Indeed the Davinson robust linear regulator [2] incorporates dynamical systems that are copies of the one generating the exogenous signal (the exosyst...

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[4] I. Kanellakopoulos, P. V. Kokotovic, and A. S. Morse, “Systematic design of adaptive controllers for feedback linearizable systems,” IEEE Trans. Automat. Contr., vol. 36, pp. 1241–1253, Nov. 1991. [5] M. Krstic and P. V. Kokotovic, “Adaptive nonlinear design with controller-identifier separation and swapping,” IEEE Trans. Automat. Contr., vol. 40, pp. 426–440, Mar. 1995. [6] D. Seto, A. M. Annaswamy, and J. Baillieul, “Adaptive control of nonlinear systems with triangular structure,” IEEE Trans. Automat. Contr., vol. 39, pp. 1411–1428, July 1994. [7] S. S. Ge, “Adaptive control of uncertain lorenz system using decoupled backstepping,” Int. J. Bifurcation Chaos, 2004, to be published. [8] Z. P. Jiang and D. J. Hill, “A robust adaptive backstepping scheme for nonlinear systems with unmodeled dynamics,” IEEE Trans. Automat. Contr., vol. 44, pp. 1705–1711, Sept. 1999. [9] M. M. Plycarpou and P. A. Ioannou, “A robust adaptive nonlinear control design,” Automatica, vol. 31, pp. 423–427, 1995. [10] B. Yao and M. Tomizuka, “Adaptive robust control of siso nonlinear systems in a semi-strict feedback form,” Automatica, vol. 33, pp. 893–900, 1997. [11] R. Marino and P. Tomei, “Robust adaptive state-feedback tracking for nonlinear systems,” IEEE Trans. Automat. Contr., vol. 43, pp. 84–89, Jan. 1998. [12] S. S. Ge, C. C. Hang, and T. Zhang, “Stable adaptive control for nonlinear multivariable systems with a triangular control structure,” IEEE Trans. Automat. Contr., vol. 45, pp. 1221–1225, June 2000. [13] S. S. Ge and C. Wang, “Adaptive nn control of uncertain nonlinear purefeedback systems,” Automatica, vol. 38, pp. 671–682, 2002. [14] R. D. Nussbaum, “Some remarks on the conjecture in parameter adaptive control,” Syst. Control Lett., vol. 3, pp. 243–246, 1983. [15] B. Martensson, “Remarks on adaptive stabilization of first-order nonlinear systems,” Syst. Control Lett., vol. 14, pp. 1–7, 1990. [16] Z. Ding, “Adaptive control of nonlinear systems with unknown virtual control coefficients,” Int. J. Adapt. Control Signal Processing, vol. 14, pp. 505–517, 2000. [17] X. Ye and J. Jiang, “Adaptive nonlinear design without a priori knowledge of control directions,” IEEE Trans. Automat. Contr., vol. 43, pp. 1617–1621, Nov. 1998. [18] S. S. Ge and J. Wang, “Robust adaptive neural control for a class of perturbed strict feedback nonlinear systems,” IEEE Trans. Neural Networks, vol. 13, pp. 1409–1419, Nov. 2002. [19] X. D. Ye, “Asymptotic regulation of time-varying uncertain nonlinear systems with unknown control directions,” Automatica, vol. 35, pp. 929–935, 1999. [20] J. B. Pomet and L. Praly, “Adaptive nonlinear regulation: estimation from the lyapunov equation,” IEEE Trans. Automat. Contr., vol. 37, pp. 729–740, June 1992. [21] E. P. Ryan, “A universal adaptive stabilizer for a class of nonlinear systems,” Syst. Control Lett., vol. 16, pp. 209–218, 1991. [22] H. Khalil, Nonlinear Systems, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 1996. [23] P. A. Ioannou and J. Sun, Robust Adaptive Control. Upper Saddle River, NJ: Prentice-Hall, 1996. [24] A. Isidori, Nonlinear Control Systems. New York: Springer-Verlag, 1999, vol. II.

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Corrections to “Robust Control for Linear Discrete-Time Systems With Norm-Bounded Nonlinear Uncertainties” Yoonsun Kim and Youngjin Park Abstract—This note provides some corrections to the aforementioned paper.

In [1, eq. (7)], C and aMk E1 cannot be added because the dimensions of the matrices are different. Therefore, the consequent derivations have to be adjusted. We propose two methods that can be used to correct this problem without harming the main idea of [1]. The first method is zero padding to zk 2

zk

=

zk 0

;

0

2

(1)

The over bar represents a modification and the modified vector or matrix replace the original vector or matrix. Consequently, the matrices C , D1 , D2 are also changed as follows:

C

=

C

D1

0

=

D1 0

D2

=

D2 0

;

0

2

)

(2)

However, this method can introduce unnecessary conservatism. The second method is as follows. [1, eq. (7)] can be modified as

zk

= (C + aM k E1 )xk + (D1 + bM k E2 )wk

+ (D2 + cM k E3 )uk r2n M k = HMk ; H

2<

(3)

(H ) 1:

(4)

Consequently, [1, eq. (9)] is modified as

z~k

C D1 "H xk + 01 w ~k : "01 aE1 " bE2 0

=

(5)

With (5), [1, eqs. (10), (11), and (13)] have to be written as (6), (7), and (8), respectively

0P T 0

Y

=

0

AT B T "I <0 0P T

S YS

=

I~ =

0I CT D1T

0

0

"I "H T <0 I H : 0

0

0

(6) 0

AT B T

0

T

A B "I "H C D1 0 0I "01 aE1 "01 bE2 0 T 0 1 C " aE1T 0P 0 0 T D1 "01 bE2T 0 0I 0 0 0 0 0I "H T 0

0I

A C 0P

B D1

0

0

0 0

"01 aE1 0 0I "01 bE2T 0 1 T 0 0I " aE1 " 1 bE2T 0

0

"I "H 0 0 0

0I (7) (8)

0

Manuscript received June 15, 2002. Recommended by Associate Editor P. A. Iglesias. The authors are with the Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology (KAIST), Taejon 305-701, Korea. Digital Object Identifier 10.1109/TAC.2003.815765 0018-9286/03$17.00 © 2003 IEEE

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 8, AUGUST 2003

Finally, for the same reason, [1, eq. (15)] has to be changed to

~ = "01CaE1

zk

xk

"H D2 1 + "0D1bE 0 w~k + "01cE3 2

uk : (9)

This second method does not introduce unnecessary conservatism as in the first method, and it corrects the matrix dimension unmatching problem without harming the main idea of [1]. REFERENCES

H

[1] P. Shi and S.-P. Shue, “Robust control for linear discrete-time systems with norm-bounded nonlinear uncertainties,” IEEE Trans. Automat. Contr., vol. 44, pp. 108–111, Jan. 1999.

Comments on “Nonlinear Repetitive Control” Pasquale Lucibello

Abstract—The purpose of this note is to draw the reader’s attention to a central question of closed-loop stabilization of repetitive controllers through a brief focused excursus of the literature on this subject and some comments on a paper which recently appeared in these TRANSACTIONS. Index Terms—Repetitive control, robust control.

I. INTRODUCTION The name of repetitive control has been introduced by Nakano et al. (see [5] and the references therein). These authors addressed the problem of robustly regulating the output of a linear control system on a periodic signal. In order to build up such a robust control, Hara et al. [5] have proposed to introduce in the closed control loop a simple delay, i.e., a system capable of generating an arbitrary periodic signal. Unfortunately, in [5, Prop. 2], it is proven that if the transfer matrix of the linear finite dimensional system to be stabilized is strictly proper, the system obtained by adding a pure delay is not exponentially stabilizable through output feedback. The idea of including in the control loop a model of the disturbance signal and/or of the signal to be tracked has been proven to be very fruitful in the robust control theory of linear finite dimensional systems. Indeed the Davinson robust linear regulator [2] incorporates dynamical systems that are copies of the one generating the exogenous signal (the exosystem). The fact that a finite-dimensional, pure output feedback, linear robustly regulated system must incorporate suitable redundant copies of the exosystem has been cast by Francis and Wonham [3] in a principle known as the internal model principle. In case of plant perturbations relevant to each control input, according to this principle, a suitable copy of the exosystem has to be placed in each control channel in such

Manuscript received January 5, 2001; revised March 4, 2002 and August 29, 2002. Recommended by Associate Editor Y. Yamamoto. The author is with the Sogin—Società Gestione Impianti Nucleari, 6-00184 Rome, Italy (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2003.815047

a way that the extended system is exponentially stabilizable via output feedback. A different version of this principle has been later formulated for a class of nonlinear finite dimensional systems by Hepburn and Wonham [4], where it is proven that a feedback controller must incorporate a copy of the exosystem dynamics. A program similar to that of [2] has been carried out in [5] for linear repetitive control, while the internal model principle of [3] has been extended to pseudorational linear systems, a class of infinite dimensional systems, by Yamamoto and Hara [12]. However, as already said, in [5] it has been proven that exponential stabilization is not achievable for repetitive control systems with strictly proper transfer functions. This result is consistent with the general observation made in [12], where it is pointed out that stabilization of these systems is not a trivial problem since they turn out to be delay-differential systems of neutral type (see, e.g., [8]). In spite of this negative result, the use of a simple delay with a low-pass filter has been shown to be useful, once the idea of zero steady state error is removed and on the contrary a small steady state error is accepted [5]. The usefulness of this approach has been also shown by Omata et al. [6]. Claim of a good performance of a digital implementation of repetitive control has also been reported in [13]. Also, in this case, a small steady-state error is expected, since, owing to the sampling procedure, the internal model adopted is not a pure delay. In other words, a good performance has been claimed if an approximate model of the exogenous signal is incorporated in the control loop. The idea of using an approximate model has been also explicitly considered by Lucibello [7], even if by taking a different point of view. In [7], the model to be incorporated in the closed loop is a system capable of generating the control needed to track the desired output trajectory for all admissible plant perturbations (see [9] for a formulation of this approach), rather than a model of the exogenous signal. In particular if the steady-state control belongs to a class of periodic signal, like in repetitive control, the dynamical model to be incorporated must be able to generate any signal of that class. From this point of view, an approximate model is a system capable of generating input signals belonging to a subset of the class of the controls needed to track the desired output trajectory for all admissible plant perturbations. Also, the repetitive control scheme proposed by Messner et al. [10] is an example of a controller built up having in mind a model of the input rather than a model of the output. The model that Messner et al. [10] uses is infinite dimensional, but the problem of the stabilization of the closed loop is less severe since they suppose the availability of the full-state error and the perfect matching between input and disturbance. The previous excursus points out that closed-loop stabilization is a central question of repetitive control, as long as an infinite dimensional system (the internal model) is incorporated in the control loop. In other words, apparently, the more reach the model incorporated, the more difficult the stabilization of the closed loop. More recently, the repetitive control of the “beam and ball” nonlinear system has been considered in [1], by analyzing the effectiveness of a regulator incorporating harmonic oscillators with frequencies multiple of the fundamental frequency of the periodic signals to be rejected and/or tracked. The theoretical conclusion reported in [1] on the possibility of arbitrarily decrease the output error by increasing the number of oscillators incorporated in the controller is related to the above conjecture and in particular to the stability result presented in [5] previously recalled. These aspects are commented in the next section.

0018-9286/03$17.00 © 2003 IEEE

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