Documentation

lqi

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

$u=-K\left[x;{x}_{i}\right]$

where xi 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;xi] minimizes the following cost functions (for r = 0)

• $J\left(u\right)={\int }_{0}^{\infty }\left\{{z}^{T}Qz+{u}^{T}Ru+2{z}^{T}Nu\right\}dt$ for continuous time

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

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

${x}_{i}\left[n+1\right]={x}_{i}\left[n\right]+Ts\left(r\left[n\right]-y\left[n\right]\right)$

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 (Aa,Ba) is stabilizable.

• R > 0 and $Q-N{R}^{-1}{N}^{T}\ge 0$.

• $\left(Q-N{R}^{-1}{N}^{T},{A}_{a}-{B}_{a}{R}^{-1}{N}^{T}\right)$ has 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

 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.