control.lqr

control.lqr(A, B, Q, R[, N])[source]

Linear quadratic regulator design.

The lqr() function computes the optimal state feedback controller u = -K x that minimizes the quadratic cost

J = \int_0^\infty (x' Q x + u' R u + 2 x' N u) dt

The function can be called with either 3, 4, or 5 arguments:

  • K, S, E = lqr(sys, Q, R)

  • K, S, E = lqr(sys, Q, R, N)

  • K, S, E = lqr(A, B, Q, R)

  • K, S, E = lqr(A, B, Q, R, N)

where sys is an LTI object, and A, B, Q, R, and N are 2D arrays or matrices of appropriate dimension.

Parameters
A, B2D array_like

Dynamics and input matrices.

sysLTI StateSpace system

Linear system.

Q, R2D array

State and input weight matrices.

N2D array, optional

Cross weight matrix.

integral_actionndarray, optional

If this keyword is specified, the controller includes integral action in addition to state feedback. The value of the integral_action keyword should be an ndarray that will be multiplied by the current state to generate the error for the internal integrator states of the control law. The number of outputs that are to be integrated must match the number of additional rows and columns in the Q matrix.

methodstr, optional

Set the method used for computing the result. Current methods are ‘slycot’ and ‘scipy’. If set to None (default), try ‘slycot’ first and then ‘scipy’.

Returns
K2D array

State feedback gains.

S2D array

Solution to Riccati equation.

E1D array

Eigenvalues of the closed loop system.

See also

lqe, dlqr, dlqe

Notes

If the first argument is an LTI object, then this object will be used to define the dynamics and input matrices. Furthermore, if the LTI object corresponds to a discrete-time system, the dlqr function will be called.

Examples

>>> K, S, E = lqr(sys, Q, R, [N])                           
>>> K, S, E = lqr(A, B, Q, R, [N])