Multiple model reduction approach using gap metric and stability margin for control nonlinear systems

This paper deals with the control of nonlinear systems using multimodel approach. The main idea of this work consists on the association of the gap metric and the stability margin tools to reduce the number of models constituting the multimodel bank. In fact, the self-organisation map (SOM) algorithm is used, firstly, to develop a preliminary multimodel bank. Then, the gap metric and the stability margin are computed to determine the redundancy of the initial multimodel bank. So, the multimodel controller is elaborated based on the reduced model bank. Simulations confirm the method for selecting the appropriate number of local models which should be used in the controller design.

     

A gap metric based weighting method for multimodel predictive control of MIMO nonlinear systems

A novel weighting method is proposed for multimodel predictive control of nonlinear systems with multiple scheduling variables (MIMO nonlinear systems), in which the gap metric is employed to formulate weighting functions for local controller combination. Compared to existent weighting functions, the proposed weighting method has two major advantages: firstly, there is only one tuning parameter, which makes it simpler. Secondly, the weights depend only on the scheduling vector and can be calculated off-line and stored in a look-up table. Therefore, the computational load can be reduced, especially for nonlinear systems with multiple scheduling variables. A MIMO CSTR system is studied to demonstrate the effectiveness of the proposed weighting method.     

A polynomial approach for optimal control of switched nonlinear systems

Optimal control problems for switched nonlinear systems are investigated. We propose an alternative approach for solving the optimal control problem for a nonlinear switched system based on the theory of moments. The essence of this method is the transformation of a nonlinear, nonconvex optimal control problem, that is, the switched system, into an equivalent optimal control problem with linear and convex structure, which allows us to obtain an equivalent convex formulation more appropriate to be solved by high‐performance numerical computing. Consequently, we propose to convexify the control variables by means of the method of moments obtaining semidefinite programs. Copyright © 2013 John Wiley & Sons, Ltd.     

A multiple model approach for predictive control of nonlinear hybrid systems

This paper presents modeling and control of nonlinear hybrid systems using multiple linearized models. Each linearized model is a local representation of all locations of the hybrid system. These models are then combined using Bayes theorem to describe the nonlinear hybrid system. The multiple models, which consist of continuous as well as discrete variables, are used for synthesis of a model predictive control (MPC) law. The discrete-time equivalent of the model predicts the hybrid system behavior over the prediction horizon. The MPC formulation takes on a similar form as that used for control of a continuous variable system. Although implementation of the control law requires solution of an online mixed integer nonlinear program, the optimization problem has a fixed structure with certain computational advantages. We demonstrate performance and computational efficiency of the modeling and control scheme using simulations on a benchmark three-spherical tank system and a hydraulic process plant.     

LMI-based model predictive control for switched nonlinear systems

This paper proposes an LMI approach to model predictive control of nonlinear systems with switching between multiple modes. In this approach, at each mode, the nonlinear system is divided to a linearized model in addition to a nonlinear term. A sum of squares (SOS) optimization problem is presented to find a quadratic bound for the nonlinear part. The stability condition of the switching system is obtained by using a discrete Lyapunov function and then the sufficient state feedback control law is achieved so that guarantees the stability of the system and also minimizes an infinite prediction horizon performance index. Moreover, two other LMI optimization problems are solved at each mode in order to find the maximum area region of convergence of the nonlinear system inscribed in the region of stability. The performance and effectiveness of the proposed MPC approach are illustrated by two case studies.     

A poly-quadratic stability based approach for linear switched systems

In this paper, a link between poly-quadratic stability and stability of arbitrary switching systems is established. Polyquadratic stability aims to check asymptotic stability of a polytopic system by means of polytopic quadratic Lyapunov functions. The necessary and sufficient condition of poly-quadratic stability proposed in Daafouz and Bernussou (2001) is shown to be immediately applicable to a class of switched control and observer design problems. Chaos synchronization for which the transmitter is described by a piecewise linear map is presented as an application.     

Region stability analysis for switched nonlinear systems with multiple equilibria

This paper investigates the region stability analysis for switched nonlinear systems whose subsystems have different equilibria, which are referred to multi-equilibrium switched nonlinear systems, and proposes a number of new results on the region stability analysis. Through investigating the dynamical behavior of the multi-equilibrium switched system, two methods are established, based on which several new results are then obtained for the region stability. It is shown that the main results obtained in this paper not only guarantee the region stability of the multi-equilibrium switched nonlinear system under arbitrary switching, but also provide several new approaches to determine the corresponding regions of convergence. Moreover, an improved average dwell time (ADT) method is proposed for the stability of switched nonlinear systems whose subsystems do not share a common Lyapunov function. Finally, an illustrative example with numerical simulation is studied by using the obtained results. The study of example shows that our results work very well in analyzing the region stability of some multi-equilibrium switched nonlinear systems.     

A sliding mode observer for uncertain nonlinear systems based on multiple models approach

This paper presents a method of state estimation for uncertain nonlinear systems described by multiple models approach. The uncertainties, supposed as norm bounded type, are caused by some parameters’ variations of the nonlinear system. Linear matrix inequalities (LMIs) have been established in order to ensure the stability conditions of the multiple observer which lead to determine the estimation gains. A sliding mode gain has been added in order to compensate the uncertainties. Numerical simulations through a state space model of a real process have been realized to show the robustness of the synthesized observer.     

A converse Lyapunov theorem for nonlinear switched systems

In this paper we present a converse Lyapunov theorem for uniform asymptotic stability of switched nonlinear systems. Its proof is a simple consequence of some results on converse Lyapunov theorems for systems with bounded disturbances obtained by Lin et al. (SIAM J. Control Optim. 34 (1996) 124–160), once an association of the switched system with a nonlinear system with disturbances is established.     

切换非线性系统的分布式模型预测控制

针对一类具有预先指定切换序列的切换非线性系统,研究了具有通信信道干扰和时滞测量的分布式模型预测控制问题.在每个子系统都存在镇定控制器的假设下,利用基于Lyapunov函数的模型预测控制器设计了分布式模型预测控制器,并给出了闭环切换非线性系统最终有界的充分条件.最后,通过仿真结果表明了分布式模型预测控制策略的有效性.     

Stochastic input-to-state stability for impulsive switched stochastic nonlinear systems with multiple jumps

In this study, we investigate the stochastic input-to-state stability (SISS) of impulsive switched stochastic nonlinear systems. In this model, the impulse jumps are component multiple maps that depend on time. Thus the model differs from traditional impulsive systems with single impulse between two adjacent switching times. We provide sufficient conditions in three cases with the SISS system by using the Lyapunov function and average impulsive interval approach. The destabilising impulses cannot destroy the SISS properties if the impulses do not occur too frequently when all the subsystems that control the continuous dynamics are SISS. In other words, the average impulsive interval satisfies a lower bound restraint. Conversely, when all subsystems that control the continuous dynamics are not SISS, impulses can contribute to stabilising the system in the SISS sense when the average impulsive interval satisfies an upper bound. Then, we investigate the SISS property of impulsive switched stochastic nonlinear systems with some subsystems that are not SISS under certain conditions such that the property remains obtained. Finally, we show three examples to demonstrate the validity of the main result.     

A condition for the stability of switched nonlinear systems

In this paper, we present a sufficient condition for the global asymptotic stability of a switched nonlinear system composed of a finite family of subsystems. We show that the global asymptotic stability of each subsystem and the pairwise commutation of the vector fields that define the subsystems (i.e., the Lie bracket of any pair of them is zero) are sufficient for the global asymptotic stability of the switched system. We also show that these conditions are sufficient for the existence of a common Lyapunov function.     

Application of gap metric to model bank determination in multilinear model approach

This paper presents a gap metric based method which aims to perform the operating range decomposition and the minimum linear model bank determination of a nonlinear system when multilinear model approach is employed to design a controller for this nonlinear system. For a prescribed distance level, the minimum linear model bank determined by the proposed method can provide sufficient information for multilinear model controller design of the nonlinear system. To illustrate the usefulness of the proposed method, two examples of nonlinear systems are presented. Moreover, a mixed logical dynamical model-based MPC (MLD–MPC) controller is designed based on the minimum model bank. Simulations confirm the method for selecting linear model bank in multilinear model approach.     

A switching robust model predictive control approach for nonlinear systems

This work considers enhancing the stability and improving the economic performance of nonlinear model predictive control in the presence of disturbances or model uncertainties. First, a robust control Lyapunov function (RCLF)-based predictive control strategy is proposed. Second, the approximate dynamic programming (ADP) is employed to further improve regulation performance. Finally, the ADP and RCLF-MPC are combined to provide a switching control scheme, which is illustrated on a CSTR example to show its effectiveness.     

Output feedback control of switched nonlinear systems using multiple Lyapunov functions

This work presents a hybrid nonlinear control methodology for a broad class of switched nonlinear systems with input constraints. The key feature of the proposed methodology is the integrated synthesis, via multiple Lyapunov functions, of “lower-level” bounded nonlinear feedback controllers together with “upper-level” switching laws that orchestrate the transitions between the constituent modes and their respective controllers. Both the state and output feedback control problems are addressed. Under the assumption of availability of full state measurements, a family of bounded nonlinear state feedback controllers are initially designed to enforce asymptotic stability for the individual closed-loop modes and provide an explicit characterization of the corresponding stability region for each mode. A set of switching laws are then designed to track the evolution of the state and orchestrate switching between the stability regions of the constituent modes in a way that guarantees asymptotic stability of the overall switched closed-loop system. When complete state measurements are unavailable, a family of output feedback controllers are synthesized, using a combination of bounded state feedback controllers, high-gain observers and appropriate saturation filters to enforce asymptotic stability for the individual closed-loop modes and provide an explicit characterization of the corresponding output feedback stability regions in terms of the input constraints and the observer gain. A different set of switching rules, based on the evolution of the state estimates generated by the observers, is designed to orchestrate stabilizing transitions between the output feedback stability regions of the constituent modes. The differences between the state and output feedback switching strategies, and their implications for the switching logic, are discussed and a chemical process example is used to demonstrate the proposed approach.     

Stabilization for nonlinear fractional-order time-varying switched systems

This paper deals with the global asymptotic stabilization problem for discontinuous nonlinear time-varying switched systems in the sense of fractional-order. Via the designed periodic switching law, some criteria are firstly developed. It is shown that the derivatives of the constructed Lyapunov functions are allowed to be indefinite. Several corollaries are further derived, and some existing results can be regarded as special cases of our obtained results. Finally, our findings are verified by three numerical examples and one experiment.     

An invariance principle for nonlinear switched systems

In this paper we address the problem of extending LaSalle Invariance Principle to switched system. We prove an extension of the invariance principle relative to dwell time switched solutions, and a second one relative to constrained switched systems.     

Output tracking control of switched nonlinear systems with multiple time-varying delays

In this paper, the output tracking control problem for a class of switched nonlinear systems with multiple time-varying delays is studied based on equivalent-input-disturbance (EID) approach. More precisely, with the use of suitable Lyapunov-Krasovskii functional together with average dwell-time technique, an output feedback tracking controller is designed which makes that the states of resulting system can asymptotically track the desired trajectory. Further, the EID estimator is implemented to reject both matched and unmatched disturbances effectively without requiring any prior knowledge of the disturbances. Simulation results are presented to illustrate the effectiveness and potential of the developed EID-based output tracking control design technique. The results reveal the fact that the tracking controller based on EID provides a better tracking performance than the feedback controller based on sliding mode technique.     

Invertibility of switched nonlinear systems

This article addresses the invertibility problem for switched nonlinear systems affine in controls. The problem is concerned with reconstructing the input and switching signal uniquely from given output and initial state. We extend the concept of switch-singular pairs, introduced recently, to nonlinear systems and develop a formula for checking if the given state and output form a switch-singular pair. A necessary and sufficient condition for the invertibility of switched nonlinear systems is given, which requires the invertibility of individual subsystems and the nonexistence of switch-singular pairs. When all the subsystems are invertible, we present an algorithm for finding switching signals and inputs that generate a given output in a finite interval when there is a finite number of such switching signals and inputs. Detailed examples are included to illustrate these newly developed concepts.