been paid to the design problems of reliable linear con- trol systems, and a number of ... and design method for linear systems subject to both. * This work is supported in ..... a class of open-loop unstable linear systems: theory and application.
Physics Department, Research unit Constrained and Robust Regulation,. P.O.Box 2390, Marrakesh, Morocco (e-mail: [email protected]). âââ Dpto. Ingenieria de Sistemas y Automatica,. Universidad de Valladolid, 47005 Valladolid, Spain (e-mail: ai
Abstract: The design of a repetitive control system for linear systems with time-varying norm- bounded uncertainties is presented. Using a Lyapunov functional for the time-delay systems, a sufficient condition is derived in terms of either an algebra
adding damping mechanisms to model-reference adaptive control. The robustness ... Optimal Control Modification for Linear Time Invariant Uncertain Systems.
symbolic models for the class of linear control systems with politopically bounded states and disturbances. We show that under an asymptotic stabilizability assumption, it is always possible to construct a symbolic model that approximates the control
May 15, 2002 - ing material for the course 'Control Theory for Linear Systems', given within the framework of the national Dutch graduate school of systems and control, in the pe- riod from 1987 to 1999. The aim of this course is to provide an extens
Abstract. The subject area of this paper is the application of systems theory developed for linear repetitive processes, a distinct class of 2D linear systems, to linear iterative learn- ing control schemes. A unique feature is the inclusion of exper
For discrete Ã¸me linear system: The magnitude of the real part of the eigenvalues of the state transiÃ¸on matrix should be less than 1. E.g., For A = 1, and B = 1,.
linear system involving some norm-bounded uncertainty in the state-space matrices, the ..... Therefore, by the well known bounded real lemma. 11], A is ...
time invariant systems subject to input constraints. ... LMI constraints, e.g. (Hindi and Boyd, 1998; Kapila ... of the domain of attraction when compared with the.
Assuming null controllability of the pair (A, B), several authors have proposed the quadratic family V (x, Î») = xT P(Î»)x generated from the Riccati equation. P(Î»)A + ...
Linear Quadratic Control for Sampled-data Systems with Stochastic Delays. Masashi Wakaiki1, Masaki Ogura2, and JoËao P. Hespanha3. AbstractâWe study optimal control for sampled-data sys- tems with stochastic delays. Assuming that the delays can be
Robust predictive control of non-linear systems under state estimation errors and input and state constraints is a challenging problem, and solutions to it have generally involved solving computationally hard non-linear optimizations. Feedback linear
This paper presents the optimal regulator for a linear system with mul- tiple time delays in control input and a quadratic criterion. The optimal regu- lator equations are obtained using the duality principle, which is applied to the optimal filter f
Linear matrix inequalities (LMIs) prove to be a useful tool for solving several analysis and design problems in linear systems control. However, LMI techniques have mainly been used conjointly with state-space methods. The aim of this paper is to sur
Sep 20, 2009 - This article presents and describes the new Lin-. earSystems and Controller libraries which are de- veloped to enhance analysis, design and simula- tion of linear control systems in Modelica. The. LinearSystems library contains basic f
Guanghui Wena, Zhongkui Lib, Zhisheng Duana* and Guanrong Chenac. aDepartment of ... and flocking (Su, Wang, and Lin 2009; Wen, Duan, Li, and Chen ...
system for both of state and output feedback and 2. design of stabilizing nonlinear feedback law. .... Proof The proof is based on the theory of linear matrix.
When planning trajectories for linear control systems, a demand that arises naturally in, for instance, air tra c control, noise contaminated data interpolation, and plan- ning for switched control systems, is that the curve inter- polate through giv
Aug 1, 2008 - From classical control theory, it is well-known that state-derivative feedback can be very ... control design is the use of state-derivative and state feedback. ...... The title of the book System, Structure and Control encompasses ...
Abstract. This paper gives a very brief overview of material of the short course entitled âNumerical. Methods for Linear Control Systemsâ delivered by Biswa Nath Datta and Daniel Boley at the. MTNS '96. As with the short course, the paper has two
Observer Design for Linear Switched Control Systems. Weitian Chen and Mehrdad Saif. AbstractâFull and reduced order observers for a class of linear switched control systems (LSCS) are studied in this paper. A âsub-observerâ is first designed fo
Aug 29, 2014 - Abstract: Event-triggered control is developed to reduce the communication load in networked control systems. This means that output or actuator signals are only transmitted over the network when an event-triggering condition is violat
On Self-Triggered Control for Linear Systems: Guarantees and Complexity. Manuel Mazo Jr., Adolfo Anta and Paulo Tabuada. AbstractâTypical digital implementations of feedback con- trollers periodically measure the state, compute the control law, and
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 8, AUGUST 2003
 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.  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.  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.  S. S. Ge, “Adaptive control of uncertain lorenz system using decoupled backstepping,” Int. J. Bifurcation Chaos, 2004, to be published.  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.  M. M. Plycarpou and P. A. Ioannou, “A robust adaptive nonlinear control design,” Automatica, vol. 31, pp. 423–427, 1995.  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.  R. Marino and P. Tomei, “Robust adaptive state-feedback tracking for nonlinear systems,” IEEE Trans. Automat. Contr., vol. 43, pp. 84–89, Jan. 1998.  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.  S. S. Ge and C. Wang, “Adaptive nn control of uncertain nonlinear purefeedback systems,” Automatica, vol. 38, pp. 671–682, 2002.  R. D. Nussbaum, “Some remarks on the conjecture in parameter adaptive control,” Syst. Control Lett., vol. 3, pp. 243–246, 1983.  B. Martensson, “Remarks on adaptive stabilization of first-order nonlinear systems,” Syst. Control Lett., vol. 14, pp. 1–7, 1990.  Z. Ding, “Adaptive control of nonlinear systems with unknown virtual control coefficients,” Int. J. Adapt. Control Signal Processing, vol. 14, pp. 505–517, 2000.  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.  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.  X. D. Ye, “Asymptotic regulation of time-varying uncertain nonlinear systems with unknown control directions,” Automatica, vol. 35, pp. 929–935, 1999.  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.  E. P. Ryan, “A universal adaptive stabilizer for a class of nonlinear systems,” Syst. Control Lett., vol. 16, pp. 209–218, 1991.  H. Khalil, Nonlinear Systems, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 1996.  P. A. Ioannou and J. Sun, Robust Adaptive Control. Upper Saddle River, NJ: Prentice-Hall, 1996.  A. Isidori, Nonlinear Control Systems. New York: Springer-Verlag, 1999, vol. II.
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 . The first method is zero padding to zk 2
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:
However, this method can introduce unnecessary conservatism. The second method is as follows. [1, eq. (7)] can be modified as
= (C + aM k E1 )xk + (D1 + bM k E2 )wk
+ (D2 + cM k E3 )uk r2n M k = HMk ; H
(H ) 1:
Consequently, [1, eq. (9)] is modified as
C D1 "H xk + 01 w ~k : "01 aE1 " bE2 0
With (5), [1, eqs. (10), (11), and (13)] have to be written as (6), (7), and (8), respectively
0P T 0
AT B T "I <0 0P T
0I CT D1T
"I "H T <0 I H : 0
AT B 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
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
"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 . REFERENCES
 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  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.  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  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  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 , where it is proven that a feedback controller must incorporate a copy of the exosystem dynamics. A program similar to that of  has been carried out in  for linear repetitive control, while the internal model principle of  has been extended to pseudorational linear systems, a class of infinite dimensional systems, by Yamamoto and Hara . However, as already said, in  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 , 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., ). 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 . The usefulness of this approach has been also shown by Omata et al. . Claim of a good performance of a digital implementation of repetitive control has also been reported in . 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 , even if by taking a different point of view. In , 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  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.  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.  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 , 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  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  previously recalled. These aspects are commented in the next section.