State-transition matrix

In control theory, the state-transition matrix is a matrix whose product with the state vector [math]\displaystyle{ x }[/math] at an initial time [math]\displaystyle{ t_0 }[/math] gives [math]\displaystyle{ x }[/math] at a later time [math]\displaystyle{ t }[/math]. 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

[math]\displaystyle{ \dot{\mathbf{x}}(t) = \mathbf{A}(t) \mathbf{x}(t) + \mathbf{B}(t) \mathbf{u}(t) , \;\mathbf{x}(t_0) = \mathbf{x}_0 }[/math],

where [math]\displaystyle{ \mathbf{x}(t) }[/math] are the states of the system, [math]\displaystyle{ \mathbf{u}(t) }[/math] is the input signal, [math]\displaystyle{ \mathbf{A}(t) }[/math] and [math]\displaystyle{ \mathbf{B}(t) }[/math] are matrix functions, and [math]\displaystyle{ \mathbf{x}_0 }[/math] is the initial condition at [math]\displaystyle{ t_0 }[/math]. Using the state-transition matrix [math]\displaystyle{ \mathbf{\Phi}(t, \tau) }[/math], the solution is given by:

[math]\displaystyle{ \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 }[/math]

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 the Peano–Baker series

[math]\displaystyle{ \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 + ... }[/math]

where [math]\displaystyle{ \mathbf{I} }[/math] is the identity matrix. This matrix converges uniformly and absolutely to a solution that exists and is unique.

Other properties

The state transition matrix [math]\displaystyle{ \mathbf{\Phi} }[/math] satisfies the following relationships:

1. It is continuous and has continuous derivatives.

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

3. [math]\displaystyle{ \mathbf{\Phi}(t, t) = I }[/math] for all [math]\displaystyle{ t }[/math] .

4. [math]\displaystyle{ \mathbf{\Phi}(t_2, t_1)\mathbf{\Phi}(t_1, t_0) = \mathbf{\Phi}(t_2, t_0) }[/math] for all [math]\displaystyle{ t_0 \leq t_1 \leq t_2 }[/math].

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

6. The state-transition matrix [math]\displaystyle{ \mathbf{\Phi}(t, \tau) }[/math], given by

[math]\displaystyle{ \mathbf{\Phi}(t, \tau)\equiv\mathbf{U}(t)\mathbf{U}^{-1}(\tau) }[/math]

where the [math]\displaystyle{ n \times n }[/math] matrix [math]\displaystyle{ \mathbf{U}(t) }[/math] is the fundamental solution matrix that satisfies

[math]\displaystyle{ \dot{\mathbf{U}}(t)=\mathbf{A}(t)\mathbf{U}(t) }[/math] with initial condition [math]\displaystyle{ \mathbf{U}(t_0) = I }[/math].

7. Given the state [math]\displaystyle{ \mathbf{x}(\tau) }[/math] at any time [math]\displaystyle{ \tau }[/math], the state at any other time [math]\displaystyle{ t }[/math] is given by the mapping

[math]\displaystyle{ \mathbf{x}(t)=\mathbf{\Phi}(t, \tau)\mathbf{x}(\tau) }[/math]

Estimation of the state-transition matrix

In the time-invariant case, we can define [math]\displaystyle{ \mathbf{\Phi} }[/math], using the matrix exponential, as [math]\displaystyle{ \mathbf{\Phi}(t, t_0) = e^{\mathbf{A}(t - t_0)} }[/math].

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

See also

• Magnus expansion
• Liouville's formula

Further reading

• Baake, M.; Schlaegel, U. (2011). "The Peano Baker Series". Proceedings of the Steklov Institute of Mathematics 275: 155–159. doi:10.1134/S0081543811080098. 
• Brogan, W.L. (1991). Modern Control Theory. Prentice Hall. ISBN 0-13-589763-7. https://archive.org/details/moderncontrolthe00brog.