Slack Matrices Research Articles

Global non-quadratic Lyapunov-based stabilization of T–S fuzzy systems: A descriptor approach

This article studies the problem of global stability of the Takagi–Sugeno fuzzy systems based on a novel descriptor-based non-quadratic Lyapunov function. A modified non-quadratic Lyapunov function, which comprises an integral term of the membership functions, and a modified non-parallel distributed controller constructed by constant delayed premise variables are considered that assure the global stability of the closed-loop T–S fuzzy system. The special structure of the used non-quadratic Lyapunov function results in time-delayed terms of the membership functions, instead of appearing their time derivatives, which is the well-known issue of the common non-quadratic Lyapunov functions in the literature. Also, the memory fuzzy controller is chosen such that the artificial constant delay-dependent stability analysis conditions for a non-delayed closed-loop T–S fuzzy system are formulated in terms of linear matrix inequalities. To further reduce the conservatives, some slack matrices are introduced by deploying the descriptor representation and decoupling lemmas. Moreover, the design of the robust fuzzy controller is studied through the performance criteria. The main advantages of the proposed approach are its small conservatives and the global stability analysis, which distinguish it from the state-of-the-art methods. To show the merits of the proposed approach, comparison results are provided, and two numerical case studies, namely, flexible joint robot and two-link joint robot are considered.

Delay-Dependent $H_\infty$ Guaranteed Cost Control for Uncertain Switched T–S Fuzzy Systems With Multiple Interval Time-Varying Delays

This article deals with delay-dependent H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> guaranteed cost control for uncertain switched Takagi-Sugeno (T-S) fuzzy systems with multiple interval time-varying delays. It is the first time that the fault tolerant control (FTC) of nonlinear systems subject to measurement noise, external disturbances, nonlinear functions, and intermittent faults (IFs) in sensors and actuators is investigated. A passive dynamic full-order output feedback controller is first addressed for the system to guarantee the exponential stability. Compared with the existing results, the restrictions on the multiple time-varying delays are relaxed in the derivation of this work. The suggested controller has a wider application. Moreover, as the piecewise fuzzy Lyapunov function is introduced with some slack matrices, the delay-dependent sufficient conditions are provided in the way of a cluster of linear matrix inequalities (LMIs) and then less conservatism is attained by our design contrast to the existing fault tolerant controllers. At last, two simulation examples and some comparisons are achieved to illustrate the novelty and feasibility of the method gathered in the article.

Robust over finite frequency ranges H∞ model reduction for uncertain 2D continuous systems

This paper deals with the problem of robust H∞ model reduction for two-dimensional (2D) continuous systems described by Roesser model with polytopic uncertainties, over finite frequency (FF) domain. The problem to be solved in the paper is to find a reduced-order model such that the approximation error system is asymptotically stable, which is able to approximate the original continuous systems system with comparatively small and minimised H∞ performance when frequency ranges of noises are known beforehand. Via the use of the generalised Kalman Yakubovich Popov (gKYP) lemma, homogeneous polynomially parameter-dependent matrices, Finsler's lemma and we introduce many slack matrices, sufficient conditions for the existence of H∞ model reduction for different FF ranges are proposed and then unified in terms of solving a set of linear matrix inequalities (LMIs). To highlight the effectiveness of the proposed H∞ model reduction design, two illustrative examples are given.

Novel Equalities for Stability Analysis of Asynchronous Sampled-Data Systems

This paper is concerned with the stability analysis problems of asynchronous sampled-data systems with use of a input-delay approach, where a sampled-data system can be reformulated into a time-delay system having incremental delays. For these systems, this paper introduces a new looped-functional to utilize both integral states and their interval-normalized ones. Utilization of these two types of integral state variables in a construction of stability conditions has been effective in reducing the conservatism. Further, generalized equalities are proposed for the case utilizing these two types of integral states. The sampled-data system formulation consists of a sampled state, a system state and its derivative. Here, the sampled state between two consecutive sampling instants is a constant value and thus can be eliminated by an integration with Legendre polynomials of positive degrees. Consequently, the system formulation turns into the equalities consisting of state variables, sampled-state variables, integral state variables and their interval-normalized ones without additional slack matrices. Based on the proposed equalities, a large computational burden can be reduced while reducing the conservatism. The effectiveness of the proposed approaches is demonstrated via three numerical examples for the stability analysis of asynchronous sampled-data systems.

Stability analysis of linear systems with time-varying delay via intermediate polynomial-based functions

This note is devoted to stability analysis for linear system with time-varying delay. By advisably introducing slack matrices, novel fractional order intermediate polynomial-based functions (IPFs) are proposed. Then, the stability condition is derived for the time delay system. From configuration on fractional order polynomials, the relationships among system states are taken into account and consolidated via slack variables, and the characteristics for integral inequality are synthetically considered, while avoiding higher order time delays. More remarkably, adjusting tunable parameters also contributes to reduction of conservatism. The comparisons of computational complexity and stability region on a well known numerical example are provided to validate the advantages of the resulting stability criterion.

Extended dissipativity asynchronous static output feedback control of Markov jump systems

This paper is concerned with both continuous- and discrete-time extended dissipativity-based control problems for Markov jump systems. Hidden Markov model is employed to describe the jump asynchronization behavior between the original system and the controller. Then, an asynchronous static output feedback controller is designed. By using the mode-dependent Lyapunov function technique with slack matrices, several sufficient conditions are obtained to ensure that the closed-loop systems are stochastically stable with extended dissipativity performance. Based on these conditions, solutions to the controller gains are derived. Finally, simulation results are presented to illustrate the feasibility and the effectiveness of the proposed approaches.

A New Integral Inequality Approach for Extended Dissipative Filters Design of Singular Markovian Jump Systems with Discrete and Distributed Delays

This paper studies the problem of reduced-order extended dissipative filters design for continuous-time singular Markovian jump systems with discrete and distributed delays. By proposing a new integral inequality, a sufficient condition based on linear matrix inequalities (LMIs) is presented to ensure the existence of the extended dissipative filters. Compared with the previous ones, the proposed method can sufficiently exploit the information on discrete and distributed delays. A set of slack matrices instead of fixed ones are introduced in the LMIs to find all solutions of the reduced-order filters, which can enhance the flexibility of the obtained result. Two numerical examples are provided to illustrate the effectiveness and advantages of the proposed filters design condition.

Finite-Frequency H-/H∞ Fault Detection for Discrete-Time T-S Fuzzy Systems With Unmeasurable Premise Variables.

This paper investigates a finite-frequency H-/H∞ fault detection method for discrete-time T-S fuzzy systems with unmeasurable premise variables. To minimize the effect of uncertainties on system performance and maximize that of actuator faults on the generated residual, both the H∞ disturbance attenuation index and finite-frequency H- fault sensitivity index are utilized. Since the premised variables are unmeasurable, the existing generalized Kalman-Yakubovich-Popov lemma cannot be directly extended to these nonlinear systems. In this paper, the conditions of allowing one to design the proposed H-/H∞ fault detection observer are established and transformed into linear matrix inequalities. Some scalars and slack matrices are introduced to bring extra degrees of freedom in observer design. Finally, a single-link robotic manipulator model is utilized to illustrate that the proposed technique can detect faults with smaller amplitude than that required by a normal H∞ observer technique.

New results for sampled-data control of interval type-2 fuzzy nonlinear systems

This paper is devoted to the investigation of the interval type-2 (IT2) fuzzy sampled-data stabilization problem for the controlled plant subject to nonlinearities and parameter uncertainties. Some free-weighting matrices, slack matrices, and the bound information in membership functions are used to improve the stability analysis. Based on the Lyapunov–Krasovskii functional (LKF) theory, a new relaxed sufficient condition with fewer linear matrix inequality (LMI) constraints is derived. According to this criterion, the IT2 fuzzy sampled-data controller is devised to ensure the closed-loop system is asymptotically stable. Finally, three practical examples are provided to demonstrate the effectiveness and efficiency of the proposed design. Some comparisons show that the proposed algorithm is more simple and practical.

Asynchronous H∞ fixed-order filtering for LPV switched delay systems with mode-dependent average dwell time

This paper investigates the problem of robust H∞ fixed-order filtering for a class of linear parameter-varying (LPV) switched delay systems under asynchronous switching that the system parameter matrices and the time delays are dependent on the real-time measured parameters. The so-called asynchronous switching means that there are time delays between the switching of filters and the switching of system modes. By constructing the parameter-dependent and mode-dependent Lyapunov-Krasovskii functional which is allowed to increase during the running time of active subsystem with the mismatched filter, and using the mode-dependent average dwell time (MDADT) switching method, the sufficient conditions for exponential stability and satisfying a novel weighted H∞ criterion are derived. As there exist couplings between Lyapunov-Krasovskii functional matrices and system parameter matrices, we utilize slack matrices to decouple them. Based on the above results, a suitable weighted H∞ fixed-order filter can be obtained in the form of the parameter linear matrix inequalities (PLMIs). By virtue of approximate basis function and gridding technique, the design of weighted H∞ fixed-order filter can be transformed into the solution of the finite dimensional LMIs. Finally, a numerical example is presented to verify both the effectiveness and the low conservatism of the parameter-dependent and mode-dependent fixed-order filtering method proposed in this paper.

Event-triggered reliable dissipative filtering for nonlinear networked control systems

This paper investigates the event-triggered reliable dissipative filtering problem for nonlinear networked control systems (NCSs) described by interval type-2 T-S fuzzy models. First, a novel adaptive event-triggered scheme is proposed to save the limited network resources. Then, a new event-triggered reliable dissipative filter is designed subject to mismatched membership functions and asynchronous constraints. Furthermore, sufficient conditions for ensuring asymptotic stability and strict dissipativity of the filtering error system are derived by utilizing slack matrices and the information of the membership functions. Compared with the existing event-triggered schemes, it is shown that the proposed one can more effectively reduce the utilization of network resources while ensuring the better estimation performance. Finally, a simulation example is given to illustrate the advantages of the proposed method.

Stability and Stabilization With Additive Freedom for Delayed Takagi–Sugeno Fuzzy Systems by Intermediary-Polynomial-Based Functions

This paper is devoted to the stability and stabilization for Takagi-Sugeno fuzzy systems with time-varying delays. First, an improved matrix inequality is presented to bound both strictly and nonstrictly proper rational functions, which is more general than the existing versions of reciprocally convex lemmas. Second, by suitable operations on parameter-dependent polynomial multiplied by state rate, a couple of novel intermediary-polynomial-based functions (IPFs) are developed in delay-product types. Benefitting from slack matrices of IPFs, a certain degree of flexibility is furnished. More importantly than that, by feat of adjustment of the variable parameter, the resulting conditions will be further endowed with additive freedom, which relaxes the feasible space in a distinctive manner. Third, by utilizing IPFs along with triple integrals, the stability criteria and the controller design approach are derived by some advanced integral inequalities. Resorting to elaborate construction of IPFs, the strengths of bounding techniques are sufficiently exploited, and the information on delay derivative is adequately reflected. Consequently, more desirable performances are achieved, while without excessive computational complexity. Finally, the effectiveness of the proposed methods is verified by numerical examples.

Polynomial size linear programs for problems in P

A perfect matching in an undirected graph G=(V,E) is a set of vertex disjoint edges from E that include all vertices in V. The perfect matching problem is to decide if G has such a matching. Recently Rothvoß proved the striking result that the Edmonds’ matching polytope has exponential extension complexity. In this paper for each n=|V| we describe a polytope for the perfect matching problem that is different from Edmonds’ polytope and define a weaker notion of extended formulation. We show that the new polytope has a weak extended formulation (WEF) Q of polynomial size. For each graph G with n vertices we can readily construct an objective function so that solving the resulting linear program over Q decides whether or not G has a perfect matching. With this construction, a straightforward O(n4) implementation of Edmonds’ matching algorithm using O(n2) bits of space would yield a WEF Q with O(n6logn) inequalities and variables. The construction is uniform in the sense that, for each n, a single polytope is defined for the class of all graphs with n nodes. The method extends to solve polynomial time optimization problems, such as the weighted matching problem. In this case a logarithmic (in the weight of the optimum solution) number of optimizations are made over the constructed WEF.The method described in the paper involves the construction of a compiler that converts an algorithm given in a prescribed pseudocode into a polytope. It can therefore be used to construct a polytope for any decision problem in P which can be solved by a well defined algorithm. Compared with earlier results of Dobkin–Lipton–Reiss and Valiant our method allows the construction of explicit linear programs directly from algorithms written for a standard register model, without intermediate transformations. We apply our results to obtain polynomial upper bounds on the non-negative rank of certain slack matrices related to membership testing of languages in P/poly.

Interval type-2 fuzzy sampled-data control of time-delay systems

The problem of interval type-2 (IT2) fuzzy sampled-data stabilization is addressed for nonlinear continuous-time systems with time-varying delay and parameter uncertainties. The input-delay approach together with free-weighting and slack matrices is utilized to enforce the stability analysis, and a sufficient condition is formulated. Based on the condition, an IT2 fuzzy sampled-data controller is developed such that the closed-loop system is asymptotically stable. Illustrative examples are used to manifest the merits of this design.

New relaxed stabilization conditions for discrete‐time Takagi–Sugeno fuzzy control systems

AbstractThis paper develops relaxed stabilization conditions for discrete‐time Takagi–Sugeno fuzzy control systems based on the extended nonquadratic Lyapunov function, the nonparallel distributed compensation law, and the convexity of the fuzzy blending coefficients. Three new main results are proposed to further reduce the conservatism by fully exploring the slack matrix technique and introducing new slack matrices and extra collection matrices. The new stabilization conditions are gradually less and less conservative, and more advantageous than the existing results by overall consideration of the conservatism and the computational efforts. A well‐known numerical case and a practical case are carried out to demonstrate the effectiveness of the proposed stabilization conditions.

Design of robust &lt;i&gt;H&lt;/i&gt;&lt;SUB align="right"&gt;∞ fuzzy output feedback controller for affine nonlinear systems: fuzzy Lyapunov function approach

In this paper, we propose a new systematic approach based on non-quadratic Lyapunov function and technique of introducing slack matrices, for a class of affine nonlinear systems with disturbance. To achieve the goal, first, the affine nonlinear system is represented via Takagi-Sugeno (T-S) fuzzy bilinear model. Subsequently, the robust H∞ controller is designed based on parallel distributed compensation (PDC) scheme. Then, the stability conditions are derived in terms of linear matrix inequalities (LMIs) by utilising Lyapunov function. Moreover, some slack matrices are proposed to reduce the conservativeness of the LMI stability conditions. Finally, for illustrating the merits and verifying the effectiveness of the proposed approach, the application of an isothermal continuous stirred tank reactor (CSTR) for Van de Vusse reactor is discussed in details.

Design of robust H_ fuzzy output feedback controller for affine nonlinear systems: Fuzzy Lyapunov function approach

In this paper, we propose a new systematic approach based on non-quadratic Lyapunov function and technique of introducing slack matrices, for a class of affine nonlinear systems with disturbance. To achieve the goal, first, the affine nonlinear system is represented via Takagi-Sugeno (T-S) fuzzy bilinear model. Subsequently, the robust H∞ controller is designed based on parallel distributed compensation (PDC) scheme. Then, the stability conditions are derived in terms of linear matrix inequalities (LMIs) by utilising Lyapunov function. Moreover, some slack matrices are proposed to reduce the conservativeness of the LMI stability conditions. Finally, for illustrating the merits and verifying the effectiveness of the proposed approach, the application of an isothermal continuous stirred tank reactor (CSTR) for Van de Vusse reactor is discussed in details.

A Novel Approach to Delay-Variation-Dependent Stability Analysis of 2-D Discrete-Time Systems With Mixed Delays

In this paper, a stability analysis problem is studied for a class of two-dimensional (2-D) discrete-time systems with time-varying and distributed delays described by the second Fornasini-Marchesini (FM) model. First, new 2-D polynomials-based summation inequalities are proposed to estimate summation terms in the forward difference of Lyapunov-Krasovskii functional (LKF). The inequalities can reduce to 2-D Jensen inequalities and 2-D finite-sum inequalities by designing slack matrices and arbitrary vectors. Second, a new augment LKF is constructed, which makes full use of the delay changing information. By the Lyapunov stability theory, sufficient conditions for asymptotic stability of 2-D discrete-time systems are derived in the form of linear matrix inequalities. Finally, two simulation examples are given to demonstrate the effectiveness of the proposed methods.

State Feedback Controller Design for General Polynomial-Fuzzy-Model-Based Systems with Time-Varying Delay

This paper mainly investigates the state feedback controller design and the stability analysis for the general polynomial-fuzzy-model-based systems with time-varying delay. A Lyapunov–Krasovskii (L–K) function with the delay information in integral parts is used in the stability analysis process, and the time-delay terms are solved by an introduced lemma. Here, the assumption that the input matrices and the delay matrices have zero rows at the same time is not needed. A novel square of sum (SOS) optimization method successfully solves the non-convex problem by transforming the nonlinear terms and the time-varying delay matrix items into indices, which are optimized to zero by a semi-definite programming. The resultant SOS conditions are delay dependent, and more flexible stability conditions are obtained by introducing the slack matrices. Numerical simulation proves the effectiveness of the method.

Event-Triggered Fault Detection Filter Design for Nonlinear Networked Systems

This paper investigates the problem of event-triggered fault detection (FD) filter design for nonlinear networked systems in the framework of interval type-2 fuzzy systems. In the system model, the parameter uncertainty is captured effectively by the membership functions (MFs) with upper and lower bounds. For reducing the utilization of limited communication bandwidth, an event-triggered communication mechanism is applied. A novel FD filter subject to event-triggered communication mechanism, data quantization, and communication delay is designed to generate a residual signal and detect system faults, where the premise variables are different from those of the system model. Consequently, the augmented FD system is with imperfectly matched MFs, which hampers the stability analysis and FD. To relax the stability analysis and achieve a better FD performance, the information of MFs and slack matrices are utilized in the stability analysis. Finally, two examples are employed to demonstrate the effectiveness of the proposed scheme.