control.matlab.lqr

control.matlab.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 (2D array_like) – Dynamics and input matrices

  • B (2D array_like) – Dynamics and input matrices

  • sys (LTI StateSpace system) – Linear system

  • Q (2D array) – State and input weight matrices

  • R (2D array) – State and input weight matrices

  • N (2D array, optional) – Cross weight matrix

  • integral_action (ndarray, 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.

  • method (str, 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

  • K (2D array (or matrix)) – State feedback gains

  • S (2D array (or matrix)) – Solution to Riccati equation

  • E (1D 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])