Variable structure system

A variable structure system, or VSS, is a discontinuous nonlinear system of the form

{\dot {\mathbf {x} }}=\varphi (\mathbf {x} ,t)

where $\mathbf {x} \triangleq [x_{1},x_{2},\ldots ,x_{n}]^{\operatorname {T} }\in \mathbb {R} ^{n}$ is the state vector, $t\in \mathbb {R}$ is the time variable, and $\varphi (\mathbf {x} ,t)\triangleq [\varphi _{1}(\mathbf {x} ,t),\varphi _{2}(\mathbf {x} ,t),\ldots ,\varphi _{n}(\mathbf {x} ,t)]^{\operatorname {T} }:\mathbb {R} ^{n+1}\mapsto \mathbb {R} ^{n}$ is a piecewise continuous function.^[1] Due to the piecewise continuity of these systems, they behave like different continuous nonlinear systems in different regions of their state space. At the boundaries of these regions, their dynamics switch abruptly. Hence, their structure varies over different parts of their state space.

The development of variable structure control depends upon methods of analyzing variable structure systems, which are special cases of hybrid dynamical systems.

References

^ Edwards, Cristopher; Fossas Colet, Enric; Fridman, Leonid, eds. (2006). Advances in Variable Structure and Sliding Mode Control. Lecture Notes in Control and Information Sciences. Vol. 334. Berlin: Springer-Verlag. ISBN 978-3-540-32800-1.

2. Emelyanov, S.V., ed. (1967). Variable Structure Control Systems. Moscow: Nauka.

3. Emelyanov S, Utkin V, Tarin V, Kostyleva N, Shubladze A, Ezerov V, Dubrovsky E. 1970. Theory of Variable Structure Control Systems (in Russian). Moscow: Nauka.

4. Variable Structure Systems: From Principles to Implementation. A. Sabanovic, L. Fridman and S. Spurgeon (eds.), IEE, London, 2004, ISBN 0863413501.

5. Advances in Variable Structure Systems and Sliding Mode Control—Theory and Applications. Li, S., Yu, X., Fridman, L., Man, Z., Wang, X.(Eds.), Studies in Systems, Decision and Control, v. 115, Springer, 2017, ISBN 978-3-319-62895-0

6.Variable-Structure Systems and Sliding-Mode Control. M. Steinberger, M. Horn, L. Fridman.(eds.), Studies in Systems, Decision and Control, v.271, Springer International Publishing, Cham, 2020, ISBN 978-3-030-36620-9.

Khalil, H.K. (2002). Nonlinear Systems (3rd ed.). Upper Saddle River, NJ: Prentice Hall. ISBN 0-13-067389-7.
Utkin, V.I. (1992). Sliding Modes in Control and Optimization. Springer-Verlag. ISBN 978-0-387-53516-6.
Zinober, A.S.I., ed. (1990). Deterministic control of uncertain systems. London: Peter Peregrinus Press. ISBN 978-0-86341-170-0.
Zinober, Alan S.I., ed. (1994). Variable Structure and Lyapunov Control. Lecture Notes in Control and Information Sciences. Vol. 193. London: Springer-Verlag. doi:10.1007/BFb0033675. ISBN 978-3-540-19869-7.

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