kalmd

Design discrete Kalman estimator for continuous plant

Syntax

[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

$\begin{matrix} \dot{x} = A x + B u + G w & (state equation) \\ y_{v} = C x + D u + v & (measurement equation) \end{matrix}$

with process noise w and measurement noise v satisfying

$\begin{matrix} E (w) = E (v) = 0, & E (w w^{T}) = Q_{n}, & E (v v^{T}) = R_{n}, & E (w v^{T}) = 0 \end{matrix}$

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 Q_n and R_n are replaced by their discrete equivalents

$\begin{array}{l} Q_{d} = \int_{0}^{T_{s}} e^{A τ} G Q_{n} G^{T} e^{A^{T} τ} d τ \\ R_{d} = R_{n} / T_{s} \end{array}$

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.

Version History

Introduced before R2006a