# lqrd

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

## Syntax

```lqrd [Kd,S,e] = lqrd(A,B,Q,R,Ts) [Kd,S,e] = lqrd(A,B,Q,R,N,Ts) ```

## Description

`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

`$u\left[n\right]=-{K}_{d}x\left[n\right]$`

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

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

The matrices `A` and `B` specify the continuous plant dynamics

`$\stackrel{˙}{x}=Ax+Bu$`

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.

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

## Limitations

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

## Algorithms

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

`$\begin{array}{cc}\Phi \left(\tau \right)={e}^{A\tau },& {A}_{d}=\Phi \left({T}_{s}\right)\\ \Gamma \left(\tau \right)={\int }_{0}^{\tau }{e}^{A\eta }Bd\eta ,& {B}_{d}=\Gamma \left({T}_{s}\right)\end{array}$`

the discretized plant has equations

`$x\left[n+1\right]={A}_{d}x\left[n\right]+{B}_{d}u\left[n\right]$`

and the weighting matrices for the equivalent discrete cost function are

`$\left[\begin{array}{cc}{Q}_{d}& {N}_{d}\\ {N}_{d}^{T}& {R}_{d}\end{array}\right]={\int }_{0}^{{T}_{s}}\left[\begin{array}{cc}{\Phi }^{T}\left(\tau \right)& 0\\ {\Gamma }^{T}\left(\tau \right)& I\end{array}\right]\left[\begin{array}{cc}Q& N\\ {N}^{T}& R\end{array}\right]\left[\begin{array}{cc}\Phi \left(\tau \right)& \Gamma \left(\tau \right)\\ 0& I\end{array}\right]d\tau$`

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 `dlqr`.

## References

 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.