Bounded Real Lemma Research Articles

Bounded real lemma and H∞ control for BAM neural networks with unbounded time‐varying delays

AbstractThis article investigates the exponential control issue of bidirectional associative memory neural network (BAMNN) with unbounded time‐varying delays. A bounded real lemma (BRL) is first established via a direct method, which is on the basis of the solutions of BAMNN. Second, based on the obtained BRL, the state feedback controller is designed to guarantee the global exponential stability of the resulting closed‐loop BAMNN with an performance index. Since no Lyapunov–Krasovskii functionals is constructed in the proposed method, the computation burden and complexity are reduced. Lastly, the effectiveness of the theoretical results is illustrated through two numerical examples.

Finite frequency H ∞ control for singular systems with application to a 2-machine 6-bus power network

This paper is concerned with the H ∞ control problem for singular systems over finite frequency ranges. The disturbance is assumed to be within low-/middle-/high-frequency ranges. Both the state feedback and static output feedback controllers are designed. First, by combining the Lyapunov function method and the generalized Kalman–Yakubovich–Popov lemma, a new finite frequency bounded real lemma (BRL) is proposed, which ensures the resulting closed-loop system to be regular, impulse free and stable with given finite frequency H ∞ performance bound. It is shown by theoretical analysis and numerical simulation that the proposed BRL has less conservatism than the existing ones in the literature. With the aid of the Finsler's Lemma, two improved finite frequency BRLs are further presented for the design of controllers. Then, by utilizing convexification techniques, sufficient conditions for solving the desired state feedback and static output feedback gains are presented in terms of linear matrix inequalities, respectively. Finally, the applicability and effectiveness of the proposed control methods are verified by a 6-bus power network system.

Asynchronous adaptive event-triggered fault detection for delayed Markov jump neural networks: A delay-variation-dependent approach

This paper presents a delay-variation-dependent approach to fault detection of a discrete-time Markov jump neural network (MJNN) with a time-varying delay and mismatched modes. The goal is to detect the potential fault of delayed MJNNs by constructing an appropriate adaptive event-triggered and asynchronous H∞ filter. By choosing a delay-product-type Lyapunov–Krasovskii (L–K) functional with a delay-dependent matrix and exploiting some matrix polynomial inequalities, bounded real lemmas (BRLs) are obtained on the existence of suitable adaptive event generator and filters. These BRLs are dependent not only on the delay bounds but also on the delay variation rate. Simulation results are given to show the validity of the proposed theoretical method.

On bounded realness for ℋ∞ control design in discrete‐time descriptor systems

AbstractThis article suggests a new necessary and sufficient condition of bounded real lemma (BRL) for discrete‐time descriptor systems (DTDSs). First, a Lyapunov function for DTDSs is designed by using a positive definite matrix whose the dimension equals to the rank of a singular matrix in DTDSs, whereas the existing work in the literature has considered the dimension which equals to the length of the state. By considering the zero constraint which comes from the singular matrix in DTDSs, two slack variables are introduced into the difference of Lyapunov function. Then, a set of linear matrix inequalities (LMIs) is provided to ensure the necessary and sufficient condition of the BRL for DTDSs. Next, an inversion formula is given to apply the proposed BRL to a stabilization problem of DTDSs. Based on the inversion formula, the BRL for the closed‐loop system with state‐feedback control is obtained in terms of non‐convex conditions. Therefore, a sufficient condition of the non‐convex conditions is provided in terms of LMIs. Three numerical examples show the effectiveness of the proposed approaches.

Bounded real lemmas and exponential [formula omitted] control for memristor-based neural networks with unbounded time-varying delays

This paper focuses on developing a bounded real lemma (BRL) and designing a state-feedback controller which guarantees a prescribed H∞ performance level for a class of memristor-based neural networks (MNNs) with unbounded time-varying delays. Firstly, a BRL for MNNs is presented by taking a new approach based on system solutions. This approach requires neither transformation of the model nor construction of Lyapunov–Krasovskii functionals, thereby reducing computational effort and complexity. In addition, the obtained BRL contains only a few simple inequalities, which can be easily solved by using MATLAB. Secondly, the condition for the existence of exponential H∞ controller is given based on the obtained BRL. Finally, two simulation examples demonstrate the validity of the theoretical results.

The Infinite-Dimensional Standard and Strict Bounded Real Lemmas in Continuous Time: The Storage Function Approach

The bounded real lemma (BRL) is a classical result in systems theory, which provides a linear matrix inequality criterium for dissipativity, via the Kalman-Yakubovich-Popov (KYP) inequality. The BRL has many applications, among others in H^infty control. Extensions to infinite dimensional systems, although already present in the work of Yakubovich, have only been studied systematically in the last few decades. In this context various notions of stability, observability and controllability exist, and depending on the hypothesis one may have to allow the KYP-inequality to have unbounded solutions which forces one to consider the KYP-inequality in a spatial form. In the present paper we consider the BRL for continuous time, infinite dimensional, linear well-posed systems. Via an adaptation of Willems’ storage function approach we present a unified way to address both the standard and strict forms of the BRL. We avoid making use of the Cayley transform and work only in continuous time. While for the standard bounded real lemma, we obtain analogous results as there exist for the discrete time case, when treating the strict case additional conditions are required, at least at this stage. This might be caused by the fact that the Cayley transform does not preserve exponential stability, an important property in the strict case, when transferring a continuous-time system to a discrete-time system.

An efficient model for robust load frequency control in multi-area power systems with communication delays

This paper proposes an effective method such that the robust load frequency control (LFC) scheme can be designed efficiently for the large-scale power system with time delay. A novel constraint time-delayed ordinary differential equation (CTODE) model is proposed, based on which a new bounded real lemma (BRL) is established for the H∞ performance analysis. The CTODE model is investigated considering the small number of remote signals influenced by delays in the LFC scheme. It consists of three parts, i.e., a delayed part includes the remote states, whose order is far less than that of the original system and remains unchanged with the increased scale of the power system, and a delay-free (related) part involves the local signals irrelevant (subjected) to the delayed states. Then, the BRL is established by constructing a new Lyapunov functional where the delayed part is employed to address the delay information. Case studies are implemented on the three-area LFC scheme and 39 bus New England systems. Compared with the existing methods, the developed BRL is available for the stability analysis with minor conservatism, but the calculation complexity is greatly reduced in large-scale power systems. Moreover, based on the CTODE model and original model, the robust controllers are designed and compared in robustness against load variations, time delays, and parameter uncertainties. Similar dynamic performance is verified for these two kinds of controllers through robust performance analysis, system eigenvalue analysis, and simulation studies.

New bounded real lemma for singular Markovian jump systems: Application to H∞ control

AbstractThis article proposes a new strictly bounded real lemma (BRL) for continuous‐time singular Markovian jump systems (SMJSs) in terms of strict linear matrix inequalities (LMIs). While the previous studies have been done for SMJSs that do not have a path from disturbances to desired output, the proposed BRL covers SMJSs with disturbance‐contaminated output by introducing additional free matrix variables. Based on the proposed BRL, the sufficient condition for state‐feedback stabilization is successfully obtained in terms of strict LMIs. The result leads to find the minimal performance by solving the optimization problem subject to the suggested LMIs. Two examples show the less conservatism of proposed method compared with the previous work.

Multiple delay-dependent H ∞ guaranteed cost control for uncertain semi-Markovian jump random nonlinear systems with intermittent actuator and sensor faults and input constraint

In this study, multiple delay-dependent guaranteed cost control is first investigated for uncertain semi-Markovian jump random nonlinear systems with intermittent actuator and sensor faults and input constraint. First, a semi-Markovian jump dynamic output feedback controller is first designed for this model. Multiple delay-dependent noise-to-state stochastic stability conditions are acquired by involving slack matrices by means of bounded real lemma (BRL) as well as linear matrix inequalities (LMIs), leading to the less conservatism. control and guaranteed cost control are achieved simultaneously. Finally, the effectiveness and novelty of the approach are demonstrated through three examples.

Distributed filtering for Markov jump systems with randomly occurring one-sided Lipschitz nonlinearities under Round-Robin scheduling

This paper focuses on distributed filter design for Markov jump one-sided Lipschitz nonlinear systems over a sensor network by applying Round-Robin scheduling scheme. The nonlinear perturbations occur randomly and satisfy the so-called one-sided Lipschitz constraint that is more general than the conventional Lipschitz restraint. Two categories of exogenous noises, i.e., the persistent bounded noise and square additive noise in l2 space, are taken into investigation, respectively. Meanwhile, the well-known Round-Robin scheduling protocol is employed to regulate the data transmission through the shared communication media. The exponential mean-square ultimate boundedness and stochastic bounded real lemma (BRL) are obtained for the filtering error dynamics, and then two distributed filters are correspondingly designed. Finally, an illustrative example shows the effectiveness of the proposed results.

Fault detection for a class of uncertain Lipschitz systems with time delays in finite‐frequency domain

This study investigates the problem of fault detection (FD) filter design for a class of uncertain Lipschitz systems with time delays, unknown faults, parameter uncertainties and bounded disturbances. A novel FD filter is proposed and the augmented filter system is converted to a linear parameter varying (LPV) system by the reformulated Lipschitz property. Then, two bounded real lemmas (BRLs) are given for the generated residual signal to achieve the fault sensitivity performance in low-/middle-/high-frequency domains and disturbance attenuation performance in the full-frequency domain. Next, the obtained BRLs are further converted into a set of linear matrix inequalities (LMIs) to implement the filter design. Finally, two illustrative simulations including a numerical example and Chua's circuit system clearly demonstrate the advantages of the developed FD scheme.

On estimating neuronal states of delayed neural networks based on canonical Bessel-Legendre inequalities

In this paper, the dissipativity state estimation problem is addressed for neural networks with time-varying delays. Applying the N-order Bessel-Legendre inequality, we formulate a hierarchical criterion on the existence of state estimators for the delayed neural networks. First, an augmented Lyaponov–Krasovskii functional is introduced to incorporate the vector information of the N-order Bessel-Legendre inequality. Second, a novel bounded real lemma (BRL) is established for the estimation error system. In particular, the quadratic delay coefficients are absorbed by introducing some new vectors, which, together with smaller upper bounds for certain integral items, leads to the BRL less conservative than some existing results. As such, the state estimators are designed via the linear matrix inequality approach. Third, the obtained criterion is proven to form a hierarchy, which means that the conservatism reduces with the increasing of the order N of the Bessel-Legendre inequality. Finally, the validity of our results is demonstrated through three examples.

Distributed output feedback control of decomposable LPV systems with delay and switching topology: application to consensus problem in multi-agent systems

This paper presents distributed output feedback control of a class of distributed linear parameter varying systems with switching topology and parameter varying time delay. To formulate the synthesis conditions for the distributed controller in terms of LMIs, the delay dependent bounded-real lemma based on parameter-dependent Lyapunov–Krasovskii functionals is used. The efficacy of the result is illustrated by applying it to two real-world examples pertaining to the consensus problem of multi-agent systems.

Fuzzy Output Tracking Control and Filtering for Nonlinear Discrete-Time Descriptor Systems Under Unreliable Communication Links.

In this paper, the problems of output tracking control and filtering are investigated for Takagi-Sugeno fuzzy-approximation-based nonlinear descriptor systems in the discrete-time domain. Especially, the unreliability of the communication links between the sensor and actuator/filter is taken into account, and the phenomenon of packet dropouts is characterized by a binary Markov chain with uncertain transition probabilities, which may reflect the reality more accurately than the existing description processes. A novel bounded real lemma (BRL), which ensures the stochastic admissibility with H∞ performance for fuzzy discrete-time descriptor systems despite the uncertain Markov packet dropouts, is presented based on a fuzzy basis-dependent Lyapunov function. By resorting to the dual conditions of the obtained BRL, a solution for the designed fuzzy output tracking controller is given. A design method for the full-order fuzzy filter is also provided. Finally, two examples are finally adopted to show the applicability of the achieved design strategies.

Robust state-feedback control for discrete-time descriptor systems with norm-bounded parametric uncertainties

ABSTRACTThis paper deals with a state feedback control problem for linear time-invariant discrete-time descriptor systems with norm-bounded parametric uncertainties. To this end, bounded real lemma (BRL) is extended on the class of uncertain descriptor systems. The control design procedure based on the conditions of BRL for uncertain descriptor systems is proposed. Numerical examples are included to illustrate the effectiveness of the present result.

Finite Horizon Worst Case Analysis of Launch Vehicles

Abstract This paper presents a robust linear time varying (LTV) analysis of a space launcher. It evaluates the worst case aerodynamic loads during the atmospheric flight phase under wind turbulence. The results of the LTV analysis are evaluated against a Monte Carlo simulation of the respective industry-sized nonlinear launcher model over a set of allowable uncertainties and disturbances. The recent extension of the strict/ finite horizon bounded real lemma (BRL) for integral quadratic constraints is applied, covering the robustness analysis of finite horizon linear time varying (LTV) systems. Here, an equivalent formulation of the strict BRL in the form of a Riccati differential equality (RDE) is used. The RDEs direct dependence on the IQC multiplier leads to a nonlinear optimization problem, which can be solved efficiently. Using this, it is possible to explicitly respect the time varying dynamics of the space launcher in the worst case analysis. Furthermore, the incorporation of IQCs opens the analysis to a broader range of analyzable perturbations.

On the Optimal Control of Passive or Nonexpansive Systems

The positive-real and bounded-real lemmas solve two important linear-quadratic optimal control problems for passive and nonexpansive systems, respectively. The lemmas assume controllability, yet a passive or nonexpansive system can be uncontrollable. In this paper, we solve these optimal control problems without making any assumptions. In particular, we show how to extract the greatest possible amount of energy from a passive but not necessarily a controllable system (e.g., a passive electric circuit) using state feedback. A complete characterization of the set of solutions to the linear matrix inequalities in the positive-real and bounded-real lemmas is also obtained. In addition, we obtain necessary and sufficient conditions for a system to be nonexpansive that augment the bounded-real condition with new conditions relevant to uncontrollable systems.

Delay-Dependent Robust H∞ Performance for Uncertain Neutral Systems with Mixed Time-Varying Delays and Nonlinear Perturbations

This paper deals with the problems of delay-dependent stability and H∞ performance for uncertain neutral systems with time-varying delays, and nonlinear perturbations. The time-varying delays are neutral, discrete, and distributed time-varying delays that the upper bounds for the delays are available. The restrictions on the derivatives of the discrete and distributed time-varying delays are removed, which mean that a fast discrete time-varying delay is allowed. The uncertainties under consideration are nonlinear time-varying parameter perturbations and norm-bounded uncertainties, respectively. Firstly, by applying a novel Lyapunov-Krasovskii functional approach, Wirtinger-based integral inequality, Peng-Park’s integral inequality, decomposition technique of constant matrix, descriptor model transformation, Leibniz Newton formula and utilization of zero equation, and improved delay-dependent bounded real lemmas (BRL) for systems are established in terms of linear matrix inequalities (LMIs). Then, based on the obtained BRL, some less conservative delay-dependent stability criteria of uncertain neutral systems with mixed time-varying delays and nonlinear perturbations are obtained and improved H∞ performance criterion with the framework of LMIs is introduced. Finally, some numerical examples are given to illustrate that the presented method is effective.

Full- and Reduced-Order Fault Detection Filter Design With Application in Flow Transmission Lines

The full- and reduced-order fault detection filter design is examined for fault diagnosis in linear time-invariant (LTI) systems in the presence of noise and disturbances. The fault detection filter design problem is formulated as an H∞ problem using a linear fractional transformation (LFT) framework and the solution is based on the bounded real lemma (BRL). Necessary and sufficient conditions for the existence of the fault detection filter are presented in the form of linear matrix inequalities (LMIs) resulting in a convex problem for the full-order filter design and a rank-constrained nonconvex problem for the reduced-order filter design. By minimizing the sensitivity of the filter residuals to noise and disturbances, the fault detection objective is fulfilled. A reference model can be incorporated in the design in order to shape the desired performance of the fault detection filter. The proposed fault detection and isolation (FDI) framework is applied to detect instrumentation and sensor faults in fluid transmission and pipeline systems. To this end, a lumped parameter framework for modeling infinite-dimensional fluid transient systems is utilized and a low-order model is obtained to pursue the instrumentation fault diagnosis objective. Full- and reduced-order filters are designed for sensor FDI. Simulations are conducted to assess the effectiveness of the proposed fault detection approach.

Multivariable Proportional-Integral-Derivative Controller Tuning Via Linear Matrix Inequalities Based on Minimizing the Nonconvexity of Linearized Bilinear Matrix Inequalities

This paper proposes an iterative linear matrix inequality (LMI) method for tuning the parameters of multi-input multi-output (MIMO) proportional–integral–derivative (PID) controllers. The proposed method calculates the parameters of controller such that the singular values (SVs) of the sensitivity function are shaped according to the given weight function. For this purpose, first using bounded real lemma (BRL), this problem is represented as a bilinear matrix inequality (BMI) where one of the matrix variables is the variable of Lyapunov equation and the other is structured and contains the matrices of the state-space representation of the controller. This BMI is solved approximately using a novel iterative procedure which linearizes the BMI around an initial solution to arrive at an LMI. The point around which the BMI is linearized is updated automatically at each iteration and the linearized BMI has the nice property that it is obtained by minimizing the amplitude of nonconvex terms.