多时滞线性切换系统的稳定性及其反馈镇定

研究了具有多时滞线性切换系统的稳定性及其反馈镇定问题,利用完备性条件、矩阵分解与二次Lyapunov泛函,给出了多时滞切换系统渐近稳定的充分条件和切换律设计方法.在此基础上,研究了这类系统的镇定控制问题,设计了保证系统时滞独立渐近镇定的控制器.     

A Practical Method for Designing Linear Quadratic Regulator for Commensurate Fractional-Order Systems

The methods currently available for designing a linear quadratic regulator for fractional-order systems are either based on sufficient-type conditions for the optimality of functionals or generate very complicated analytical solutions even for simple systems. It follows that the use of such methods is limited to very simple problems. The present paper proposes a practical method for designing a linear quadratic regulator (assuming linear state feedback), Kalman filter, and linear quadratic Gaussian regulator/controller for commensurate fractional-order systems (in Caputo sense). For this purpose, considering the fact that in dealing with fractional-order systems the cost function of linear quadratic regulator has only one extremum, the optimal state feedback gains of linear quadratic regulator and the gains of the Kalman filter are calculated using a gradient-based numerical optimization algorithm. Various fractional-order linear quadratic regulator and Kalman filter design problems are solved using the proposed approach. Specifically, a linear quadratic Gaussian controller capable of tracking step command is designed for a commensurate fractional-order system which is non-minimum phase and unstable and has seven (pseudo) states.     

基于开关控制的线性奇异系统的二次状态反馈镇定问题

主要讨论基于开关控制的线性奇异系统的二次状态反馈镇定问题.利用二次反馈镇定的概念,给出了线性奇异系统基于异步开关控制的二次状态反馈镇定问题可解的两个充分条件.进一步,对于带有范数有界的不确定项的奇异线性系统,给出了其可以基于异步开关控制的二次状态反馈鲁棒镇定的可解性条件.     

A piecewise ellipsoidal reachable set estimation method for continuous bimodal piecewise affine systems

In this work, the issue of estimation of reachable sets in continuous bimodal piecewise affine systems is studied. A new method is proposed, in the framework of ellipsoidal bounding, using piecewise quadratic Lyapunov functions. Although bimodal piecewise affine systems can be seen as a special class of affine hybrid systems, reachability methods developed for affine hybrid systems might be inappropriately complex for bimodal dynamics. This work goes in the direction of exploiting the dynamical structure of the system to propose a simpler approach. More specifically, because of the piecewise nature of the Lyapunov function, we first derive conditions to ensure that a given quadratic function is positive on half spaces. Then, we exploit the property of bimodal piecewise quadratic functions being continuous on a given hyperplane. Finally, linear matrix characterizations of the estimate of the reachable set are derived.     

A numerical approach to utility functions in risk theory

Given the standard equilibrium model for an insurance market and sharing rules defining a feasible risk-exchange, we want to determine numerically the utility functions leading to the equilibrium. In the special case of two companies we approximate the sharing rules by piecewise linear functions and give an algorithm to compute piecewise quadratic utility functions which are solutions of the equilibrium model. We apply our method to compare some insurance contracts. For this we introduce the notion of acceptability of an insurance contract and a risk equivalence property based on utility theory. The numerical examples lead to interesting interpretations which give some insight in the considered insurance contracts.     

Superconvergence analysis of fully discrete finite element methods for semilinear parabolic optimal control problems

We study the superconvergence property of fully discrete finite element approximation for quadratic optimal control problems governed by semilinear parabolic equations with control constraints. The time discretization is based on difference methods, whereas the space discretization is done using finite element methods. The state and the adjoint state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions. First, we define a fully discrete finite element approximation scheme for the semilinear parabolic control problem. Second, we derive the superconvergence properties for the control, the state and the adjoint state. Finally, we do some numerical experiments for illustrating our theoretical results.     

Decentralized Guaranteed Cost Control for Uncertain Large-Scale Systems Using Delayed Feedback: LMI Optimization Approach

In this paper, we propose a design method of guaranteed cost controllers for uncertain large-scale systems with time delays in subsystem interconnections using delayed feedback. Using the Lyapunov method, a linear matrix inequality (LMI) optimization problem is formulated to design a delayed feedback controller which minimizes the upper bound of a given quadratic cost function. A numerical example is included to illustrate the design procedures.Communicated by Q. C. ZhaoThe authors thank the Associate Editor and three anonymous referees for careful reading and useful suggestions.     

Expansion over a system of shifts of pyramidal functions with a square base

A method for approximation of functions of two variables by a linear combination of non-negative piecewise linear functions with a compact support is presented. Two quadratic pyramids are used as generating functions for the system of shifts. The accuracy of this local method is proved to have the same order as the best approximation by piecewise linear functions.     

Optimal NPID Stabilization of Linear Systems

We consider the problem of finding the optimal, robust stabilization of linear systems within a family of nonlinear feedback laws. Investigation of the efficiency of full-state based and partial-state based so-called NPID feedback schemes proposed for the stabilization of systems in robotic applications has provided the motivation for our work. We prove that, for a given quadratic Lyapunov function and a given family of nonlinear feedback laws, there exist optimal piecewise linear feedbacks that make the generalized Lyapunov derivative of the closed-loop system minimal. The result provides improved stabilization over the nonlinear stabilizing feedback law proposed in Ref. 1 as demonstrated in simulations of the Sarcos Dextrous Manipulator.     

Continuous piecewise linear finite elements for the Kirchhoff–Love plate equation

A family of continuous piecewise linear finite elements for thin plate problems is presented. We use standard linear interpolation of the deflection field to reconstruct a discontinuous piecewise quadratic deflection field. This allows us to use discontinuous Galerkin methods for the Kirchhoff–Love plate equation. Three example reconstructions of quadratic functions from linear interpolation triangles are presented: a reconstruction using Morley basis functions, a fully quadratic reconstruction, and a more general least squares approach to a fully quadratic reconstruction. The Morley reconstruction is shown to be equivalent to the basic plate triangle (BPT). Given a condition on the reconstruction operator, a priori error estimates are proved in energy norm and L ² norm. Numerical results indicate that the Morley reconstruction/BPT does not converge on unstructured meshes while the fully quadratic reconstruction show optimal convergence.     

Fuzzy guaranteed cost controller for trajectory tracking in nonlinear systems

In this paper, we propose a fuzzy logic based guaranteed cost controller for trajectory tracking in nonlinear systems. Takagi–Sugeno (T–S) fuzzy model is used to represent the dynamics of a nonlinear system and the controller design is carried out using this fuzzy model. State feedback law is used for building the fuzzy controller whose performance is evaluated using a quadratic cost function. For designing the fuzzy logic based controller which satisfies guaranteed performance, linear matrix inequality (LMI) approach is used. Sufficient conditions are derived in terms of matrix inequalities for minimizing the performance function of the controller. The performance function minimization problem with polynomial matrix inequalities is then transformed into a problem of minimizing a convex performance function involving standard LMIs. This minimization problem can be solved easily and efficiently using the LMI optimization techniques. Our controller design method also ensures that the closed-loop system is asymptotically stable. Simulation study is carried out on a two-link robotic manipulator tracking a reference trajectory. From the results of the simulation study, it is observed that our proposed controller tracks the reference trajectory closely while maintaining a guaranteed minimum cost.     

Controller design for stabilizing bilinear systems with state-based switching

Some nonlinear systems can be approximated by switching bilinear systems. In this paper, we proposed a method to design state-based stabilizing controller for switching bilinear systems. Based on the similarity between switching bilinear systems and switching linear systems, corresponding switching linear systems are obtained for switching bilinear systems by applying state-based feedback control laws. Instead, we consider asymptotically stabilizing the corresponding switching linear system through solving a number of relaxed LMI conditions. Stabilizing controllers for switching bilinear systems can be derived based on the results of the corresponding switching linear systems. The stability of the controller is proved step by step through the decreasing of the multiple Lyapunov functions along the state trajectory. The effectiveness of the method is demonstrated by both a theoretical example and an example of urban traffic network with traffic signals.     

Generation of multi-wing chaotic attractor in fractional order system

In this paper, a novel approach is proposed for generating multi-wing chaotic attractors from the fractional linear differential system via nonlinear state feedback controller equipped with a duality-symmetric multi-segment quadratic function. The main idea is to design a proper nonlinear state feedback controller by using four construction criterions from a fundamental fractional differential nominal linear system, so that the controlled fractional differential system can generate multi-wing chaotic attractors. It is the first time in the literature to report the multi-wing chaotic attractors from an uncoupled fractional differential system. Furthermore, some basic dynamical analysis and numerical simulations are also given, confirming the effectiveness of the proposed method.     

Template-Based Stabilization of Relative Equilibria in Systems with Continuous Symmetry

We present an approach to the design of feedback control laws that stabilize relative equilibria of general nonlinear systems with continuous symmetry. Using a template-based method, we factor out the dynamics associated with the symmetry variables and obtain evolution equations in a reduced frame that evolves in the symmetry direction. The relative equilibria of the original systems are fixed points of these reduced equations. Our controller design methodology is based on the linearization of the reduced equations about such fixed points. We present two different approaches of control design. The first approach assumes that the closed loop system is affine in the control and that the actuation is equivariant. We derive feedback laws for the reduced system that minimize a quadratic cost function. The second approach is more general; here the actuation need not be equivariant, but the actuators can be translated in the symmetry direction. The controller resulting from this approach leaves the dynamics associated with the symmetry variable unchanged. Both approaches are simple to implement, as they use standard tools available from linear control theory. We illustrate the approaches on three examples: a rotationally invariant planar ODE, an inverted pendulum on a cart, and the Kuramoto-Sivashinsky equation with periodic boundary conditions.     

Guaranteed cost control for uncertain large-scale systems with time-delays via delayed feedback

In this paper, we propose a design method for guaranteed cost controllers for uncertain large-scale systems with time-delays in subsystem interconnections using delayed feedback. Based on the Lyapunov method, an LMI (Linear Matrix Inequality) optimization problem is formulated to design the delayed feedback controller which minimizes the upper bound of a given quadratic cost function. A numerical example is included to illustrate the design procedures.     

Non-fragile guaranteed cost control of uncertain large-scale systems with time-varying delays

The robust non-fragile guaranteed cost control problem is studied in this paper for a class of uncertain linear large-scale systems with time-varying delays in subsystem interconnections and given quadratic cost functions. The uncertainty in the system is assumed to be norm-bounded and time-varying. Also, the state-feedback gains for subsystems of the large-scale system are assumed to have norm-bounded controller gain variations. The problem is to design state feedback control laws such that the closed-loop system is asymptotically stable and the closed-loop cost function value is not more than a specified upper bound for all admissible uncertainties. Sufficient conditions for the existence of such controllers are derived based on the linear matrix inequality (LMI) approach combined with the Lyapunov method. A parameterized characterization of the robust non-fragile guaranteed cost controllers is given in terms of the feasible solutions to a certain LMI. Finally, in order to show the application of the proposed method, a numerical example is included.     

Vibration control of micro-scale structures using their reduced second order bilinear models based on multi-moment matching criteria

This paper deals with vibration control of micro-scale structures; i.e. MEMS devices. For modeling of the structures, finite element method which is a distinguished and accurate technique will be used. This method, however, leads to a model with high number of degrees of freedom which may cause computational costs especially for control problems. Hence, we will apply the second order Krylov subspace method based on multi-moment matching to obtain a reduced order model which is in the form of a second order bilinear system. For vibration suppression of the corresponding micro-structure, a quadratic feedback controller and also a linear state feedback controller using linear matrix inequality (LMI) will be designed. Finally, a micro-cantilever beam will be considered as a practical case study and simulation results of applying the proposed method will be presented.     

Method for Constructing Piecewise Quadratic Value Functions in a Control Problem for a Switched System

The problem of constructing internal approximations to solvability sets and the control synthesis problem for a piecewise linear system with control parameters and disturbances (uncertainties) are solved. The solution is based on the comparison principle and piecewise quadratic value functions of a special form. Relations defining such functions and, in particular, “continuous binding conditions” for the functions and their first derivatives are obtained. The results are used to construct numerical methods for solving the control synthesis problem for the class of switched systems under study. An example of approximate solution of the control synthesis problem in a target control problem for a nonlinear mathematical model of a pendulum with a flywheel is considered.     

The design of a linear regulator based on an integral-equation representation

This paper presents an alternative on-line algorithm for calculating regulators of linear deterministic dynamical systems which minimize quadratic cost functions employing the invariant-imbedding method. The design scheme used for the optimum linear regulator is based on the integral-equation representation, which enables one to obtain the solution to the corresponding two-point boundary-value problem. The algorithm can be implemented in forward time without memory, unlike the conventional one which uses the Kalman gain function to calculate the feedback gain.     

Euler-Bernoulli Beam with Boundary Control: Stability and FEM

We consider a model for the time evolution of a piezoelectric cantilever with tip mass. With appropriately shaped actuator and sensor electrodes, boundary control is applied and a passivity based feedback controller is designed to include damping into the system. Assuming that the cantilever can be modeled by the Euler-Bernoulli beam equation, we obtain a coupled PDE-ODE system. First we discuss its dissipativity, and its asymptotic but non-exponential stability. Next we derive a FEM using piecewise cubic Hermitian shape functions that is still dissipative. This is illustrated on a numerical simulation. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)