Comparison with Stabilizing Terminal Constraints

Even though being able to guarantee stability a priori is a strong motivation to consider this analysis tool, the result would not be attractive if it is considerably more conservative than what is considered standard in the MPC literature, that is, using stabilizing terminal constraints as presented in Chapters 3-5. In this section, a brief comparison is made in terms of the region of attraction and minimum prediction horizon. To this end, consider again the model from Example 3.2 with tuning matrices as selected above and admissible parameter region

P₂ (

𝜌1 ∈ [−0.2 0.2] 𝜌2 ∈ [−0.2 0.2]

chosen as such to avoid limiting the size of the ellipsoid (recall Figure 3.4). Assuming a constraint

|𝑢| ≤ 0.1 (necessary to compute the ellipsoidal terminal region), and for simplicity constant terminal ingredients, solution of the LMI problem in Theorem 3.1 yields the ellipsoid shown in Figure 7.6a; for comparison purposes, the region of attraction (corresponding to P) from the approach in this chapter is also illustrated in the figure. Under these conditions, the gap between

both approaches in terms of feasibility and minimal horizon is small. Indeed, the optimization problem implementing stabilizing terminal constraints based on the computed ellipsoid with an initial condition𝑥(0) = [−0.05 −0.1] becomes feasible for 𝑁 ≥ 6 (Figure 7.6b). It is worth mentioning that, although the result is comparable, the dissipativity-based approach gives a priori stability guarantees, meaning that the control law is simpler to implement as no additional constraints need to be introduced.

x2

x₁

0 -0.02 -0.04 -0.1

-0.05 0 0.05 -0.1

0.02 0.04 0.06 -0.06

(a) Terminal constraint set ( ), region of attrac-tion from dissipativity approach ( )

x2

x₁

-0.05 0 -0.1

-0.1 -0.05 0 0.05

(b) Predicted trajectory with stabilizing terminal constraint and𝑁 =6

Figure 7.6: Comparison between dissipativity-based result and stabilizing terminal constraints The approach presented throughout this chapter appears to be superior in this scenario. However, one disadvantage is that there is no way to include the input saturation level ¯𝑢 in the analysis.

As a consequence, if the saturation level is greater, say ¯𝑢 =1, this approach seems to be overly conservative. Indeed, there would be no change in minimum horizon or region of attraction (as neither depend on the saturation level, and they hold for all possible saturation levels), but the ellipsoidal terminal region obtained with the standard approach would be much larger. On the other hand, a change in the opposite direction, i.e. 𝑢¯ = 0.01 would further benefit the dissipativity-based approach.

7.5 Summary

The analysis tool developed in this chapter provides the means to guarantee stability of nonlinear and parameter-dependent predictive control laws a priori, and without the need to implement stabilizing terminal constraints. This means that feasibility and performance are in no way affected by guaranteeing stability of the closed-loop. Analysis is based on the deemed parameter-dependent quadratic constraints, which, alternative to integral quadratic constraints, prove useful when dealing with parameter-dependent systems. The use of 𝐷/𝐺-scalings and assuming infinitely fast changing parameters are the main sources of conservatism of the approach, yet as seen in the example section, conservatism could potentially be low, under some conditions.

Important assumptions made are that the plant (more specifically the equilibrium of interest) is open-loop stable and that the constraints of the problem are such that the affine term of

the inequality constraint 𝐴𝑥 ≤ 𝑏 is piece-wise nonnegative, that is 𝑏 ≥ 0. This might prove prohibitive if there are state constraints.

In the example it was highlighted that under some conditions, the approach is superior to the standard stabilizing MPC (i.e. with stabilizing terminal constraints) but under other conditions it might prove overly conservative. As expected, there is no one superior approach that is better than others for all possible scenarios and having alternative tools at one’s disposal seems to be the best bet. Indeed, although the dissipativity-based approach seems to be restrictive, should all conditions hold, it becomes extremely convenient, given that the control law is simple to implement, and both feasibility and stability are guaranteed.

Conclusions and Outlook

8.1 Conclusions

The interest in fast Nonlinear Model Predictive Control stems from the desire to take advantage of the attractive features offered by MPC, particularly anticipative behavior and hard constraint satisfaction, to control complex nonlinear systems with fast dynamics, e.g. mechatronic, electri-cal systems, etc. However, there are two main challenges faced by predictive control laws, and they represent the main motivations for the proposed framework, these are:

1. Predictive controllers require the online solution of a constrained optimization problem within a sampling interval. In the nonlinear case, this optimization problem is non-convex and hard to solve.

2. Deriving tractable stability conditions for nonlinear systems is in general a difficult task, in the context of nonlinear MPC this is often done by linearizing the nonlinear system around the desired equilibrium and solving a convex optimization problem to derive the terminal ingredients.

To a somewhat lesser extent (at least in comparison to the other two), an additional challenge faced in nonlinear MPC is

3. NMPC requires an accurate yet as simple as possible model.

In order for predictions to be meaningful and in the interest of point number 2. above (i.e. to establish stability) an accurate model is needed; however, simpler models are preferred due to point number 1. above, i.e. complexity of the computations scale with model complexity. In practice a trade-off has to be made.

The quasi-Linear Model Predictive Control framework presented in this thesis addresses items 1.-3. in the following ways:

i. It provides an easy to implement, highly efficient algorithm for nonlinear Model Predictive Control by means of an iterative algorithm which exploits the similarity of two subsequent MPC optimization problems; this is achieved by freezing the parameter trajectory to the

one given by the (shifted) parameter trajectory from the previous time step (or iteration), thereby turning the nonlinear optimization problem linear, but time-varying and making it easy to solve.

ii. Conservatism of the stability conditions often encountered in MPC is substantially reduced by considering quasi-LPV models in lieu of a linearization around equilibrium points.

Further reduction in conservatism can be achieved by using parameter dependent terminal ingredients.

iii. Quasi-LPV models enable exact representations of nonlinear systems in a simplified manner. Compared to Jacobian linearization, for example, quasi-LPV models are generally more accurate and simpler (as they are linear and not affine).

More specifically, there are several other important aspects that make qLMPC an attractive alternative to other efficient algorithms for implementation of nonlinear MPC laws. An extension to input-output models enables to straightforwardly implement a predictive control law which does not need observers. Stability conditions are adapted to the IO setting and the complexity of the resulting LMI problems is the same as in the state space case. Addressing item 3. above, if no model is available, IO-LPV system identifications techniques are simpler and similarly accurate than their state space counterparts and the IO-qLMPC framework allows to directly utilize the resulting model thus avoiding complications entailed by conversions between IO and SS models [118].

Back to item 2., the novel velocity algorithm for NMPC guarantees stability of the closed-loop without the need to compute terminal ingredients, thereby simplifying design. This is achieved by using terminal equality constraints in the velocity space, which have the advantage (compared to using terminal equality constraints in standard MPC) that no parameterization of equilibrium steady states as a function of the desired set point is necessary (which in the nonlinear case can potentially be a complex task). Furthermore, for the same reason, intermediate way points need not be explicitly computed as all equilibria are mapped to the origin. Under mild assumptions (the system having a continuous locus of equilibria), recursive feasibility of the control law is guaranteed even for short horizons. Short horizons could admittedly have an adverse effect on closed-loop performance since the terminal velocity constraints would to an extent preclude fast trajectories. The velocity algorithm is built upon the use of velocity-based linearization, the resulting model is such that a quasi-LPV representation follows straightforwardly thereby making qLMPC a prime candidate to be used together with the proposed velocity algorithm.

Under somewhat more restrictive conditions (equilibrium of interest being open-loop stable and 𝑢 =0 being always feasible) stability can be established a priori using the stability analysis tool developed in Chapter 7. In this case, no artificially imposed stabilizing constraints are necessary.

Finally item 3. can be addressed by data-driven techniques in conjunction with predictive control schemes. Indeed, in this thesis a Koopman operator-based qLMPC is proposed in which the dynamics of the system are discovered in real-time and the resulting model is used in a velocity-based qLMPC. In most practical applications a short training experiment would likely be needed to bootstrap the predictive controller (i.e. to give an initial model to work with).