Model predictive control for switched systems with a novel mixed time/event-triggering mechanism

This paper investigates observer-based model predictive control (MPC) for switched systems with a mixed time/event-triggering mechanism. The problem of predictive control that can achieve receding horizon optimization is considered and solved by minimizing an upper bound of the quadratic cost function. Since the system state may not be fully measured in practice, state observers are employed to estimate. A mixed mechanism including adaptive event-triggering and time-triggering is proposed, which can be switched determined by a threshold describing system performance to better balance system resource utilization and performance requirements. Then, a closed-loop switched system subject to networked-time-delay is modeled. Piecewise Lyapunov function technique and average dwell time approach are utilized to ensure asymptotical stability. Afterwards, MPC controller construction problem is turned into a LMIs feasibility problem. A new solving method of sufficient conditions for co-design of the state observers, feedback controllers and mixed triggering mechanism is derived. Lastly, simulation examples illustrate the correctness and advantages of research content.     

Robust tube-based MPC of constrained piecewise affine systems with bounded additive disturbances

In this article, we propose a robust tube-based MPC formulation for a class of hybrid systems, namely autonomously switched PWA systems, with bounded additive disturbances. The term tube-based refers to those control techniques whose objective is to maintain all possible trajectories of the uncertain system inside a tube which is a set around the nominal (or reference) system trajectory, that is free from disturbances. Common methods in tube-based control systems consider an error dynamical system as the difference between the state of the nominal system and the state of the perturbed system. However, this definition of the error dynamical system leads to a complicated switched affine system for PWA systems. Therefore, we use a new notion of the reference system similar to the nominal system except that the switching between the various modes of the PWA system is driven by the state of the real system. Using this reference system instead of the nominal system leads us to an error dynamical system that can be modeled as a switched linear system. We employ a switched linear controller to stabilize this error system under arbitrary switching. This auxiliary controller forces the states of the uncertain system to remain in a tube confined to the invariant set around the state of the reference system. We add new constraints and tighten some other constraints of the nominal hybrid MPC for the reference system, in order to ensure convergence of the uncertain system and to guarantee robust exponential stability of the closed-loop system.     

A computationally efficient robust tube based MPC for linear switched systems

This article considers the robust regulation problem for a class of constrained linear switched systems with bounded additive disturbances. The proposed solution extends the existing robust tube based model predictive control (RTBMPC) strategy for non-switched linear systems to switched systems. RTBMPC utilizes nominal model predictions, together with tightened sets constraints, to obtain a control policy that guarantees robust stabilization of the dynamic systems in presence of bounded uncertainties. In this work, similar to RTBMPC for non-switched systems, a disturbance rejection proportional controller is used to ensure that the closed loop trajectories of the switched linear system are bounded in a tube centered on the nominal system trajectories. To account for the uncertainty related to all sub-systems, the gain of this controller is chosen to simultaneously stabilize all switching dynamics. The switched system RTBMPC requires an on-line solution of a Mixed Integer Program (MIP), which is computationally expensive. To reduce the complexity of the MIP, a sub-optimal design with respect to the previous formulation is also proposed that uses the notion of a pre-terminal set in addition to the usual terminal set to ensure stability. The RTBMPC design with the pre-terminal set aids in determining the trade-off between the complexity of the control algorithm with the performance of the closed-loop system while ensuring robust stability. Simulation examples, including a Three-tank benchmark case study, are presented to illustrate features of the proposed MPC.     

Model predictive control of nonlinear hybrid systems with discrete inputs employing a hybrid fuzzy model

The paper deals with model predictive control (MPC) of nonlinear hybrid systems with discrete inputs based on reachability analysis. In order to implement a MPC algorithm, a model of the process that we are dealing with is needed. In the paper, a hybrid fuzzy modelling approach is proposed. The hybrid system hierarchy is explained and the Takagi–Sugeno fuzzy formulation for hybrid fuzzy modelling purposes is tackled. An efficient method of identification of the hybrid fuzzy model is also discussed.

An algorithm that is–due to its MPC nature–suitable for controlling a wide spectrum of systems (provided that they have discrete inputs only) is presented.

The benefits of the algorithm employing a hybrid fuzzy model are verified on a batch reactor example. The results suggest that by suitably determining the cost function, satisfactory control can be attained, even when dealing with complex hybrid–nonlinear–stiff systems such as the batch reactor.

Finally, a comparison between MPC employing a hybrid linear model and a hybrid fuzzy model is carried out. It has been established that the latter approach clearly outperforms the approach where a linear model is used.     

LMI Approach to Robust Model Predictive Control

This paper introduces a new approach to robust model predictive control (MPC) based on conservative approximations to semi-infinite optimization using linear matrix inequalities (LMIs). The method applies to problems with convex quadratic costs, linear and convex quadratic constraints, and linear predictive models with bounded uncertainty. If the MPC optimization problem is feasible at the initial control step (the first application of the MPC optimization), it is shown that the MPC optimization problems will be feasible at all future time steps and that the controlled system will be closed-loop stable. The method is illustrated with a solenoid control example. The authors thank the anonymous reviewers for suggestions that improved the presentation of this work. The work was supported in part by the EPRI/DoD Complex Interactive Networks/Systems Initiative under Contract EPRI-W08333-05 and by the US Army Research Office Contract DAAD19-01-1-0485.     

CFD-based predictive control of melt temperature in plastic injection molding

A unique method of coupling computational fluid dynamics (CFD) to model predictive control (MPC) for controlling melt temperature in plastic injection molding is presented. The methodology is based on using CFD to generate, via open-loop testing, a temperature and input dependent system model for multi-variable control of a three-heater barrel on an injection molding machine. Results clearly show the benefit of temperature and input dependent system models for MPC control, and that CFD can be used to dramatically reduce the time associated with open-loop testing through physical experiments.     

A Fast MPC Algorithm Using Nonfeasible Active Set Methods

Model predictive control (MPC) is an optimization-based control framework which is attractive to industry both because it can be practically implemented and it can deal with constraints directly. One of the main drawbacks of MPC is that large MPC horizon times can cause requirements of excessive computational time to solve the quadratic programming (QP) minimization which occurs in the calculation of the controller at each sampling interval. This motivates the study of finding faster ways for computing the QP problem associated with MPC. In this paper, a new nonfeasible active set method is proposed for solving the QP optimization problem that occurs in MPC. This method has the feature that it is typically an order of magnitude faster than traditional methods. This work has been supported by the Canadian NSERC under Grant A4396.     

Multivariable RBF-ARX model-based robust MPC approach and application to thermal power plant

For a class of smooth nonlinear multivariable systems whose working-points vary with time and the future working-points knowledge are unknown, a combination of a local linearization and a polytopic uncertain linear parameter-varying (LPV) state-space model is built to approximate the present and the future system’s nonlinear behavior, respectively. The combination models are constructed on the basis of a matrix polynomial multi-input multi-output (MIMO) RBF-ARX model identified offline for representing the underlying nonlinear system. A min–max robust MPC strategy is designed to achieve the systems’ output-tracking control based on the approximate models proposed. The closed loop stability of the MPC algorithm is guaranteed by the use of time-varying parameter-dependent Lyapunov function and the feasibility of the linear matrix inequalities (LMIs). The effectiveness of the modeling and control methods proposed in this paper is illustrated by a case study of a thermal power plant simulator.     

A Model Predictive Control Algorithm Applied To Non-Linear Processes

Model predictive control (MPC) is an optimization-based approach that has been successfully applied to a wide variety of control problems. In most of nonlinear strategies, the controllers are based on linear models with fixed parameters so that the vast body of linear control theory can be applied. Other solutions include the use of a nonlinear analytical model, combinations of linear empirical models, etc. This paper presents an MPC algorithm which uses on-line simulation and rule-based control. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Reduced-order model predictive control of a fish schooling model

We study the problem of model predictive control (MPC) for the fish schooling model proposed by Gautrais et al. (2008). The high nonlinearity of the model attributed to its attraction/alignment/repulsion law suggests the need to use MPC for controlling the fish schooling’s motion. However, for large schools, the hybrid nature of the law can make it numerically demanding to perform finite-horizon optimizations in MPC. Therefore, this paper proposes reducing the fish schooling model for numerically efficient MPC; the reduction is based on using the weighted average of the directions of individual fish in the school. We analytically show how using the normalized eigenvector centrality of the alignment-interaction network can yield a better reduction by comparing reduction errors. We confirm this finding on the weight and numerical efficiency of the MPC with the reduced-order model by numerical simulations. The proposed reduction allows us to control a school with up to 500 individuals. Further, we confirm that reduction with the normalized eigenvector centrality allows us to improve the control accuracy by factor of five when compared to that using constant weights.     

MPC for the Burgers Equation Based on an LQG Design

We consider optimal control problems for semilinear parabolic PDEs where process and measurement noise can occur. We discuss the solution of such problems by using a Model Predictive Control (MPC) strategy. For the resulting sub-problems we will use a Linear Quadratic Gaussian (LQG) design. Thus we will discuss the efficient implementation of the LQG approach since it is the major computational part in the MPC scheme for this class of optimal control problems. We will present some numerical results for the Burgers equation. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Robust stability and H∞ control for nonlinear discrete‐time switched systems with interval time‐varying delay

This paper considers the problems of the robust stability analysis and H_∞ controller synthesis for uncertain discrete‐time switched systems with interval time‐varying delay and nonlinear disturbances. Based on the system transformation and by introducing a switched Lyapunov‐Krasovskii functional, the novel sufficient conditions, which guarantee that the uncertain discrete‐time switched system is robust asymptotically stable are obtained in terms of linear matrix inequalities. Then, the robust H_∞ control synthesis via switched state feedback is studied for a class of discrete‐time switched systems with uncertainties and nonlinear disturbances. We designed a switched state feedback controller to stabilize asymptotically discrete‐time switched systems with interval time‐varying delay and H_∞ disturbance attenuation level based on matrix inequality conditions. Examples are provided to illustrate the advantage and effectiveness of the proposed method.     

Controllability of switched time-delay systems under constrained switching

Different from the existing mathematical models for switched systems, where the switching from one subsystem to another subsystem is finished instantly, in this paper it is assumed that the switching is a transfer process. Moreover, there exists a basic transfer subsystem such that in the transfer process, the transfer subsystem is active. Based on the model of switched systems under constrained switching, this paper studies the controllability of such systems with time delay in the control function. A necessary and sufficient condition for controllability of such systems is established. Finally, an example is given to illustrate the utility of our results.     

Observer-based reliable exponential stabilization and H∞ control for switched systems with faulty actuators: An average dwell time approach

This paper deals with the problem of reliable stabilization and $H_{\infty }$ control for a class of continuous-time switched Lipschitz nonlinear systems with actuator failures. We consider the case that actuators suffer “serious failure”—the never failed actuators cannot stabilize the given system. The differential mean value theorem (DMVT) allows transforming the switched Lipschitz nonlinear systems into switched linear parameter varying (LPV) systems. Based on average dwell time scheme and under the condition that activation time ratio between stabilizable subsystems and unstabilizable ones is not less than a specified constant, sufficient conditions for reliable exponential stabilization of the switched systems are derived by hybrid observer-based output feedback control. The result is also extended to the reliable $H_{\infty }$ control problem.     

Theoretical and algorithmic advances in multi-parametric programming and control

This paper presents an overview of recent theoretical and algorithmic advances, and applications in the areas of multi-parametric programming and explicit/multi-parametric model predictive control (mp-MPC). In multi-parametric programming, advances include areas such as nonlinear multi-parametric programming (mp-NLP), bi-level programming, dynamic programming and global optimization for multi-parametric mixed-integer linear programming problems (mp-MILPs). In multi-parametric/explicit MPC (mp-MPC), advances include areas such as robust multi-parametric control, multi-parametric nonlinear MPC (mp-NMPC) and model reduction in mp-MPC. A comprehensive framework for multi-parametric programming and control is also presented. Recent applications include a hydrogen storage device, a fuel cell power generation system, an unmanned autonomous vehicle (UAV) and a hybrid pressure swing adsorption (PSA) system.     

Stability analysis of switched impulsive systems with time delays

In this paper, a new stability analysis of switched impulsive systems with time delays whose subsystem is not necessarily stable is presented. A sufficient condition on uniformly asymptotical stability for nonlinear switched impulsive systems is obtained. Using the result obtained and the minimum (maximum) holding time, an easily verifiable condition on uniformly asymptotical stability for linear switched impulsive systems with time delays is derived. The control synthesis is also discussed. Finally, two examples with simulation results are given to validate the results.     

Optimal Prioritized Infeasibility Handling in Model Predictive Control: Parametric Preemptive Multiobjective Linear Programming Approach

All practical implementations of model-based predictive control (MPC) require a means to recover from infeasibility. We propose a strategy designed for linear state-space MPC with prioritized constraints. It relaxes optimally an infeasible MPC optimization problem into a feasible one by solving a single-objective linear program (LP) online in addition to the standard online MPC optimization problem at each sample. By optimal, it is meant that the violation of a lower prioritized constraint cannot be made less without increasing the violation of a higher prioritized constraint. The problem of computing optimal constraint violations is naturally formulated as a parametric preemptive multiobjective LP. By extending well-known results from parametric LP, the preemptive multiobjective LP is reformulated into an equivalent standard single-objective LP. An efficient algorithm for offline design of this LP is given, and the algorithm is illustrated on an example.     

Necessary and sufficient conditions for quadratic stabilizability of switched systems on non-uniform time domains

In this paper, we consider the quadratic stabilizability via state feedback for a particular class of switched systems that evolve on a non-uniform time domain by introducing time scales theory. The system considered switches between a continuous-time subsystem with variable lengths and a discrete-time subsystem with variable discrete step sizes. Necessary and sufficient conditions are derived to guarantee the quadratic stability of this class of switched systems via a switching state feedback law based on the existence of a common positive definite matrix satisfying the quadratic stabilizability condition by considering that the two subsystems are unstable. By state feedback, we mean that the switching among subsystems depends on the system states. Current results for this kind of state switching feedback control are derived only for switched systems evolving on a continuous time domain or a discrete time domain with fixed step’s size. These results are not applicable for the particular class of switched systems where there is a mixing between the continuous and discrete dynamics. This motivates the derivation of a new and more general state feedback control law for switched systems in this work. A numerical example illustrating the results is presented.     

Equivalence between the Lyapunov–Krasovskii functionals approach for discrete delay systems and that of the stability conditions for switched systems

The stability of discrete-time systems with time varying delay in the state can be analyzed by using a discrete-time extension of the classical Lyapunov–Krasovskii approach. In the networked control systems domain a similar delay stability problem is treated using a switched system transformation approach. The paper aims to establish a relation between the switched system transformation approach and the classical Lyapunov–Krasovskii method. It is shown that using the switched systems transformation is equivalent to using a general delay dependent Lyapunov–Krasovskii functionals. This functional represents the most general form that can be obtained using sums of quadratic terms. Necessary and sufficient LMI conditions for the existence of such functionals are presented.     

Event-triggered sliding mode control for switched genetic regulatory networks with persistent dwell time

This paper focus on the event-triggered sliding mode controller design for discrete-time switched genetic regulatory networks (GRNs) with persistent dwell time (PDT) switching. Firstly, the observation error dynamics of switched GRNs with PDT is constructed in the light of event-triggered sliding mode control (SMC) scheme. Next, sufficient conditions are derived to ensure the exponential stability of the augmented plant. Moreover, an event-triggered SMC law is synthesized to impel the system trajectories onto the sliding surface in a finite time. Finally, a verification example is provided to illustrate the effectiveness and potential of the proposed method.