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.

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. For continuous-time systems with nonzero feedthrough, covar returns Inf for the output covariance P.

References

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