Iterative Linear Matrix Inequality Algorithm Research Articles

H∞ observer-based controller synthesis for fractional order systems over finite frequency range

This paper focuses on the H∞ observer-based control problem for fractional order systems (FOSs) over finite frequency range. Firstly, as an extension to the KYP lemma, the structure singular value–μ and its upper bound are analyzed and determined. The necessary and sufficient conditions of the H∞ performance for FOSs over finite frequency range are given through the generalized μ analysis method. This method is different from the traditional μ-analysis method, which is only suitable for integer order systems. Secondly, according to these obtained LMI-based criteria for H∞ performance index and a key projection lemma, we design an H∞ observer-based controller to stabilize the estimation error and decrease the H∞ norm for the FOS simultaneously. With the help of the matrix congruence transformation, the feedback gain matrix is parameterized by a scalar matrix. Thirdly, to deal with the nonlinear terms in matrix inequalities, an iterative linear matrix inequality (ILMI) algorithm is developed. Finally, two numerical examples are provided to explain the correctness of the established results.

Stability analysis for a class of switched systems with state constraints via time-varying Lyapunov functions

This article investigates the stability analysis for a class of continuous-time switched systems with state constraints under pre-specified dwell time switchings. The state variables of the studied system are constrained to a unit closed hypercube. Firstly, based on the definition of set coverage, the system state under saturation is confined to a convex polyhedron and the saturation problem is converted into convex hull. Then, sufficient conditions are derived by introducing a class of multiple time-varying Lyapunov functions in the framework of pre-specified dwell time switchings. Such a dwell time is an arbitrary pre-specified constant which is independent of any other parameters. In addition, the proposed Lyapunov functions can efficiently eliminate the “jump” phenomena of adjacent Lyapunov functions at switching instants. The feature of this paper is that the definition of set coverage is utilized to replace the restriction on the row diagonally dominant matrices with negative diagonal elements to analyze stability. The other feature of the constructed time-varying Lyapunov functions is that there are two time-varying functions. One of the two time-varying functions contains the jump rate, which will present a certain degree of freedom in designing the dwell time switching signal. An iterative linear matrix inequality (LMI) algorithm is presented to verify the sufficient conditions. Finally, two examples are presented to show the validity of the method.

Stability analysis and stabilisation of continuous-discrete fractional-order 2D Fornasini–Marchesini first model

This paper focuses on the structural stability and stabilisation of the continuous-discrete fractional-order two-dimensional Fornasini-Marchesini first model. The sufficient and necessary stability conditions are given in polynomial form at first. Secondly, to make the problem solvable, based on the property of Kronecker product and the Generalized–Kalman–Yakubovich–Popov Lemma, the new sufficient and necessary stability conditions are established in the form of linear matrix inequalities. Thirdly, with the help of the Projection Lemma, the stabilisation conditions are obtained after introducing a state feedback controller, which can be solved with an iterative linear matrix inequality algorithm. In the end, the effectiveness of the proposed results is verified by two numerical examples.

Filter Design for Positive T–S Fuzzy Continuous-Time Systems With Time Delay Using Piecewise-Linear Membership Functions

This article focuses on the filtering problem and stability analysis for positive Takagi-Sugeno (T-S) fuzzy systems with time delay under L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -induced performance. Due to the importance of estimation of system states but the few filter design results on positive nonlinear systems, it is an attractive and meaningful topic well worth studying. In order to fully exploit and take advantage of the positivity of positive T-S fuzzy systems, many commonly used methods, for instance free-weighting matrix approach and similarity transformation are probably not suitable for positive systems. To address the hard-nut-to-crack problem, an auxiliary variable is introduced so that the augmentation approach can be employed to carry out the positivity and stability analysis of filtering error systems. In addition, another obstacle that cannot be ignored is the existence of nonconvex terms in the stability and positivity conditions. For getting around this barrier, some iterative linear matrix inequality algorithms have been proposed in the literature. However, considering the weakness that these methods cannot guarantee the convergence to a numerical solution and the iterative process is exhaustive, we present an effective matrix decoupling method to convert the nonconvex conditions into convex ones in this article. Furthermore, a linear copositive Lyapunov function, which incorporates the positivity of system states and time delay at the same time is chosen so that the positivity characteristic of filtering error systems can be captured further. However, because of plenty of valuable information of membership functions being ignored, hence, the obtained results are conservative. For the sake of relaxing the conservativeness, the advanced piecewise-linear membership functions approximate method is utilized to facilitate the stability and positivity analysis. Therefore, the relaxed stability and positivity conditions, which are cast as sum of squares (SOS) are obtained and can be solved numerically. Finally, the effectiveness of the designed fuzzy filtering strategy with satisfying L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -induced performance are demonstrated by a simulation example.

Event-triggered controller design for linear systems with saturated input and bounded disturbance

This paper deals with the stabilisation problem of linear systems subject to saturated input and bounded disturbance via event-triggered control strategy. In a hybrid systems framework, both a state feedback law and a static output feedback law are proposed to design the event-triggered controllers, which provide solutions to the event-triggered control collaborative design problem. For the case of using static output feedback, an iterative linear matrix inequality algorithm is proposed to obtain the feedback gain. In addition, the local uniform ultimate boundedness of the closed-loop system is ensured. Finally, two examples are given to illustrate the effectiveness of the proposed controllers.

Spatially Local Piecewise Fuzzy Control for Nonlinear Delayed DPSs With Random Packet Losses

This article introduces a spatially local piecewise fuzzy control (SLPFC) for nonlinear delayed distributed parameter systems (DPSs) with random packet losses. Due to the limited bandwidth of the channels, the random packet losses could occur simultaneously in the communication channels from the sensor to the controller and from the controller to the actuator, which are assumed to obey the Bernoulli random binary distribution. An SLPFC design is given such that, for all possible random packet losses, the closed-loop delayed DPS is mean square exponentially stable. Delay-dependent conditions are obtained and then two procedures are given for designing the spatially local piecewise fuzzy controller: one casts the controller design into an iterative linear matrix inequality (LMI) algorithm and the other casts the controller design into a parameter-dependent LMI problem. Two examples are provided to illustrate the usefulness of the presented design approach.

Finite frequency fault estimation observer design for T-S Fuzzy discrete-time descriptor systems

This paper deals with the problem of fault estimation observer design for T-S fuzzy nonlinear discrete-time descriptor systems in the finite frequency domain. First, a novel fault estimation observer for the augmented descriptor system is given. Based on Parseval's theorem and Lyapunov theory, the performance in the finite frequency domain is derived via time-domain method. Then, new sufficient conditions of admissibility and robust performance index for the dynamic error system are obtained as bilinear matrix inequalities. Meanwhile, an iterative linear matrix inequality algorithm is given to obtain the optimal solution. Finally, an example is given to demonstrate the effectiveness of our method.

Delay‐dependent robust control of stochastic systems with convex polynomial uncertainty

SummaryThis article addresses a robust control problem of time‐varying stochastic systems with time‐delays. Through Linear Parameter Varying (LPV) modeling approach, the time‐varying parameters can be described via the convex combination. Therefore, the LPV stochastic system is interpreted by a weighting function and multiplicative noised linear systems. Furthermore, stabilization problem for the systems is investigated via Gain‐Scheduled (GS) technique to increase robustness. For the problem, some sufficient conditions are derived by Integral Lyapunov‐Krasovskii Function (ILKF) such that the conservatism caused by derivative of weighting function is ignored. Besides, an Iterative Linear Matrix Inequality algorithm is used to determine the auxiliary variable such that the problem can be solved via the convex calculation. Finally, some simulations are provided to demonstrate effectiveness and applicability of this article.

Decentralized Switching Control Strategy for Load Frequency Control in Multi-Area Power Systems With Time Delay and Packet Losses

With the deep coupling between the electrical power systems and cyber systems, there is a increasingly detrimental effect of the communication delays and packet losses on the load frequency control (LFC) performance. Traditional methods mainly focus on guaranteeing the asymptotic stability in maximum delay case but ignore improving the system dynamic performance. The resulting designed controller cannot satisfy the actual operational requirements completely. Hence, this paper investigates a novel decentralized LFC strategy based on switching control theory. First, the networked LFC system is modelled as a series of subsystems with transmission delay as the switching decision variable. Compared with the existing modelling methods, our modelling approach can depict the influences of time delay and packet loss on system dynamics more accurately. Second, a memoryless state-feedback control design scheme is proposed. Design constraints are deduced by constructing a Lyapunov function and then presented in a bilinear matrix inequality form. To improve the system dynamic performance in power mismatch and communication change case, an iterative linear matrix inequality (LMI) algorithm is proposed. Finally, simulation results illustrate that our proposed method can restore the frequency deviation to zero quickly.

An enhanced state estimation for linear parabolic PDE systems with mobile sensors

SummaryThis paper studies an enhanced state estimation problem of distributed parameter processes modeled by a linear parabolic partial differential equation using mobile sensors. The proposed estimation scheme contains a state estimator and the guidance of mobile sensors, where the spatial domain is decomposed into multiple subdomains according to the number of sensors and each sensor is capable of moving within the respective subdomain. The state estimator is desired to make the state estimation error system exponentially stable while providing an performance bound. The mobile sensor guidance is used to enhance the transient performance of the error system. By the Lyapunov direct technique, an integrated design of state estimator and mobile sensor guidance laws is developed in the form of bilinear matrix inequalities (BMIs) to meet the desired design objectives. Moreover, to make the performance bound as small as possible, a suboptimal enhanced state estimation problem is formulated as a BMI optimization one, which can be solved via an iterative linear matrix inequality algorithm. Finally, numerical simulations are given to show the effectiveness of the proposed method.

Robust consensus for linear multi-agent systems with structured uncertainties

ABSTRACT This paper addresses a robust consensus problem for linear multi-agent systems subject to structured uncertainty and external disturbances under the leaderless framework. A distributed dynamic output-feedback protocol is proposed, which utilises not only relative output information of neighbouring agents but also relative state information of neighbouring controllers. Through model transformations, the robust consensus control problem of multi-agents network is reduced to a set of independent stabilisation problems for single linear subsystems. For robust consensus, it is shown that the analysis and full state-feedback synthesis conditions for such subsystems can be formulated as linear matrix inequality (LMI) optimisation problems. On the other hand, the synthesis condition for dynamic output-feedback protocol is formulated as non-convex bilinear matrix inequality (BMI) optimisation problem. An iterative LMI algorithm is then presented to solve the resulting BMI optimisation problem. An example of multiple mass-spring-damper systems has been used to demonstrate theoretical results.

Simultaneous design of AWC and nonlinear controller for uncertain nonlinear systems under input saturation

SummaryThis article addresses a novel technique for the simultaneous design of a robust nonlinear controller and static anti‐windup compensator (AWC) for uncertain nonlinear systems under actuator saturation and exogenous bounded input. The system is presumed to have locally Lipschitz nonlinearities, time‐varying uncertainties (appearing both in the linear as well as nonlinear dynamics and both in the state in addition to the output equations), and external norm‐bounded inputs both in the state and the output equations. Several bilinear matrix inequality–based conditions are derived to simultaneously design the robust nonlinear controller and AWC gains for uncertain nonlinear systems by employing the Lyapunov functional, reformulated Lipschitz property, uncertainty bounds, linear parameter‐varying approach, modified local and global sector conditions, iterative linear matrix inequality algorithm, convex optimization procedure, and gain minimization. The proposed multiobjective AWC‐based dynamic robust nonlinear controller guarantees the mitigation of saturation effects, robustness against time‐varying parametric norm‐bounded uncertainties, the asymptotic stability of the closed‐loop nonlinear system under zero external disturbances, and the attenuation of disturbance effects under nonzero external disturbances. The effectiveness of the proposed AWC‐based dynamic robust nonlinear controller synthesis scheme is illustrated by simulation examples.

Power Allocation Based on Proportional-Integral Controller in Femtocell Networks With Consideration of Maximum Power Constraint

In this paper, a two-tier uplink femtocell network operating in orthogonal frequency-division multiple access is investigated. Femtocell base stations endowed with cognitive capacity can sense the void spectrum of macrocell. We aim at an energy-saving scheme of power allocation for femtocell users. The new idea that performance of Femtocell networks can be analyzed using control theory is proposed in this paper. A simple and novel power control algorithm is designed based on the virtual Proportional-Integral (PI) controller. When considering the maximum power constraints, different from the proof of the convergence via standard interference functions, we establish a linear system with state saturation (maximum power constraint) for femtocell networks, and the power's convergence is verified in theory with iterative linear matrix inequality algorithm. Two introduced control parameters not only guarantee the stability of power control algorithm, but also can adjust the transient response and save energy. Numerical results illustrate the convergence performance and advantages of the PI algorithm.

Strong γc-γcl H∞ stabilization for networked control systems under denial of service attacks

This paper is concerned with the strong γc-γcl H∞ stabilization problem for networked control systems (NCSs) subject to denial of service (DoS) attacks, which are common attack behaviors that affect the packet transmission of measurement or control signals. The purpose of the problem under consideration is to design a stable dynamic output feedback (DOF) controller (strong stabilizing controller) with the prescribed H∞ performance norm bound γc to tolerate multiple packet dropouts caused by DoS attacks, such that, the closed-loop system is mean-square stable and captures the H∞ disturbance attenuation norm bound γcl. Based on the Lyapunov functional and the stochastic control approach, some sufficient conditions with the form of matrix inequalities for the existence of the desired stable DOF controller are established. Then, by an orthogonal complement space technique, the controller gain is parameterized. Next, an iterative linear matrix inequality (LMI) algorithm is developed to obtain the controller gain. Finally, the usefulness of the proposed method is indicated by a numerical simulation example.

Static Output Feedback Stabilization of a Class of Switched Linear Systems with State Constraints

This paper will research the problem of static output feedback (SOF) stabilization of state-constrained switched linear systems via an improved average dwell time method (ADT). Firstly, an improved ADT method is adopted to establish sufficient conditions for SOF of the state-constrained switched linear systems in the form of matrix inequality. It has been shown that this method is less conservative than traditional ADT, which in view of different decay rates of a Lyapunov function related to an active subsystem on the basis of whether the saturations occur or not. Then, a new iterative algorithm is designed to solve the matrix inequality and a SOF controller can be added. In the iterative linear matrix inequality (ILMI) algorithm, it is important not only to overcome the typical bilinear matrix inequality (BMI) problem of SOF, but also to solve the non-convex problem caused by state constraints. Finally, the availability and the applicability of the proposed method is shown by the application of a boost converter.

An iterative LMI algorithm for quantization noise reduction in ΔΣ modulators

In this paper, we design noise shaping filters for quantization noise reduction in ΔΣ modulators. The design problem for the noise shaping infinite impulse response (IIR) filter in the feedback of the ΔΣ modulator is in non-convex form, which is hard to solve in general. We propose an iterative linear matrix inequality (LMI) algorithm to find a near optimal noise shaping IIR filter by solving the non-convex design problem. The least-squares and the hybrid design are two methods, either of which can serve as a starting point for our iterative LMI algorithm to obtain a near optimal solution. We also compare and analyze the performances of some other noise shaping techniques with our proposed algorithm. A design example for a bandpass ΔΣ modulator based radio frequency transmitter is provided which takes into account the non-ideal behavior of the output analog filter.

Exponential Stability and Stabilization for Quadratic Discrete‐Time Systems with Time Delay

AbstractIn this note, we deal with the exponential stability and stabilization problems for quadratic discrete‐time systems with time delay. By using the quadratic Lyapunov function and a so called ‘Finsler's lemma', delay‐independent sufficient conditions for local stability and stabilization for quadratic discrete‐time systems with time delay are derived in terms of linear matrix inequalities (LMIs). Based on these sufficient conditions, iterative linear matrix inequality algorithms are proposed for maximizing the stability regions of the systems. Finally, two examples are given to illustrate the effectiveness of the methods presented in this paper.

Robust overlapping load frequency output feedback control of multi-area interconnected power systems

This paper presents a new approach to the design of robust proportional-integral (PI) load frequency controller (LFC) with overlapping structure in the presence of generation rate constraint (GRC) and time-varying communication delays. The power system is decomposed into overlapping areas with tie-lines being the overlapping (shared) parts between the areas. The overlapping LFC design is based on the expansion-contraction principles, where the power system is first expanded in such a way as to decouple the areas from each other. Then, for obtaining an appropriate level of performance, robust local stabilising PI controllers are designed for each of the decoupled areas using an iterative linear matrix inequality (ILMI) algorithm such that asymptotic stability with a minimal H∞ performance level is achieved. Finally, the designed robust local H∞ controllers are contracted (transformed) to a robust overlapping LFC for implementation on the multi-area power system. Although the approach is applicable to multi-area power systems, a time-delay three-area interconnected power system experiencing communication delays is considered in this paper. Simulation results under different scenarios clearly illustrate an improved performance obtained with the overlapping structured LFC of this paper compared with existing decentralized LFCs.

A new iterative approach for multi-objective fault detection observer design and its application to a hypersonic vehicle

ABSTRACTThis paper addresses the multi-objective fault detection observer design problems for a hypersonic vehicle. Owing to the fact that parameters’ variations, modelling errors and disturbances are inevitable in practical situations, system uncertainty is considered in this study. By fully utilising the orthogonal space information of output matrix, some new understandings are proposed for the construction of Lyapunov matrix. Sufficient conditions for the existence of observers to guarantee the fault sensitivity and disturbance robustness in infinite frequency domain are presented. In order to further relax the conservativeness, slack matrices are introduced to fully decouple the observer gain with the Lyapunov matrices in finite frequency range. Iterative linear matrix inequality algorithms are proposed to obtain the solutions. The simulation examples which contain a Monte Carlo campaign illustrate that the new methods can effectively reduce the design conservativeness compared with the existing methods.

Method for improving the bank‐to‐turn missile manoeuvrability

This study addresses the autopilot design problem for a bank-to-turn missile to improve the manoeuvrability. A modified dynamic model is introduced by adding new reaction jet inputs to provide extra manouvrability. The H_ index, which is defined as the minimum singular value of the transfer function, is utilised to assess the manoeuvrability and the autopilot design problem is then transformed to a problem of H_ index controller design. An iterative linear matrix inequality algorithm is presented to obtain the control law. Simulation results illustrate the effectiveness of the proposed approach.