Stochastically switched control systems

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. DESIGN OF A SIMPLE SWITCHED SYSTEM WITH A NON-CONVEX UNSTABLE REGION The central task of control theory [1,2] is to direct and regulate the behaviour of a system in order to make it conform to a specified set of standards. Our immediate aim is to design a simple "toy" system in the s domain 2 which has two unstable modes of operation that can be combined, using switching, to create a single stable mode of operation. If a linear plant is placed inside a feedback control loop then a new system with new properties is created. A simple system topology is shown in Figure 1. We can write the equations for this new system as: F(3r = G(3)+ K (1) ) In control theory the system is often called the "plant". ) The variable, s, is a generalised complex frequency. It is used in the Laplace transform. Multiplication by s in this domain corresponds to differentiation in the time domain [3] as long as we have zero initial conditions. CP511, Unsolved Problems of Noise and Fluctuations, edited by D. Abbott and L. B. Kish © 2000 American Institute of Physics 1-56396-826-6/007$ 17.00