Periodic optimal control, dissipativity and MPC

Recent research has established the importance of dissipativity for proving stability of economic MPC in the case of a steady state. In many cases, though, steady state operation is not economically optimal and periodic operation of the system yields a better performance. In this paper, we propose three different ways of extending the notion of dissipativity for periodic systems and illustrate them with three examples

Robust economic model predictive control: recursive feasibility, stability and average performance

This thesis is mainly concerned with designing algorithms for Economic Model Predictive Control (EMPC), and analysis of its resulting recursive feasibility, stability and asymptotic average performance. In particular, firstly, in order to extend and unify the formulation and analysis of economic model predictive control for general optimal operation regimes, including steady-state or periodic operation, we propose the novel concept of a “control storage function” and introduce upper and lower bounds to the best asymptotic average performance for nonlinear control systems based on suitable notions of dissipativity and controlled dissipativity. As a special case, when the optimal operation is periodic, we present a new approach to formulate terminal cost functions. Secondly, aiming at designing a robust EMPC controller for nonlinear systems with general optimal regimes of operation, we present a tube-based robust EMPC algorithm for discrete-time nonlinear systems that are perturbed by disturbance inputs. The proposed algorithm minimizes a modified economic objective function, which considers the worst cost within a tube around the solution of the associated nominal system. Recursive feasibility and an a-priori upper bound to the closed-loop asymptotic average performance are ensured. Thanks to the use of dissipativity of the nominal system with a suitable supply rate, the closed-loop system under the proposed controller is shown to be asymptotically stable, in the sense that it is driven to an optimal robust invariant set. Thirdly, for the purpose of combining robust EMPC design with a state observer in a single pure economic optimization problem, we consider homothetic tube-based EMPC synthesis for constrained linear discrete time systems. Since, in practical systems, full state measurement is seldom available, the proposed method integrates a moving horizon estimator to achieve closed-loop stability and constraint satisfaction despite system disturbances and output measurement noise. In contrast to existing approaches, the worst cost within a single homothetic tube around the solution of the associated nominal system is minimized, which at the same time tightens the bound on the set of potential states compatible with past output and input data. We show that the designed optimization problem is recursively feasible and adoption of homothetic tubes leads to less conservative economic performance bounds. In addition, the use of strict dissipativity of the nominal system guarantees asymptotic stability of the resulting closed-loop system. Finally, to deal with the unknown nonzero mean disturbance and the presence of plant-model error, we propose a novel economic MPC algorithm aiming at achieving optimal steady-state performance despite the presence of plant-model mismatch or unmeasured nonzero mean disturbances. According to the offset-free formulation, the system's state is augmented with disturbances and transformed into a new coordinate framework. Based on the new variables, the proposed controller integrates a moving horizon estimator to determine a solution of the nominal system surrounded by a set of potential states compatible with past input and output measurements. The worst cost within a single homothetic tube around the nominal solution is chosen as the economic objective function which is minimized to provide a tightened upper bound for the accumulated real cost within the prediction horizon window. Thanks to the combined use of the nominal system and homothetic tube, the designed optimization problem is recursively feasible and less conservative economic performance bounds are achieved.Open Acces

Integral and measure-turnpike properties for infinite-dimensional optimal control systems

We first derive a general integral-turnpike property around a set for infinite-dimensional non-autonomous optimal control problems with any possible terminal state constraints, under some appropriate assumptions. Roughly speaking, the integral-turnpike property means that the time average of the distance from any optimal trajectory to the turnpike set con- verges to zero, as the time horizon tends to infinity. Then, we establish the measure-turnpike property for strictly dissipative optimal control systems, with state and control constraints. The measure-turnpike property, which is slightly stronger than the integral-turnpike property, means that any optimal (state and control) solution remains essentially, along the time frame, close to an optimal solution of an associated static optimal control problem, except along a subset of times that is of small relative Lebesgue measure as the time horizon is large. Next, we prove that strict strong duality, which is a classical notion in optimization, implies strict dissipativity, and measure-turnpike. Finally, we conclude the paper with several comments and open problems

A generalized approach to Economic Model Predictive Control with terminal penalty functions

In this paper, we first introduce upper and a lower bounds of the best asymptotic average performance for nonlinear control systems based on the concepts of dissipativity and control storage functions. This allows to extend the formulation and analysis of Economic Model Predictive Control to more general optimal operation regimes, such as periodic solutions. A performance and stability analysis is carried out within this generalized framework. Finally two examples are proposed and discussed to show the merits of the proposed approach