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[n]=Kdx[n]

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

J=0(xTQx+uTRu)dt

The matrices A and B specify the continuous plant dynamics

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=0(xTQx+uTRu+2xTNu)dt

Limitations

The discretized problem data should meet the requirements for dlqr.

More About

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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

Φ(τ)=eAτ,Ad=Φ(Ts)Γ(τ)=0τeAηBdη,Bd=Γ(Ts)

the discretized plant has equations

x[n+1]=Adx[n]+Bdu[n]

and the weighting matrices for the equivalent discrete cost function are

[QdNdNdTRd]=0Ts[ΦT(τ)0ΓT(τ)I][QNNTR][Φ(τ)Γ(τ)0I]dτ

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.

See Also

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