Mechanics Based Design of Structures and Machines An International Journal
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Nonlinear identification of inverted pendulum system using Volterra polynomials G. Ronquillo-Lomeli, G. J. Ríos-Moreno, A. Gómez-Espinosa, L. A. MoralesHernández & M. Trejo-Perea To cite this article: G. Ronquillo-Lomeli, G. J. Ríos-Moreno, A. Gómez-Espinosa, L. A. MoralesHernández & M. Trejo-Perea (2016) Nonlinear identification of inverted pendulum system using Volterra polynomials, Mechanics Based Design of Structures and Machines, 44:1-2, 5-15 To link to this article: http://dx.doi.org/10.1080/15397734.2015.1028551
Published online: 06 Apr 2016.
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Date: 07 April 2016, At: 09:37
MECHANICS BASED DESIGN OF STRUCTURES AND MACHINES 2016, VOL. 44, NOS. 1–2, 5–15 http://dx.doi.org/10.1080/15397734.2015.1028551
Nonlinear identification of inverted pendulum system using Volterra polynomials G. Ronquillo-Lomelia , G. J. Ríos-Morenob , A. Gómez-Espinosaa , L. A. Morales-Hernándezb , and M. Trejo-Pereab Investigación Aplicada, Centro de Ingeniería y Desarrollo Industrial, Santiago de Querétaro, México; b Facultad de Ingeniería, Universidad Autónoma de Querétaro, Santiago de Querétaro, México
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a
ABSTRACT
ARTICLE HISTORY
The inverted pendulum is one of the most significant problems in control theory and has been studied in control literatures. The nonlinear model is useful in the control design. In the present work, Volterra polynomial basis function networks have been used to identify a single inverted pendulum on a moving cart system. Here, the nonlinear model of the inverted pendulum has been implemented. The offline structure selection through orthogonal least square algorithm is used for the nonlinear system identification via the basis function selection of Volterra polynomial networks. The results show good matching between predicted and actual outputs.
Received 19 October 2014 Accepted 6 March 2015 KEYWORDS
Basis function; identification; inverted pendulum; nonlinear systems; Volterra polynomials
1. Introduction Nonlinear systems have been away from recent engineering research. This is partly because nonlinear systems have been perceived as difficult. The reason for this was that there were not many good analytical tools similar to the ones that have been developed for linear and, time-invariant systems during the same period of time. Linear systems are well understood and can be easily analyzed. It is well known that the inverted pendulum can describe a variety of inherently unstable systems and has wide applications in many areas, such as, human body self-balancing explanation (Bowden et al., 2012; Font-Llagunes and Kövecses, 2009; Kuo, 2007; Milton et al., 2009; Wojtyra, 2003; Young-Dae and Jong-Hwan, 2013) and robot design technology (Díaz-Rodríguez et al., 2008; Erbatur and Kurt, 2009; Vanderborght, 2010). Lately, due to the wide range of applications of the inverted pendulum, the development of its control strategy has become more attractive, but in order to make a good model-based controller a good model of the system is needed. Although there are many techniques available for identification of linear dynamic systems, in general, dynamic systems are complex and nonlinear (Todorovic and Klan, 2006). The main problem for using linearization techniques is that the resulting model is valid only in a certain operating range; consequently, the use of nonlinear identification methods has become necessary, but nonlinear system identification is much more difficult than linear system identification (Liu, 2001). The Volterra polynomial basis function (VPBF) neural networks based identification approach is good option for nonlinear modeling, due its single structure and reduced computational complexity (Minu and Jessy, 2012), besides being simpler than others neural networks (Liu et al., 1998). Volterra polynomials can be expressed for a single layer neural network in a lineal combination of basis function, these characteristics enable the VPBF for using the lineal theory advantages to solve nonlinear system identification problems.
CONTACT G. Ronquillo-Lomeli
[email protected] Centro de Ingeniería y Desarrollo Industrial, Investigación Aplicada, Playa Pie de la Cuesta 702, Santiago de Querétaro, Querétaro 76130, Mexico. Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/lmbd. Communicated by Marco Ceccarelli. © 2016 Taylor & Francis
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Historically, it is easy to realize that the artificial neural networks have been used for nonlinear identification for many years due to their capabilities, being the most important their ability to learn and their good performance for approximation of nonlinear functions (Purwar et al., 2007). The use of Neural Networks (NNs) for identification of nonlinear models has already been explored (Rankovic and Nikolic, 2008; Ruan et al., 2007; Xing et al., 2011). Sutradhar et al. (2010) demonstrate that identification of nonlinear dynamical systems using static and dynamic back-propagation methods. More recently, Wang (2011) studied the problem in two axis movement with a computer aided simulation. Some other works have been devoted to friction modeling and parameter identification. Finally, Zupanˇcíˇc and Sodja (2013) made physical modeling of various simulated systems using computer software for educational purposes. However, the results mainly concern for control problems of inverted pendulums. Few results have been obtained as a solution to the problem of modeling structure for such systems, especially expansionlike solution, which usually plays an important role in the engineering calculus. In the present work, a complex nonlinear mechanic system with VPBF network through data acquired directly from real plant is modeled. The experimental data are generally classified as input and output, measured at discrete instants of time and collected as an array of finite duration data. VPBF neural networks have been constructed from experimental data instead of mathematical model. The models were trained through orthogonal least square algorithm and individual VPBF approximation errors were calculated in order to select the best structure for the actual plant identification. Experimental works to identification of inverted pendulum on a moving cart system driven by a DC motor system have been carried out to illustrate the effectiveness of this approach.
2. Methodology 2.1. Inverted pendulum system: Mathematical model The system to identify consists of a cart driven by a direct current (DC) motor coupled to a mechanical traction system to generate the force applied, which is the input to the system. An inverted pendulum is mounted over the cart as it is shown in Fig. 1. The system modeling is done in two parts: First, the translational force is considered as the system input for the nonlinear dynamics of the single inverted pendulum on a moving cart (SIPC) and second, the lineal system existing between the actuator, the DC motor, and the generated force is analyzed.
Figure 1. Block diagram of the system.
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MECHANICS BASED DESIGN OF STRUCTURES AND MACHINES
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Figure 2. SIPC free body diagram.
2.2. Dynamics of inverted pendulum system Consider the SIPC system, Fig. 2, given by d ∂L ∂L − = Bu + D dt ∂ q˙ ∂q
(1)
where q = (qa , qu )T ∈ R × R and q˙ = (qa , qu )T ∈ R × R vectors of generalized coordinates and velocities, respectively; qa and qu are actuated and unactuated variables, respectively; B = (1, 0)T and u ∈ R is a control input variable; the function 1 (2) L q, q˙ = q˙ T M q q˙ − V q 2 is the Lagrangian; M(q) is a symmetric positive-definite inertia matrix; and V(q) is the potential energy of the system. By introducing a lumped disturbance vector D ∈ R2 , the system Eq. (1) can also be rewritten in a matrix form as M q q¨ + C q, q˙ q˙ + G q = Bu + D (3) or, equivalently
m11 (q) m12 (q) q¨a n (q, q˙ ) u + d1 + 1 = (4) m21 (q) m22 (q) q¨u n2 (q, q˙ ) 0 where C(q, q˙ )˙q is the Coriolis and centrifugal loading vector, G q is the gravitational loading vector, and D = (d1 , 0)T is assumed to satisfy the classical matching condition. d1 includes parameter uncertainties, external disturbance, and unmodeled dynamics, such as viscous and Coulomb’s friction forces exerted on the actuated joint, and it is assumed that |d1 | ≤ d1 for a known constant bound d1 . In the case of the SIPC in Fig. 2, according to Fantoni and Lozano (2002), the equations of motion in Eq. (4) can be rewritten as γ βcos(θ ) x¨ u + d1 −β sin(θ )θ˙ 2 + = (5) βcos(θ ) α 0 θ¨ −η sin(θ ) where α = ml2 , β = ml, γ = M + m, and η = mgl; M and m are masses of the cart and pendulum, respectively. l, θ , x are length of pendulum, angle of pendulum with respect to vertical line, and moving
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distance of cart from initial position, respectively; g is a gravitational acceleration and u is a control force applied to the cart.
2.3. Modeling of the mechanical system for the DC motor The DC motor is coupled via a mechanical system band–pulley–wheel which pushes the cart generating the force u. The motor is driven through the armature voltage vin (t). The torque generated by the DC motor is proportional to the armature current as indicated in the equation tm (t) = Kt ia (t)
(6)
where tm (t) is the torque developed by the motor, Kt is the torque constant of the motor, and ia (t) is the armature current of the motor. The electrical circuit equations for the motor are
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vin (t) − vemf = La
dia (t) + Ra ia (t) dt
vemf (t) = Kv θ˙m (t) = Kv ω¯ m (t)
(7) (8)
where vin (t) is the motor voltage, vemf is the electromotive voltage, Ra is the armature resistance, La is the motor winding inductance, Kv is the back electromotive force’s constant, θm (t) is the motor’s shaft angular position, and ω¯ m (t) is the motor’s shaft angular velocity. The torque equation for motor output shaft is tm (t) − tL (t) = Jm θ¨m (t) + Bm θ˙m (t)
(9)
where Jm is inertia moment of the rotor with load and Bm is the damping coefficient. The motor is coupled to the cart with a radius wheel rw and a coupling gear ratio n, tL (t) is the required torque to generate the linear force u (Fig. 1). Converting the rotational variables into translational variables using x tL (t) = rw nu and θm (t) = (10) rw n The equilibrium torque is obtained from Eqs. (9) and (10), tm (t) = Jm
x¨ x˙ + Bm + nrw u nrw nrw
(11)
Assuming zero inductance and using Eqs. (6)–(8), tm (t) = Ki
Vin − Kv nrx˙w Vin − KV θ˙m = Ki Ra Ra
(12)
Equations (5), (11), and (12) describe the nonlinear model of the SIPC impulse by a DC motor.
2.4. Nonlinear modeling by VPBF networks The nonlinear discrete system described by X t+1 = G (X t , ut )
(13)
yt = h (X t , ut )
(14)
where G(·) is a nonlinear function vector, h(·) is a nonlinear function, X t is the state vector, yt is the output, and ut is the input. Based on the input and output relation of a system, the above nonlinear discrete system can also be expressed by a nonlinear auto-regressive moving average (NARMA) model
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Figure 3. Neural network based identification.
(Leontaritis and Billings, 1985), that is, yt = f yt−1 , yt−2 , . . . , yt−ny , ut−1, ut−2 , . . . , ut−nu
(15)
where f (·) is a nonlinear function, ny and nu are the corresponding maximum delays. It is well known that NNs provide good nonlinear function approximation techniques. A nonlinear identification structure by neural networks is shown in Fig. 3. Here it assumes that the nonlinear function f (·) in the NARMA model is approximated by a single layer neural network, which consists of a linear combination of basic functions. N
fˆ (xt ) =
f X
wk ϕk (xt )
(16)
k=1
where ϕk (xt ) is the basis function, xt = [yt−1 , yt−2 , . . . , yt−ny , ut−1 , ut−2 , . . . , ut−nu ], and wk is the weight. The representation of the nonlinear function f (xt ) is given by fˆ (xt ) = w1 + w2 yt−1 + · · · + wny +1 yt−ny + wny +2 ut−1 + · · · + wny +nu +1 ut−nu 2 + wny +nu +2 yt−1 + wny +nu +3 yt−1 yt−2 + · · · + wNf u0t−nu
where o is the system order. The number of polynomial basis function is given by nu + ny + o ! Nf = o!(nu + ny )! Using the VPBF network, the nonlinear function f (·) can be obtained by f (xt ) = fˆ (xt ) − ε xot
(17)
(18)
(19)
where ε(xot ) is the approximation error.
It is assumed that a set of input–output data (yt , ut , t = 1, 2, . . . , Mt ) of the system is given. Based on Eq. (17) the input–output relation may compactly be written in the following vector form: Y = φ (x) W + E xo (20)
where the output vector is Y ∈ RMt ×1 , the weight vector is W ∈ RNf ×1 , the approximation error vector is E(xo ) ∈ RMt ×1 , and the basis function matrix is φ(x) ∈ RMt ×Nf . The weight vector W is usually found minimizing the Euclidean norm, i.e., b = argmin Y − φ(x)W 2 W W
which is a least squares solution. Figure 4 shows a block diagram of VPBF neural network.
(21)
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Figure 4. VPBF neural network block diagram.
The classical Gram Schmidt method can be used to transform from the set of basis vectors {φi }, into a set of orthogonal basis vectors and thus makes it possible to calculate the individual contribution to the desired output from each basis vector. An orthogonal decomposition of the matrix φ(x) gives φ (x) = PQ
(22)
where P = [P1 , P2 , . . . , PNf ] is an Mt × Nf matrix with orthogonal columns and Q is an Nf × Nf unit upper triangular matrix with 1 on the diagonal and 0 below the diagonal. The corresponding optimal weight vector is b = [ˆv1 , vˆ 2 , . . . , vˆ N ]T ∈ RNf ×1 where V f b vi =
b = Q−1 V b W Y T Pi , PTi Pi
(23)
for i = 1, 2, . . . , Nf
(24)
vˆ i2 PTi Pi is the increment to the desired output variance introduced by Pi , the error reduction ratio due to Pi may be defined by ri =
vˆ i2 PTi Pi
(25) YTY This ratio offers a simple and effective means of seeking a subset of significant basis function. This implementation based on the classical Gram Schmidt is given by Billings et al. (1988) and Billings et al. (1989). Changing the order of the VPBFs will lead to a change in the error reduction ratio ri . The normalized residual sum of squares (NRSSs) is given by NRSS = 1 −
Ls X
rj
(26)
j=1
The procedure is terminated when NSSR < e0 , where e0 is a chosen tolerance. This gives a subset model containing Ls significant terms. For more information on offline structure selection see Liu (2001) and Luo and Billings (1995).
2.5. Experiment description The results of the SIPC experiment are presented in this section. SIPC parameters are shown in Table 1. The SIPC is inherently unstable, which means that in an open loop the pendulum will fall. One of the requirements for the systems identification is that the data must satisfy the persistent excitation condition
MECHANICS BASED DESIGN OF STRUCTURES AND MACHINES
Table 1. Parameters from the real SIPC system. Parameter
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M l m r g Bm Ra Kt Kv Jm n Ps E
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Description
Value
Mass of the cart Length of the pendulum Mass of the pendulum rod Radius of the wheel Gravity Damping coefficient Armature resistance Torque constant Back EMF constant Moment of inertia Gear ratio Potentiometer for angle position Encoder for linear position
2.00 Kg 1.00 m 0.10 Kg 0.0335 m 9.81 m/s2 1.568 × 10−4 Nm/rad/sec 1.8 0.0168 Nm/A 0.0168 V/rad/sec 0.000011 Kgm2 0.2 5 k 50 pulses per revolution
which has been obtained within an operation interval; for this, it is necessary to stabilize the inverted pendulum using a controller. To stabilize the inverted pendulum a Fuzzy controller was used, it was implemented on a SIPC didactic system (equipment for engineering education) which has an encoder, for measuring the linear position and linear velocity, and a potentiometer, for measuring the position and velocity angular. The experimental prototype is shown in Fig. 5. The controlled system was continuously disturbed in order to accomplish the persistent excitation condition. A compact real-time input–output (cRIO) system from National Instruments with encoder and analog input cards was used to acquire the inverted pendulum experimental data with a sampling time of Ts = 10 ms which meet the Shannon theorem taking the mechanical time constant. Experimental data were acquired during 75 sec which are 7500 samples, the first 60 sec were used to train the network, and the last 15 sec were used for test and validation of the network. The experimental data were obtained using LabView real time software. All signals were saved to a file in cRIO. The experimental data are shown in Fig. 6.
Figure 5. Experiment system. Carrier vehicle with inverted pendulum.
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Figure 6. SIPC experimental data acquired: (a) Pendulum angular position, (b) Pendulum angular velocity, (c) Cart displacement, (d) Cart velocity, (e) Force.
Several VPBF NN algorithms were built, using MATLAB with training data, using different VPBF architectures, changing maximum delays nu , ny and system’s order o, in order to determine the architecture model that best fits the experimental data.
3. Results and discussion We assume that the model structure has the same delay in the input and output. Several models with different structures were built, which are combinations between order (o) and delays of the inputs/outputs (nu = ny ) being varied between 2 and 4. This was done for each output variable of the SIPC, i.e., x, x˙ , θ and θ˙ using training experimental data (0–60 sec).
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Table 2. NRSS train results. nu,y 2 3 4 2 3 4 2 3
o 2 2 2 3 3 3 4 4
x 0.0092 0.0081 0.0076 0.0082 0.0069 0.0059 0.0078 0.0062
x˙
θ
θ˙
0.1856 0.1759 0.1657 0.1704 0.1489 0.1293 0.1617 0.1289
3.06 × 10−4 1.87 × 10−4 1.78 × 10−4 1.86 × 10−4 1.62 × 10−4 8.18 × 10−5 1.80 × 10−4 1.47 × 10−4
0.0492 0.0444 0.0419 0.047 0.0396 0.0333 0.0457 0.0359
Bold values indicate the best models in the training process.
Table 3. NRSS test results.
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nu,y
o
x
2 2 0.0121 3 2 0.011 4 2 0.0095 2 3 0.0113 3 3 0.0089 4 3 0.0052 2 4 0.0107 33 4 0.0066 Bold values indicate the best models in the testing process.
x˙
θ
θ˙
0.3463 0.3167 0.2668 0.305 0.2476 0.1251 0.2839 0.1629
63 × 10−4 62 × 10−4 58 × 10−4 62 × 10−4 61 × 10−4 15 × 10−4 62 × 10−4 49 × 10−4
0.0484 0.0421 0.0372 0.0424 0.0329 0.016 0.0414 0.0236
Table 2 shows the results of all built models with classical Gram Schmidt method to get orthogonal decomposition. The model’s structure that best fits the experimental training data is marked in bold. The VPBF NNs were built with the first 20 VPBFs and their respective weights, the basic functions were reordered from highest to lowest contribution to the outputs using ri from Eq. (25). These models were tested with experimental test data (60–75 sec) and their respective calculated NRSSs, which were different but consistent with the training NRSSs. The model test results are shown in Table 3. It is clear here that NRSSs from the model’s test are not equal to NRSSs from the model’s training. The model structure with better performance is nu = ny = 4 maximum delays, o = 3 system order. The difference between normalized residuals in the training process and normalized residuals in the testing process is due to the model information from experimental data test which is not incorporated into the model. Hence, the NRSSs from the model with training data and NRSSs from the model with test data are different but very close. This means that the model has enough information to emulate the real system in states near to the experimental training data. The graphs in Fig. 7 show the experimental test and estimated output signals by selected models of VPBF NNs. In this figure it can be seen that the selected models fit well to the experimental data test but these have not been used for training the network, which means that the model is well trained and has the actual system information. The structures of identification algorithm based on VPBF neural networks with the same number of basis function have different accuracy for the same dynamic system. It means that structure of the model in important but different for each dynamical system. This identification model is valid only for invariant time dynamical systems; if the system will change could be necessary to retrain the model. In applications where the mechanism is time variant, different approach to fit structure and parameters online will be necessary in order to guarantee optimal modeling.
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Figure 7. Validation curves for: (a) Pendulum angular position, (b) Pendulum angular velocity, (c) Cart displacement, (d) Cart velocity.
4. Conclusions The complex dynamic systems containing nonlinear relations are difficult to model with conventional techniques. The inverted pendulum is a nonlinear dynamic system with nontypical uncertainty. Although the uncertainty is not dealt with, it affects the performance. For identification of the present problem, an offline nonlinear identification scheme based on VPBF neural networks and orthogonal least-squares has been applied. The orthogonal least squares algorithm was used to find a set of VPBF terms which were ordered according to the reduction in the approximation error. The structure selection in the VPBF neural network was established with the first 20 best nonlinear polynomials and the lowest approximation error obtained from different combinations of polynomial order with input–output signal delays. In this work, the VPBF neural network model has been developed using experimental data from real inverted pendulum on a moving cart system. The VPBF neural networks present computational complexity reduced, rate of convergence, and good model accuracy. The VPBF networks approach may be used in offline identification of complex mechanisms due to its good performance.
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