lqi

Linear-Quadratic-Integral control

Syntax

[K,S,e] = lqi(SYS,Q,R,N)


Description

lqi computes an optimal state-feedback control law for the tracking loop shown in the following figure.

For a plant sys with the state-space equations (or their discrete counterpart):

$$\begin{array}{l}\frac{dx}{dt}=Ax+Bu\\ y=Cx+Du\end{array}$$

the state-feedback control is of the form

$$u=-K[x;x_i]$$

where x_i is the integrator output. This control law ensures that the output y tracks the reference command r. For MIMO systems, the number of integrators equals the dimension of the output y.

[K,S,e] = lqi(SYS,Q,R,N) calculates the optimal gain matrix K, given a state-space model SYS for the plant and weighting matrices Q, R, N. The control law u = –Kz = –K[x;x_i] minimizes the following cost functions (for r = 0)

• $$J(u)={\displaystyle {\int }_0^{\infty }\{z^{T}Qz+u^{T}Ru+2z^{T}Nu\}dt}$$ for continuous time

• $$J(u)={\displaystyle \sum _{n=0}^{\infty }\{z^{T}Qz+u^{T}Ru+2z^{T}Nu\}}$$ for discrete time

In discrete time, lqi computes the integrator output x_i using the forward Euler formula

$$x_i[n+1]=x_i[n]+Ts(r[n]-y[n])$$

where Ts is the sample time of SYS.

When you omit the matrix N, N is set to 0. lqi also returns the solution S of the associated algebraic Riccati equation and the closed-loop eigenvalues e.

Limitations

For the following state-space system with a plant with augmented integrator:

$$\begin{array}{l}\frac{\delta z}{\delta t}=A_{a}z+B_{a}u\\ y=C_{a}z+D_{a}u\end{array}$$

The problem data must satisfy:

• The pair (A,B) must be stabilizable.

• R must be positive definite.

• $$\left[\begin{array}{cc}Q& N\\ N^T& R\end{array}\right]$$ must be positive semidefinite (equivalently, $$Q-N{R}^{-1}{N}^T\ge 0$$).

• $$\left(Q-N{R}^{-1}{N}^T,A_a-B_a{R}^{-1}{N}^T\right)$$ must have no unobservable mode on the imaginary axis (or unit circle in discrete time).

Tips

lqi supports descriptor models with nonsingular E. The output S of lqi is the solution of the Riccati equation for the equivalent explicit state-space model

$$\frac{dx}{dt}=E^{-1}Ax+E^{-1}Bu$$

References

[1] P. C. Young and J. C. Willems, "An approach to the linear multivariable servomechanism problem", International Journal of Control, Volume 15, Issue 5, May 1972 , pages 961–979.

Version History

Introduced in R2008b