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):

$\begin{array}{l} \frac{d x}{d t} = A x + B u \\ y = C x + D u \end{array}$

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)

$J (u) = \int_{0}^{\infty} {z^{T} Q z + u^{T} R u + 2 z^{T} N u} d t$ for continuous time
$J (u) = \sum_{n = 0}^{\infty} {z^{T} Q z + u^{T} R u + 2 z^{T} N u}$ for discrete time

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

$x_{i} [n + 1] = x_{i} [n] + T s (r [n] - y [n])$

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:

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

The problem data must satisfy:

The pair (A,B) must be stabilizable.
R must be positive definite.
$[\begin{matrix} Q & N \\ N^{T} & R \end{matrix}]$ must be positive semidefinite (equivalently, $Q - N R^{- 1} N^{T} \geq 0$ ).
$(Q - N R^{- 1} N^{T}, A_{a} - B_{a} R^{- 1} N^{T})$ 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

$\frac{d x}{d t} = E^{- 1} A x + E^{- 1} B u$

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

lqi