Robustness of Stochastic Discrete-time Switched Linear Systems with Application to Control with Shared Resources (original) (raw)
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Stability is a global property of a system. It is concerned with the behaviour of whole systems over indefinitely long periods of time, for all admissible inputs and uncertainties. Stability and instability are ultimately topological properties. They depend on the topology of the space defined by the equations that govern the system. It follows that that instability is not linear. It is possible to construct a linear combination of two unstable systems which will be stable. The operation of linear combination can be performed using time averaging. The switching can be periodic or stochastic. In the stochastic case, the variance plays an important role. It is possible to drive a system into instability by making the variance sufficiently large. The behaviour near the limit of stability is quite complex, even for very simple "toy" systems. The stochastically switched system is not the same as a stationary linear filter, although we show that the power spectral densities of the two systems can appear to be very similar. We show that variation in the strength of a feedback loop is a new mechanism introducing noise into a system.
SIAM Journal on Control and Optimization, 2006
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Topological formulation of discrete-time switched linear systems and almost sure stability
2008
In this paper, we study the stability of discretetime switched linear systems via symbolic topology formulation and the multiplicative ergodic theorem. A sufficient and necessary condition for µA-almost sure stability is derived, where µA is the Parry measure of the topological Markov chain with a prescribed transition (0,1)-matrix A. The obtained µAalmost sure stability is invariant under small perturbations of the system. The topological description of stable processes of switched linear systems in terms of Hausdorff dimension is given, and it is shown that our approach captures the maximal set of stable processes for linear switched systems. The obtained results cover the stochastic Markov jump linear systems, where the measure is the natural Markov measure defined by the transition probability matrix.
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