kalmd

Design discrete Kalman estimator for continuous plant

Syntax

kalmd
[kest,L,P,M,Z] = kalmd(sys,Qn,Rn,Ts)

Description

kalmd designs a discrete-time Kalman estimator that has response characteristics similar to a continuous-time estimator designed with kalman. This command is useful to derive a discrete estimator for digital implementation after a satisfactory continuous estimator has been designed.

[kest,L,P,M,Z] = kalmd(sys,Qn,Rn,Ts) produces a discrete Kalman estimator kest with sample time Ts for the continuous-time plant

x˙=Ax+Bu+gw(state equation)yv=Cx+Du+v(measurement equation)

with process noise w and measurement noise v satisfying

E(w)=E(v)=0,E(wwT)=Qn,E(vvT)=Rn,E(wvT)=0

The estimator kest is derived as follows. The continuous plant sys is first discretized using zero-order hold with sample time Ts (see c2d entry), and the continuous noise covariance matrices Qn and Rn are replaced by their discrete equivalents

Qd=0TseAτGQGTeATτdτRd=R/Ts

The integral is computed using the matrix exponential formulas in [2]. A discrete-time estimator is then designed for the discretized plant and noise. See kalman for details on discrete-time Kalman estimation.

kalmd also returns the estimator gains L and M, and the discrete error covariance matrices P and Z (see kalman for details).

Limitations

The discretized problem data should satisfy the requirements for kalman.

References

[1] Franklin, G.F., J.D. Powell, and M.L. Workman, Digital Control of Dynamic Systems, Second Edition, Addison-Wesley, 1990.

[2] Van Loan, C.F., "Computing Integrals Involving the Matrix Exponential," IEEE® Trans. Automatic Control, AC-15, October 1970.

See Also

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