zbMATH — the first resource for mathematics

Control of nonlinear variable structure systems. (English) Zbl 0622.93026
This paper is devoted to the research of different nonlinear control problems with variable structure described by \(\dot x=f(t,x,u)\), \(x(t)\in R^ n\), \(u(t)\subset V\in R^ m\) with sliding manifold \(S(x)=0,S(x)\in R^ m\). Controllability conditions based on Filipov’s definition of the solution of ordinary differential equations with discontinuous right-hand sides are derived. The derived conditions are illustrated via particular control problems with variable structure.
Reviewer: K.Aida-Zade

93C10 Nonlinear systems in control theory
49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
93B05 Controllability
93B03 Attainable sets, reachability
93C15 Control/observation systems governed by ordinary differential equations
Full Text: DOI
[1] Utkin, V.I, Sliding modes and their application in variable structure systems, (1978), MIR Publishers Moscow · Zbl 0398.93003
[2] Filippov, A.V, Differential equations with discontinuous right-hand side, (), 199-232 · Zbl 0148.33002
[3] Slotine, J.J; Sastry, S.S, Tracking control of nonlinear systems using sliding surfaces, with application to robot manipulators, Internat. J. control, 38, 465-492, (1983) · Zbl 0519.93036
[4] Utkin, V.I, Variable structure systems with sliding modes, IEEE trans. automat. control, 22, 212-220, (1977) · Zbl 0382.93036
[5] Piccinini, L.C, ()
[6] Filippov, A.V, Application of the theory of differential equations with discontinuous right-hand sides to non-linear problems in automatic control, (), 923-927
[7] Aizerman, M.A; Pyatniski, E.S; Aizerman, M.A; Pyatniski, E.S, Foundations of a theory of discontinuous systems II, Automat. remote control, Automat. remote control, 35, 1242-1262, (1975) · Zbl 0299.93003
[8] Partasarathy, T, On global univalence theorems, ()
[9] Rockafellar, R.T, Convex analysis, (1970), Princeton Univ. Press Princeton · Zbl 0202.14303
[10] Clarke, F.H, Optimization and nonsmooth analysis, (1983), Wiley-Interscience New York · Zbl 0727.90045
[11] Grego, P.Patuzzo, Sulla G-convergenza delle equazioni differenziali ordinarie nel caso di domini illimitati, Boll. un. mat. ital. B, 16, 466-479, (1979) · Zbl 0427.34016
[12] Jarnik, J; Kurczweil, J, Continuous dependence on a parameter, (), 25-35 · Zbl 0102.07901
[13] Bartolini, G; Zolezzi, T, Variable structure systems nonlinear in the control law, IEEE trans. automat. control, 30, 681-684, (1985) · Zbl 0568.93034
[14] Haddad, G, Monotone trajectories of differential inclusions and functional differential inclusions with memory, Israel J. math., 39, 83-100, (1981) · Zbl 0462.34048
[15] Cellina, A; Aubin, J.P, Differential inclusions and viability theory, () · Zbl 0538.34007
[16] Cellina, A; Aubin, J.P, Differential inclusions, (1984), Springer-Verlag Berlin
[17] Hajek, O; Hajek, O, Discontinuous differential equations II, J. differential equations, J. differential equations, 32, 171, (1979) · Zbl 0681.34009
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.