mpcrl.util.control.dlqr#
- mpcrl.util.control.dlqr(A, B, Q, R, M=None)[source]#
Computes the solution to the discrete-time LQR problem.
The LQR problem is to solve the following optimization problem
\[\min_{u} \sum_{t=0}^{\infty} x_t^\top Q x_t + u_t^\top R u_t + 2 x_t^\top M u_t\]for the linear time-invariant discrete-time system
\[x_{t+1} = A x_t + B u_t.\]The (famous) solution takes the form of a state feedback law
\[u_t = -K x_t\]with a quadratic cost-to-go function
\[V(x_t) = x_t^\top P x_t.\]The function returns the optimal state feedback matrix \(K\) and the quadratic terminal cost-to-go matrix \(P\). If not provided,
Mis assumed to be zero.- Parameters:
- Aarray
State matrix.
- Barray
Control input matrix.
- Qarray
State weighting matrix.
- Rarray
Control input weighting matrix.
- Marray, optional
Mixed state-input weighting matrix, by default
None.
- Returns:
- tuple of two arrays
Returns the optimal state feedback matrix \(K\) and the quadratic terminal cost-to-go matrix \(P\).
