11)

which satisfies F (Uk+1 (t − 1)) = F (Uk (t − 1) + H ) = R(t).

(12)

Now expanding the nonlinear function F (Uk (t − 1) + H ) using the Taylor series expansion and considering only the linear first order terms [21], F (Uk (t − 1) + H ) ≈ F (Uk (t − 1)) + F (Uk (t − 1)) H. (13) The term F (Uk (t − 1)) is the J acobian matrix with elements ∂ f i /∂u k j (t − 1), corresponding to the jth input and ith output. Using Eq. 12 in Eq. 13, the value of the correction vector H can be obtained as, H = F (Uk (t − 1))−1 (R(t) − F (Uk (t − 1)))

(14)

or H = F (Uk (t − 1))−1 (R(t) − Ym (t)))

(15)

Hence, the Newton–Raphson solution for the controller will be, Uk+1 (t −1) =Uk (t − 1)+ F (Uk (t −1))−1 (E(t)−Ym (t)). (16) Remark The Newton–Raphson solution is conditioned with the existence of the inverse of the J acobian in Eq. 16. It is possible during the update process to have a singular J acobian matrix. This situation can be avoided using one of the following techniques: 1. Employing pseudoinverse, 2. or using the inverse of J acobian matrix from the previous instant, 3. or adding a small number to the J acobian matrix to avoid singularity. 3.3 Algorithm summary The proposed algorithm can be implemented as follows: 1. Measure plant output Y (t) and compute model output Ym (t) using Eq. 4. 2. Calculate mismatch Err (t), 3. Update the weights W(t) and A1 (t) using Eq. 7 and Eq. 8, 4. Synthesize control move U (t − 1) using the updated values of W(t) and A1 (t) in Eq. 16, 5. Go back to step 1.

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3.5 Adaptation algorithm and error quantities Consider a SISO plant having a set of input vectors {u(t)} with the corresponding desired set of output vectors {y(t)} and assuming the plant can be represented as a U-model of form given by Eq. 3. For convenience in future analysis, we express the U-model in vector notation as, y(t) = A(t)U (t − 1) + v(t),

(17)

where A(t) = [α0 (t) α1 (t) . . . α M (t)] and U (t − 1) = [ 1 u(t − 1) u 2 (t − 1) . . . u M (t − 1)]T . The noisy perturbations v(t) in the desired signal can be due to model mismatch or to measurement noise. Therefore, the vector A(t) can be regarded as the true parameter vector to be identified. This converts the problem of identifying the plant into finding the estimates of the U-model parameters α j (t), for 0 ≤ j ≤ M, such that ˆ (t − 1), ym (t) = A(t)U

(18)

ˆ is an estimate of A(t) at time instant t. where the vector A(t) ˆ Starting with an initial guess A(0), the parameters are updated recursively based on the least mean square (LMS) principle as, ˆ + 1) = A(t) ˆ + µ(t)e(t)U T (t − 1) A(t

(19)

where µ(t) is the learning rate and the error e(t) is defined as, e(t) = y(t) − ym (t) + v(t) ˆ (t − 1) + v(t) e(t) = A(t)U (t − 1) − A(t)U

(20)

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Defining a priori and a posteriori error quantities as ˜ ea (t) = A(t)U (t − 1) ˜ ep (t) = A(t + 1)U (t − 1)

q-1

(21)

˜ where A(t) is the parameter error vector symbolizing the difference between the actual parameter and its estimate as ˜ = A(t) − A(t). ˆ A(t) Therefore, ˆ ea (t) = A(t) − A(t) U (t − 1),

~ A(t + 1)

A˜ (t )

(22)

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η (t )ea (t )

ˆ = A(t)U (t − 1) − A(t)U (t − 1) = A(t)U (t − 1) − ym (t).

(23)

and the parameter error update equation satisfies the following recursion, ˜ + 1) = A(t) ˜ + µ(t)e(t)U T (t − 1). A(t

(24)

1−

µ (t ) y '(τ ) η (t )

Fig. 3 A lossless mapping in feedback structure for the learning algorithm of U-model

Note that the Eq. 23 depicts how far is the U-model output from a model with true parameter vector A(t). Contractive Mapping: In this continuation, we will develop a contractive mapping from the tth instant to t + 1th instant of the recursion.

Hence, the recursive parameter update Eq. 19 can be written as ˆ + 1) = A(t) ˆ + η(t) ea (t) − ep (t) U T (t − 1). A(t

Definition A linear map that transforms x to y, as y = T [x], is said to be contractive mapping, if ∀x we have T [x]2 ≤ x2 .

Similarly, the parameter error recursive Eq. 24 can be reformulated as, ˜ = A(t) ˜ − η(t)U T (t − 1) ea (t) − ep (t) (30) A(t)

This means the output energy does not exceed the input energy. The contractive mapping will relate the energies in such a way that the energy at t + 1th instant of the recursion becomes the same as the energy at tth instant, i.e., output energy =1 input energy

(25)

More specifically, the Euclidean norm of the parameter error vector and the a priori estimation errors at the t + 1th instant is compared with the Euclidean norms of the parameter error vectors and the a posteriori estimation errors at the tth instant. Now using Eq. 24, the a posteriori error in Eq. 22 becomes,

2 ˜ + 1)2 = A(t) ˜ A(t − 2η(t)ea2 (t)+η(t)ea2 (t)+η(t)ep2 (t)

˜ − µ(t)U T (t − 1)e(t)]U (t − 1) = [ A(t) (26)

Introducing a parameter η(t) as 1 η(t) = U (t − 1)2

(27)

Therefore, Eq. 26 becomes, µ(t) e(t) η(t) η(t)ep (t) = η(t)ea (t) − µ(t)e(t) η(t) ea (t) − ep (t) = µ(t)e(t).

ep (t) = ea (t) −

1 2 1 ea (t) − 2η(t)2 ea (t)ep (t) η(t) η(t) 1 2 e (t) + η(t)2 η(t) p 2 ˜ − 2η(t)ea2 (t) + 2η(t)ea (t)ep (t) = A(t) + η(t)2

+ η(t)ea2 (t) − 2η(t)ea (t)ep (t) + η(t)ep2 (t)

˜ + 1)U (t − 1) ep (t) = A(t = ea (t) − µ(t)U (t − 1)2 e(t).

To obtain the energy relation, the squared norm of Eq. 30 is found out as follows, 2 ˜ ˜ ˜ + 1)2 = A(t) −2η(t) A(t)U (t − 1) ea (t)−ep (t) A(t 2 + η(t)2 U (t − 1)2 ea (t) − ep (t) 2 ˜ = A(t) − 2η(t)ea (t) ea (t) − ep (t)

(28)

(29)

2 ˜ ˜ + 1)2 + η(t)ea2 (t) = A(t) + η(t)ep2 (t) A(t

(31)

˜ + 1)2 + η(t)ea2 (t) A(t = 1. 2 + η(t)e2 (t) ˜ A(t) p

(32)

Hence, the energy ratio given in Eq. 32 holds for all possible choices of the learning rate. √ This implies that the mapping ˜ ˜ + T i from the signals { A(t), η(t)ep (t)} to the signals { A(t √ 1), η(t)ea (t)} is lossless. The lossless mapping T i can be represented as a feedback structure shown in Fig. 3. (y (τ ) is the derivative of the output y(t) at the instant τ ). Now, if

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we further apply the mean-value theorem to the output of the U-model y(t), we can write ˆ A(t)U (t − 1) − A(t)U (t − 1) = y (τ )ea (t)

(33)

for some point τ along the segment connecting A(t)U (t − 1) ˆ (t − 1) during estimation procedure. Therefore, and A(t)U combining Eq. 20 and Eq. 28, µ(t) e(t) ep (t) = ea (t) − η(t) µ(t) A(t)U (t − 1) ep (t) = ea (t) − η(t) ˜ (t − 1) + v(t) − A(t)U µ(t) y (τ )ea (t) + v(t) ep (t) = ea (t) − η(t) µ(t) µ(t) y (τ ) ea (t) − v(t) ep (t) = 1 − η(t) η(t) µ(t) − η(t)ep (t) = √ v(t) η(t) µ(t) y (τ ) η(t)ea (t) − 1− η(t)

0 < µ(t)y (τ ) < 2η(t) =

2 u(t)2

(36)

In [22], the authors have presented a number of choices for learning rate. They based the selection of learning rate on the availability of the derivative function y (τ ). For the case of U-model, it is straight forward to obtain the estimate of the derivative function as, (34)

This relation shows that the overall √ mapping from the original (weighted) disturbances η(t)v(t) √ to the resulting a priori (weighted) estimation errors η(t)ea (t) can be expressed in terms of a feedback structure, as shown in Fig. 3. The stability of such structures can be studied via tools that are by now standard in system theory (e.g. the small gain theorem). Conditions on the learning rate µ(t) will be derived in order to guarantee a robust training algorithm, as well as faster convergence speeds. This will be achieved by establishing conditions under which the feedback configuration is l2 stable in the sense that it should map a finite-energy input noise sequence (which √ η(t)v(t)} to a include the noiseless case a special case) { √ finite-energy a priori error sequence { η(t)ea (t)} 3.6 Optimal learning rate for U-model via small gain theorem In order to make use of the tools from the system theory, such as the l2 stability and small gain theorem, define µ(t) (35) (N ) = max 1 − y (τ ) . 0≤t≤N η(t) According to the definition in Eq. 35, (N ) is the maximum absolute gain of the feedback loop over the interval 0 ≤ t ≤ N. The small gain theorem states that the l2 stability of a feedback configuration such as the configuration in Fig. 3 as special case requires that the product of norms of the feedforward and feedback maps be strictly bounded by one.

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In our case, the norm of the feedforward map is equal to one (since it is lossless) while the norm of the feedback map is defined in Eq. 35 as (N ). Hence, the condition (N ) < 1 guarantees an overall contractive map. Therefore, for (N ) < 1 to hold, we need to choose the learning rate such that, for all t

y(τ ) =

M

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j=0

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(37)

j=1

Hence, the derivative in Eq. 37 can be used to find the optimal learning rate to speed up the convergence as, µ(t) < 2η(t)y (τ ), µ(t) < η(t)

M

jα j (τ )u j−1 (t − 1).

(38)

j=1

Remark The inequality given by Eq. 38 not only ensures the stability of the update recursion of the U-model parameters but also guarantees faster convergence speeds.

4 Simulation results A Hammerstein system is considered to verify the performance of the proposed LFFC scheme with a constant and with an adaptive learning rate obtained in Eq. 38. The Hammerstein system considered is a 2-input 2-output heat exchanger. The two similar static nonlinearities in heat exchanger are defined as [24]. x(t) = −31.549u(t) + 41.732u 2 (t) + 24.201u 3 (t) + 68.634u 4 (t).

(39)

The linear dynamic block is considered as, y1 (t) = −0.82y1 (t − 1) − 0.75y1 (t − 2) + 0.81x1 (t) + 0.53x2 (t),

(40)

y2 (t) = −0.61y2 (t − 1) − 0.53y2 (t − 2) + 0.62x1 (t) + 0.22x2 (t).

(41)

The reference signal is set to be a random piece-wise continuous signal with a step time of 3 s. A third order U-model

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with M = 3 is used in the LFFC based tracking. The learning rate is set to 0.05 after few trial runs. The output is made noisy with additive white gaussian noise of 20dB SNR. The tracking behavior is shown in Fig. 4. It can be seen that a constant learning behavior is observed. However, The tracking behavior can be different for a different learning rate. Now, the same system is used and the learning is made adaptive using the criteria in Eq. 38. The learning rate µ(t) is kept 95% of the right hand side. The tracking behavior is shown in Fig. 5. It can be seen that although initially the performance was not acceptable, the adaptive learning rate eventually set the learning process in stable region and finer tracking behavior is followed after some time.

The first link (named primar y) has a length L 1 30 cm and the second link (named secondar y) has a length L 2 19 cm. The primary link is made of Aluminium and the secondary link is made of Plastic material. The primary link is actuated by a geared motor HN-GH 27, with a gear ratio of 50:1 and an allowable maximum torque of 2.3 KG cm. The secondary link is actuated by another geared motor HN-GH 12, with a gear ratio of 188:1 and an allowable maximum torque of 1.4 KG-cm. The geometry of the 2-link robot is shown in Fig. 7. To add the effect of load variation, the link is connected by elastic strings on both sides, such that the tension in the string is variable according to the angular position of the link. Tension in the string increases with increasing rotation angle. The feedback signals i.e. the angles of the links are measured by two 0–50 k potentiometers. Due to the physical limitations, the primary link is constrained to have a maximum rotation of ±60◦ from the central position. However, the secondary link can manoeuver the whole ±180◦ rotation. The workspace for the developed 2-link robot is shown in Fig. 8.

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M(θ )θ¨ + C(θ, θ˙ )θ˙ = τ

y1

(43)

Ø

The element of inertia matrix M(θ ) are, M11 (θ ) = m1L 2c1 +m 2 (L 21 + L 2c2 +2L 1 L 2 cos(θ2 ))+ I1 + I2

L1

M12 (θ ) = M21 (θ ) = m 2 (L 2c2 + 2L 1 L 2 cos(θ2 ))

y2

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Ø

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The elements of the centrifugal and Coriolis matrix C(θ, θ˙ ) are,

(xeff,yeff )

C11 (θ, θ˙ ) = −m 2 L 1 L c2 sin(θ2 )θ˙2 C12 (θ, θ˙ ) = −m 2 L 1 L c2 sin(θ2 )(θ˙1 + θ˙2 ) C21 (θ, θ˙ ) = m 2 L 1 L c2 sin(θ2 )θ˙1

x1

C22 (θ, θ˙ ) = 0 where L i , m i , L ci and Ii are the length, mass, distance to the center of gravity and moment of inertia of the ith link, respectively.

x2 Fig. 7 The 2 link robot geometry

5.1.3 Kinematics and inverse kinematics

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5.1.2 The dynamic model of the 2-link robot arm The dynamics of the 2-link robot can be written as [25]: M(θ )θ¨ + C(θ, θ˙ )θ˙ + g(θ ) = τ

(42)

where θ is the 2 × 1 vector of angular positions of links, τ is the 2 × 1 vector of applied torques, M(θ ) is the 2 × 2 symmetric positive definite inertia matrix, C(θ, θ˙ )θ˙ is the 2 × 1 vector of centrifugal and Coriolis torques, and g(θ ) is the 2 × 1 vector of gravitational torques. Gravitational torques can be eliminated, since the robot arm is set for horizontal motion. Therefore, the dynamic equation can be simplified as,

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The objective of the 2-link robot manipulator is; given any coordinates in the workspace, the end − e f f ector will be driven to those desired coordinates in the robot workspace within finite time, practically in shortest time. This is achieved by rotating the robot links to corresponding angles. The problem of finding the angles of the robot links given any coordinates in the workspace is called inverse kinematics. The control algorithm treats the values of the angles in radians as the setpoints and attempts to track the angles of the links to those setpoints. The actual position of the end effector can be obtained using the forward kinematics. Given the angles of the joints, the problem of finding the position of the end effector is called forward kinematics. The inverse kinematics problem is defined as, given the desired xd and yd coordinate of the end effector, find the angles for the primary and secondary links. This can be achieved by the following equations. Defining B as the distance of the end effector from the origin of the base frame.

(44) B = x 2 + y2 The angles θ1 and θ2 are calculated as, θ1 = Atan2(yd /xd )+cos −1 [(L 21 − L 22 + B 2 )/2L 1 B],

θ2 = cos −1 [(L 21 + L 22 − B 2 )/2L 1 L 2 ],

(45) (46)

The function Atan2(Y/ X ) finds the proper quadrant for the angle (There could be more than one solution to even a single link as the inverse of cosine generates ± angles, so it is necessary to find the correct quadrant). The position of the end effector is calculated using the forward kinematics of the 2-link robot. Given the angles θ1

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and θ2 , the end effector coordinates xeff and yeff are, xeff = L1 cos(θ1 ) + L2 cos(θ1 + θ2 ),

(47)

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(48)

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5.2 Tracking at no load conditions For initial comparison, a PID controller is tuned manually by intuition at no load to achieve best possible tracking. The tracking behavior is shown in Fig. 9. The figure shows an acceptable steady-state tracking even though there are high overshoots at the transition and occasional mismatch in the tracking. The proposed U-model based LFFC scheme is applied to the 2 link robot using a third order U-model and a 2 input 2 output RBFNN with 2 neurons for the A0 . The width of the Gaussian basis functions is kept as 1 to cover a large input range. The randomly initialized weights of the RBFNN and the matrix parameters A j are updated using the nLMS principle with a learning rate of 0.05. In the real-time experiments,

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The real-time code is built in SIMULINK on Intel Pentium III 933MHz Computer with 256 MB RAM with a sample time of 0.01 s. The interfacing is performed using the Advantech PCI-1711 I/O card. Analogue-to-Digital conversion to provide the control signal, and the Digital-to-Analogue conversion to get the feedback signal is performed using the Feed Back 33-301 interface module. The control signals are powered by Feed Back Power Supply PS150E and amplified using the Servo Amplifiers Feed Back SA150D. The reference signal is set to be a random piece-wise continuous signal with a step time of 3 s.

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the learning rate is set as a small enough constant after few initial trial runs. The tracking is shown in Fig. 10. Fine tracking performance can be observed with no overshoot and mistracking as compared to the standard PID controller. 5.3 Tracking at varying load conditions Using the same setup, with a varying load, the performance of the PID controller tuned at no load is shown in Fig. 11 and it can be clearly seen that the PID controller tuned at no load was not able to track the reference trajectory with load variations. However, no degradation in the performance is observed when the proposed U-model based scheme is applied under varying load conditions. The U-model based adaptive scheme compensated for the varying load conditions and tracking

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A new technique is introduced for the learning feedforward control of unknown MIMO nonlinear systems. MIMO U-model is proposed in the control scheme for the online identification of the unknown MIMO plant. The controller is developed based on the U-model methodology using the Newton–Raphson method. The proposed technique adequately simplifies the synthesis of control law that is directly derived from the model. The proposed scheme is tested on a case study of a 2 link robot manipulator and is compared with several schemes. The comparison depicted better tracking performance using the U-model based LFFC under varying load conditions. Acknowledgments The authors acknowledge the support of Iqra University, Karachi, King Fahd University of Petroleum and Minerals and SABIC for funding this work under project SABIC 2006-11.

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behavior similar to no load conditions is observed as depicted in Fig. 12. This shows the tracking robustness of the adaptive scheme that is able to perform even with load variation which is a common practice in robotics. 5.4 Performance comparison The performance of the proposed U-model based scheme is compared with a number of schemes; such as, nonlinear PID controller, adaptive PD controller, and adaptive inverse control scheme. Figure 13 shows a comparison in the mean squared errors sense. The comparison shows that the proposed scheme has performed better under varying load conditions. The U-model based LFFC scheme is also compared with the U-model based IMC. An initial faster convergence was observed using the U-model based IMC scheme, how-

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ISA TRANSACTIONS® ISA Transactions 44 共2005兲 353–362

Internal model control structure using adaptive inverse control strategy Muhammad Shafiq* Systems Engineering Department, King Fahd University of Petroleum & Minerals, Dharan 31261, Saudi Arabia

共Received 5 March 2003; accepted 19 November 2004兲

Abstract In this paper, we propose a new adaptive internal model control scheme based on adaptive finite impulse response filters. This scheme provides the same design procedure for both minimum and nonminimum phase plants. The plants under consideration may contain time delay. The tracking objective is accomplished for both invertible and noninvertible stable plants. The internal model of the plant and its inverse are estimated by recursive least-squares and leastmean-squares algorithms, respectively. The closed loop is designed such that the system from the reference input to the plant output can be approximately represented by a pure delay. The effect of the process zeros on the output is compensated by using adaptive finite impulse response filters. This avoids the cancellation of noncancellable zeros of the plant and forces the plant output to track the reference input with a delay. The stability of the closed loop for both minimum and nonminimum phase systems is guaranteed. Computer simulation and laboratory scale experimental results are included in the paper to demonstrate the effectiveness of the proposed method. © 2005 ISA—The Instrumentation, Systems, and Automation Society. Keywords: Adaptive control; Internal model control; Nonminimum phase systems; Real time

1. Introduction Internal model control 共IMC兲 structure has long been successfully used for controlling open-loop stable plants. Most of the industrial processes are open-loop stable. The IMC structure is composed of an explicit model of the plant and a stable feedforward controller. The IMC controller guarantees the internal stability of the closed loop and parameters of the controller can be easily tuned online without disturbing the stability of the closed-loop system 关1兴. Asymptotic tracking in the IMC structure is achieved by incorporating the inverse of the plant model in the feed-forward path. But, the inverse of the nonminimum phase plant is unstable. The use *E-mail address: [email protected]

of this inverse in the IMC control loop gives rise to instability in the system. There are many plants with nonminimum phase behavior, such as dc motors with field regulation, blast furnaces, hydraulic pumps, distillation columns, and so on. In this situation it becomes very important to obtain the stable inverse of the plant model to use in the IMC scheme to accomplish the tracking objective. When the plant parameters are not exactly known or changing slowly, then IMC controllers can be designed online using adaptive control strategies 关2兴. The discrete-time model of the plant is identified online and then stable controllers are designed 关3兴. The effect of numerator polynomial in the discrete-time model of the plant can be compensated by approximate inverse systems 关4兴. The identified discrete-time transfer function usually has zeros outside the unit circle and so the

0019-0578/2005/$ - see front matter © 2005 ISA—The Instrumentation, Systems, and Automation Society.

354

Muhammad Shafiq / ISA Transactions 44 (2005) 353–362

inverse of the system becomes unstable 关5,6兴. The use of such inverse systems gives rise to unstable modes in the closed loop as unstable poles of the controller cancel with the noncancellable zeros of the plant 关7兴. To overcome this problem approximate inverse systems are designed online to compensate the effect of zeros of the plant on the output 关8兴. These approximate inverse systems are implemented as adaptive finite impulse response 共FIR兲 filters 关9,10兴. These filters are stable and also the cancellation of noncancellable zeros can be avoided and hence the closed-loop system will not have unstable hidden modes. The design of these approximate inverse systems is computationally expensive as p⫹m equations have to be solved online in the adaptive case, where p represents the order of the approximate inverse systems and m is the degree of numerator polynomial of the plant transfer function. Introduction of these FIR filters in the feed-forward path of the plant gives linear phase FIR filter behavior to the overall loop, that is the delay of the system becomes independent of the frequency of the excitation signal 关10兴. Adaptive inverse control 共AIC兲 is one of the strategies, which implements the inverse of the plant as adaptive FIR filters 关11兴. In this scheme the inverse of discrete-time model of the plant is estimated using the least-mean-squares 共LMS兲 algorithm as an adaptive FIR filter. In this paper, we implement AIC to synthesize the approximate inverse system and the recursive least-mean-squares 共RLS兲 method to identify the plant model. It is demonstrated that the tracking objective can be achieved approximately by using the estimated plant model and approximate inverse system of the plant in the IMC structure. The rest of the paper is organized as follows. In Section 2 design of the controller is discussed. Section 3 discusses adaptive controller design. Section 4 is devoted to the stability and robustness analysis. Computer simulation results are described in Section 5. In Section 6, we illustrate results of real-time experiments. Section 7 concludes the paper by summarizing the results.

2. Controller design Consider the continuous-time stable plant transfer function P ( s ) . Let h 0 ( s ) denote the zero-order

Fig. 1. Discrete-time IMC configuration.

hold and ␥ ( s ) is an antialiasing pre-filter. The discrete-time version of the plant is

P 1 共 z 兲 ⫽ZL⫺1 兵 h 0 共 s 兲 P 共 s 兲 ␥ 共 s 兲其 , where Z and L represent the z and Laplace transforms, respectively. P 1 ( z ) in the context of IMC is known as the internal model of the model. The discrete-time internal model in shift operator form can be given by

A 共 q 兲 yˆ 共 k 兲 ⫽B 共 q 兲 u 共 k⫺d 兲 ,

共1兲

where

A 共 q 兲 ⫽1⫹a 1 q ⫺1 ⫹¯⫹a n q ⫺n , B 共 q 兲 ⫽b 0 ⫹b 1 q ⫺1 ⫹¯⫹b m q ⫺m . b 0 ⫽0 and polynomials A ( q ) and B ( q ) are coprime polynomials with unknown coefficients. q ⫺1 is a backward shift operator and d is a known delay of the system. u ( k ) is the control input to the plant and y ( k ) is the output of the plant. In this paper backshift operator functions are represented by f ( q ) , z transforms of the functions are given by f ( z ) , and ˆf stands for the estimated value of f. The control objective is to synthesize u ( k ) such that y ( k ) tracks some bounded piecewise continuous desired trajectory r ( k ) , while the plant parameters are unknown or slowly time varying. The discretetime IMC configuration is depicted in Fig. 1. If P 1 ( z ) is an exact representation of the stable plant, then stability of the closed loop is assured by designing stable discrete-time IMC controller Q ( z ) to minimize some given performance index 关1兴. In the tracking problem, Q ( z ) is designed to minimize the l 2 norm of r ( k ) ⫺y ( k ) , provided 关 r⫺y 兴 ⑀ l 2 . This is an H 2 optimization problem. Let us define the tracking error e ( k ) by e 共 k 兲 ⫽r 共 k 兲 ⫺y 共 k 兲 .

共2兲

It is straightforward to write the following from Fig. 1:


冋

e 共 k 兲 ⫽ 1⫺q ⫺d

册

B共 q 兲 Q共 q 兲 r共 k 兲. A共 q 兲

共3兲

Now, using Parseval’s theorem one may write

兺兩 e 共 k 兲兩 2 ⫽ 2 ␲ 冕⫺ ␲ 兩关 1⫺ P 1共 e j ␻ 兲 Q 共 e j ␻ 兲兴 k⫽0 ⬁

1

␲

⫻R 共 e j ␻ 兲兩 2 d ␻ .

共4兲

Let H 2 norm of the discrete-time transfer function be represented by 储 • ( z ) 储 2 and R ( z ) denote the z transform of r ( k ) , then one may write from Eq. 共4兲储 e 储 2 ⫽ 储关 1⫺ P 1 共 z 兲 Q 共 z 兲兴 R 共 z 兲储 2 .

共5兲

Thus the problem of minimizing the l 2 norm of e ( k ) is reduced to the minimization of 储关 1 ⫺ P 1 ( z ) Q ( z ) 兴 R ( z ) 储 2 over Q ( z ) . When P 1 ( z ) has all zeros inside the unit circle then choice Q ( z ) ⫽1/P 1 ( z ) minimizes the criterion. It has long been known that most of the industrial processes have nonminimum phase behavior and very often a discrete-time transfer function of the continuous-time plant becomes a nonminimum phase even if the original plant is a minimum phase system 关5兴. In this case, zeros outside the unit circle are not cancelled but their effect is recompensed by obtaining all pass filter behavior of the closed-loop system 关1兴. This means delay of the closed-loop system depends on the frequency of the input signal as the gain of an all pass system is unity but the phase response depends on the frequency. Therefore the tracking error will change as the frequency of the exciting signal changes. Further, if the input signal has more than one frequency component, each component will have a different delay, and so the tracking problem becomes much more complex. This problem can be overcome if a controller can be designed such that the closed-loop dynamics of the system matches the characteristics of the linear phase all pass filter at frequencies of interest. This process is equivalent to finding the inverse of the system at given frequencies. This means that the controller is a function of the frequency of the desired trajectory. Using FIR filters it is possible to design the controller for the IMC structure such that the closed loop behaves like an approximate linear phase FIR all pass filter. These FIR filters can be

355

designed by a variety of procedures, such as concept of L-delay approximate inverse systems 关4,8 – 10兴, least-mean-squares based adaptive inverse filter 关11兴, and adaptive zero phase error tracking filters 关12兴. Using FIR filters, it is possible to minimize the tracking error arbitrarily by choosing suitable parameters and order of the filter 关8兴. To the author’s knowledge and survey none of these methods has been used and investigated for the design of adaptive tracking in the internal model structure. Here for the sake of clarity of presentation and taking into account the page limitation, we discuss only adaptive inverse filters based adaptive tracking in IMC structure, which can be designed for both minimum and nonminimum phase systems by using the same methodology. 3. Adaptive controller design The inverse of the system is designed online in the IMC structure for the estimated model of the unknown or slowly time varying plants 关13,14兴. If the plant is minimum phase and is known in advance then the inverse of the estimated transfer function can be used as a stable controller. This cannot be done for the nonminimum phase systems because the inverse of the transfer function will be unstable and so the controller will be unstable. Further, there will be unstable pole-zero cancellation between the controller poles and the plant zeros. Therefore stability of the closed-loop and boundedness of the control input signal cannot be assured. It has been suggested to decompose a numerator polynomial into cancellable and noncancellable parts, and compensate the cancellable part by the infinite impulse response 共IIR兲 filter and noncancellable by FIR filters. Here the IIR filter is a stable inverse of the cancellable polynomial and the FIR filter is the stable approximate inverse system of the noncancellable polynomial. The decomposition of the estimated numerator polynomial of the plant into stable invertible and unstable invertible becomes computationally expensive 关4兴. Therefore use of methods treating the compensation of stable invertible zeros and unstable invertible zeros in distinct fashion is not feasible. The methods based on FIR filters do not consider the type of zeros for the design of compensating filters 关9兴. Using FIR filters the decomposition of the numerator polynomial of the plant can be avoided

356


but the order of the FIR filter becomes high. These compensators introduce extra delay; the maximum value of this delay is p⫹m steps, where p is the order of the compensating FIR filter and m is the degree of numerator polynomial of the plant transfer function. This delay is very small as compared to the process delay and it can easily be compensated using the Smith predictor techniques. Adaptive inverse control 共AIC兲 has been successfully used for control of stable plants. In this approach the plant model and inverse of the plant are estimated as adaptive FIR filters to accomplish output tracking, and simultaneously the disturbances are cancelled using adaptive noise cancellers 关13兴. The main contribution of the paper is that the autoregressive moving average 共ARMA兲 process is used to represent the internal model of the plant. The properties of the AIC for IMC can be improved by incorporating the internal model of the plant as an ARMA process, which can be implemented as an IIR filter. Least-mean-squares methods are suitable for the parameter estimation of the moving average 共MA兲 process, which are implemented by FIR filters. LMS algorithms are not suitable for online identification of the ARMA process in adaptive control 关7,15兴. RLS methods for parameter estimation have successfully been used for the online identification of plants in several adaptive control schemes 关14,16 –18兴. It is briefly described in Appendix A. The availability of high-speed processors has made it possible to implement RLS based plant identification, especially in the process control area where the sampling frequency is in the kH range. The parameters of the approximate inverse filter are estimated using normalized least-mean-squares 共NLMS兲 based estimator. The method is briefly explained in Appendix B. The adaptive IMC scheme is depicted in Fig. 2. The ARMA model of the plant is estimated using the robust RLS algorithm and it is introduced in the loop as the internal model of the plant. Then control input u ( k ) is synthesized using IMC structure control. The IMC controller is implemented by using estimated inˆ ( z ) and internal model verse model Q z ⫺d 关 Bˆ ( z ) /Aˆ ( z ) 兴 of the system. Therefore the control input u ( k ) can be given by

Fig. 2. AIC based discrete-time IMC configuration.

ˆ ( z ) to minimize the l In this case, we choose Q 2 norm of the tracking error e ( k ) ⫽r ( k⫺L ) ⫺y ( k ) , where L is a delay. The problem is translated to minimize ˆ 共 z 兲兴 R 共 z 兲储 . 储关 z ⫺L ⫺ Pˆ 1 共 z 兲 Q 2 The objective can be accomplished by estimating the plant parameters by the robust recursive least-squares algorithm and the approximate inverse system by the normalized least-meansquares 共NLMS兲 technique as explained in Appendixes A and B 关15兴. The order of the polynomial ˆ ( q ) is discussed in Section 6. Q 4. Stability analysis The stability and convergence of parameters estimation laws have been discussed in the literature extensively 关14 –18兴. Here, we discuss the stability of the closed loop. The transfer function of the closed loop from reference input to plant output can be obtained as

T共 z 兲⫽

P 1共 z 兲 Q 共 z 兲

. 1⫹Q 共 z 兲关 P 1 共 z 兲 ⫺ Pˆ 1 共 z 兲兴

共6兲

If Pˆ 1 ( z ) is an exact replica of the plant that is Pˆ 1 ( z ) ⫽ P 1 ( z ) , then the closed loop will remain stable, as the plant is considered to be stable system and Q ( z ) is the representation of a FIR filter. Using P 1 ( z ) ⫽z ⫺d 关 B ( z ) /A ( z ) 兴 , Pˆ 1 ( z ) ⫺d ˆ ⫽z 关 B ( z ) /Aˆ ( z ) 兴 and Eq. 共6兲, it is easy to show that the characteristic polynomial of the closed loop is given by

A 共 z 兲 Aˆ 共 z 兲 ⫹z ⫺d Q 共 z 兲关 Aˆ 共 z 兲 B 共 z 兲 ⫺A 共 z 兲 Bˆ 共 z 兲兴 ˆ 共 q 兲关 r 共 k 兲 ⫺y 共 k 兲 ⫹yˆ 共 k 兲兴 . u 共 k 兲 ⫽Q

⫽D 共 z 兲 .


Fig. 3. Desired output and the plant output.

357

Fig. 4. Control input to the plant.

In case of mismatch between the original and estimated transfer functions, the closed-loop system will remain stable if it can be shown that

ˆ 共 z 兲关 Aˆ 共 z 兲 B 共 z 兲 ⫺A 共 z 兲 Bˆ 共 z 兲兴兩兩 A 共 z 兲 Aˆ 共 z 兲兩 ⬎ 兩 z ⫺d Q 共7兲 for all z on the unit circle 关19–21兴. The robust recursive least-mean-squares estimators ensure the convergence of Bˆ ( z ) to B ( z ) and Aˆ ( z ) to A ( z ) 关14兴. This means that 兩 Aˆ ( z ) B ( z ) ⫺A ( z ) Bˆ ( z ) 兩 is very small.

5. Computer simulation results

Fig. 5. Estimated parameters.

Let us consider a plant with parameters

A 共 q 兲 ⫽1⫹0.5q ⫺1 ⫹0.1q ⫺2 ,

共8兲

B 共 q 兲 ⫽1⫹1.2q ⫺1 .

共9兲

This plant has poles at ⫺0.2500⫾0.1936i and a zero at ⫺1.2. This is a nonminimum phase stable plant and the zero is noncancellable. The simulation is carried out in SIMULINK. Fig. 3 shows that plant output converges to the desired output. Control input to the plant is bounded and the result is depicted in Fig. 4. Figs. 5 and 6 indicate that plant parameters converge to the true parameters and parameters of the inverse system also converge, respectively. Initial parameters are chosen randomly. The zeros of the characteristic polynomial are shown in Fig. 7. This figure shows that poles of the closed loop remain within the unit circle.

Fig. 6. Estimated parameters of the inverse system.

358


Fig. 7. Zeros of the characteristics polynomial.

6. Real-time implementation The proposed scheme is implemented in real time on the temperature control of a heating process, speed control of a direct current motor and rate of flow of liquid drawn by a pump. In these experiments, a standard IBM PC-type Pentium III is used for the computation in real time. Data acquisition is accomplished by Advantech card PCI-1711 and the controller is implemented in SIMULINK real-time windows target environment. The computations are performed in floating-point format and the sampling interval for the temperature of the heating system selected is 0.1 sec and for the other two experiments 0.01 sec. Setup for the real-time control of temperature of a heating process is shown in Fig. 8. The process is composed of a blower, a heating grid, tube, and temperature sensor 共bead thermistor兲. A variable

Fig. 8. Experimental setup for heating process.

Fig. 9. Desired temperature and the process output.

power supply provides power to the heater. This power can be controlled by initiating an appropriate controlling signal from the computer. The process can be considered as a second-order time delay system. Input of the process is power and output is the temperature of air at some desired location in the process tube. Fig. 9 shows that output 共temperature兲 of the process converges to the desired temperature quickly and it can be confirmed from Fig. 10 that control input to the plant remains bounded. Plant parameters are shown in Fig. 11. It is obvious that parameters of the system change as the reference input to the system varies. The parameters of the approximate inverse filter are depicted in Fig. 12. In this experiment, the order of the approximate inverse system is chosen as 9. It can be verified

Fig. 10. Control input to heating process.


359

Fig. 13. Experimental setup for motor speed control.

Fig. 11. Estimated model of heating process.

from Fig. 12 that the estimated parameters converge. Variation in the parameters of the inverse system is small as compared to the changes in the plant parameters. This property depends on the number of parameters of the inverse system. The larger the number of parameters of inverse system the smaller will be the variation in the estimated parameters compared to the fluctuation in plant properties. It is observed, when the number of parameters is small, that there is a large change in the estimated parameters when the reference input is changed, while the tracking error does not change significantly. When the number of parameters is large as in this example 共about 10兲 little change in the estimated parameters is observed due to the variation in the reference input. Choosing more than ten parameters of the inverse system in this example gives similar results to that of the ten parameters inverse system. There exists no

Fig. 12. Estimated inverse of heating process.

simple way to choose the number of the parameters of the inverse system. A rough estimate for the number of parameters of the inverse system can be obtained offline by analyzing the step response of the system. It is worth mentioning that the plant output converges to the desired trajectory while the estimated parameters of the plant and the approximate inverse system are still in the convergence process. This behavior of the RLS and NLMS estimators is discussed excellently in detail in Ref. 关7兴. Now, we discuss the implementation of the proposed algorithm for the real-time speed control of a brush dc motor 共Cruzet 8285002兲. This motor has a maximum speed of 3200 revolutions per minute, which can be achieved on exciting the motor by 24 V dc. Speed is measured by using a tachometer, which produces a voltage proportional to the speed of the motor. A servo amplifier is used to provide variable voltage 共control input兲 for the excitation of the motor. The setup for the experiment is shown in Fig. 13. It can be confirmed from Figs. 14 and 15 that the speed of the motor con-

Fig. 14. Desired speed and the motor output.

360


Fig. 15. Control input to motor.

verges to the desired speed and the control input is bounded. The flow rate of a liquid drawn by diaphragm pump 共Shurflo 8000-243-332兲 is also controlled by using the proposed algorithm. A servo amplifier is used to excite the pump electrically. The flow rate is measured by a turbine-type flow sensor 共RS 256-225兲, which produces a frequency proportional to the flow rate, this frequency is converted to proportional dc voltage by a frequency to voltage converter. Fig. 16 indicates that the flow rate converges to the desired flow rate quickly and Fig. 17 shows that the control input is bounded. The convergence behavior of the plant and approximate inverse system parameters are similar to that of the heating process experiment. Initial parameters of the plant and inverse system are chosen randomly.

Fig. 17. Control input to pump.

7. Conclusion An adaptive inverse control algorithm based on internal model control structure has been proposed in this paper. The inverse of the plant has been implemented as an adaptive FIR filter. The internal model of the zero-order hold, plant, and the antialiasing filter is considered as an autoregressive moving average model. This model is estimated using standard robust recursive least-squares estimation. It has been shown that the closed loop remains stable for both minimum and nonminimum phase systems and in both cases output of the plant tracks the desired trajectory with a delay. Computer simulation results and real-time experiments show the effectiveness of the proposed method. Acknowledgment The author acknowledges King Fahd University of Petroleum & Minerals, Dhahran 31261 Saudi Arabia, for supporting this research. Appendix A: Estimation of plant parameters The parameters of an unknown plant can be estimated in the presence of disturbances using the least-squares parameter adjustment law 关16兴. Let us introduce a filter to eliminate the disturbances such that the filtered input u f ( k ) and output y f ( k ) are defined by

Fig. 16. Desired flow rate and the pump output.

F 共 q 兲 u f 共 k 兲 ⫽D 共 q 兲 u 共 k 兲 ,

共A1兲


F 共 q 兲 y f 共 k 兲 ⫽D 共 q 兲 y 共 k 兲 ,

共A2兲

m

variation in the system excitation signal. The method is briefly described below. A comprehensive discussion could be found in Refs. 关15兴, 关11兴. Let y 1 ( k ) be required to track u ( k⫺L ) and the discretized plant output be y ( k ) , then y 1 ( k ) is given by

j⫽0

y 1 共 k 兲 ⫽W T 共 k 兲 ␾ 共 k 兲 ,

where F ( q ) is a Hurwitz polynomial and D ( q ) /F ( q ) is a low pass filter. The system can be given by n

y f 共 k 兲 ⫽⫺ 兺 a i q y f 共 k 兲 ⫹ 兺 b j q ⫺d⫺ j u f 共 k 兲 ⫺i

i⫽1

⫽ ␣ TX f 共 k 兲,

共A3兲

where

W T 共 k 兲 ⫽ 关 qˆ 0 ,qˆ 1 ,...,qˆ p 兴 ,

where T stands for the transpose,

␾ T 共 k 兲 ⫽ 关 y 共 k 兲 ,y 共 k⫺1 兲 ,...,y 共 k⫺p 兲兴 .

␣ T ⫽ 关 a 1 ,a 2 ,...,a n ,b 0 ,b 1 ,...,b m 兴 ,

The parameter estimation law for an approximate inverse filter based on NLMS is given by

and

X Tf ⫽ 关 ⫺y f 共 k⫺1 兲 ,⫺y f 共 k⫺2 兲 ,..., ⫺y f 共 k⫺n 兲 ,u f 共 k⫺d 兲 ,u f 共 k⫺d⫺1 兲 ,...,

W 共 k 兲 ⫽W 共 k⫺1 兲 ⫹ ␮

u f 共 k⫺d⫺m 兲兴 . The vector ␣ represents the unknown parameters of the plant and X f ( k ) is a vector consisting of filtered input and output signals. The parameter adjustment law which ensures that the estimated parameters converge to their true values can be given by 关14,16 –18兴

␣ˆ 共 k 兲 ⫽ ␣ˆ 共 k⫺1 兲 ⫺

⌫ 共 k⫺1 兲 X f 共 k 兲 E f 共 k 兲 1⫹X Tf 共 k 兲 ⌫ 共 k⫺1 兲 X f 共 k 兲

,

E f 共 k 兲 ⫽ 关 ␣ˆ T 共 k⫺1 兲 X f 共 k 兲 ⫺y f 共 k 兲兴 , ⌫共 k 兲⫽ ⌿共 k 兲⫽

1 关 ⌫ 共 k⫺1 兲 ⫺⌿ 共 k 兲兴 , ␴

␭⌫ 共 k⫺1 兲 X f 共 k 兲 X Tf 共 k 兲 ⌫ 共 k⫺1 兲

␴ ⫹␭X Tf ⌫ 共 k⫺1 兲 X f 共 k 兲 ⌫ 共 0 兲 ⫽ ␦ I,

361

共A4兲

,

␦ ⬎0,

where 0⬍␴⭐1 and 0⬍␭⬍2.

Appendix B: Estimation of parameters of inverse system NLMS is used to estimate the parameters of the approximate inverse system. This approximate inverse system is the local inverse of the system. The parameters of this inverse may change with

␾共 k 兲 ⑀ ⫹ ␾ T共 k 兲 ␾ 共 k 兲

⑀共 k 兲, 共B1兲

where ⑀ ( k ) ⫽u ( k⫺L ) ⫺y 1 ( k ) and ⑀⬎0. ⑀ is introduced in the estimator to avoid division by zero when ␾ T ( k ) ␾ ( k ) is very small and 1⬎␮⬎0 is the learning rate.

References 关1兴 Morari, M. and Zaferiou, E., Robust Process Control. Prentice-Hall, Englewood Cliffs, NJ, 1989. 关2兴 Datta, A., Adaptive Internal Model Control. SpringerVerlag, London, 1998. 关3兴 Silva, G. J. and Datta, A., Adaptive internal model control: The discrete-time case. Int. J. Adapt. Control Signal Process. 15共1兲, 15–36 共2001兲. 关4兴 Lu, J. and Yahagi, T., A new design method for model reference adaptive control of non-minimum phase discrete-time systems with disturbances. IEE Proc.-D: Control Theory Appl. 140共1兲, 34 – 40 共1993兲. ˚ stro¨m, K. J. and Wittenmark, B., Self-tuning control关5兴 A lers based on pole-zero placement. IEE Proc.-D: Control Theory Appl. 127共5兲, 120–130 共1980兲. ˚ stro¨m, K. J., Hagander, P., and Sternby, J., Zeros of 关6兴 A sampled system. Automatica 20共1兲, 31–38 共1984兲. ˚ stro¨m, K. J. and Wittenmark, B., Adaptive Control. 关7兴 A Addison-Wesley, Reading, MA, 1995. 关8兴 Yahagi, T. and Lu, J., On self-tuning control of nonminimum phase discrete time systems using approximate inverse systems. J. Dyn. Syst., Meas., Control 115共3兲, 12–18 共1993兲. 关9兴 Shafiq, M., Lu, J., and Yahagi, T., On self-tuning control of non-minimum phase discrete-time stochastic systems. IEICE Trans. Fundamentals 79共12兲, 2176 – 2184 共1996兲. 关10兴 Shafiq, M., Lu, J., and Yahagi, T., A new method for self-tuning control of non-minimum phase continuous-

362

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time systems based on pole-zero placement. IEICE Trans. Fundamentals 79共4兲, 578 –584 共1996兲. Widrow, B. and Walach, E., Adaptive Inverse Control. Prentice Hall, Englewood Cliffs, NJ, 1996. Tsao, T. C. and Tomizuka, M., Adaptive zero phase error tracking algorithm for digital control. J. Dyn. Syst., Meas., Control 109共4兲, 349–354 共1987兲. Plett, G. L., Adaptive inverse control of unmodeled stable SISO and MIMO linear systems. Int. J. Adapt. Control Signal Process. 16共4兲, 243–272 共2002兲. Landau, I. D., Lozano, R., and M’Saad, M., Adaptive Control. Springer, UK, 1998. Haykin, S., Adaptive Filter Theory. Prentice-Hall, NJ, 2002. Ljung, L., System Identification: Theory for Users. Prentice Hall, Englewood Cliffs, NJ, 1987. So¨derstro¨m, T. and Stoica, P., Systems Identification. Prentice-Hall, UK, 1989. Gupta, M. M., Adaptive Methods for Control System Design. IEEE Press, New York, USA, 1986. Jury, E. I., Robustness of a discrete-time system. Autom. Remote Control 共Engl. Transl.兲 51共5兲, 571–592 共1990兲.

关20兴 Yeung, K. S., Linear discrete-time system stablity under parameter variations. Int. J. Control 40共4兲, 855– 862 共1984兲. 关21兴 Mastorpakis, N. E., Robust stability of polynomials: New approach. J. Optim. Theory Appl. 93共3兲, 635– 638 共1997兲.

Muhammad Shafiq received the Ph.D. degree in information and computer sciences from Chiba University Japan in 1997. He was with SES Pakistan as an electronics engineer from 1991 to 1997. He worked as Technical Manager in STESA, Saudi Arabia from 1998 to 1999. He was a Visiting Assistant Professor at Pakistan Institute of Engineering and Applied Sciences, Islamabad from 1997 to 1998. Since 1999, he has been with systems engineering department, King Fahd University of Petroleum and Minerals, Saudi Arabia. His current research interests are in the area of adaptive control, neuroadaptive control, and mechatronics. He is a member of IEEE 共USA兲, a member of the Control System Society of IEEE, and a member of IEICE, Japan.

Neural networks for tracking of unknown SISO

Neural networks for tracking of unknown SISO

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