Control System Toolbox™

covar - Output and state covariance of system driven by white noise

Syntax

P = covar(sys,W) P(:,:,i1,...iN) Q(:,:,i1,...iN) sys(:,:,i1,...iN)

Description

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

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

given the noise intensity

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

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).

Example

Compute the output response covariance of the discrete SISO system

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.

Algorithm

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

For continuous-time state-space models

is obtained by solving the Lyapunov equation

The output response covariance is finite only when and then .

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

and is given by

Note that is well defined for nonzero in the discrete case.

Limitations

The state and output covariances are defined for stable systems only. For continuous systems, the output response covariance is finite only when the matrix is zero (strictly proper system).

References

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