23. Linear systems

23.1. General solution

The general solution of the initial value problem

(23.1)\[\begin{split}\begin{aligned} & \dot{\mathbf{x}} = \mathbf{A} \mathbf{x} + \mathbf{B} \mathbf{u} \\ & \mathbf{x}(t_0) = \mathbf{x}_0 \end{aligned}\end{split}\]

with no impulsive forcing in \(t_0\) reads

(23.2)\[\begin{split} \mathbf{x}(t) & = \boldsymbol\Phi(t,t_0) \mathbf{x_0} + \int_{\tau=t_0}^{t} \boldsymbol\Phi(t,\tau) \mathbf{B}(\tau) \mathbf{u}(\tau) d \tau \\ \end{split}\]

with

\[\begin{split}\begin{cases} \frac{\partial}{\partial t} \boldsymbol\Phi(t,t_0) = \mathbf{A}(t) \boldsymbol\Phi(t,t_0) \\ \boldsymbol\Phi(t,t_0) = \mathbf{I} \ , \end{cases}\end{split}\]

or alternatively, \(\partial_{t_0} \boldsymbol\Phi(t,t_0) = \boldsymbol\Phi(t,t_0) \mathbf{A}(t_0)\).

General solution - details

Pre-multiplication of equation (23.1) by \(\boldsymbol\phi(t)\), with \(\dot{\boldsymbol\phi}(t) = - \boldsymbol\phi(t) \mathbf{A}(t)\) gives

\[\begin{split}\begin{aligned} \boldsymbol\phi(t) \dot{\mathbf{x}} - \boldsymbol\phi(t) \mathbf{A} \mathbf{x} & = \boldsymbol\phi(t) \mathbf{B} \mathbf{u} \\ \boldsymbol\phi(t) \dot{\mathbf{x}} + \dot{\boldsymbol\phi}(t) \mathbf{x} & = \boldsymbol\phi(t) \mathbf{B} \mathbf{u} \\ \frac{d}{dt} \left( \boldsymbol\phi(t) \mathbf{x} \right) & = \boldsymbol\phi(t) \mathbf{B} \mathbf{u} \\ \end{aligned}\end{split}\]

and integration - without impulsive forces in \(t = t_0\), that would cause jump in initial conditions, see section about impulsive force -

\[\begin{split}\begin{aligned} \boldsymbol\phi(t) \mathbf{x} - \boldsymbol\phi(t_0) \mathbf{x_0} & = \int_{\tau=t_0}^{t} \boldsymbol\phi(\tau) \mathbf{B}(\tau) \mathbf{u}(\tau) d \tau \\ \end{aligned}\end{split}\]

and thus

\[\begin{split}\begin{aligned} \mathbf{x}(t) & = \boldsymbol\phi^{-1}(t) \boldsymbol\phi(t_0) \mathbf{x_0} + \int_{\tau=t_0}^{t} \boldsymbol\phi^{-1}(t)\boldsymbol\phi(\tau) \mathbf{B}(\tau) \mathbf{u}(\tau) d \tau \\ \end{aligned}\end{split}\]

or, defining \(\boldsymbol\Phi(a,b) := \boldsymbol\phi^{-1}(a) \boldsymbol\phi(b)\),

\[\begin{split}\begin{aligned} \mathbf{x}(t) & = \boldsymbol\Phi(t,t_0) \mathbf{x_0} + \int_{\tau=t_0}^{t} \boldsymbol\Phi(t,\tau) \mathbf{B}(\tau) \mathbf{u}(\tau) d \tau \\ \end{aligned}\end{split}\]

with \(\boldsymbol\Phi(t,t_0)\) satisfying the initial value problem

\[\begin{split}\begin{cases} \frac{\partial}{\partial t} \boldsymbol\Phi(t,t_0) = \mathbf{A}(t) \boldsymbol\Phi(t,t_0) \\ \boldsymbol\Phi(t,t_0) = \mathbf{I} \ . \end{cases}\end{split}\]

The initial conditions follows from the solution, as the integral identically vanishes as \(t \rightarrow t_0\) if there’s no impulsive force in \(t_0\). The matrix differential equation follows from the definition of the function \(\boldsymbol\Phi(t,\tau)\) in terms of \(\boldsymbol\phi(t)\) and from the time derivative of the inverse of a matrix

\[\frac{d}{dt} \boldsymbol\phi^{-1} = - \boldsymbol\phi^{-1} \dot{\boldsymbol\phi} \boldsymbol\phi^{-1} \ ,\]

as \(\mathbf{0} = \frac{d}{dt} \mathbf{I} = \frac{d}{dt} \left( \boldsymbol\phi^{-1} \boldsymbol\phi \right)\).

Time derivative of \(\boldsymbol\Phi(t,\tau)\) w.r.t. the first argument reads

\[\begin{split}\begin{aligned} \partial_t \boldsymbol\Phi(t,\tau) & = \partial_t \left( \boldsymbol\phi^{-1}(t) \boldsymbol\phi(\tau) \right) = \\ & = \frac{d}{dt} \boldsymbol\phi^{-1}(t) \, \boldsymbol\phi(\tau) = \\ & = - \boldsymbol\phi^{-1}(t) \dot{\boldsymbol\phi}(t) \boldsymbol\phi^{-1}(t) \, \boldsymbol\phi(\tau) = \\ & = \underbrace{\boldsymbol\phi^{-1}(t) \boldsymbol\phi(t)}_{=\mathbf{I}} \mathbf{A}(t) \underbrace{\boldsymbol\phi^{-1}(t) \, \boldsymbol\phi(\tau)}_{\boldsymbol\Phi(t,\tau)} = \\ & = \mathbf{A}(t) \boldsymbol\Phi(t, \tau) \ . \end{aligned}\end{split}\]

Derivative w.r.t. the second argument simply reads

\[\begin{split}\begin{aligned} \partial_{\tau} \boldsymbol\Phi(t,\tau) & = \boldsymbol\phi^{-1}(t) \frac{d}{d\tau} \boldsymbol\phi(\tau) = \\ & = \boldsymbol\phi^{-1}(t) \boldsymbol\phi(\tau) \mathbf{A}(\tau) = \\ & = \boldsymbol\Phi(t,\tau) \mathbf{A}(\tau) \ . \end{aligned}\end{split}\]

23.2. Stochastic response

23.2.1. Correlation matrix

Using the general expression of the solution (23.2) of the linear system (23.1), the following relation holds for the correlation \(\mathbf{R}_{\mathbf{x}\mathbf{x}}(t,\tau) := \mathbb{E}[ \mathbf{x}(t) \mathbf{x}^*(\tau) ]\),

\[\mathbf{R}_{\mathbf{x}\mathbf{x}}(t, \tau) = \boldsymbol\Phi(t,t_0) \mathbf{P}_0 \boldsymbol\Phi^*(t,t_0) + \int_{\lambda=t_0}^{t} \int_{\chi=t_0}^{\tau} \boldsymbol\Phi(t,\lambda) \mathbf{B}(\lambda) \mathbf{R}_{\mathbf{u}\mathbf{u}}(\lambda, \chi) \mathbf{B}^*(\chi) \boldsymbol\Phi^*(\tau,\chi) d \lambda \, d \chi \ ,\]

or for \(\mathbf{P}_{\mathbf{x}\mathbf{x}}(t) := \mathbf{R}_{\mathbf{x} \mathbf{x}}(t,t)\)

\[\mathbf{P}_{\mathbf{x}\mathbf{x}}(t) = \boldsymbol\Phi(t,t_0) \mathbf{P}_0 \boldsymbol\Phi^*(t,t_0) + \int_{\lambda=t_0}^{t} \int_{\chi=t_0}^{t} \boldsymbol\Phi(t,\lambda) \mathbf{B}(\lambda) \mathbf{R}_{\mathbf{u}\mathbf{u}}(\lambda, \chi) \mathbf{B}^*(\chi) \boldsymbol\Phi^*(t,\chi) d \lambda \, d \chi \ .\]

Details

\[\begin{split}\begin{aligned} \mathbf{R}_{\mathbf{x} \mathbf{x}}(t, \tau) & = \mathbb{E}\left[ \mathbf{x}(t) \mathbf{x}^*(\tau) \right] = \\ & = \mathbb{E} \left[ \boldsymbol\Phi(t,t_0) \mathbf{x_0} \mathbf{x_0}^* \boldsymbol\Phi^*(t,t_0) \right] + \\ & + \mathbb{E}\left[ \int_{\lambda=t_0}^{t} \boldsymbol\Phi(t,\lambda) \mathbf{B}(\lambda) \mathbf{u}(\lambda) d \lambda \, \mathbf{x_0}^* \boldsymbol\Phi^*(t,t_0) \right] + \\ & + \mathbb{E}\left[ \boldsymbol\Phi(t,t_0) \mathbf{x_0} \int_{\lambda=t_0}^{t} \mathbf{u}^*(\lambda) \mathbf{B}^*(\lambda) \boldsymbol\Phi^*(t,\lambda) d\lambda \right] + \\ & + \mathbb{E}\left[ \int_{\lambda=t_0}^{t} \boldsymbol\Phi(t,\lambda) \mathbf{B}(\lambda) \mathbf{u}(\lambda) d \lambda \left( \int_{\chi=t_0}^{\tau} \boldsymbol\Phi(\tau,\chi) \mathbf{B}(\chi) \mathbf{u}(\chi) d \chi \right)^* \right] \ , \end{aligned}\end{split}\]

and applying the expectation opearator only on stochastic functions,

\[\begin{split}\begin{aligned} \mathbf{R}_{\mathbf{x} \mathbf{x}}(t, \tau) & = \boldsymbol\Phi(t,t_0) \mathbb{E} \left[ \mathbf{x_0} \mathbf{x_0}^* \right] \boldsymbol\Phi^*(t,t_0) + \\ & + \int_{\lambda=t_0}^{t} \boldsymbol\Phi(t,\lambda) \mathbf{B}(\lambda) \mathbb{E} \left[ \mathbf{u}(\lambda) \mathbf{x}^*_0 \right] d \lambda \boldsymbol\Phi^*(t,t_0) + \\ & + \boldsymbol\Phi(t,t_0) \int_{\lambda=t_0}^{t} \mathbb{E}\left[ \mathbf{x_0} \mathbf{u}^*(\lambda) \right] \mathbf{B}^*(\lambda) \boldsymbol\Phi^*(t,\lambda) d\lambda + \\ & + \int_{\lambda=t_0}^{t} \int_{\chi=t_0}^{\tau} \boldsymbol\Phi(t,\lambda) \mathbf{B}(\lambda) \mathbb{E}\left[ \mathbf{u}(\lambda) \mathbf{u}^*(\chi) \right] \mathbf{B}^*(\chi) \boldsymbol\Phi^*(\tau,\chi) d \lambda \, d \chi \ . \end{aligned}\end{split}\]

Assuming uncorrelated input noise and uncertainty on the initial conditions, \(\mathbb{E}[ \mathbf{u}(t) \mathbf{x}^*_0 ] = \mathbf{0}\), and thus

\[\mathbf{R}_{\mathbf{x}\mathbf{x}}(t, \tau) = \boldsymbol\Phi(t,t_0) \mathbf{P}_0 \boldsymbol\Phi^*(t,t_0) + \int_{\lambda=t_0}^{t} \int_{\chi=t_0}^{\tau} \boldsymbol\Phi(t,\lambda) \mathbf{B}(\lambda) \mathbf{R}_{\mathbf{u}\mathbf{u}}(\lambda, \chi) \mathbf{B}^*(\chi) \boldsymbol\Phi^*(\tau,\chi) d \lambda \, d \chi \ ,\]

with the definition

\[\mathbf{P}_{\mathbf{x}\mathbf{x}}(t) := \mathbf{R}_{\mathbf{x}\mathbf{x}}(t,t) \ .\]

If the input is a white noise, \(\mathbf{R}_{\mathbf{u} \mathbf{u}}(t,\tau) = \mathbf{U}(t) \delta(t - \tau)\),

\[\mathbf{P}_{\mathbf{x}\mathbf{x}}(t) = \boldsymbol\Phi(t,t_0) \mathbf{P}_0 \boldsymbol\Phi^*(t,t_0) + \int_{\lambda=t_0}^{t} \boldsymbol\Phi(t,\lambda) \mathbf{B}(\lambda) \mathbf{U}(\lambda) \mathbf{B}^*(\lambda) \boldsymbol\Phi^*(\tau,\lambda) d \lambda \]

Details

\[\begin{split}\begin{aligned} & \int_{\lambda=t_0}^{t} \int_{\chi=t_0}^{\tau} \boldsymbol\Phi(t,\lambda) \mathbf{B}(\lambda) \mathbf{R}_{\mathbf{u}\mathbf{u}}(\lambda, \chi) \mathbf{B}^*(\chi) \boldsymbol\Phi^*(\tau,\chi) d \lambda \, d \chi = \\ & = \int_{\lambda=t_0}^{t} \int_{\chi=t_0}^{\tau} \boldsymbol\Phi(t,\lambda) \mathbf{B}(\lambda) \mathbf{U}(\lambda) \delta(\lambda-\chi) \mathbf{B}^*(\chi) \boldsymbol\Phi^*(\tau,\chi) d \lambda \, d \chi = \\ & = \int_{\lambda=t_0}^{\min\{ t, \tau\}} \boldsymbol\Phi(t,\lambda) \mathbf{B}(\lambda) \mathbf{U}(\lambda) \mathbf{B}^*(\lambda) \boldsymbol\Phi^*(\tau,\lambda) d \lambda \ , \end{aligned}\end{split}\]

as

\[\begin{split}\int_{\chi=t_0}^{\tau} \delta(\lambda-\chi) \mathbf{B}^*(\chi) \boldsymbol\Phi^*(\tau,\chi) d \chi = \left\{ \begin{aligned} \mathbf{B}^*(\lambda) \boldsymbol\Phi^*(\tau,\lambda) & \quad , \quad \text{if } \ \lambda \in [ t_0, \tau ] \\ 0 & \quad , \quad \text{otherwise} \\ \end{aligned} \right.\end{split}\]

23.2.1.1. Lyapunov equation of the correlation matrix

Matrix \(\mathbf{P}(t)\) satisfies the following Lyapunov equation,

(23.3)\[\dot{\mathbf{P}} = \mathbf{A} \mathbf{P} + \mathbf{P} \mathbf{A}^* + \mathbf{B} \mathbf{U} \mathbf{B}^* \ .\]

Details

\[\begin{split}\begin{aligned} \frac{d \mathbf{P}}{d t} & = \partial_t \boldsymbol\Phi \mathbf{P}_0 \boldsymbol\Phi + \boldsymbol\Phi \mathbf{P}_0 \partial_t \boldsymbol\Phi + \underbrace{\boldsymbol\Phi(t,t)}_{\mathbf{I}} \mathbf{B}(t) \mathbf{U}(t) \mathbf{B}^*(t) \underbrace{\boldsymbol\Phi(t,t)}_{=\mathbf{I}} + \\ & + \int_{\lambda=t_0}^{t} \left\{ \partial_t \boldsymbol\Phi(t,\lambda) \mathbf{B}(\lambda) \mathbf{U}(\lambda) \mathbf{B}^*(\lambda) \boldsymbol\Phi^*(\tau,\lambda) + \boldsymbol\Phi(t,\lambda) \mathbf{B}(\lambda) \mathbf{U}(\lambda) \mathbf{B}^*(\lambda) \partial_t \boldsymbol\Phi^*(\tau,\lambda) \right\} d \lambda = \\ & = \mathbf{B}(t) \mathbf{U}(t) \mathbf{B}^*(t) + \mathbf{A}(t) \underbrace{\left[ \boldsymbol\Phi(t,t_0) \mathbf{P}_0 \boldsymbol\Phi^*(t,t_0) + \int_{\lambda=t_0}^{t} \boldsymbol\Phi(t,\lambda) \mathbf{B}(\lambda) \mathbf{U}(\lambda) \mathbf{B}^*(\lambda) \boldsymbol\Phi(t,\lambda) \right]}_{=\mathbf{P}(t)} + \left[ \dots \right]^* \mathbf{A}^*(t) = \\ & = \mathbf{A} \mathbf{P} + \mathbf{P} \mathbf{A}^* + \mathbf{B} \mathbf{U} \mathbf{B}^* \ . \end{aligned}\end{split}\]