In this section it is described how the input parametrization for infinite receding horizon control discussed in the previous section can be applied in an efficient model predictive control algorithm. The properties of this algorithm are investigated where special atten- tion is paid to nominal performance and constrained closed-loop stability.
5.3.1
Infinite horizon model predictive control
If input and state constraints are incorporated in infinite receding horizon control, a finite parametrization of the input space is needed to obtain a finite dimensional constrained optimization problem. In section 2.4.7 several approaches to this problem are discussed that can be found in the literature. An important approach is the one described in [140] where the input is parametrized freely over the first P samples and is fixed to the LQ optimal state feedback for the tail to infinite time.
In this section the parametrization of lemma 5.2.1 is applied to obtain an alternative parametrization for model predictive control. This controller and some of its properties is given in the following proposition.
Proposition 5.3.1 Let the linear discrete-time system and the model both be given by
(2.20) subject to input and state constraints (2.22). Let the receding horizon controller
cost function be given by (2.23) with P =∞ and let x0 be either the measured state vector (full information case) or a prediction thereof (partial information case). Let the input over the infinite horizon be parametrized as u(t, θ) = F (A− BF )tθ. Then
1. the optimal control input is given by u(0, θ∗) = F θ∗ with θ∗ the solution to the finite dimensional quadratic programming problem
min θ∈IRn θT xT 0 Y θ x0 (5.10)
subject to: KuF (A− BF )tθ < ku and
0 Kx ˜ At θ x0 < kx, t = 0, 1, . . . , Nc
where ˜A = A− BF 0 BF A , ˜C = √ Q1F 0 0 √Q2
and Nc is the constraint horizon
that is chosen such that after this time instant no constraints are active. Finally, Y is the solution of the Lyapunov equation
˜
ATY ˜A + ˜CTC = Y˜
2. if no constraints are active this controller is equivalent to LQ control with state feedback F .
Proof: The input and state trajectories over the infinite horizon are given by
u(t, θ) x(t, θ, x0) = ˜At θ x0 (5.11)
Substituting this in the cost function yields
J (θ, x0) = θT xT 0 ,∞ t=0 ˜ AT tC˜TC ˜˜At - θ x0
The matrix in this expression can be calculated with the Lyapunov equation in statement 1. Statement 2 can be proven by the fact that the unconstrained optimal solution is given by θ∗ = −x0 which is the state measurement or prediction. From the description (5.11) it follows that this yields a state feedback control with state feedback F which is
LQ-optimal. ✷
As described in the proposition the quadratic programming problem is either initialized with a state measurement or a state estimate. The latter is more likely because usually measurement of all the state variables is too costly or even impossible. It is well known in literature that an LQ optimal state feedback combined with a Kalman filter gives an LQG controller [3][72]. This dynamic output feedback controller is an optimal controller with respect to white noise disturbances on the outputs and states with known covariance matrices.
With the approach discussed in this section the optimization problem can be built up by solving one Riccati equation for the solution of the LQ control problem and a Lya- punov equation to specify the cost function. This can be done quickly because good software tools are available for solving Riccati and Lyapunov equations, also for large scale problems. Therefore, the procedure is flexible for on-line changes in the internal model, the parametrization and the controller cost function. This flexibility of the pro- posed method can be a successful property for constrained control of nonlinear systems
with switching linear predictive controllers such as nonlinear quadratic dynamic matrix control (NLQDMC, [46]).
Another property is that the tuning of the proposed algorithm is simple. A standard LQG control design is needed. The only additional choice that has to be made is the number of samples in the future over which the constraints are evaluated. This is given by the con- straint horizon Nc which must be chosen such that possible constraint activation can be detected sufficiently long in advance. The parameter Nc is no tuning variable for nominal unconstrained performance as it has no influence on the closed-loop performance. The constraints horizon can also be chosen automatically as in [128].
The parametrization is based on the observation that in the unconstrained case the op- timization problem with cost function (5.8) has a closed-form solution if x0 is known. The proposed parametrization contains the optimal solutions for all x0. This amounts to a subspace Ur which is also used to parametrize input profiles for the constraint case. Therefore the input profiles that are obtained in the constrained case are suboptimal. Because the parametrization is based on unconstrained observations the constrained per- formance may in some situations not be good. This issue will be addressed in section 5.5. But for many situations this parametrization provides a controller that attains optimal performance in the unconstrained case and good performance in the constrained case.
5.3.2
Nominal stability under constraints
In this section constrained stability properties are analyzed of the controller described in the previous section. In the next proposition it is proven that under mild conditions the controller provides a stable closed loop, also if constraints are active.
Proposition 5.3.2 The predictive control strategy given in proposition 5.3.1 is globally
asymptotically stable if and only if the optimization problem (5.10) is feasible
Proof: The global time index is denoted with t and the local time index within the
optimization is denoted with k. Let the input trajectory u∗t(k) = −F (A − BF )kθ∗ t be a feasible but possibly not optimal solution at time t. Let the corresponding cost be given by J (t). The first sample of this trajectory is applied as current input u(t) = F θ∗. This yields a state x(t) which is equal to the predicted state if no disturbances are present and the model and plant are equal. Then a feasible trajectory for t + 1 is given by
u∗t+1(k) = −F (A − BF )kθ∗
t+1 with θt∗+1 = (A− BF )θ∗t as this is equivalent with the previous trajectory without the first sample. Denote the corresponding cost function with J (t + 1). This performance cost level need not be optimal therefore it holds that
Because Q1, Q2 > 0 the sequence J (t) is decreasing. It is bounded from below by zero
and therefore J (t) converges to zero, hence x(t), u(t) also converge to zero. Therefore the
nonlinear state feedback is stabilizing. ✷
A similar result holds for the partial information case. Due to the separation principle this stabilizing state feedback combined with a stable observer yields a stabilizing dynamic output feedback [174]. The proposition implies that also in the presence of constraints the closed loop system remains stable if and only if the optimization problem is feasible. Feasibility can only be lost if hard state (output) constraints are used: only then is it possible that there is no input trajectory in the set of feasible input trajectories that renders the state (output) inside the feasible set of states (outputs). In [139] it is described how the problem of feasibility can be avoided. Often applied methods are constraint softening [174] or discarding constraints that are not crucial until the problem becomes feasible [41]. In both cases the proposed algorithm is also stabilizing in the presence of constraints.