LMI Conditions Research Articles

Evaluating regions of attraction of LTI systems with saturation in IQS framework

The integral quadratic separation (IQS) framework is a specialization of topological separation theory to the case where operators are described by (or embedded in) integral quadratic constraints. IQS applies both to regular and descriptor system representations. Thanks to this feature, IQS provides a methodology for deriving sequences of LMI conditions with reducing conservatism. It has been applied for robust performance analysis of systems with uncertainties and delays. In this paper the methodology is adapted for the first time for region of attraction evaluation in the case of systems with saturations. Results are illustrated on a launcher attitude control example with saturated inputs.

Observer-based anti-windup compensator design for saturated control systems using an LMI approach

In practical control systems, actuator saturation usually induces a windup phenomenon and potentially results in system instability. Accordingly, this paper develops an observer-based auxiliary anti-windup compensation scheme to mitigate the effects of actuator limitations on the performance and stability of the controlled system. In the proposed approach, the controller output signal passing through the saturation element is treated as an external disturbance imported to the designed controller and an auxiliary controller is designed to minimize the difference between the controller output signal and the system input signal. The conditions required to maintain the system performance in the presence of actuator saturation are formulated as an LMI criterion. The L2-stability criterion of the anti-windup compensator design is also formulated as an LMI condition. It is shown that by integrating the two LMI conditions and solving the resulting optimization problem, the resulting anti-windup controller both minimizes the performance attenuation of the saturated control system and guarantees its L2-stability.

Robust synchronization of fractional-order unified chaotic systems via linear control

A new scheme for accomplishing synchronization between two fractional-order unified chaotic systems is proposed in this paper. The scheme does not require that the nonlinear dynamics of the synchronization error system must be eliminated. Moreover, the parameter of the systems does not have to be known. A controller is a linear feedback controller, which is simple in implementation. It is designed based on an LMI condition. The LMI condition guarantees that the synchronization between the slave system and the master system is achieved. Numerical simulations are performed to demonstrate the effectiveness of the proposed scheme.

H∞ Static Output Feedback Control of 2‐D Discrete Systems in FM Second Model

AbstractThe problem of H∞ static output feedback (SOF) control of two‐dimensional (2‐D) discrete systems described by the Fornasini‐Marchesini (FM) second model is investigated in this paper. First, by applying the 2‐D Bounded Real Lemma, the 2‐D H∞ SOF control problem is formulated in terms of a bilinear matrix inequality (BMI). Then, by combining the slack variable technique with two kinds of existing LMI methods, respectively, less conservative sufficient LMI conditions are proposed for the BMI formulation. The relation of these two kinds of LMI conditions are revealed by analyzing the choices of coordinate transformation matrices involved in the first kind of LMI conditions. Finally, a numerical example is provided to demonstrate the effectiveness and merits of the proposed methods.

New LMI-based condition on global asymptotic stability concerning BAM neural networks of neutral type

In this paper, we discuss global asymptotic stability to a BAM neural networks of neutral type with delays. Under the assumptions that the activation functions only satisfy global Lipschitz conditions, a new and complicated LMI condition is established on global asymptotic stability for the above neutral neural networks by means of using Homeomorphism theory, matrix and Lyapunov functional. In our result, the hypotheses for boundedness in [20,21] and monotonicity in [20] on the activation functions are removed. On the other hand, the LMI condition is also different from those in [20,21]. Finally, an example is given to show the effectiveness of the theoretical result.

Formula omitted] filtering for nonuniformly sampled systems: A Markovian jump systems approach

This paper is concerned with the design of an H ∞ filter for a sampled-data system, where the measurement is sampled nonuniformly and randomly but the input is updated uniformly at a fast rate. The process of nonuniform measurement sampling is modeled using a Markov chain; therefore, the discrete-time system together with the Markov chain is a Markovian jump system. A mode-dependent, full-order H ∞ filter is constructed. Sufficient LMI conditions are derived to ensure stochastic stability and H ∞ disturbance attenuation for the error system. The case of unknown measurement sampling probabilities is also considered. Finally, the effectiveness of the proposed approach is demonstrated through simulation results and a comparison with a general time-varying H ∞ filter.

Síntese de controladores repetitivos chaveados: uma aplicação à fontes ininterruptas de energia (UPS)

Este trabalho apresenta uma metodologia de síntese de controladores repetitivos visando o seguimento assintótico de sinais senoidais e a rejeição de perturbações periódicas em fontes ininterruptas de energia (UPS). Para obter uma boa relação entre desempenho transitório e em regime permanente, emprega-se uma representação no espaço de estados do controlador repetitivo com dois modos distintos de operação. Dessa forma, permite-se que o controlador repetitivo chaveie entre as seguintes situações: regime transitório (responsável por melhorar o tempo de recuperação quando a carga é alterada) e regime permanente (responsável pela rejeição da distorção harmônica do sinal de saída quando alimentando cargas não-lineares). A partir da representação do sistema e do controlador repetitivo no espaço de estados, utiliza-se uma realimentação de estados para a garantia da estabilidade do sistema em malha fechada quando este é sujeito a uma função de chaveamento arbitrária. A partir de um funcional de Lyapunov-Krasovskii, obtém-se um conjunto de condições na forma de desigualdades matriciais lineares (LMIs) para a determinação da realimentação de estados. O desempenho dinâmico do esquema de controle proposto é ilustrado através da simulação de modelos reais de sistemas UPS sujeitos a variações de cargas não-lineares.

영구자석 동기전동기의 퍼지 속도제어기 및 퍼지 각가속도 관측기 설계

This paper proposes a new fuzzy speed controller for the precise speed control of a permanent magnet synchronous motor(PMSM). The proposed control system needs the information of the angular acceleration instead of the load torque, so the third-order fuzzy acceleration observer estimates it. Moreover, the LMI conditions are derived for the existence of the fuzzy acceleration observer and fuzzy speed controller, and the gain matrices of the observer and controller are obtained. It is analytically proven that the proposed observer-based fuzzy speed regulator is exponentially stable. To evaluate the performance of the proposed control algorithm, experimental results as well as simulation results are provided under the conditions of motor parameter and load torque variations. Finally, it is clearly confirmed that the proposed control method can accurately control the speed of a PMSM.

LMI conditions for time-varying uncertain systems can be non-conservative

Establishing robust asymptotic stability of uncertain systems affected by time-varying uncertainty is a key problem. LMI sufficient conditions have been proposed in the literature for addressing this problem based on homogeneous polynomial Lyapunov functions. Unfortunately, till now it has been unclear whether these conditions are also necessary. This paper proposes a proof in order to show that one of these conditions is not only sufficient but also necessary for a sufficiently large degree of the Lyapunov function.

영구자석 동기전동기를 위한 퍼지 관측기 기반의 퍼지 제어기의 설계 및 안정도 해석

본 논문에서는 영구자석 동기전동기(PMSM)의 강인한 속도 제어를 위하여 퍼지 부하 토크관측기와 퍼지 속도제어기를 제안한다. 또한, 퍼지 부하 토크관측기와 퍼지 속도제어기가 존재하기 위한 LMI 조건을 유도하고, 관측기 및 제어기의 이득을 계산한다. 제안된 제어시스템의 안정도가 보장됨을 해석적으로 증명한다. 본 연구에서 제안된 관측기 기반의 퍼지 속도제어기의 성능을 검증하기 위하여, 시뮬레이션 및 실험 결과가 주어졌다. 본 결과를 통하여 제안된 제어 알고리즘은 영구자석 동기전동기의 속도를 정확하게 제어할 수 있음을 검증하였다. This paper proposes a new fuzzy load torque observer and a fuzzy speed regulator to guarantee a robust speed control of a permanent magnet synchronous motor (PMSM). Also, the LMI conditions are given for the existence of the fuzzy load torque observer and fuzzy speed controller, and the gains of the observer and controller are calculated. The stability of the proposed control system is analytically proven. To validate the effectiveness of the proposed observer-based fuzzy speed controller, the simulation and experimental results are presented. Finally, it is definitely demonstrated that the proposed control algorithm can be used to accurately control the speed of a PM synchronous motor.

State feedback stabilization of a class of uncertain periodic systems

AbstractThis paper deals with the robust stability and stabilization problem for a class of uncertain linear periodic discrete time system. This class is defined as the class of systems subject, at first, to LFR (Linear fractional representation)-based uncertainty and then to LFR and polytopic uncertainties together. The LFR-based uncertainty encompasses the case of norm bounded uncertainty as a special case. The use of a full block S-procedure allows to reformulate the stability LMI conditions of an uncertain system through a dual form of quadratic Lyapunov inequality in equivalent efficient LMI conditions that can be easily checked by any LMI solver.The obtained conditions are then used to propose a periodic state feedback controller ensuring the robust stabilizability of the obtained closed loop system. The use of the PDLF (parameter-dependent Lyapunov functions) technique enables one to take account in the same time of polytopic uncertainties. Numerical examples are provided to illustrate the proposed results.

H.INF. Output Feedback Control for A Class of General Fuzzy Systems

This paper is concerned with H∞ output feedback control design for a general class of fuzzy systems. When the premise variable is the measurement output, the control design is easy but a fuzzy system representation is limited. On the other hand, when it is the state variable, a fuzzy system describes a general class of nonlinear systems but the control design becomes difficult. This is because a conventional parallel distributed compensator (PDC) is not feasible any more. In this paper, we introduce an output feedback controller whose premise variable is independent of the premise variable of an original fuzzy system. Then, we formulate an H∞ control problem for a general class of fuzzy system where the premise variable is not always available. Our control design method is based on a set of strict LMI conditions. No scaling parameter is necessary a priori to solve LMI conditions for designing an H∞ controller. Our method includes matrices that can tune control gain matrices in an H∞ controller and hence it can reflect the control performance of the resulting closed-loop system. A numerical example is given to illustrate our H∞ output feedback control design.

Delay‐Dependent Stability Analysis and Synthesis for Uncertain Impulsive Switched System with Mixed Delays

This paper studies the asymptotic stability problem for a class of uncertain impulsive switched systems with discrete and distributed delays. Based on Lyapunov functional theory, delay‐dependent sufficient LMI conditions are established for the asymptotic stability of the considered systems. Moreover, an appropriate feedback controller is constructed for stabilizing the corresponding closed‐loop system. The results are illustrated to be efficient through an example.

Stability analysis and observer design for string-connected TS systems

AbstractDistributed systems consist of interconnected, lower-dimensional subsystems. For such systems, distributed analysis and design present several advantages, such as modularity, easier analysis, and reduced computational complexity. Applications include distributed process control, traffic and communication networks, irrigation systems, hydropower valleys, etc. A special case of distributed systems is when the subsystems are connected in a string. By exploiting such a structure, in this paper, we propose conditions for the distributed stability analysis of Takagi-Sugeno fuzzy systems connected in a string. These conditions are extended to observer design. Sufficient LMI conditions, which are easy to solve are also provided. The approach is illustrated on a simulation example.

Further Results on Sufficient LMI Conditions for H_infinity Static Output Feedback Control of Discrete-Time Systems

Further Results on Sufficient LMI Conditions for H_infinity Static Output Feedback Control of Discrete-Time Systems

Model Predictive Control based on LMIs Applied to an Omni-Directional Mobile Robot

AbstractThe paper presents a methodology for state feedback MPC synthesis applied to the trajectory tracking control problem of a three wheeled omnidirectional mobile robot. The MPC design used here is based on a cost function developed over finite horizon and LMI framework. It is shown that MPC concepts well established for robot applications, for instance, the use of open loop predictions, receding horizon control, constraints manipulation, are preserved in this new formulation. The stability of the closed loop system is guaranteed by LMI conditions related with the cost function monotonicity. Simulation results of navigation are provided to demonstrate the performance of the proposed control strategy.

Nonlinear and locally optimal controllers design for input affine systems

AbstractAssuming that a globally stabilizing nonlinear controller is available, the aim is to modify the local behavior of the trajectories in order to get local optimality with respect to a given quadratic cost. A sufficient condition is given in terms of linear matrix inequalities to design a locally optimal and globally stabilizing control law. This approach is illustrated on an inverted pendulum model in order to stabilize this upper equilibrium point. In addition, a stronger sufficient condition is then given (which is not anymore linear) but allows to address the cases in which the previous LMI condition failed to be satisfied.

Simply structured controllers for parameter varying distributed systems

This paper presents a systematic approach to the synthesis of simply structured controllersfor large scale parameter varying distributed systems. The design and optimizationprocedure consists of two loops. In an inner loop, an efficient LMI-based optimizationalgorithm is used to evaluate the closed-loop performance. The outer loop uses evolutionaryoperators to iteratively search for the parameters of a controller network that achieves thebest performance.For the purpose of performance evaluation, new sufficient LMI conditions are presented tofind the induced norm of such systems. The conservatism of the analysis is reduced by the use ofparameter dependent Lyapunov functions and by taking bounds on the parameter variationrates into account. Since the control algorithms are implemented on digital computers, theresults are presented for discrete time and discrete space systems. The problem posed inthis form is infinite dimensional, a gridding algorithm is also proposed to convert it into afinite dimensional problem.The approach is illustrated with two applications: active vibration control of a cantilever,and position control of a seismic cable. These examples illustrate that the performanceanalysis condition presented here is less conservative than previously presented LMIconditions. It is also demonstrated that the approach can be used to systematically designsimply structured low-order controllers, while satisfying all performance requirements.

Studies on linear matrix inequality relaxations for fuzzy control systems via homogeneous polynomials

In this study, the authors investigate a relaxed condition characterised by parameter-dependent linear matrix inequality (PD-LMI) in terms of firing strength belonging to the unit simplex, exploiting the algebraic property of Polyas theorem to construct a family of finite-dimensional LMI relaxations. The main contribution of this study is that sets of relaxed LMI are parameterised in terms of the polynomial degree d . As d increases, progressively less conservative LMI conditions are generated, being easier satisfied owing to more freedom provided by new variables involved. Another protruding feature is that a verifiable necessary condition is derived. Furthermore, the new relaxation results for PD-LMI is shown to include and generalise all previous results on quadratic (common P ) stability approach. Lastly, numerical experiments for illustrating the advantage of relaxation, being less conservative and effective, are provided.

퍼지제어기를 이용한 영구자석 동기전동기의 강인한 속도제어

본 논문에서는 영구자석 동기전동기(PMSM)의 강인한 속도 제어를 위하여 Takagi-Sugeno 퍼지 기법을 이용한 퍼지 속도제어기를 제안한다. 또한 제안된 퍼지 속도제어기는 부하 토크의 정보를 필요로 하므로 2차 부하 토크관측기를 이용하여 부하 토크를 추정한다. 퍼지 속도제어기가 존재하기 위한 LMI 조건을 유도하고, 제어기의 이득이 주어진다. 제안된 퍼지 속도제어기와 부하 토크관측기를 포함하는 확장된 제어시스템의 안정도가 보장됨을 보인다. 본 연구에서 제안된 퍼지 속도제어기의 성능을 검증하기 위하여, 전동기 파라미터 변동 하에서 시뮬레이션 및 실험결과가 주어졌다. 본 결과를 통하여 제안된 제어 기법은 영구자석 동기전동기의 속도를 정확하게 제어할 수 있음을 검증하였다. This paper proposes a new fuzzy speed controller based on the Takagi-Sugeno fuzzy method to achieve a robust speed control of a permanent magnet synchronous motor (PMSM). The proposed controller requires the information of the load torque, so the second-order load torque observer is used to estimate it. The LMI condition is derived for the existence of the proposed fuzzy speed controller, and the gains of the controller are provided. It is proven that the augmented control system including the fuzzy speed controller and the load torque observer is exponentially stable. To evaluate the performance of the proposed fuzzy speed controller, the simulation and experimental results are presented under motor parameter variations. Finally, it is clearly verified that the proposed control method can accurately control the speed of a permanent magnet synchronous motor.