Design discrete linear-quadratic (LQ) regulator for continuous plant
[Kd,S,e] = lqrd(A,B,Q,R,Ts)
[Kd,S,e] = lqrd(A,B,Q,R,N,Ts)
lqrd designs a discrete full-state-feedback
regulator that has response characteristics similar to a continuous state-feedback
regulator designed using
lqr. This command is useful to design a gain
matrix for digital implementation after a satisfactory continuous state-feedback gain
has been designed.
[Kd,S,e] = lqrd(A,B,Q,R,Ts) calculates the
discrete state-feedback law
that minimizes a discrete cost function equivalent to the continuous cost function
B specify the continuous
Ts specifies the sample time of the discrete regulator. Also
returned are the solution
S of the discrete Riccati equation for the
discretized problem and the discrete closed-loop eigenvalues
[Kd,S,e] = lqrd(A,B,Q,R,N,Ts) solves the more
general problem with a cross-coupling term in the cost function.
The discretized problem data should meet the requirements for
The equivalent discrete gain matrix
Kd is determined by
discretizing the continuous plant and weighting matrices using the sample time
Ts and the zero-order hold approximation.
With the notation
the discretized plant has equations
and the weighting matrices for the equivalent discrete cost function are
The integrals are computed using matrix exponential formulas due to Van Loan (see
). The plant is discretized using
c2d and the gain
matrix is computed from the discretized data using
 Franklin, G.F., J.D. Powell, and M.L. Workman, Digital Control of Dynamic Systems, Second Edition, Addison-Wesley, 1980, pp. 439-440.
 Van Loan, C.F., "Computing Integrals Involving the Matrix Exponential," IEEE® Trans. Automatic Control, AC-23, June 1978.