Random State Space Systems

We already know how a deterministic state-space model looks like

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And if, $x(0)=x_0$, gives

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However, if $u$ and $x_0$ are random, we get the Uncertain state-space model

$$\begin{array}{rl}\dot{x}& =Ax+Bu,x(0)=x_0\\ y& =Cx\end{array}$$

With the following uncertain solution

$$\begin{array}{rl}x(t)& =e^{At}{x}_0+{\int }_0^t{e}^{A(t-\tau )}Bu(\tau )d\tau \\ y(t)& =Cx(t)\end{array}$$

We can find out what is likely to happen. We can find stuff like Expected Value, Variance, Covariance etc.

Covariance Matrix

Mean of Random Process

$$m_x(t)=E[x(t)]=e^{At}E[x_0]+{\int }_0^t{e}^{A(t-\tau )}BE[u(\tau )]d\tau$$

Where $E[x_0]=m_x(0)$ and $E[u(\tau )]=m_u(\tau )$

We can therefore derive (For calculations, see lecture 10, page 10→12.)

$$\begin{array}{rl}{\dot{m}}_x(t)& =A{m}_x(t)+B{m}_u(t)\\ m_y(t)& =C{m}_x(t)\end{array}$$

The mean depends on on the preceding equation, and this results implies that deterministic models are found in the limit $var[x]\stackrel{t\to 0}{\to }0$