covar

Output and state covariance of system driven by white noise

Syntax

`P = covar(sys,W) [P,Q] = covar(sys,W)`

Description

`covar` calculates the stationary covariance of the output y of an LTI model `sys` driven by Gaussian white noise inputs w. This function handles both continuous- and discrete-time cases.

`P = covar(sys,W) `returns the steady-state output response covariance

$P=E\left(y{y}^{T}\right)$

given the noise intensity

`[P,Q] = covar(sys,W)` also returns the steady-state state covariance

$Q=E\left(x{x}^{T}\right)$

when `sys` is a state-space model (otherwise `Q` is set to `[]`).

When applied to an `N`-dimensional LTI array `sys`, `covar` returns multidimensional arrays P, Q such that

`P(:,:,i1,...iN)` and `Q(:,:,i1,...iN)` are the covariance matrices for the model `sys(:,:,i1,...iN)`.

Examples

Compute the output response covariance of the discrete SISO system

$\begin{array}{cc}H\left(z\right)=\frac{2z+1}{{z}^{2}+0.2z+0.5},& {T}_{s}\end{array}=0.1$

due to Gaussian white noise of intensity `W = 5`. Type

```sys = tf([2 1],[1 0.2 0.5],0.1); p = covar(sys,5) ```

These commands produce the following result.

```p = 30.3167 ```

You can compare this output of `covar` to simulation results.

```randn('seed',0) w = sqrt(5)*randn(1,1000); % 1000 samples % Simulate response to w with LSIM: y = lsim(sys,w); % Compute covariance of y values psim = sum(y .* y)/length(w); ```

This yields

```psim = 32.6269 ```

The two covariance values `p` and `psim` do not agree perfectly due to the finite simulation horizon.

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Algorithms

Transfer functions and zero-pole-gain models are first converted to state space with `ss`.

For continuous-time state-space models

$\begin{array}{l}\stackrel{˙}{x}=Ax+Bw\\ y=Cx+Dw,\end{array}$

the steady-state state covariance Q is obtained by solving the Lyapunov equation

$AQ+Q{A}^{T}+BW{B}^{T}=0.$

In discrete time, the state covariance Q solves the discrete Lyapunov equation

$AQ{A}^{T}-Q+BW{B}^{T}=0.$

In both continuous and discrete time, the output response covariance is given by P = CQCT + DWDT. For unstable systems, P and Q are infinite.

References

[1] Bryson, A.E. and Y.C. Ho, Applied Optimal Control, Hemisphere Publishing, 1975, pp. 458-459.