![]() If you have a theoretical or empirical state transition matrix, create a Markov chain model object by using dtmc. For details on supported forms of P, see Discrete-Time Markov Chain Object Framework Overview. These three matrices should be representative of some of the common forms of system matrices. A state transition matrix P characterizes a discrete-time, time-homogeneous Markov chain. ![]() Matrix 1 is a diagonal matrix, Matrix 2 has complex eigenvalues, and Matrix 3 is Jordan canonical form. The sections in this chapter will discuss the solutions to the state-space equations, starting with the easiest case (Time-invariant, no input), and ending with the most difficult case (Time-variant systems). The resulting equation will show the direct relationship between the system input and the system output, without the need to account explicitly for the internal state of the system. Once the state equation has been solved for x, that solution can be plugged into the output equation. ![]() Because this is a first-order equation, we can use results from Ordinary Differential Equations to find a general solution to the equation in terms of the state-variable x. The state equation is a first-order linear differential equation, or (more precisely) a system of linear differential equations. Readers should have a prior knowledge of that subject before reading this chapter. The solutions in this chapter are heavily rooted in prior knowledge of Ordinary Differential Equations.
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