Design discrete linear-quadratic (LQ) regulator for continuous plant

`lqrd `

[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

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

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

$$J={\displaystyle {\int}_{0}^{\infty}\left({x}^{T}Qx+{u}^{T}Ru\right)}dt$$

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.

$$J={\displaystyle {\int}_{0}^{\infty}\left({x}^{T}Qx+{u}^{T}Ru+2{x}^{T}Nu\right)dt}$$

The discretized problem data should meet the requirements for `dlqr`

.

[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.

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