Automatic selection of stabilizing matrices¶

The auto keyword¶

The auto keyword indicates that the terminal state weighting matrix $P$ is to be selected automatically, such that the terminal cost is a control Lyapunov function for the model predictive control setup. We consider the following special cases:

the open-loop system is stable and there are only constraints on the inputs. The solution of a matrix Lyapunov equation is returned. In this case the regulation point of the closed-loop system is globally exponentially stable.
the pair (Ad, Bd) is stabilizable and there are constraints on the inputs and states. The solution of a discrete algebraic Riccati equation is returned. The regulation point is asymptotically (exponentially) stable with region of attraction $X_N$ ( $X_S$ ).

The following conditions are expected on the MPC setup:

$Q$ is symmetric positive definite.
$R$ is symmetric positive definite.
The pair $(A_d, B_d)$ is stabilizable.
If $Q$ is positive semi-definite, the pair $(A_d, Q^{\frac{1}{2}})$ must be detectable. This is not checked at the moment.

The following conditions are checked:

Does the problem have only input constraints?
Does $A_d$ have all its eigenvalues strictly inside the unit circle?

If conditions 1. and 2. are both true, a discrete Lyapunov equation is solved. If any of the conditions 1. and 2. is false, then a discrete algebraic Ricatti equation is solved.

In this context, $X_N$ is defined as the set of states for which the MPC problem is feasible. The set $X_S$ is any sublevel set of the MPC optimal value [RM09].