Matthias Lorenzen

Constraint-Tightening and Stability in Stochastic Model Predictive Control

Constraint tightening to non-conservatively guarantee recursive feasibility and stability in Stochastic Model Predictive Control is addressed. Stability and feasibility requirements are considered separately, highlighting the difference between existence of a solution and feasibility of a suitable, a priori known candidate solution. Subsequently, a Stochastic Model Predictive Control algorithm which unifies previous results is derived, leaving the designer the option to balance an increased feasible region against guaranteed bounds on the asymptotic average performance and convergence time. Besides typical performance bounds, under mild assumptions, we prove asymptotic stability in probability of the minimal robust positively invariant set obtained by the unconstrained LQ-optimal controller. A numerical example, demonstrating the efficacy of the proposed approach in comparison with classical, recursively feasible Stochastic MPC and Robust MPC, is provided.

Distributed local stabilization in formation control

We present new results for formation control in the plane. We work with the usual assumptions stipulating that each agent in the formation has only access to the position of its leader(s) and the target edge length it has to maintain, i.e. only has partial knowledge of the global objective and global state of the formation. In this paper, we show how local stability of the formation around a target equilibrium is affected by this partial knowledge. In particular, we show why many natural control laws work for a formation of three agents but fail to work for slightly more complex, yet minimally persistent, formations.

An improved constraint-tightening approach for Stochastic MPC

The problem of achieving a good trade-off in Stochastic Model Predictive Control between the competing goals of improving the average performance and reducing conservativeness, while still guaranteeing recursive feasibility and low computational complexity, is addressed. We propose a novel, less restrictive scheme, which is based on considering stability and recursive feasibility separately. Through an explicit first step constraint we guarantee recursive feasibility. In particular we guarantee the existence of a feasible input trajectory at each time instant, but we only require that the input sequence computed at time k remains feasible at time k+1 for most disturbances, but not necessarily for all, which suffices for stability. To overcome the computational complexity of probabilistic constraints, we propose an offline constraint-tightening procedure, which can be efficiently solved to the desired accuracy via a sampling approach. The online computational complexity of the resulting Model Predictive Control (MPC) algorithm is similar to that of a nominal MPC with terminal region. A numerical example, which provides a comparison with classical, recursively feasible Stochastic MPC and Robust MPC, shows the efficacy of the proposed approach.

Robust economic Model Predictive Control using stochastic information

In this paper, we develop a new tube-based robust economic MPC scheme for linear time-invariant systems subject to bounded disturbances with given distributions. By using the error distribution in the predictions of the finite horizon optimal control problem, we can incorporate stochastic information in order to improve the expected performance while being able to guarantee strict feasibility. For this new framework, we can provide bounds on the asymptotic average performance of the closed-loop system. Moreover, a constructive approach is presented in order to find an appropriate terminal cost leading to a slight degradation of the bound on the guaranteed average performance.

Stochastic MPC with offline uncertainty sampling

For discrete-time linear systems subject to parametric uncertainty described by random variables, we develop a sampling-based Stochastic Model Predictive Control algorithm. Unlike earlier results employing a scenario approximation, we propose an offline sampling approach in the design phase instead of online scenario generation. The paper highlights the structural difference between online and offline sampling and provides rigorous bounds on the number of samples needed to guarantee chance constraint satisfaction. The approach does not only significantly speed up the online computation, but furthermore allows to suitably tighten the constraints to guarantee robust recursive feasibility when bounds on the uncertain variables are provided. Under mild assumptions, asymptotic stability of the origin can be established.

A distributed solution to the adjustable robust economic dispatch problem

The problem of maintaining balance between consumption and production of electric energy in the presence of a high share of intermittent power sources in a transmission grid is addressed. A distributed, asynchronous optimization algorithm, based on the ideas of cutting-plane approximations and adjustable robust counterparts, is presented to compute economically optimal adjustable dispatch strategies. These strategies guarantee satisfaction of the power balancing constraint as well as of the operational constraints for all possible realizations of the uncertain power generation or demand. The communication and computational effort of the proposed distributed algorithm increases for each computational unit only slowly with the number of participants, making it well suited for large scale networks. A distributed implementation of the algorithm and a numerical study are presented, which show the performance in asynchronous networks and its robustness against packet loss.

Scenario-based Stochastic MPC with guaranteed recursive feasibility

This paper addresses recursive feasibility and asymptotic stability, as well as the reduction of the online computational complexity, in scenario-based Stochastic Model Predictive Control for systems with time-varying parametric uncertainty. We propose a scheme, based on offline uncertainty sampling, which allows to suitably modify the constraints in such a way that recursive feasibility can be guaranteed robustly. The approach significantly speeds up the online computation, because no samples need to be generated online and, furthermore, unnecessary samples, which create redundant constraints, can be removed offline. Under mild additional assumptions, asymptotic stability with probability one can be proved. A numerical example, which provides a comparison with classical online sampling-based Stochastic MPC, demonstrates the efficacy of the proposed approach.

Stabilizing stochastic MPC without terminal constraints

The stability proofs of Model Predictive Control without terminal constraints and/or cost are tightly based upon the principle of optimality, which does not hold in most currently employed approaches to Stochastic MPC. In this paper, we first provide a stability proof for Stochastic Model Predictive Control without terminal cost or constraints under the assumption of optimization over feedback laws and propagation of the probability density functions of predicted states. Based thereon, we highlight why the proof does not remain valid if approximations such as parametrized feedback laws or relaxations on the chance constraints are employed and provide tightened assumptions that are sufficient to establish closed-loop stability. General statements valid for nonlinear systems are provided along with examples and computational simplifications in the case of linear systems.