Matthias Albrecht Müller

Input/output-to-state stability and state-norm estimators for switched nonlinear systems

In this paper, the concepts of input/output-to-state stability (IOSS) and state-norm estimators are considered for switched nonlinear systems under average dwell-time switching signals. We show that when the average dwell-time is large enough, a switched system is IOSS if all of its constituent subsystems are IOSS. Moreover, under the same conditions, a non-switched state-norm estimator exists for the switched system. Furthermore, if some of the constituent subsystems are not IOSS, we show that still IOSS can be established for the switched system, if the activation time of the non-IOSS subsystems is not too big. Again, under the same conditions, a state-norm estimator exists for the switched system. However, in this case, the state-norm estimator is a switched system itself, consisting of two subsystems. We show that this state-norm estimator can be constructed such that its switching times are independent of the switching times of the switched system it is designed for.

Economic model predictive control with self-tuning terminal cost ☆

Abstract In this paper, we propose an economic model predictive control (MPC) framework with a self-tuning terminal weight, which builds on a recently proposed MPC algorithm with a generalized terminal state constraint. First, given a general time-varying terminal weight, we derive an upper bound on the closed-loop average performance which depends on the limit value of the predicted terminal state. After that, we derive conditions for a self-tuning terminal weight such that bounds for this limit value can be obtained. Finally, we propose several update rules for the self-tuning terminal weight and analyze their respective properties. We illustrate our findings with several examples.

On convergence of averagely constrained economic MPC and necessity of dissipativity for optimal steady-state operation

The contribution of this paper is twofold. First, we discuss necessity of a dissipativity condition which was recently used to prove optimal operation of a system at steady-state in an economic MPC framework. We illustrate with several examples that this dissipativity condition is not necessary, but comes close to being so in a certain sense. Second, we provide a Lyapunov-like stability analysis for averagely constrained economic model predictive control. This closes a gap which was left open in a recent paper on economic MPC.

On Necessity and Robustness of Dissipativity in Economic Model Predictive Control

In this paper, we study a dissipativity property which was recently used in several results on economic model predictive control to ensure optimal operation of a system at steady-state as well as stability. In particular, we first investigate whether this dissipativity property is not only sufficient, but also necessary for optimal steady-state operation. In the most general case, this is not true; nevertheless, under an additional controllability assumption, we show that dissipativity is in fact necessary. Second, we provide a robustness analysis of the dissipativity property with respect to changes in the constraint set, which can result in a change in the considered supply rate.

Brief paper: Improving performance in model predictive control: Switching cost functionals under average dwell-time

In this paper, we propose a novel switched model predictive control (MPC) algorithm for nonlinear continuous-time systems, where we switch between different cost functionals in order to enhance performance. Thus, different performance criteria can be taken into account. In order to ensure stability of the resulting closed-loop system, we consider switching signals which exhibit a certain average dwell-time. When considering switching signals of this type, certain assumptions are common in the switched systems literature in order to ensure stability, like a matching condition for the different Lyapunov functions and the possibility to find Lyapunov functions with an exponential decay rate. In this paper, we show how these assumptions can be satisfied in the MPC context and thus stability of the proposed switched MPC algorithm can be established.

Convergence in economic model predictive control with average constraints

In this paper, we thoroughly investigate various aspects of economic model predictive control with average constraints, i.e., constraints on average values of state and input variables. In particular, we first show that a certain time-varying output constraint has to be included into the MPC problem formulation in order to ensure fulfillment of these average constraints. Optimizing a general (possibly economic) performance criterion may result in a non-converging behavior of the corresponding closed-loop system. While such a behavior might be acceptable in some cases, it may be undesirable for other types of applications. Hence as a second contribution, we provide a Lyapunov-like analysis to conclude that indeed asymptotic convergence to the optimal steady-state follows if the system satisfies a certain dissipativity condition. Finally, for the case that this dissipativity property is not satisfied but still a convergent behavior of the closed-loop is required, we examine two different methods how convergence can be enforced within an economic MPC setup by imposing additional average constraints on the system. In the first method, an additional average constraint is defined which results in the system being dissipative, while the second consists of imposing an additional even zero-moment average constraint. We illustrate our results with various examples.

On the relation between strict dissipativity and turnpike properties

Abstract For discrete time nonlinear systems we study the relation between strict dissipativity and so called turnpike-like behavior in optimal control. Under appropriate controllability assumptions we provide several equivalence statements involving these two properties. The relation of strict dissipativity to an exponential variant of the turnpike property is also studied.

Economic model predictive control without terminal constraints: Optimal periodic operation

In this paper, we analyze economic model predictive control schemes without terminal constraints, where the optimal operating regime is not steady-state operation, but periodic behavior. We first show by means of two counterexamples, that a classical 1-step receding horizon control scheme does not necessarily result in an optimal closed-loop performance. Instead, a multi-step MPC scheme may be needed in order to establish near optimal performance of the closed-loop system. This behavior is analyzed in detail, and we derive checkable dissipativity-like conditions in order to obtain closed-loop performance guarantees.

Robustness of steady-state optimality in economic model predictive control

We study a dissipativity property which was recently used in several results on economic model predictive control to ensure optimal operation of a system at steady-state as well as stability. This dissipativity property involves a supply rate which depends on the state and input constraints imposed on the system. The main contribution of this paper is to show that under certain conditions, this dissipativity property, and hence also optimal operation of a system at steady-state, is robust with respect to small changes in the constraint set. Moreover, we show that further results are possible if a certain convexity assumption is satisfied.

Transient average constraints in economic model predictive control

In this paper, an economic model predictive control algorithm is proposed which ensures satisfaction of transient average constraints, i.e., constraints on input and state variables averaged over some finite time period. We believe that this stricter form of average constraints (compared to previously proposed asymptotic average constraints) is of independent interest in various applications such as the operation of a chemical reactor, where e.g. the amount of inflow or the heat flux during some fixed period of time must not exceed a certain value. Besides guaranteeing fulfillment of transient average constraints for the closed-loop system, we show that closed-loop average performance bounds and convergence results established in the setting of asymptotic average constraints also hold in case of transient average constraints. Furthermore, we illustrate our results with a chemical reactor example.