transition matrix

Continuous-time systems

For continuous-time state equations,
continuous time state equations#Definition

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

1. $\phi(t_0,t_0) = I$
2. $\phi(t_2,t_1)\phi(t_1,t_0) = \phi(t_2,t_0)$
3. $\mathbf{x}(t_2) = \phi(t_2, t_1) \mathbf{x}(t_1) = \phi(t_2, t_1) \phi(t_1, t_2) \mathbf{x}(t_2)$
   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,

$$\begin{aligned} \dot{\phi}(t,t_0) = A(t) \phi(t,t_0) \\ \dot{\phi}(t,t_0)|_{t_0 = t} = A(t) \end{aligned}$$

Solution of forced system equations

Consider forced system with input present,

$$\dot{\mathbf{x}}(t) = A(t) \mathbf{x}(t) + B(t)\mathbf{u}(t)$$

Assume complete solution has form,

$$\mathbf{x}(t) = \phi(t,t_0) \mathbf{f}(t)$$

Seek conditions of $\mathbf{f}(t)$ to make this assumption true.

Thus,

$$\begin{aligned} \dot{\mathbf{x}}(t) & = \dot{\phi}(t,t_0)\mathbf{f}(t) + \phi(t,t_0) \dot{\mathbf{f}}(t) \\ & = A(t) \phi(t,t_0) \mathbf{f}(t) + \phi(t,t_0) \dot{\mathbf{f}}(t) \\ & = A(t) \mathbf{x}(t) + \phi(t,t_0)\dot{\mathbf{f}}(t) \end{aligned}$$

Therefore

$$\phi(t,t_0)\dot{\mathbf{f}}(t) = B(t) \mathbf{u}(t)$$

then

$$\begin{aligned} \mathbf{x} & = \phi(t,t_0) \mathbf{x}(t_0) + \int_{t_0}^t \phi(t,\lambda) B(\lambda) \mathbf{u}(\lambda) d\lambda \\ y(t) & = C(t) \phi(t,t_0) \mathbf{x}(t_0) + \int_{t_0}^t C(t) \phi(t,\lambda) B(\lambda) \mathbf{u}(\lambda) d\lambda + D(t) \mathbf{u}(t) \end{aligned}$$

where $C(t) \phi(t,t_0) \mathbf{x}(t_0)$ is the zero-input response and $\int_{t_0}^t C(t) \phi(t,\lambda) B(\lambda) \mathbf{u}(\lambda) d\lambda + D(t) \mathbf{u}(t)$ is the zero-state response.

Fixed systems

Constant $A$ matrix

General solution

$$\phi(t,t_0) = e^{A(t-t_0)}$$

note, this is a matrix exponential,
matrix exponential#Definition

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

Solution of the forced system equations

$$\begin{aligned} \mathbf{x}(t) = e^{At} \mathbf{x}(0) + \int_{t_0}^t e^{A(t-\tau)}Bu(\tau) d\tau \\ h(t) = Ce^{At} B 1(t) + D \delta(t) \end{aligned}$$

Discrete systems

In the case of discrete time state equations,
discrete time state equations#Definition

The transition matrix takes the form

$$\phi(k,i) \triangleq A(k-1) A(k-2) ... A(i) \quad \text{for } k \geq i+1$$

#incomplete

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References

1. https://crrl.poly.edu/6253/lectures/lect4.pdf
2. P. E. Sarachik, Principles of Linear Systems, Cambridge Press, 1996, pp. 60-88.
3. https://en.wikipedia.org/wiki/State-transition_matrix
4. https://people.engr.tamu.edu/spb/courses/linearsystems/ECEN605Lecture07.pdf
5. https://faculty.washington.edu/ratliffl/teaching/2020_LinearSystems.pdf p 12

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