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

$$u[n]=-{K}_{d}x[n]$$

that minimizes a discrete cost function equivalent to the continuous
cost function

The matrices A and B specify
the continuous plant dynamics

$$\dot{x}=Ax+Bu$$

and 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 e = eig(Ad-Bd*Kd).

[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 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.

The integrals are computed using matrix exponential formulas
due to Van Loan (see [2]). The
plant is discretized using c2d and the gain matrix
is computed from the discretized data using dlqr.

References

[1] Franklin, G.F., J.D. Powell, and M.L.
Workman, Digital Control of Dynamic Systems,
Second Edition, Addison-Wesley, 1980, pp. 439-440.

[2] Van Loan, C.F., "Computing Integrals Involving
the Matrix Exponential," IEEE^{®} Trans. Automatic Control,
AC-23, June 1978.