Model predictive control of hybrid systems : stability and robustness

This thesis considers the stabilization and the robust stabilization of certain classes of hybrid systems using model predictive control. Hybrid systems represent a broad class of dynamical systems in which discrete behavior (usually described by a finite state machine) and continuous behavior (usually described by differential or difference equations) interact. Examples of hybrid dynamics can be found in many application domains and disciplines, such as embedded systems, process control, automated traffic-management systems, electrical circuits, mechanical and bio-mechanical systems, biological and bio-medical systems and economics. These systems are inherently nonlinear, discontinuous and multi-modal. As such, methodologies for stability analysis and (robust) stabilizing controller synthesis developed for linear or continuous nonlinear systems do not apply. This motivates the need for a new controller design methodology that is able to cope with discontinuous and multi-modal system dynamics, especially considering its wide practical applicability. Model predictive control (MPC) (also referred to as receding horizon control) is a control strategy that offers attractive solutions, already successfully implemented in industry, for the regulation of constrained linear or nonlinear systems. In this thesis, the MPC controller design methodology will be employed for the regulation of constrained hybrid systems. One of the reasons for the success of MPC algorithms is their ability to handle hard constraints on states/outputs and inputs. Stability and robustness are probably the most studied properties of MPC controllers, as they are indispensable to practical implementation. A complete theory on (robust) stability of MPC has been developed for linear and continuous nonlinear systems. However, these results do not carry over to hybrid systems easily. These challenges will be taken up in this thesis. As a starting point, in Chapter 2 of this thesis we build a theoretical framework on stability and input-to-state stability that allows for discontinuous and nonlinear system dynamics. These results act as the theoretical foundation of the thesis, enabling us to establish stability and robust stability results for hybrid systems in closed-loop with various model predictive control schemes. The (nominal) stability problem of hybrid systems in closed-loop with MPC controllers is solved in its full generality in Chapter 3. The focus is on a particular class of hybrid systems, namely piecewise affine (PWA) systems. This class of hybrid systems is very appealing as it provides a simple mathematical description on one hand, and a very high modeling power on the other hand. For particular choices of MPC cost functions and constrained PWA systems as prediction models, novel algorithms for computing a terminal cost and a local state-feedback controller that satisfy the developed stabilization conditions are presented. Algorithms for calculating low complexity piecewise polyhedral invariant sets for PWA systems are also developed. These positively invariant sets are either polyhedral, or consist of a union of a number of polyhedra that is equal to the number of affine subsystems of the PWA system. This is a significant reduction in complexity, compared to piecewise polyhedral invariant sets for PWA systems obtained via other existing algorithms. Hence, besides the study of the fundamental property of stability, the aim is to create control algorithms of low complexity to enable their on-line implementation. Before addressing the robust stabilization of PWA systems using MPC in Chapter 5, two interesting examples are presented in Chapter 4. These examples feature two discontinuous PWA systems that both admit a discontinuous piecewise quadratic Lyapunov function and are exponentially stable. However, one of the PWA systems is non-robust to arbitrarily small perturbations, while the other one is globally input-to-state stable (ISS) with respect to disturbance inputs. This indicates that one should be careful in inferring robustness from nominal stability. Moreover, for the example that is robust, the input-to-state stability property cannot be proven via a continuous piecewise quadratic (PWQ) Lyapunov function. However, as ISS can be established via a discontinuous PWQ Lyapunov function, the conservatism of continuous PWQ Lyapunov functions is shown in this setting. Therefore, this thesis provides a theoretical framework that can be used to establish robustness in terms of ISS of discontinuous PWA systems via discontinuous ISS Lyapunov functions. The sufficient conditions for ISS of PWA systems are formulated as linear matrix inequalities, which can be solved efficiently via semi-definite programming. These sufficient conditions also serve as a tool for establishing robustness of nominally stable hybrid MPC controllers a posteriori, after the MPC control law has been calculated explicitly as a PWA state-feedback. Furthermore, we also present a technique based on linear matrix inequalities for synthesizing input-to-state stabilizing state-feedback controllers for PWA systems. In Chapter 5, the problem of robust stabilization of PWA systems using MPC is considered. Previous solutions to this problem rely without exceptions on the assumption that the PWA system dynamics is a continuous function of the state. Clearly, this requirement is quite restrictive and artificial, as a continuous PWA system is in fact a Lipschitz continuous system. In Chapter 5 we present an input-to-state stabilizing MPC scheme for PWA systems based on tightened constraints that allows for discontinuous system dynamics and discontinuous MPC value functions. The advantage of this new approach, besides being the first robust stabilizing MPC scheme applicable to discontinuous PWA systems, is that the resulting MPC optimization problem can still be formulated as mixed integer linear programming problem, which is a standard optimization problem in hybrid MPC. A min-max approach to the robust stabilization of perturbed nonlinear systems using MPC is presented in Chapter 6. Min-max MPC, although computationally more demanding, can provide feedback to the disturbance, resulting in better performance when the controlled system is affected by perturbations. We show that only input-to-state practical stability can be ensured in general for perturbed nonlinear systems in closed-loop with minmax MPC schemes. However, new sufficient conditions that guarantee inputto- state stability of the min-max MPC closed-loop system are derived, via a dual-mode approach. These conditions are formulated in terms of properties that the terminal cost and a local state-feedback controller must satisfy. New techniques for calculating the terminal cost and the local controller for perturbed linear and PWA systems are also presented in Chapter 6. The final part of the thesis focuses on the design of robustly stabilizing, but computationally friendly, sub-optimal MPC algorithms for perturbed nonlinear systems and hybrid systems. This goal is achieved via new, simpler stabilizing constraints, that can be implemented as a finite number of linear inequalities. These algorithms are attractive for real-life implementation, when solvers usually provide a sub-optimal control action, rather than a globally optimal one. The potential for practical applications is illustrated via a case study on the control of DC-DC converters. Preliminary realtime computational results are encouraging, as the MPC control action is always computed within the allowed sampling interval, which is well below one millisecond for the considered Buck-Boost DC-DC converter. In conclusion, this thesis contains a complete framework on the synthesis of model predictive controllers for hybrid systems that guarantees stable and robust closed-loop systems. The latter properties are indispensable for any application of these control algorithms in practice. In the set-ups of the MPC algorithms, a clear focus was also on keeping the on-line computational burden low via simpler stabilizing constraints. The example on the control of DC-DC converters showed that the application to (very) fast systems comes within reach. This opens up a completely new range of applications, next to the traditional process control for typically slow systems. Therefore, the developed theory represents a fertile ground for future practical applications and it opens many roads for future research in model predictive control and stability of hybrid systems as well.