4.3 Deterministic Predictive Control
4.3.4 Stability and Feasibility in Model Predictive Control
Predictive control, using the receding horizon idea, is a feedback control policy. There is, therefore, a risk that the resulting closed-loop might be unstable. Even though the performance of the plant is being optimised over the prediction horizon, and even though the optimisation keeps being repeated, each optimisation ’does not care’ about what happens beyond the prediction horizon, and so could be putting the plant into such a state that it will eventually be impossible to stabilise. One might guess that the short prediction horizon (’short-sighted’) can lead to instability. It turns out that stability can usually be ensured by making the prediction horizon long enough, or even infinite (Maciejowski,2002).
In addition, there are generally two principal methods of guaranteeing nominal stability. Terminal constraints at the end of the prediction horizon and Infinite Horizon. For the purpose of proving stability, it is enough to consider the case when the state is to be driven to the origin, from some initial condition (Maciejowski, 2002). All the ’nominal stability’ analysis of the closed-loop is under the assumption that an exact model of the plant is available. In most of the control algorithms presented in this chapter, the decrease in the optimal cost, on which the proof of stability is founded, is based on the assumption that the next state is exactly as predicted and that the global solution to the optimal control problem can be computed (James B. Rawlings and D. Q. Mayne,2009). The use of terminal penalties and/or constraints, Lyapunov functions or invariant sets has given rise to a wide family of techniques that guarantee the stability of the controlled system. Terminal constraint considering a finite horizon and ensuring stability by adding a state terminal constraint of the form xN = xs. With this constraint, the state is zero
at the end of the finite horizon and therefore the control action is also zero. Infinite horizon consists of increasing the control and prediction horizons to infinity. In this case, the objective function can be considered a Lyapunov function, providing nominal stability (Raff et al., 2007).
In order to state the sharpest results on stabilization, we require the concepts of con- trollability, stabilisability, observability, and detectability (James B. Rawlings, D. Q. Mayne, and M. M. Diehl,2015). A system is controllable if, for any pair of states (x, z) in the state space, z can be reached in finite time from x (or x controlled to z). A linear discrete-time system xi+1 = Axi+ Bui is therefore controllable if there exists a
finite time N and a sequence of inputs that can transfer the system from any x to any z. The basic idea of observability is that any two distinct states can be distinguished by applying some input and observing the two system outputs over some finite time interval (James B. Rawlings and D. Q. Mayne,2009).
Because of the finite horizon in the OCP, the predicted state can differ considerably from the actual state. Ideally, the horizon N in theOCPsolved online should be infinite in which case the predicted state would equal the actual state. A relaxed version of the dynamic programming principle, which allows proving stability and suboptimality results for nonoptimal feedback laws without using the optimal value function is the infinite horizon OCP. For practical reasons, N is usually finite with the result that the controlled system is not necessarily stable or optimal nor is recursively feasible (D. Q. Mayne, 2014). The performance below is the infinite horizon cost of the controlled system: V∞∗(x) := Jc(xi, ui) := ∞ X i=0 xTi Qxi+ uTi Rui. (4.48)
The OCPwith infinite horizon when N → ∞ provides a control law that is guaranteed to asymptotically stabilise system and is recursively feasible (J. B. Rawlings et al.,
1993). A control law κ(x) := u∗0(x) is called recursively feasible for x(0) if κ(xi) ∈ U
and xi ∈ X along the closed-loop trajectory xi+1 = Axi+ Bκ(xi) for all i ∈ Z[0,N −1].
In order to recover feasibility and stability in the finite horizonMPC there are rules to setup the problem, which are provided by D.Q. Mayne et al. (2000). However, it was also shown that near optimal infinite horizon performance is not needed for ensuring stability and admissibility (Gr¨une and Pannek, 2011). Moreover, it is shown that the stability properties of the infinite horizon OCPs are, in general, not preserved in MPC
as long as purely quadratic costs are employed. This indicates the necessity of using the stage cost as a design parameter to achieve asymptotic stability (M¨uller et al., 2017). For the purpose of proving stability e.g., Maciejowski (2002), Worthmann et al. (2017) provides the related theorems for more information.
General constrained optimisation problems can be extremely difficult to solve, and just adding a terminal constraint may not be feasible (Maciejowski,2002). In addition, there are several works of literature present stability analysis for various types of MPC. For instance, a novel NMPC scheme, which is based on the concept of passivity, was pre- sented by Raff et al. (2007). A passivity-based constraint is used to obtain a NMPC
scheme with guaranteed closed-loop stability for any, possible arbitrarily small, predic- tion horizon. Since passivity and stability are closely related, the proposed approach by Raff et al. (2007) can be seen as an alternative to NMPC schemes, which are based on the concept of control Lyapunov functions.
The fact that predictive control is usually implemented on top of traditional local con- trollers had important implications for its acceptance and development. The great ma- jority of the process plant is stable in this condition, so although this may not be the most profitable condition, it is at least a safe one. This is one of the reasons why there were many predictive controllers installed even before there was a satisfactory stability theory for predictive control, and indeed why many current installations take no account of that stability theory. In addition, the commercial predictive controllers being devel- oped almost exclusively for the stable plant. The plant ’seen’ by the predictive controller is one which is already running under closed-loop control and is almost invariably stable (Maciejowski, 2002).