Transition matrix

Continuous-time systems

For continuous-time state equations,

Definition

Suppose homogeneous system (unforced system) $\dot{\mathbf{x}} = A(t) \mathbf{x}(t)$ , then, if a solution exists such that $\dot Q(t) = A(t) Q(t)$ , then we may define transition matrix, $\phi(t,t_0) = Q(t) Q^{-1}(t)$ such that $\mathbf{x}(t) = \phi(t, t_0) \mathbf{x}(t_0)$

Properties

$\phi(t_0,t_0) = I$
$\phi(t_2,t_1)\phi(t_1,t_0) = \phi(t_2,t_0)$
𝐱(t2)=ϕ(t2,t1)𝐱(t1)=ϕ(t2,t1)ϕ(t1,t2)𝐱(t2)\mathbf{x}(t_2) = \phi(t_2, t_1) \mathbf{x}(t_1) = \phi(t_2, t_1) \phi(t_1, t_2) \mathbf{x}(t_2)
1. $\phi(t_2,t_1)\phi(t_1,t_2) = I$
2. for all finite $t_1, t_2$ on $I$ , $\phi(t_1, t_2) = \phi^{-1} (t_2, t_1)$

Evaluation

Given transition matrix $\phi(t,t_0)$ , $A(t)$ can be evaluated as follows $\dot{\mathbf{x}}(t) = \dot{\phi}(t,t_0)\mathbf{x}(t_0)$ Also, $\dot{\mathbf{x}}(t) = A(t)\mathbf{x}(t)= A(t) \phi(t,t_0) \mathbf{x}(t_0)$ Furthermore, $\begin{eqnarray} \dot{\phi}(t,t_0) = A(t) \phi(t,t_0) \\ \dot{\phi}(t,t_0)|_{t_0 = t} = A(t) \end{eqnarray}$

Solution of forced system equations

Assume complete solution has form, $\mathbf{x}(t) = \phi(t,t_0) \mathbf{f}(t)$ then $\begin{eqnarray} \mathbf{x} = \phi(t,t_0) \mathbf{x}(t_0) + \int_{t_0}^t \phi(t,\lambda) B(\lambda) \mathbf{u}(\lambda) d\lambda \\ y(t) = C(t) \phi(t,t_0) \mathbf{x}(t_0) + \int_{t_0}^t C(t) \phi(t,\lambda) B(\lambda) \mathbf{u}(\lambda) d\lambda + D(t) \mathbf{u}(t) \end{eqnarray}$

Fixed systems

Constant $A$ matrix

General solution

$\phi(t,t_0) = e^{A(t-t_0)}$ note, this is a Matrix exponential,

$\mathbf{x}(t) = e^{A(t-t_0)} \mathbf{x}(t_0)$

Solution of the forced system equations $\begin{eqnarray} \mathbf{x}(t) = e^{At} \mathbf{x}(0) + \int_{t_0}^t e^{A(t-\tau)}Bu(\tau) d\tau \\ h(t) = Ce^{At} B 1(t) + D \delta(t) \end{eqnarray}$

Discrete systems

In the case of discrete time state equations,

The transition matrix takes the form $\phi(k,i) \triangleq A(k-1) A(k-2) ... A(i) \quad \text{for } k \geq i+1$

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