The Concept of State 2.9 2001/01/17 2002/10/24 Thanos Antoulas aca@rice.edu John Paul Slavinsky jps@alumni.rice.edu Elizabeth Chan lizychan@rice.edu Thanos Antoulas aca@rice.edu John Paul Slavinsky jps@alumni.rice.edu memory state Memory and state in systems. In order to characterize the memory of a dynamical system, we use a concept known as state. A system's state is defined as the minimal set of variables evaluated at tt0 needed to determine the future evolution of the system for tt0, given the excitation ut for tt0. We are given the following differential equation describing a system. Note that ut 0 . t1 y t y t 0 Using the Laplace transform techniques described in the module on Linear Systems with Constant Coefficients, we can find a solution for yt: y t y t0 t0 t As we need the information contained in yt0 for this solution, yt defines the state. The differential equation describing an unforced system is: t2 y t 3 t1 y t 2 y t 0 Finding the qs function, we have q s s 2 3 s 2 The roots of this function are λ1 -1 and λ2 -2 . These values are used in the solution to the differential equation as the exponents of the exponential functions: y t c1 t c2 -2 t where c1 and c2 are constants. To determine the values of these constants we would need two equations (with two equations and two unknowns, we can find the unknowns). If we knew y0 and t1 y 0 we could find two equations, and we could then solve for yt. Therefore the system's state, xt, is x t y t t1 y t In fact, the state can also be defined as any two non-trivial (i.e. independent) linear combinations of yt and t1 y t . Basically, a system's state summarizes its entire past. It describes the memory-side of dynamical systems.