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):

dxdt=Ax+Buy=Cx+Du

the state-feedback control is of the form

u=K[x;xi]

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(u)=0{zTQz+uTRu+2zTNu}dt for continuous time

  • J(u)=n=0{zTQz+uTRu+2zTNu} for discrete time

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

xi[n+1]=xi[n]+Ts(r[n]y[n])

where Ts is the sampling 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:

δzδt=Aaz+Bauy=Caz+Dau

The problem data must satisfy:

  • The pair (Aa,Ba) is stabilizable.

  • R > 0 and QNR1NT0.

  • (QNR1NT,AaBaR1NT) has no unobservable mode on the imaginary axis (or unit circle in discrete time).

More About

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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

dxdt=E1Ax+E1Bu

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.

See Also

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