State-transition matrix

In control theory, the state-transition matrix is a matrix whose product with the state vector $$x$$ at an initial time $$t_0$$ gives $$x$$ at a later time $$t$$. The state-transition matrix can be used to obtain the general solution of linear dynamical systems.

Linear systems solutions
The state-transition matrix is used to find the solution to a general state-space representation of a linear system in the following form
 * $$\dot{\mathbf{x}}(t) = \mathbf{A}(t) \mathbf{x}(t) + \mathbf{B}(t) \mathbf{u}(t),    \;\mathbf{x}(t_0) = \mathbf{x}_0 $$,

where $$\mathbf{x}(t)$$ are the states of the system, $$\mathbf{u}(t)$$ is the input signal, $$\mathbf{A}(t)$$ and $$\mathbf{B}(t)$$ are matrix functions, and $$\mathbf{x}_0$$ is the initial condition at $$t_0$$. Using the state-transition matrix $$\mathbf{\Phi}(t, \tau)$$, the solution is given by:
 * $$\mathbf{x}(t)= \mathbf{\Phi} (t, t_0)\mathbf{x}(t_0)+\int_{t_0}^t \mathbf{\Phi}(t, \tau)\mathbf{B}(\tau)\mathbf{u}(\tau)d\tau$$

The first term is known as the zero-input response and represents how the system's state would evolve in the absence of any input. The second term is known as the zero-state response and defines how the inputs impact the system.

Peano–Baker series
The most general transition matrix is given by a product integral, referred to as the Peano–Baker series
 * $$\begin{align}

\mathbf{\Phi}(t,\tau) = \mathbf{I} &+ \int_\tau^t\mathbf{A}(\sigma_1)\,d\sigma_1 \\ &+ \int_\tau^t\mathbf{A}(\sigma_1)\int_\tau^{\sigma_1}\mathbf{A}(\sigma_2)\,d\sigma_2\,d\sigma_1 \\ &+ \int_\tau^t\mathbf{A}(\sigma_1)\int_\tau^{\sigma_1}\mathbf{A}(\sigma_2)\int_\tau^{\sigma_2}\mathbf{A}(\sigma_3)\,d\sigma_3\,d\sigma_2\,d\sigma_1 \\ &+ \cdots \end{align}$$ where $$\mathbf{I}$$ is the identity matrix. This matrix converges uniformly and absolutely to a solution that exists and is unique. The series has a formal sum that can be written as
 * $$\mathbf{\Phi}(t,\tau) = \exp \mathcal{T}\int_\tau^t\mathbf{A}(\sigma)\,d\sigma$$

where $$\mathcal{T}$$ is the time-ordering operator, used to ensure that the repeated product integral is in proper order. The Magnus expansion provides a means for evaluating this product.

Other properties
The state transition matrix $$ \mathbf{\Phi}$$ satisfies the following relationships. These relationships are generic to the product integral.

1. It is continuous and has continuous derivatives.

2, It is never singular; in fact $$\mathbf{\Phi}^{-1}(t, \tau) = \mathbf{ \Phi}(\tau, t)$$ and $$\mathbf{\Phi}^{-1}(t, \tau)\mathbf{\Phi}(t, \tau) = I$$, where $$I$$ is the identity matrix.

3. $$\mathbf{\Phi}(t, t) = I$$ for all $$t$$.

4. $$\mathbf{\Phi}(t_2, t_1)\mathbf{\Phi}(t_1, t_0) = \mathbf{\Phi}(t_2, t_0)$$ for all $$t_0 \leq t_1 \leq t_2$$.

5. It satisfies the differential equation $$\frac{\partial \mathbf{\Phi}(t, t_0)}{\partial t} = \mathbf{A}(t)\mathbf{\Phi}(t, t_0)$$ with initial conditions $$\mathbf{\Phi}(t_0, t_0) = I$$.

6. The state-transition matrix $$\mathbf{\Phi}(t, \tau)$$, given by
 * $$\mathbf{\Phi}(t, \tau)\equiv\mathbf{U}(t)\mathbf{U}^{-1}(\tau)$$

where the $$n \times n$$ matrix $$\mathbf{U}(t)$$ is the fundamental solution matrix that satisfies
 * $$\dot{\mathbf{U}}(t)=\mathbf{A}(t)\mathbf{U}(t)$$ with initial condition $$\mathbf{U}(t_0) = I$$.

7. Given the state $$\mathbf{x}(\tau)$$ at any time $$\tau$$, the state at any other time $$t$$ is given by the mapping
 * $$\mathbf{x}(t)=\mathbf{\Phi}(t, \tau)\mathbf{x}(\tau)$$

Estimation of the state-transition matrix
In the time-invariant case, we can define $$ \mathbf{\Phi}$$, using the matrix exponential, as $$\mathbf{\Phi}(t, t_0) = e^{\mathbf{A}(t - t_0)}$$.

In the time-variant case, the state-transition matrix $$\mathbf{\Phi}(t, t_0)$$ can be estimated from the solutions of the differential equation $$\dot{\mathbf{u}}(t)=\mathbf{A}(t)\mathbf{u}(t)$$ with initial conditions $$\mathbf{u}(t_0)$$ given by $$[1,\ 0,\ \ldots,\ 0]^T$$, $$[0,\ 1,\ \ldots,\ 0]^T$$, ..., $$[0,\ 0,\ \ldots,\ 1]^T$$. The corresponding solutions provide the $$n$$ columns of matrix $$\mathbf{\Phi}(t, t_0)$$. Now, from property 4, $$\mathbf{\Phi}(t, \tau) = \mathbf{\Phi}(t, t_0)\mathbf{\Phi}(\tau, t_0)^{-1}$$ for all $$t_0 \leq \tau \leq t$$. The state-transition matrix must be determined before analysis on the time-varying solution can continue.