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

$$P=E(y{y}^T)$$

given the noise intensity

$$\begin{array}{cc}E(w(t)w{(\tau )}^T)=W\delta (t-\tau )& \text{(continuous time)}\\ E(w\left[k\right]w{\left[l\right]}^T)=W{\delta }_{kl}& \text{(discrete time)}\end{array}$$

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

$$Q=E(x{x}^T)$$

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

$$\begin{array}{cc}H(z)=\frac{2z+1}{z^2+0.2z+0.5},& T_s\end{array}=0.1$$

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.

Algorithms

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

For continuous-time state-space models

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

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 = CQC^T + DWD^T. For unstable systems, P and Q are infinite. For continuous-time systems with nonzero feedthrough, covar returns Inf for the output covariance P.

References

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

Version History

Introduced before R2006a