National Instruments 370753C-01

Chapter 6 State-Space Design

all the eigenvalues of the system A matrix are negative. The discrete

Lyapunov equation is:

(6-12)

Analogously, the preceding equation has a unique solution X when

λi(A)λj(A) ≠1 for all i and j. Again, this means that a unique X exists for a

stable discrete-time system matrix A, because all eigenvalues of A have

absolute value less than 1 in this case.

You can use the Lyapunov equation to compute the state covariance matrix

of a stable system with white noise input, as illustrated in [BH75]. For a

continuous-time state-space system described by

(6-13)

and supplied with zero-mean white noise ω(t) having covariance Q:

the state covariance X=E[xx'] is given by the differential Lyapunov

equation:

(6-14)

For the discrete-time system described by

the white noise input covariance is defined as in the continuous case, using

a Kronecker rather than a Dirac delta.

For this case, the state covariance matrix X arises from the solution of the

discrete Lyapunov equation:

(6-15)

After you have obtained the state covariance, you can obtain the output

covariance Y easily. Whether you are using the following equation for the

continuous case:

AXA'C+X=

·Ax Bu+=

Eωt()ω'τ()[]Qδtτ–()=

·AX XA'BQB'++=

xk1+AxkBuk

XAXA'BQB'+=

yCxDu+=