×

High-gain robust adaptive controllers for multivariable systems. (English) Zbl 0632.93046

The time-varying linear system \[ (1)\quad dx/dt=({\mathcal A}-k(t){\mathcal D})x(t),\quad t\geq 0 \] is considered. Here \({\mathcal A}\), \({\mathcal D}\in {\mathbb{R}}^{n^ 2}\), \(k(t)\in{\mathcal K}\), where \({\mathcal K}\) is a set of all piecewise continuous functions monotonically non-decreasing, \(k(t)\to +\infty\) as \(t\to \infty\). The following conditions are equivalent:
(i) Re \(\lambda_ j({\mathcal D})>0\), \(j=1,...,n\), where \(\lambda_ j({\mathcal D})\) are eigenvalues of \({\mathcal D},\)
(ii) Re \(\lambda_ j[{\mathcal A}-k(t){\mathcal D}]\to -\infty\), \(j=1,...,n,\)
(iii) For some \(M>0\), \(\omega\) (\(\cdot)\in {\mathcal K}:\| \Phi (t,t_ 0)\| \leq Me^{-\omega (t_ 0)(t-t_ 0)}\), for all \(t\geq t_ 0\geq 0\), where \(\Phi (t,t_ 0)\) is the evolution matrix of (1).
Let the following conditions be satisfied: (a) There exist \(k^*\), \(\epsilon >0\) such that \(\lambda_ j[{\mathcal A}-k{\mathcal D}]<-\epsilon\) for all \(k>k^*,\)
(b) If \({\mathfrak O}\in \{\lambda_ j({\mathcal D})\}\) then \({\mathfrak O}\) is semisimple.
Then (1) is exponentially stable for every k(\(\cdot)\in {\mathcal K}\). Moreover, there exists \(M'>0\) such that for all \(\epsilon >0\) and all k(\(\cdot)\in {\mathcal K}\) we can find \(T=T(\epsilon,k(\cdot))\) such that for all \(t\geq t_ 0\geq T:\| \Phi (t,t_ 0)\| \leq M'e^{-\epsilon (t- t_ 0)}.\)
This result can be used for the construction of the adaptive regulator \(u(\cdot)=\psi [\sigma (\cdot)]\sigma (t)\) of the system \[ (2)\quad dx/dt=Ax+Bu,\quad \sigma =Cx. \] The matrices A,B,C are unknown, \(A,D=BC\) satisfy the conditions (a),(b); u(t), \(\sigma (t)\in {\mathbb{R}}^ 1\). The robustness of this adaptive controller with respect to small nonlinear perturbations of (2) is proved.
Reviewer’s remark. For stabilization problems of minimum phase objects, the adaptive regulators of the above type were widely used in the Russian literature, see, e.g., the book [V. N. Fomin, A. L. Fradkov, and V. A. Yakubovich, Adaptive control of dynamical objects (1981; Zbl 0522.93002) (Russian), Ch. 7 and references to Ch. 7].
Reviewer: Y.Yakubovich

MSC:

93C40 Adaptive control/observation systems
93C35 Multivariable systems, multidimensional control systems
93B35 Sensitivity (robustness)
93C05 Linear systems in control theory

Keywords:

time-dependent

Citations:

Zbl 0522.93002
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Byrnes, C. I.; Isidori, A., A frequency domain philosophy for nonlinear systems with applications to stabilization and to adaptive control, (Proc. of the 23rd IEEE Conf on Decision and Control. Proc. of the 23rd IEEE Conf on Decision and Control, Las Vegas, NV (1984)), 1569-1573
[2] Byrnes, C. I.; Willems, J. C., Adaptive stabilization of multivariable linear systems, (Proc. of the 23rd Conf. on Decision and Control. Proc. of the 23rd Conf. on Decision and Control, Las Vegas, NV (1984)), 1574-1577
[3] Copper, W. A., Dichotomics in Stability Theory, (Lecture Notes in Mathematics No. 629 (1978), Springer: Springer Berlin)
[4] Heymann, M.; Lewis, J. H.; Meyer, G., Remarks on the adaptive control of linear plants with unknown high-frequency gain, Systems Control Lett., 5, 6, 357-362 (1985) · Zbl 0571.93036
[5] Mårtensson, B., Adaptive Stabilization, (Doctoral Dissertation (1986), Lund Institute of Technology) · Zbl 0651.93042
[6] Mårtensson, B., The order of any stabilizing regulator is sufficient a priori information for adaptive stabilizing, Systems Control Lett., 6, 2, 87-91 (1985) · Zbl 0564.93055
[7] Morse, A. S., Recent problems in parameter adaptive control, (Landau, I. D., Outils et Modèles Mathématiques pour l’Automatique. Outils et Modèles Mathématiques pour l’Automatique, l’Analyse de Système et le Traitment du Signal, Vol. 3 (1983), Editions du CNRS: Editions du CNRS Paris), 733-740
[8] Nobel, B.; Daniel, J. W., Applied Linear Algebra (1977), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ
[9] Nussbaum, R. D., Some remarks on a conjecture in parameter adaptive control, Systems Control Lett., 3, 243-246 (1983) · Zbl 0524.93037
[10] Owens, D., Feedback and Multivariable Systems (1978), Peter Peregrinus: Peter Peregrinus Stevenage · Zbl 0446.93001
[11] Willems, J. C.; Byrnes, C. I., Global adaptive stabilization in the absence of information on the sign of the high frequency gain, (Lecture Notes in Control and Information Sciences No. 62 (1984), Springer: Springer Berlin) · Zbl 0549.93043
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.