Marcus Reble

Unconstrained model predictive control and suboptimality estimates for nonlinear continuous-time systems

This paper presents a continuous-time version of recent results on unconstrained nonlinear model predictive control (MPC) schemes. Based on a controllability assumption and a corresponding infinite-dimensional optimization problem, performance estimates and stability conditions are derived in terms of the prediction horizon and the sampling time of the MPC controller. Moreover, improved estimates for small sampling times are discussed and a comparison to the application of the discrete-time results in a sampled-data context is provided.

Model predictive control of constrained nonlinear time-delay systems

This paper proposes a model predictive control scheme for a class of constrained nonlinear time-delay systems with guaranteed closed-loop asymptotic stability. Asymptotic stability of the closed-loop is guaranteed by utilizing an appropriate terminal cost functional and an appropriate terminal region. A novel structured procedure is derived to determine the terminal cost and the terminal region offline. For this purpose, a combination of Lyapunov-Krasovskii and Lyapunov-Razumikhin arguments is used to compute a locally stabilizing controller. The resulting conditions are formulated in terms of linear matrix inequalities.

The Role of Sampling for Stability and Performance in Unconstrained Nonlinear Model Predictive Control

We investigate the impact of sampling on stability and performance estimates in nonlinear model predictive control without stabilizing terminal constraints or costs. Interpreting the sampling period as a discretization parameter, the relation between continuous and discrete time estimates depending on this parameter is analyzed. The technique presented in this paper allows us to determine the sampling rate required in order to approximate the continuous time suboptimality bound arbitrarily well and, thus, gives insight into the trade-off between sampling time and guaranteed performance.

LMI-Based Model Predictive Control for Linear Discrete-Time Periodic Systems

This paper presents a newmodel predictive control (MPC) scheme for linear constrained discrete-time periodic systems. In each period of the system, a new periodic state feedback control law is computed via a convex optimization problem that minimizes an upper bound on an infinite horizon cost function subject to state and input constraints. The performance of the proposed model predictive controller, that stabilizes the discrete-time periodic system if it is initially feasible, is illustrated via an example.

General design parameters of model predictive control for nonlinear time-delay systems

This work presents new results on model predictive control (MPC) for nonlinear time-delay systems. In the first part, a general scheme is presented for calculating stabilizing design parameters based on the Jacobi linearization of the system. It is proven that for each system with stabilizable linearization there exist a quadratic terminal cost functional and a finite terminal region which guarantee asymptotic stability of the closed-loop. This allows the calculation of suitable design parameters for MPC based on any method available for the control of linear time-delay systems. In contrast to the delay-free case, the terminal region obtained is not characterized as sublevel set of the terminal cost functional. In the second part, a new type of terminal region is presented. Based on additional Lyapunov-Razumikhin conditions on the linear local controller it is shown that the terminal region can indeed be formulated as sublevel set of a particular terminal cost functional.

A General Distributed MPC Framework for Cooperative Control

Abstract In this paper, we consider a general framework for distributed model predictive control (DMPC) of discrete-time nonlinear systems with decoupled dynamics, but subject to coupled constraints and a common, cooperative task. In contrast to most of the existing DMPC schemes in the literature, we do not necessarily consider the stabilization of an a priori known setpoint, but also other cooperative tasks like consensus and synchronization problems can be handled within the proposed framework. In order to ensure recursive feasibility and convergence to the desired cooperative goal, the systems optimize a local cost function in a sequential order, communicating their planned trajectories only to their neighbors. We exemplarily show how the proposed DMPC algorithm can be used for achieving consensus and synchronization between the systems, and we illustrate the results with a simulation example.

Unconstrained Nonlinear Model Predictive Control and Suboptimality Estimates for Continuous-Time Systems

Abstract This paper presents a continuous-time version of recent results on unconstrained nonlinear model predictive control (MPC) schemes. Based on a controllability assumption and a corresponding infinite-dimensional optimization problem, performance estimates and stability conditions are derived in terms of the prediction horizon and the sampling time of the MPC controller. Moreover, improved estimates for small sampling times are discussed and a comparison to the application of the discrete-time results in a sampled-data context is provided.

Inherent Robustness Properties of Quasi-infinite Horizon MPC

Abstract We consider inherent robustness properties of model predictive control (MPC) for continuous-time nonlinear systems with input constraints and terminal constraints. We show that when the linear quadratic control law is chosen as the terminal control law, and the related Lyapunov matrix is chosen as the terminal penalty matrix, MPC with nominal prediction model and bounded disturbances has some degree of inherent robustness. We emphasize that the input constraint sets can be any compact set rather than convex sets, and our results do not rely on the continuity of the optimal cost functional or control law in the interior of the feasible region.

Model Predictive Control for Nonlinear Time-Delay Systems without Terminal Constraint

Abstract In this work we present new results on model predictive control (MPC) for nonlinear time-delay systems. In the first part we derive a novel scheme for determining a suitable terminal cost and terminal region based on the Jacobi linearization of the nonlinear system. The main advantage of the proposed scheme compared to previous results is that the terminal region is defined as a sublevel of the terminal cost functional without any restrictive requirements on the sampling time of the MPC. Based on this result, we present an MPC scheme without terminal constraint in the second part. The result extends existing results for delay-free systems and guarantees asymptotic stability of the closed-loop. The main difficulty in the derivation is to show that the integral over the stage cost has a lower bound if the state is outside of a certain region. This is directly satisfied for delay-free finite-dimensional systems, but requires additional arguments for time-delay systems.

Stabilizing design parameters for model predictive control of constrained nonlinear time-delay systems

Abstract In this work a novel procedure for the calculation of stabilizing design parameters for model predictive control of nonlinear time-delay systems is presented. In contrast to previous results, the conditions derived for the local control law and the terminal region are based only on Lyapunov Krasovskii arguments and do not require any Lyapunov Razumikhin arguments. Therefore, the conditions are less restrictive, however a more complicated terminal region is obtained. The applicability of the proposed scheme is demonstrated for the numerical model of a continuous stirred tank reactor with recycle stream.