Nonlinear quadratic Gaussian control with loop transfer recovery

Mechatronics 13 (2003) 273–293

Technical Note

Nonlinear quadratic Gaussian control with loop transfer recovery Seong Ik Han a, Jong Shik Kim a

b,*

Department of Control and Instrumentation, Suncheon First College, Suncheon 540-744, Republic of Korea b School of Mechanical Engineering and RIMT, Pusan National University, Pusan 609-735, Republic of Korea Received 4 September 2000; accepted 28 June 2001

Abstract A nonlinear quadratic Gaussian with loop transfer recovery control method called NQG/ LTR method is presented, which is the integration of statistical linearization, loop-shaping and loop transfer recovery techniques. The proposed NQG/LTR method is a powerful one for designing controllers of nonlinear systems with hard nonlinearities such as Coulomb friction, backlash and dead-zone. In order to show the effectiveness of the proposed method, the LQG/ LTR and NQG/LTR methods are applied to a timing-belt driving cart system with Coulomb friction and dead-zone. The results of simulation and experiment for a cart system show that the performance and stability robustness of the NQG/LTR control system are insensitive to the effects of the nonlinear elements of a cart system. Ó 2003 Elsevier Science Ltd. All rights reserved.

1. Introduction In general, nonlinear effects such as Coulomb friction, backlash and dead-zone are neglected in the design of controllers. However, sometimes the desirable performance and stability robustness cannot be obtained in linear control systems which neglected nonlinear effects. Also, stability robustness problem is important in real control systems. Most control designs are based on the use of a design plant model,

*

Corresponding author. Fax: +51 512 9835. E-mail address: [email protected] (J.S. Kim).

0957-4158/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. PII: S 0 9 5 7 - 4 1 5 8 ( 0 1 ) 0 0 0 5 3 - 8

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in which modelling errors always exist. Therefore, a robust control system should be designed in the presence of system uncertainties. In previous researches, Taylor and Strobel [1] designed a fully nonlinear PID compensator for hard nonlinear systems via sinusoidal input describing function (SIDF). Also using the SIDF method, Olsson [2] researched the prediction of limit cycles in control system with the stick–slip friction. Chen et al. [3] and Shahruz [4] developed the control method for the system with nonsmooth nonlinearities such as dead-zone, Coulomb friction by disturbance observer. Kwatny et al. [5] and Ha et al. [6] developed the method of variable structure system control for dry friction. These techniques, however, are useful for the single-input single-output case, but it is not suitable for the multi-input multi-output case. Beaman [7] suggested the nonlinear quadratic Gaussian (NQG) control for hard nonlinear multivariable systems. The NQG control method includes the optimal estimation and control for statistically linearized systems. It has some numerical problem because the NQG compensator should be synthesized by solving the modified control algebraic Riccati equation (CARE), filter algebraic Riccati equation and Lyapunov equation for the compensated plant, simultaneously. Furthermore, the NQG control method cannot address the performance and stability robustness problem completely. The nonlinear quadratic Gaussian with loop transfer recovery (NQG/LTR) control method is presented for solving these issues in nonlinear multivariable systems with hard nonlinearities. The NQG/LTR method is the integration of statistical linearization [8,9] of nonlinear systems, target filter loop design [10,11], loop transfer recovery (LTR) using the cheap control nonlinear quadratic regulator (NQR) problem (7) and inverse random input describing function (IRIDF) method [12]. The NQG/LTR control method uses loop-shaping techniques to address the performance and stability robustness problem. By considering the fictitious process and measurement noises instead of the real driving noises, the desirable loop transfer functions that satisfy the performance and stability robustness requirements can be achieved. The structure of NQG/LTR control system is exactly the same as that of NQG control system, but the method of choosing the design matrices, control gain G and filter gain H, is different. In the NQG control, the design matrices are chosen for minimizing a least-square error. And, in the NQG/LTR control, they are chosen for loop shaping to achieve desirable loop transfer functions. In addition, the required computation to choose design matrices is much simpler in the NQG/LTR control. Because, the design matrices can be chosen separately by using the separation property [13] of the statistical linearized eigenvalues and by neglecting the correction term in the modified CARE [14]. In this paper, the NQG/LTR method is applied to a timing-belt driving cart system with Coulomb friction and dead-zone as a design example. In order to show the effectiveness of the NQG/LTR method, the linear control systems using the LQG/LTR method and the nonlinear control system using the NQG/ LTR method are synthesized and compared through both simulation and experiment.

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275

2. NQG/LTR control method The following nonlinear plant dynamics are considered: x_ ðtÞ ¼ f ðxðtÞÞ þ BuðtÞ þ CwðtÞ;

ð1Þ

where xðtÞ is the ðn  1Þ plant state vector, f ðxðtÞÞ is an ðn  1Þ vector, uðtÞ is the ðm  1Þ control input vector, and wðtÞ is the ðp  1Þ disturbance input vector. And it is assumed that all the nonlinearities are symmetric and single-valued. Then, Eq. (1) can be linearized via statistical linearization techniques. If the statistically linearized system is stabilizable and detectable, then the nonlinear model based compensator can be designed by the NQG/LTR method. The NQG/LTR control system, which is shown in Fig. 1, can be represented as follows. Statistically linearized plant x_ ðtÞ ¼ Nðrx ÞxðtÞ þ BuðtÞ þ CwðtÞ;

ð2Þ

where Nðrx Þ is the ðn  nÞ statistically linearized plant matrix, and rx is the standard deviation of the plant states Measurement:

yðtÞ ¼ CxðtÞ þ vðtÞ;

ð3Þ

where yðtÞ is the ðm  1Þ measured output vector, and vðtÞ is the ðm  1Þ measurement noise vector Control:

uðtÞ ¼ GzðtÞ

ð4Þ

where G is the ðm  nÞ control gain matrix, and zðtÞ is the ðn  1Þ compensator state vector. Model based compensator: z_ ðtÞ ¼ Nðrz ÞzðtÞ þ BuðtÞ þ HðyðtÞ  CðtÞ  rðtÞÞ;

Fig. 1. Structure of the NQG/LTR control system.

ð5Þ

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where Nðrz Þ is the ðn  nÞ statistically linearized compensator matrix, rz is the standard deviation of the compensator states, H is the ðn  mÞ filter gain matrix, and rðtÞ is the ðm  1Þ command input vector. By combining (2) and (5), the statistically linearized compensated plant dynamics can be represented as follows: 8 9
  
  < rðtÞ = BG x_ ðtÞ Nðrx Þ xðtÞ 0 C 0 wðtÞ : ¼ þ HC Nðrz Þ  BG  HC z_ ðtÞ zðtÞ H 0 H : ; vðtÞ ð6Þ In (6), the statistically linearized values of the elements of the plant ðNðrx ÞÞ and compensator ðNðrz ÞÞ are the same since the compensator is a model of the statistically linearized plant. However, since the statistics of the plant and compensator states are different it follows that the nonlinear functions which produce these statistically linearized elements, in general, will be different. The design procedure of the NQG/LTR control system is as follows: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

15. 16.

Determine a mathematical model for the nonlinear plant to be controlled. Analyse the linearized system via statistical linearization techniques. Determine the design specifications. Determine the several zero mean white noise inputs which should represent an operating range of interest. Select an operating point to design a linear controller. Estimate the describing function (DF) gains for nonlinearities at the selected operating point. Do loop shaping of the target filter loop. Accomplish the loop transfer recovery using the cheap control NQR problem. Solve the Lyapunov equation for the compensated plant. Calculate the DF gains for nonlinearities. Compare the estimated DF gains with the computed ones and repeat steps (6)– (11) until the difference between them is small enough. Store the gains (filter, control and DF) and the standard deviations (compensator states and filter innovations). Repeat the design procedure from steps (5)–(12) for each operating point. Determine the relationships between the gains (filter, control and DF) and the stationary statistics of the system, i.e., Hðrf Þ, Gðrz Þ and Nðrz Þ where rf and rz are the standard deviations of the filter innovations and compensator states, respectively. Synthesize the desired nonlinear functions via the IRIDF techniques. Implement the final nonlinear controller and evaluate the time responses of outputs and controls.

Now, the major design procedure of the NQG/LTR method will be discussed in detail.

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2.1. Design of the target filter loop If the target filter loop is broken at the output or, equivalently, at the error signal, the target filter loop transfer function matrix Gf ðsÞ is described as follows: 1

Gf ðsÞ ¼ CðsI  NÞ H:

ð7Þ

For the loop shaping of Gf ðsÞ, fictitious process and measurement noises are considered and the Kalman filter frequency domain equality (10) is used. Then, the statistically linearized design plant dynamics with fictitious noises are given by x_ ðtÞ ¼ Nðrx ÞxðtÞ þ BuðtÞ þ LnðtÞ;

yðtÞ ¼ CxðtÞ þ hðtÞ;

ð8Þ

where nðtÞ is the fictitious process white noise and hðtÞ is the fictitious measurement white noise, i.e., E½nðtÞ ¼ 0;

E½nðtÞnT ðsÞ ¼ Idðt  sÞ;

E½hðtÞ ¼ 0;

E½hðtÞhT ðsÞ ¼ lIdðt  sÞ:

The matrix L and scalar l are available as design parameters for the target filter loop design. And the filter gain matrix based on the fictitious noises is given by 1 H ¼ PCT ; l

ð9Þ

where P is the solution of the filter algebraic Riccati equation 1 NP þ PNT þ LLT  PCT CP ¼ 0: l

ð10Þ

By using the Kalman frequency domain equality, the following result can be obtained: 1 1 Gf ðsÞ ffi pffiffiffi CðsI  NÞ L: l

ð11Þ

The above approximation provides valuable insight into the selection of the design parameters L and l. 2.2. LTR using the cheap control NQR problem The NQR problem combines the use of statistical linearization and linear quadratic regulator optimal control theory. From Beaman’s results [7], the NQR problem is summarized as follows: Cost:

1 J ¼ E½xT Qx þ quT u; 2

ð12Þ

where Q is the state weighting matrix, and q is the control weighting parameter State: Control:

x_ ¼ Nx þ Bu þ Cw; u ¼ Gx;

ð13Þ ð14Þ

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Control gain:

1 G ¼ BT S; q

ð15Þ

where S is the Riccati matrix. Modified CARE 1 Q þ SN þ NT S  SBBT S þ WðS; N; XÞ ¼ 0; q where X is the state covariance matrix

oN Wij ðS; N; XÞ ¼ 2tr S X : oXij

ð16Þ

ð17Þ

Lyapunov equation T

ðN  BGÞX þ XðN  BGÞ þ CWCT ¼ 0;

ð18Þ

where W is the disturbance covariance matrix. The LTR is accomplished by solving the cheap control NQR problem to recover the target filter loop shape. If the solution of this problem is to be valid, ½N; B must be stabilizable and ½N; C must be detectable. For the LTR, we examine the limiting behaviour of the modified CARE (16) with Q ¼ CT C as q ! 0. 1 CT C þ SN þ NT S  SBBT S þ WðS; N; XÞ ¼ 0: q

ð19Þ

Now let us check the order of magnitude of each term in (19) as q ! 0. kWk is approximately kSNk from (17) and kCT Ck and kBBT k are finite. For the good LTR, the following conditions should be satisfied: 1 1 kCT Ck ffi kSBBT Sk 6 kBBT k kSk2 q q

ð20Þ

or kSk P

q kCT Ck kBBT k

1=2 ð21Þ

and kSNk 6 kSk kNk  kCT Ck

ð22Þ

or kNk 

kBBT k kCT Ck q

1=2 :

ð23Þ

For convenience, let us express (23) in another form which is called the LTR index(a) as follows:

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a ¼ kNk

q kBB k kCT Ck T

279

1=2  1:

ð24Þ

In order to check if a approaches 0 as q ! 0, the LTR condition is checked for a scalar case. We assume B ¼ C ¼ 1 for simplicity. Then, a scalar control gain G is calculated from Eqs. (15) and (16) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G ¼ ð1 þ kÞN þ ð1 þ kÞ2 N 2 þ 1=q; ð25Þ where k is a constant that depends on the nonlinearity. For example, k ¼ 1 if f ðxÞ ¼ x3 and k ¼ 1=2 if f ðxÞ ¼ sgnðxÞ. A scalar filter gain H is calculated from (9) and (10) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H ¼ N þ N 2 þ 1=l: ð26Þ A scalar variance of state X is calculated from (36) X ¼

RG2 H 2 ; ð2N  G  H ÞN ðN  G  H Þ þ GH

ð27Þ

where R is the white noise intensity of the input. Now let us check the order of magnitude of G and H, i.e., oðGÞ and oðH Þ ( oð2N Þ for jN j  p1ffiffiq ;   oðGÞ ¼ ð28Þ o p1ffiffiq for jN j  p1ffiffiq ; ( oð2N Þ for jN j  p1ffiffil ;   ð29Þ oðH Þ ¼ o p1ffiffil for jN j  p1ffiffil : From the above statements, we will prove that if the LTR condition is satisfied, the LTR index a approaches 0. Theorem 1. If q approaches 0 in the cheap control NQR problem, then a approaches 0. Proof 1. The procedures of proof are divided into four cases: Case (1): 1 jN j  pffiffiffi q

ða  1Þ:

Since oðGÞ and oðH Þ are oð2N Þ, the order of magnitude of X is oðRN Þ. If the input R is finite, X is finite in the stable system, and then N has a finite value. If we select pffiffiffi q ! 0, the given condition ðjN j  1= qÞ is not satisfied. Therefore, the possibility of a  1 does not exist for a finite input R and q ! 0. Case (2): 1 jN j ffi pffiffiffi q

ða ffi 1Þ:

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Since oðGÞ and oðH Þ are oð2N Þ, the order of magnitude of X is also oðRN Þ. In a similar manner of the above case, the possibility of a ffi 1 does not exist for a finite input R and q ! 0. Case (3): 1 1 pffiffiffi  jN j  pffiffiffi ða  1Þ: l q pffiffiffi Since oðGÞ is oð1= qÞ and oðH Þ is oð2N Þ, the order of X is oðRN Þ. If the input R is finite, X is finite in the stable system, and then N has a finite value. Therefore, if q ! 0, the condition ða  1Þ is satisfied. Case (4): 1 1 jN j  pffiffiffi  pffiffiffi ða  1Þ: l q pffiffiffi pffiffiffi Since oðGÞ is oð1= qÞ and oðH Þ is oð1= lÞ, the order of magnitude of X is oðR=lN Þ. If R and l are finite and q ! 0, then the condition ða  1Þ is satisfied, and kN k is finite if the inputs are finite in the stable system,. Therefore, if q approaches 0 in the cheap control NQR problem, a approaches 0.  Theorem 1 means that the LTR condition for the statistically linearized system is basically the same as in the linear case. Note that the magnitude of the correction term kWk in the modified CARE is the order of a. Therefore, if the LTR conditions are satisfied, the correction term in the modified CARE can be neglected. Then, the modified CARE has the same form as the standard CARE. If the systems satisfy the LTR condition, we have the following limiting behaviour of the modified CARE as q ! 0



1 1 CT C  pffiffiffi SB ð30Þ pffiffiffi BT S ! 0: q q Substituting (15) into (30) pffiffiffi T pffiffiffi ð qGÞ ð qGÞ ! CT C;

ð31Þ

which implies that pffiffiffi lim qG ! UC;

ð32Þ

q!0

where U is the ðm  mÞ unitary matrix, i.e., UT U ¼ I. Now, consider the model based compensator, KðsÞ 1

KðsÞ ¼ GðsI  N þ BG þ HCÞ H:

ð33Þ pffiffiffi If Re ki ½N  BG < 0, Re ki ½N  HC < 0 and limq ! 0 qG ! UC, then the limiting behaviour of the model based compensator KðsÞ as q ! 0 is as follows [11,14]: 1

1

1

lim KðsÞ ¼ ½CðsI  NÞ B ½CðsI  NÞ H ¼ GðsÞGf ðsÞ:

q!0

ð34Þ

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281

Then, the limiting behaviour of the loop transfer function matrix at the plant output, LðsÞ is given by lim KðsÞ ¼ GðsÞG1 ðsÞGf ðsÞ ¼ Gf ðsÞ:

ð35Þ

q!0

2.3. The Lyapunov equation for the compensated plant Determining the stationary statistics of the closed-loop system requires a little more thought than in the general linear quadratic Gaussian control design situation. This is due to the fact that the fictitious (design) noise and the real driving noise are not the same, in general. The target Kalman filter is not an optimal filter for the real driving noise. The target filter loop is designed to make the desired loop shape with fictitious noise. Therefore, the Lyapunov equation for the compensated plant (36) should be solved in order to calculate the DF gains and stationary statistics of the states. The Lyapunov equation for the compensated plant is derived from (6) Nt Xt þ Xt NTt þ Ct Wt CTt ¼ 0;

ð36Þ

where 

   BG X Y ; Xt ¼ ; N  BG  HC Y Z 2 3   R 0 0 0 C 0 Ct ¼ ; Wt ¼ 4 0 W 0 5; H 0 H 0 0 V Nt ¼

N HC

T

X ¼ E½xðtÞxðtÞ ; T

R ¼ E½rðtÞrðtÞ ;

T

Y ¼ E½zðtÞ~ xðtÞ ; T

W ¼ E½wðtÞwðtÞ ;

T

Z ¼ E½zðtÞzðtÞ ; T

V ¼ E½vðtÞvðtÞ ;

~ðtÞ ¼ xðtÞ  zðtÞ: x If the correction term (17) is considered in the cheap control NQR problem, then the modified CARE (16) and Lyapunov equation for the compensated plant (36) must be solved simultaneously with the guessed unknown variables ðXt ; nð2n þ 1Þ; S; nðn þ 1Þ=2Þ, where n is the number of design plant states. It is very difficult to find a solution for high order systems. In addition, it requires a great deal of computation time. However, if the correction term is neglected in the cheap control NQR problem, the CARE and Lyapunov equation are not coupled. Then, these two equations can be solved separately. Fortunately, the correction term W is not dominant in the good LTR, i.e., lim W ! 0 [9]. Therefore, it can be neglected in this situation. Then, we can calculate

q!0

the control gains from the LTR procedure and the stationary statistics of the system from the Lyapunov equation separately.

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2.4. Inverse random input describing function techniques Linear compensators that are designed for the several fictitious operating points have to be combined to form a single compensator for the nonlinear systems. The nonlinear gain functions are synthesized from the set of statistically linearized gains obtained as linear design results. This procedure can be executed via the IRIDF techniques. The IRIDF techniques are concerned with the determination of a nonlinear function of a variable, given the statistically linearized gain of the function and the corresponding stationary statistics of the variable.

Fig. 2. A visualization of the IRIDF techniques.

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283

Mathematically, the IRIDF problem is concerned with the determination of f ðxÞ, given the gain N ðrx Þ, where rx is the standard deviation of x, assuming zero mean. It is possible to factor the given gain into some convenient constituent gains N ðrx Þ ¼ N1 ðrx Þ þ N2 ðrx Þ þ þ Nn ðrx Þ:

ð37Þ

The corresponding inverse DF can be shown to be f ðxÞ ¼ f1 ðxÞ þ f2 ðxÞ þ þ fn ðxÞ;

ð38Þ

where each fi ðxÞ is the inverse DF of the corresponding Ni ðrx Þ. It is thus possible to factor the unknown gain into constituent gains of which the inverse describing functions are either known or easy to determine. This can be done in an iterative way, each time improving the approximation until the desired level of accuracy has been reached. The IRIDF techniques are shown schematically in Fig. 2 with a simple example.

3. Design example 3.1. Problem formulation A timing-belt driving system with Coulomb friction and dead-zone is selected as a nonlinear control system design example using the NQG/LTR method. A schematic diagram of the mechanism is shown in Fig. 3. In servomechanisms, in order to transmit power between axles or to carry a cart in a rectilinear motion, a timing-belt is often used. Timing-belt systems need no lubrication and no sliding between belt and pulley teeth and have fast transmission of power with lower noise than gear systems. Specially, when the distances between two axles are large, the timing-belt is very appropriate. Because of these properties, the timing-belt is often adapted for robot systems and many automatic mechanisms. However, if the distances between

Fig. 3. Schematic diagram of a timing-belt driving cart system.

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the axles are larger and moving loads connected to the belt exist, a guide for moving loads must be considered to prevent the deflection of the belt. In this case, the Coulomb friction between the contacting surfaces of guides and moving loads can appear. In addition, a dead-zone may be accompanied due to the loose engagements between pulley and belt. Therefore, these nonlinearities must be considered where the timing-belt is chosen as a power transmission device. In this paper, the cart is connected with a timing-belt in its lower part, and moves through two sliding guide axles. Two pulleys are connected to a timing-belt, and one side of the pulley is combined with a servo DC motor. The servo DC motor is an LG Corporation FMD-E10EA with attached incremental rotary encoder with 1000 pulse/rev resolution, driven by an FDD-102PD motor driver. The PC containing DSP system composed of TMS320C processor reads the pulse from the rotary encoder and sends the control command through D/A converter to the motor driver. The photograph of the cart system is given in Fig. 4. In this cart system, a Coulomb friction exists between the sliding guide and cart connection point and a dead-zone phenomenon is observed in the combing part of the pulley and the timing-belt. Also, the cart system has the motions of translation and rotation. In order to simplify the

Fig. 4. Photograph of a timing-belt driving cart system.

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285

problem, two motions can be transformed into an equivalent translation motion as the following dynamic equation: Meq€xðtÞ þ Ceq x_ ðtÞ þ feq sgn ð_xÞ ¼ Dz Feq ðtÞ;

ð39Þ

where xðtÞ is the position of the cart and Meq , Ceq , feq , Dz, Feq ðtÞ represent the equivalent values of mass, friction, Coulomb friction force, dead-zone of the entire system, and linear force, respectively. Table 1 shows the values of the system parameters. If the nonlinear plant is linearized via statistical linearization techniques, then the statistically linearized state space model can be described as follows: x_ ðtÞ ¼ Nðrx ÞxðtÞ þ BuðtÞ;

yðtÞ ¼ CxðtÞ;

ð40Þ

where

    0 1 0 Nðrx Þ ¼ 0  Ceq þNf ; B ¼ Kamp N ; Dz Meq Meq pffiffiffiffiffiffiffiffi feq 2=p d Nc ¼ ; Ndz ¼ 1  erf ; ru rx2

C ¼ ½1

0;

where Nf and Ndz are the DF gains for Coulomb friction, Kamp is the gain of amplifier, d is the width of dead-zone, and rx2 and ru are the standard deviations of the states x2 and control input uðtÞ, respectively. The design specifications considered are as follows: 1. Steady-state tracking errors should be zero for any constant inputs. 2. Gain crossover frequency should be about 5 rad/s. 3. The maximum singular value of the sensitivity transfer function matrix should be less than )20 dB for all x < 1 rad=s for good command following and disturbance rejection. 4. The maximum singular value of the closed-loop transfer function matrix should be less than )20 dB for all x < 100 rad=s for stability robustness to unmodelled dynamics and insensitivity to sensor noise. Table 1 The values of the system parameters System parameter

Specification

Guide length (cm) Distance between pulley (cm) Guide diameter (cm) rp (cm) Pitch of pulley (cm/rev) Meq (kgf s2 =cm) Ceq (kgf s=cm) Cfeq (kgf) d (width of dead-zone) Kamp (kgf cm=V)

100 92 2 2.83 17.78

Identified value

0.003768 0.2485 0.4271 0.2512 1.571

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3.2. Linear controller design using the LQG/LTR method A linear plant  is required to apply the LQG/LTR method. Thus, Coulomb friction feq sgnð_xÞ and dead-zone (Dz) are assumed to be neglected. Then the design plant model dynamics are represented as follows: x_ ðtÞ ¼ AxðtÞ þ BuðtÞ;

yðtÞ ¼ CxðtÞ;

ð41Þ

where  0 A¼ 0

 1 Ceq ;  Meq

 B¼

0 Kamp Meq

 ;

C ¼ ½1

0:

The design plant model is found to be completely controllable from the input uðtÞ and completely observable through the output yðtÞ, and is also a minimum phase plant. Therefore, an LQG/LTR controller can be designed with a guarantee of good LTR. The target filter loop is designed by matching the high frequency singular values. Then the design parameter L is chosen as follows (11): T

L ¼ ½ 1 0 :

ð42Þ

To determine the filter gain matrix H, the desired crossover frequency was specified as 5 rad/s. A value of 0.085 for l is found to provide a crossover frequency of about 6 rad/s for the target filter loop. This leaves us with some safety margin in the recovery phase. After choosing L and l to satisfy the desired target filter loop shaping, the filter gain matrix H is calculated from (9) and (10). The resulting filter gain matrix H is: H ¼ ½6:3246

T

0 :

LTR is attempted with the cheap control LQR problem. The target filter loop is usually recovered up to a decade beyond the crossover frequency [11]. This level of recovery is obtained with a value of 105 for q. Then the control gain matrix G is calculated from (15) and (16) without the correction term (17) as follows: G ¼ ½100

0:56701:

Let us evaluate the performance and stability robustness for the nonlinear plant with the LQG/LTR compensator. For this purpose, the frequency responses are checked for three different command inputs, which are assumed as zero mean white noises for the statistical linearization of a nonlinear plant. The white noise intensities of the selected command inputs ðRÞ are 5  104 , 107 and 109 which represent the small, medium and large input cases, respectively. Singular values of the loop transfer function matrix and target filter of the nonlinear plant with the LQG/LTR compensator are shown in Fig. 5, and the experimental normalized step responses of the nonlinear plant with the LQG/LTR compensator are shown in Figs. 6 and 7. The LQG/LTR control system maintains the stability robustness for any input magnitude and direction, but it does not meet the performance requirements for

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287

Fig. 5. Singular values of the loop transfer function matrix and target filter loop of the nonlinear plant with the LQG/LTR compensator.

Fig. 6. Normalized step responses of the LQG/LTR control system; a case of the small reference input (experiment).

small inputs. In the time response, there are some steady-state errors for most of the step inputs. This is due to the effect of the Coulomb friction and dead-zone. However, as the size of the step inputs get larger, it is seen that the steady-state errors decrease. This is because the control forces of the case in a larger step input are greater compared to the case of a small step input. Thus, the nonlinear effects on the performance of the control system vary according to the magnitude of the control

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Fig. 7. Normalized step responses of the LQG/LTR control system; a case of the large reference input (experiment).

force. In addition, some overshoot exists for large inputs due to the elastic of the timing-belt. In this paper, this problem is not dealt with. Therefore, the linear LQG/LTR compensator cannot be used for a large operating range. In order to improve the performance, a nonlinear compensator is required which can capture the effect of the Coulomb friction and dead-zone and in addition, adapt to changes in the magnitude and direction of the command input. 3.3. Nonlinear controller design using the NQG/LTR method The statistically linearized plant (40) and the selection of several operating points to cover an operating range of interest are required to apply the NQG/LTR method. The zero mean white noise intensities of the command inputs ðRÞ are selected between 5  104 and 109 . Steps (6)–(11) of the NQG/LTR design procedure are executed for a linear design at each selected operating point. Here, the final results of singular values for three input cases (R ¼ 5  104 ; 107 and 109 ) are presented. The gains (filter, control and DF) and the stationary statistics (compensator states and filter innovations) are stored for all linear designs. The control gains are almost the same and the filter gains are the same as the LQG/LTR case for any input R. Thus, the control gain matrix G and the filter gain matrix H are chosen as the LQG/ LTR case. And, the DF gains Ncz and Ndz and the standard deviations of the compensator states at all operating points are given in Table 2. The desired nonlinear functions fcz ðtÞ and fdz ðtÞ for Coulomb friction and dead-zone are obtained via the IRIDF techniques, respectively, which are shown in Fig. 8. Singular values of the loop transfer function matrix and target filter of the nonlinear plant with the nonlinear NQG/LTR compensator are shown in Fig. 9. The

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Table 2 DF gains and standard deviations of the compensator states at all operating points R

5  104

105

5  105

106

107

108

109

Nfz NDz rz2 ru

0.05859 0.9991 5.817 331.3

0.03918 0.9993 8.697 386.2

0.01622 0.9995 21.01 563.6

0.01126 0.9996 30.27 666.8

0.00345 0.00108 0.00034 0.9998 0.9999 0.9999 98.73 315.3 1000 1177 2088 3710

Fig. 8. Desired nonlinear functions via the IRIDF techniques.

Fig. 9. Singular values of the loop transfer function matrix and target filter loop of the NQG/LTR control system.

simulation results of the normalized step responses of the nonlinear plant with the NQG/LTR compensator are shown in Figs. 10 and 11, and the experimental results are shown in Figs. 12 and 13. From the simulation and experimental results, it is

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Fig. 10. Normalized step responses of the NQG/LTR control system; a case of the small reference input (simulation).

Fig. 11. Normalized step responses of the NQG/LTR control system; a case of the large reference input (simulation).

found that the NQG/LTR control system is insensitive to the magnitude and direction of the input. In addition, no steady-state error exists for constant inputs so that the precise position control of the cart can be done. But unlike simulation results, in experimental results the small overshoot and delay of the rising time can be seen by the large magnitude of the inputs like the LQG/LTR control system due to the elastic effect of the timing-belt.

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Fig. 12. Normalized step responses of the NQG/LTR control system; a case of the small reference input (experiment).

Fig. 13. Normalized step responses of the NQG/LTR control system; a case of the large reference input (experiment).

3.4. Comparison of the LQG/LTR and NQG/LTR compensators In the frequency response, the LQG/LTR control system meets the design specifications for a small operating range, but the NQG/LTR control system meets them for the entire operating range. And in the time response, the LQG/LTR control system has steady-state errors for constant inputs even if the system has free integrators,

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because the LQG/LTR compensator cannot adapt to the effect of the Coulomb friction and dead-zone which cannot be captured in the linear model. In addition, the system responses are sensitive to the input direction. However, the NQG/LTR control system meets the design specifications for all the operating ranges. The system responses are insensitive to the magnitude and direction of the command input. The response has no steady-state error and has fast settling time for all the operating ranges. This is because, the NQG/LTR compensator reflects on its model and, therefore, can adapt to the effect of the Coulomb friction and dead-zone.

4. Conclusions An NQG/LTR design methodology has been presented. The method is essentially an integration of statistical linearization, loop-shaping and loop transfer recovery techniques. By using statistical linearization techniques, nonlinear effects are considered in the design of nonlinear compensators that adapt to changes in input magnitude for nonlinear systems with hard nonlinearities such as Coulomb friction, backlash and dead-zone. In addition, by using loop-shaping techniques the performance requirements and stability robustness can be addressed simultaneously. The LTR conditions for the statistically linearized system were discussed. It is found that the LTR conditions for the statistically linearized system are basically the same as in the linear case. The only difference is that the cheap control NQR problem includes the correction term in the modified CARE. Fortunately, the correction term is not dominant in the good LTR. Therefore, it can be neglected in this situation. Then, the modified CARE has the same form as the standard CARE, and the CARE and the Lyapunov equation are not coupled. Therefore, the required computation becomes much simpler by neglecting the correction term. Finally, the NQG/LTR method is applied to a cart system with Coulomb friction and dead-zone. It is verified that the NQG/LTR compensator is insensitive to the magnitude and direction of the command input. Thus, the NQG/LTR method can be suggested to design controllers for the nonlinear multivariable systems with hard nonlinearities to meet the performance requirements and to maintain the stability robustness for a large operating range.

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