State-transition matrix

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. • It is continuous and has continuous derivatives. • 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) = \mathbf I, where \mathbf I is the identity matrix. • \mathbf{\Phi}(t, t) = \mathbf I for all t . • \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. • 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) = \mathbf I. • 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) = \mathbf I. • 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]^\mathrm{T}, [0,\ 1,\ \ldots,\ 0]^\mathrm{T}, ..., [0,\ 0,\ \ldots,\ 1]^\mathrm{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. == See also ==