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Decomposition of systems with discontinuous controls whose elements operate at different speeds. (English. Russian original) Zbl 0664.49013
Autom. Remote Control 49, No. 3, 288-292 (1988); translation from Avtom. Telemekh. 1988, No. 3, 29-34 (1988).
The authors consider the dynamical system $(*)\quad \mu {\dot \eta}=A\eta +B\sigma +Cs;\quad \mu {\dot \sigma}=F\eta +G\sigma +Ls,\quad \dot s=\sigma;\quad \dot y=Dy+\phi (z,s,y,t)$ where $$\mu$$ is a small positive parameter; $$t\in [t_ 0,\infty)$$; $$\eta \in R^ m$$; $$u,\sigma,s\in R^ 1$$; $$y\in R^ n$$; $$z=(u,\sigma)^ T$$; $$\phi$$ is of C-class; A,B,C,F,G,L,b are matrices depending continuously on a small parameter; D is a constant matrix. The coordinates of the control function are constrained: $$u_ k(s)=\beta_ k+sgn(s_ k)$$, $$| \beta_ k| \leq 1$$, $$k=1,...,l$$. A special property of (*) is that sliding can occur iff $$\sigma_ k=s_ k=0$$, $$k=1,...,l.$$
The paper deals with the decomposition of (*). A new formula describing the system dynamics enables to analyse the stability of the system.
Reviewer: W.Hejmo
MSC:
 49M27 Decomposition methods 34A60 Ordinary differential inclusions 93C10 Nonlinear systems in control theory 93C05 Linear systems in control theory 93D99 Stability of control systems