LMI-based Control Design Research Articles

Homogeneous systems stabilization based on convex embedding

The paper develops control algorithms for a class of affine nonlinear systems using the so-called canonical homogeneous representation. It is demonstrated that such a representation exists for any homogeneous vector field bounded on the unit sphere. It is shown that canonical homogeneous representation is useful for LMI-based control design and stability analysis of nonlinear systems. Theoretical results are supported by numerical examples.

LMI-based Control of Singularly Perturbed Switched Affine Systems

In this paper, we tackle the problem of stabilizing singularly perturbed switched affine systems. Even though efficient control strategies based on the solution of Linear Matrix Inequalities (LMIs) have been presented in the switched affine systems literature, the presence of the small singular perturbation parameter may introduce numerical difficulties in the solution of these LMIs due to ill conditioning. Moreover, this parameter may be uncertain or its use in the control law may not be practical. Here, we propose an LMI-based control design strategy that avoids the conditioning issues and that also provides an estimation of the domain of attraction. Furthermore, the resulting controller does not depend on the singular perturbation parameter. The proposed method is illustrated on a numerical example.

LMI-based hybrid temporal-spatial differential control design for a class of Euler-Bernoulli beam system

This paper addresses the problem of active vibration suppression for a class of Euler-Bernoulli beam system. The objective of this paper is to design a hybrid temporal-spatial differential controller, which is involved with the in-domain and boundary actuators, such that the closed-loop system is stable. The Lyapunov’s direct method is employed to derive the sufficient condition, which not only can guarantee the stabilization of system, but also can improve the spatial cooperation of actuators. In the framework of the linear matrix inequalities (LMIs) technology, the gain matrices of hybrid controller can obtained by developing a recursive algorithm. Finally, the effectiveness of the proposed methodology is demonstrated by applying a numerical simulation.

Copositive Conditions for LMI-Based Controller and Observer Design

In this report we illustrate that nonlinear control and observer design can be described as a Positivstellensatz problem, we consider a quadratic candidate function to stabilize nonlinear systems that have been modeled via the nonlinear sector methodology as an exact convex representation. New shape-independent conditions for satisfying positiveness in double sums are established based on the concept of copositivity. The conditions obtained are compared with previous approaches found in the literature via a numerical example.

Control of SISO non-affine-in-control discrete-time systems using Takagi-Sugeno models

Abstract This paper deals with the stabilization problem for single-input-single-output (SISO) non-affine-in-control nonlinear discrete-time systems. Generally, this non-affine-in-control problem leads to algebraic loops and a classical solution consists in using dynamic extensions. Based on Takagi-Sugeno (T-S) quasi-linear parameter varying (quasi-LPV) models, we propose to keep the scheme of the classical LMI-based control design framework, i.e. similar as when the membership functions of the T-S model do not depend on the control input. In particular, some properties of finite time convergence are exploited to “break” the algebraic loop problem. Numerical simulations are provided to demonstrate the interests of the proposed control approach.

Sparsely distributed sliding mode control for interconnected systems

This paper proposes a framework for the design of sparsely distributed output feedback discrete-time sliding mode control (ODSMC) for interconnected systems. The major target here is to develop an observer based discrete-time sliding mode controller employing a sparsely distributed control network structure in which local controllers exploit some other sub-systems’ information as well as its own local information. As the local controllers/observers have access to some other sub-systems’ states, the control performance will be improved and the applicability region will be widened compared to the decentralised structure. As the first step, a stability condition is derived for the overall closed-loop system obtained from applying ODSMC to the underlying interconnected system, by assuming a priori known structure for the control/observer network. The developed LMI based controller design scheme provides the possibility to employ different information patterns such as fully distributed, sparsely distributed and decentralised patterns. In the second step, we propose a methodology to identify a sparse control/observer network structure with the least possible number of communication links that satisfies the stability condition given in the first step. The boundedness of the obtained overall closed-loop system is analysed and a bound is derived for the augmented system state which includes the closed-loop system state and the switching function.

An augmented system approach for LMI-based control design of constrained Takagi-Sugeno fuzzy systems

In this paper, we address the control design of a class of constrained Takagi-Sugeno fuzzy systems. These systems are subject to both control input and state constraints. Sufficient design conditions are derived via a judicious use of an equivalent augmentation form of the closed-loop system together with a new parameter-dependent Lyapunov function. The control input nonlinearity is dealt with via a generalized sector condition. These instrumental control tools allow reducing significantly the conservatism of the results while keeping a simple control structure with a low level of numerical complexity. In particular, it has been shown that in many cases the new method can provide better results (in terms of conservatism reduction) than some recent works developed in non-quadratic Lyapunov framework. Several numerical examples are given to illustrate the effectiveness of the proposed method.

Vibration control of hysteretic base-isolated structures: an LMI approach

Seismic isolation systems are essentially designed to preserve structural safety, prevent occupants injury and properties damage. An active saturated LMI-based control design is proposed to attenuate seismic disturbances in base-isolated structures under saturation actuators. Using a mathematical model of an eight-storied building structure, an active control algorithm is designed. Performance evaluation of the controller is carried out in a simplified model version of a benchmark building system, which is recognized as a state-of-the-art model for numerical experiments of structures under seismic perturbations. Experimental results show that the proposed algorithm is robust with respect to model and seismic perturbations. Finally, the performance indices show that the proposed controller behaves satisfactorily and with a reasonable control effort.

An LMI-based controller design method for a discrete-time linear system with time-varying state delays

An LMI-based controller design method for a discrete-time linear system with time-varying state delays

LMI based control design for linear systems with distributed time delays

LMI based control design for linear systems with distributed time delaysThe paper concerns the problem of stabilization of continuous-time linear systems with distributed time delays. Using extended form of the Lyapunov-Krasovskii functional candidate, the controller design conditions are derived and formulated with respect to the incidence of structured matrix variables in the linear matrix inequality formulation. The result give sufficient condition for stabilization of the system with distributed time delays. It is illustrated with a numerical example to note reduced conservatism in the system structure.

Guaranteed Performance Level Iterative Control with Input Constraints: an LMI Approach

AbstractIn this work, a controller design methodology to be applied to iterative control schemes is presented. Iterative control requires a sequence of identification procedures followed by controllers designed on the most recent identified model. However for uncertain plants with input saturation classical iterative procedures can fail to converge. We propose an LMI based controller design which takes into account the input limitations and future variations on the process model. The controller is designed to be valid for the possible largest uncertain plant set while accomplishing the performance and control input constrains. In this way the controller design maximizes its robustness and, therefore, its validity along time. On iterative control schemes this lead to the minimization of identification and subsequent controller design procedures.

Small-Signal Stability Enhancement of Power Systems with Renewable Distributed Energy Resources

AbstractThis paper examines the application of energy storage to enhance the small-signal stability of an electrical power system with renewable power generation. In particular, a linearized model of the system and an LMI-based control design method are used to achieve frequency and voltage regulation subject to small perturbations to mechanical power inputs to synchronous generators in the system. The controller design is validated using the following simulation studies: (1) a single machine connected to an infinite bus (SMIB) (2) a two-machine (synchronous generator and doubly-fed induction generator) connected to an infinite bus. Simulation results show that the proposed controller can simultaneously achieve frequency and voltage regulation.

폴리토픽 모델을 갖는 대규모 시스템을 위한 비집중화 슬라이딩 모드 제어기 설계

Based on the sliding mode control theory, a decentralized controller design method is developed for a large scale system with a poly topic model. In terms of LMIs, we derive sufficient conditions for the existence of the decentralized controller guaranteeing a stable sliding motion. We also give an LMI-based control design algorithm. Finally, the proposed method is applied to decentralized stabilization of double-inverted pendulums. Simulation results show that our method gives not only the robust stability but perfect rejection of norm-bounded uncertainties.

A General Quadratic Performance Approach to Binary Distillation Control

High purity distillation control of a binary mixture in a tray column is considered in the paper at hand. The approach is based on an inferential control idea: dynamics within the column may be described as movements of concentration waves; the position of the wave front on the one hand side can be inferred from few temperature measurements, on the other hand the position implies the product concentrations. Dynamics of wave propagation is derived by simplification of a first principles model of the column. The resulting descriptor model is the basis for a recent LMI based controller design scheme that provides general quadratic performance for descriptor systems.

Saturated LMI Control of Hysteretic Base-Isolated Structures

The main objective of applying active control forces to base-isolated structures is to protect them in the event of an earthquake. An LMI-based control design is proposed to attenuate seismic disturbances in base-isolated structures under saturation actuators. Using a simplified model of the structure system, a control algorithm design is offered. Performance evaluation of the controller is carried out in a simplified model version of a benchmark building system, which is recognized as a state-of-the-art model for numerical experiments of structures under seismic perturbations. Experimental results show that the proposed algorithm is robust with respect to model and seismic perturbations. Finally, the performance indices show that the proposed controller behaves satisfactorily and with a reasonable control effort.

Controller Synthesis of an Uncertain Three Tank System Using Polytypic System Approach

Many of the systems in process and aerospace industry can be modeled as a three tank system. LMI and NLMI based controller design methods have been applied to three tank system in literature. Nonlinear methods suffer from lack of ease for implementation, on the other hand, LMI based methods have to be unduly robust to cater for significant uncertainties arising from linearization around an equilibrium point. This paper presents an LMI based controller design method, which specifically uses second or higher order terms of nonlinear three tank model, thus resulting in a robust controller with no compromise on performance. A second degree error model is derived using small signal linearization method about an arbitrary operating point. A set of LMI's are formulated for each polytopic region and solved to obtain the corresponding state feedback controller gains for that particular polytopic region. The series of designed controllers drives the system states from starting point to the desired state through a series of connected polytopic regions. The controller is robust with respect to parameter variations due to the combined polytopic-LMI problem formulation. The effects of parametric variations and disturbances are accounted for in such a formulation via appropriate bounds. The performance of the designed controller is also compared with LMI based H∞ controller.

Parameter-Dependent Lyapunov Function Method for a Class of Uncertain Nonlinear Systems with Multiple Equilibria

First, a new linear matrix inequality (LMI) characterization of extended strict positive realness is presented for linear continuous-time systems. Then a class of nonlinear systems with multiple equilibria subject to polytopic uncertainty is addressed by the parameter-dependent Lyapunov function method. New sufficient conditions for global convergence are presented. This allows the Lyapunov function to be parameter dependent. Furthermore, an LMI-based controller design method is also given, and reduced-order controllers can be designed by performing a structural constraint on the introduced slack variables. Several numerical examples are included to demonstrate the applicability of the proposed method.

Stability Conditions for LMI-Based Fuzzy Control From Viewpoint of Membership Functions

In this paper, we investigate the stability conditions for linear matrix inequality (LMI)-based fuzzy control design. Especially, we focus on the dependence of the stability upon membership functions. In general, the membership functions in the rule bases of Takagi-Sugeno (T-S) fuzzy model and controllers are the same and restricted between 0 and 1. In contrast to this setting, we obtain some new results when different membership functions are considered and their values lying outside the interval of [0,1] are allowed. Applying Lyapunov equation and a convex hull of fuzzy subsystems, we first establish a relationship between the stable interval characteristic polynomial and a set of feasible LMIs. Then Kharitonov's theorem gives an insight for the solvability of stabilization problems using LMI-based design and, this leads that the membership functions have an influence on stability. On the other hand, the LMI condition leads to the well-known results for LMI-based fuzzy control design. We further indicate that the different LMI conditions arise due to the same or different membership functions and find their own applications on adaptive fuzzy control. Finally, if the unit interval constraint is removed, an LMI condition for global stability is obtained

TP Model Transformation as a Way to LMI-Based Controller Design

The main objective of this paper is to propose a numerical controller design methodology. This methodology has two steps. In the first step, tensor product (TP) model transformation is applied, which is capable of transforming a dynamic system model, given over a bounded domain, into TP model form, including polytopic or Takagi-Sugeno model forms. Then, in the second step, Lyapunov's controller design theorems are utilized in the form of linear matrix inequalities (LMIs). The main novelty of this paper is the development of the TP model transformation of the first step. It does not merely transform to TP model form, but it automatically prepares the transformed model to all the specific conditions required by the LMI design. The LMI design can, hence, be immediately executed on the result of the TP model transformation. The secondary objective of this paper is to discuss that representing a dynamic model in TP model form needs to consider the tradeoff between the modeling accuracy and computational complexity. Having a controller with low computational cost is highly desired in many cases of real implementations. The proposed TP model transformation is developed and specialized for finding a complexity minimized model according to a given modeling accuracy. Detailed control design examples are given.

Robust control of dynamical systems using neural networks with input–output feedback linearization

This paper presents a control algorithm that combines three valuable features in robust and non-linear control, namely modelling using neural networks, input–output feedback linearization and LMI-based robust controller design. In the first step of the algorithm an affine description of a feedforward neural network model is derived. By performing an input–output feedback (IOF) linearization an uncertainty description of the IOF linearized system is derived based on the parametric uncertainties of the affine model. Then the LMI-based robust controller is designed by means of an optimization procedure. A key step in this procedure is the derivation of a polytopic boundary for the state-space matrices of the IOF linearized system based on the estimated parameters of the neural network and their uncertainty bounds.