Numerically generated path stabilizing controllers: Use of preliminary feedback. (English) Zbl 0859.93021

von Wissel, D.; Nikoukhah, R.; Delebecque, F.; Campbell, S. L.

Kybernetika 31, No. 6, 657-668 (1995).

The aim of this paper is to introduce a hybrid open-loop closed-loop control strategy for path following control problems, using a strategy due to Jankowski et al., but modified by a preliminary feedback.
The authors begin by considering the linear systems of the form: \[ x'=Ax+Bu, \qquad y=Cx+Du, \] for which they construct a stabilizing control \(u=F(x,\xi(t), \xi'(t),\dots, \xi^{(\nu)}(t))\), \(\nu\) being the index of the system, such that \(y(t)\) converges exponentially to the path \(\xi(t)\). Such control is called a descriptor predictive control and it gives, in a certain limiting case, the state-feedback control constructed in the context of paper, of the form: \(u(t)= K_\xi x(t)+R_\xi (d/dt)\xi(t)\). The authors state a necessary and sufficient condition for the closed-loop system \(x'=(A+BK_\xi)x\) to be stable. The extension to nonlinear systems and some tracking properties are discussed. Finally, two conclusive examples are given.

Reviewer: M.Ivanovici (Craiova)

MSC:

93B52	Feedback control
93D15	Stabilization of systems by feedback

Keywords:

differential algebraic equations; feedback stabilization; descriptor predictive control; tracking

References:

[1]	K. E. Brenan S. L. Campbell, L. R. Petzold: Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. Elsevier 1989. · Zbl 0699.65057
[2]	S. L. Campbell, C. W. Gear: The index of general nonlinear DAEs. Numer. Math., to appear. · Zbl 0844.34007
[3]	S. L. Campbell: High index differential algebraic equations. J. Mech. Structures and Machines 23 (1993), 199-222.
[4]	S. L. Campbell R. Nikoukhah, D. von Wissel: Numerically generated path stabilizing controllers I: Theoretical concerns. Proc. of ACC, 1994, pp. 1918-1920.
[5]	A. Isidori: Nonlinear Control Systems: An Introduction. Springer, Berlin 1989. · Zbl 0569.93034
[6]	R. M. Hirschorn: Invertibility of multivariable nonlinear control systems. IEEE Trans. Automat. Control AC-24 (1979), 6, 855-865. · Zbl 0427.93020
[7]	R. M. Hirschorn: Output tracking in multivariable nonlinear systems. IEEE Trans. Automat. Control AC-26 (1981), 2, 593-595. · Zbl 0477.93010
[8]	K. P. Jankowski, H. ElMaraghy: Inverse dynamics and feedforward controllers for constrained flexible joint robots. Proc. 31 Conf. Dec. Contr., 1992, pp. 317-322.
[9]	K. P. Jankowski, H. Van Brussel: Discrete-time inverse dynamics control of flexible joint robots. J. Dynamic Systems, Measurement and Control 114 (1992), 229-233. · Zbl 0775.93153
[10]	K. P. Jankowski, H. Van Brussel: An approach to discrete inverse dynamics control of flexible-joint robots. IEEE Trans. Robotics Automation 8 (1992), 651-658. · Zbl 0775.93153
[11]	L.M. Silverman: Inversion of Multivariable Linear Systems. IEEE Trans. Automat. Control AC-14 (1969), 3, 270-276.
[12]	L. M. Silverman: Discrete Riccati equations: Alternative algorithms, asymptotic properties, and system theory interpretations. Control and Dynamic Systems, Advances in Theory and Appl. 12 (1976), 313-386. · Zbl 0362.49014
[13]	W. Respondek, H. Nijmeijer: On local right-inveritibility of nonlinear control systems. Control-Theory and Advanced Technology 4 (1988), 3, 325-348.
[14]	D. von Wissel, R. Nikoukhah: Hybrid Open-Loop Closed-Loop Path-following Control with Preliminary Feedback. Research Report No. 2173, INRIA, January, 1994.
[15]	D. von Wissel R. Nikoukhah, S. L. Campbell: On a new predictive control strategy: Application to a flexible-joint robot. CDC, Florida 1994, pp. 3025-3026.
[16]	M. Wonham: Linear Multivariable Control. Springer-Verlag, New York 1972. · Zbl 0314.93008