Free Matrices Research Articles

Parametric design of reduced-order functional observers for linear time-varying delay systems

This paper investigates the design of reduced-order functional observer for linear time-invariant systems (LTI) with time-varying delay. Firstly, the sufficient condition for the existence of reduced-order delay-free functional observers is obtained. Secondly, by solving generalized Sylvester matrix equations (GSEs), completely parameterized formulas of the coefficient matrices of reduced-order function observer are established. The free parameter matrices in the formulas provide the design degrees of the freedom and improve the convergence rate of the error system. Finally, two numerical and a practical examples are provided to illustrate the effectiveness of the proposed approach.

Output feedback based poles configuration for LDP systems with varying state and input dimensions

The problem of pole configuration via periodic output feedback for linear discrete periodic systems with time-varying dimensions of states and input spaces is addressed. By some mathematical derivation, the considered problem can be reduced into solving a special Sylvester matrix equation from which the periodic output feedback gains can be separated and computed explicitly. Based on this, a detail parametric algorithm is presented, by which countless periodic output feedback gains can be generated by choosing different free parameter matrices. Furthermore, the proposed approach is used to achieve robustness performance and robust poles configuration algorithm is provided. Numerical examples are employed to illustrate the effectiveness of the proposed algorithms.

Enhanced gain‐scheduling fault estimation of discrete‐time Takagi–Sugeno nonlinear systems: A novel free matrix approach

A novel gain-scheduling fault estimation is developed on the direction of improving algorithm efficiency based on discrete-time Takagi–Sugeno fuzzy models. Different from the traditional results in which the zero-order free matrices are applied, a novel free matrix approach is proposed and the higher-order free matrices are for the first time introduced in designing conditions of the fuzzy gain-scheduling fault estimation observer. For each working mode, its proprietary properties about current-time normalized fuzzy weighting functions are made full use and thus the anti-interference ability of the developed fuzzy gain-scheduling fault estimation observer is enhanced without any increase of either the consumed time of solving the required designing criterion or the number of gain scalars needed to be stored in random access memory of the computer in real time. It does provide a good chance to obtain equal or better results of fuzzy gain-scheduling fault estimation with the usage of much less computing resource than before, which is much meaningful to the applications of theoretical methods. Finally, the effectiveness and superiority of our developed free matrix approach are tested and validated by using performance comparisons on the benchmark example.

A generalized multiple-integral inequality based on free matrices: Application to stability analysis of time-varying delay systems

This study proposed a novel generalized multiple-integral inequality based on free matrices in a quadratic form of an augmented vector stacked with a state and its derivative, with generalization realized in terms of both the integration and bounding orders. Further, additional cross information on the two entries of the augmented vector was considered using structured free matrices. Consequently, the proposed generalized multiple-integral inequality guaranteed tight bounds with various existing multiple-integral inequalities included as special cases. Moreover, it was observed that higher the orders of the generalized multiple-integral inequalities and Lyapunov–Krasovskii functions, less conservative the stability criterion obtained. Furthermore, three recognized numerical simulations were presented to confirm the improved effectiveness of the stability criterion derived through the proposed generalized multiple-integral inequality than in previous studies.

Extended dissipativity analysis for T-S fuzzy systems based on reliable memory control and aperiodic sampled-data method

This paper studies the extended dissipativity (ED) issue for T-S fuzzy systems (TSFSs) via reliable memory control scheme and aperiodic sampled-data (ASD) method. First, considering the random variation of sampling interval and the time delays (TDs) of sampling signal transmission in the communication network, a reliable aperiodic memory sampled-data control (RAMSDC) strategy is proposed. Then, the developed delay-dependent Lyapunov-Krasovskii functional (LKF) with some two-sided looped-functional (TSLF) terms is constructed to fully utilize sampled state information. The introduced free matrices in the TSLF need not to be positive definite, which reduces the conservativeness of the obtained results. Next, a sufficient condition is given to ensure the ED, and the controller gain matrix is obtained by means of linear matrix inequality (LMI) technique. At last, the effectiveness of theoretical results in practical application is verified by the use of a truck-trailer model.

Relaxed stabilization synthesis of discrete-time nonlinear systems with uplink data loss based on a novel online evaluation mechanism

The problem of promoting the adaptability to uplink data loss for nonlinear plant is addressed via developing the fuzzy gain-scheduling stabilization law based on a novel online evaluation mechanism. Different from those previous methods, the proposed online evaluation mechanism is dependent not only the current-time but also the past-time normalized fuzzy weighting functions, and thus much more information can be absorbed and utilized in the process of stabilization synthesis by introducing extra free matrices than before. More importantly, an additional constraint is for the first time proposed for updating the plant status instantly so that critical replacement operations can be implemented for reducing the conservativeness of the designing conditions. Therefore, the adaptability to uplink data loss is improved evidently and thus the worse communication quality of uplink channel is bearable in this study. Finally, the advantages relative to those reported ones are tested and validated through benchmark numerical comparisons.

Improved Delay-Dependent Stability Analysis of Fixed-Point State-Space Digital Filters With Time-Varying Delay and Generalized Overflow Arithmetic

This paper is concerned with the stability analysis of fixed-point state-space digital filters with generalized overflow arithmetic and a time-varying delay. This paper aims to derive a delay and nonlinear function bound dependent asymptotical stability criterion with less conservatism. Firstly, a new Lyapunov functional with several augmented terms, including extra free matrices and overflow nonlinear function, is constructed such that it has a relaxed positive condition. Then, for bounding the summation term arising in the forward difference of Lyapunov functional, a new lemma is developed to introduce the terms for linking the delayed states and the overflow nonlinear function, the Wirtinger-based summation inequality and several zero-value terms are applied to add more cross terms. As a result, a stability criterion with less conservatism is established and its conservatism. Finally, several numerical examples are given to illustrate the advantages of the proposed method.

A novel wheeled mobile robots control based on robust hybrid controller: Mixed H2/H∞ and predictive algorithm approach

One of the challenges facing robot tracking is the difficulty of controlling variety types of robots in known and unknown environments. This paper proposes a new strategy of path tracking for a non-holonomic mobile robot subject to kinematic disturbances using model prediction control and H2/H∞ control by Gain-scheduled control law. In this regard, the problem of tracking a mobile robot is expressed in a way that its kinematic error is in the form of linear parameter variable with bounded disturbance. The purpose of the design is to stabilize the controller by solving the linear matrix inequality (LMI) conditions for error tracking model. The following stability conditions are then formulated in such a way that the upper bounds are considered as LMI auxiliary variables. Also with the help of more free matrices, it can reduce conservatism. The target tracking is first designed with the assumption that there is no obstacle in the environment, and then the obstacle avoidance mode is added to the tracer controller, which is done by modifying the optimal state vector definition. In both cases, the system stability check is also performed. Simulation is performed on several predetermined paths, indicating that this strategy is feasible.

Stability Analysis of Drilling Inclination System with Time-Varying Delay via Free-Matrix-Based Lyapunov–Krasovskii Functional

This study provides an insight into the asymptotic stability of a drilling inclination system with a time-varying delay. An appropriate Lyapunov–Krasovskii functional (LKF) is essential for the stability analysis of the abovementioned system. In general, an LKF is constructed with each coefficient matrix being positive definite, which results in considerable conservatism. Herein, to relax the conditions of the derived criteria, a novel LKF is proposed by avoiding the positive-definite restriction of some coefficient matrices and introducing additional free matrices simultaneously. Subsequently, this relaxed LKF is applied to derive a less conservative stability criterion for the abovementioned system. Finally, the effect of reducing the conservatism of the proposed LKF is verified based on two examples.

Enhanced observer-based state estimation of discrete-time Takagi-Sugeno fuzzy systems via a distinctive multi-instant gain-scheduling law

In this paper, the observer-based state estimation of discrete-time Takagi-Sugeno fuzzy systems is enhanced markedly by developing a distinctive multi-instant gain-scheduling law. Firstly, the updated information of both the past-time and the current-time normalized fuzzy weighting functions is fused into the proposed multi-instant gain-scheduling law in a comprehensive way, and it is important to point out that those existing ones reported in the literature belong to special cases of ours. Secondly, a set of time-varying free matrices for constructing more relaxed fuzzy observers are introduced for the first time in accordance with different operating modes. Moreover, the time-varying free matrices are more general than those previous time-invariant ones and most of them no longer need to be positive or negative definite. Therefore, a significant reduction in conservatism can be achieved at the expense of a slight increase in offline computing burden over those competing approaches. Finally, a set of numerical simulations are implemented to verify the superiority of the designed approach.

Improved stability criteria for linear systems with time-varying delays

This paper is concerned with the stability analysis of linear systems with time-varying delays. First, by introducing the quadratic terms of time-varying delays and some integral vectors, a more suitable Lyapunov-Krasovskii functional (LKF) is constructed. Second, two new delay-dependent estimation methods are developed in the stability analysis of linear system with time-varying delays, which include a reciprocally convex matrix inequality and an integral inequality. More information about time-varying delays and more free matrices are introduced into the two estimation approaches, which play a key role for obtaining an accurate upper bound of the integral terms in time derivative of LKFs. Third, based on the novel LKFs and new estimation approaches, some less conservative criteria are derived in the form of linear matrix inequality (LMI). Finally, three numerical examples are applied to verify the advantages and effectiveness of the newly proposed methods.

Enhanced fuzzy state estimation of discrete‐time nonlinear systems via a multi‐instant united switch‐type observer

AbstractThe performance of state estimation of discrete‐time Takagi–Sugeno systems is enhanced by proposing a featured multiinstant united switch‐type observer. The novelty of the proposed observer lies in both the definition and application of a set of switching modes. Since different time‐variant free matrices can be for the first time introduced in accordance with each switching mode, both the current time and the past time normalized fuzzy weighting functions are together considered for designing the corresponding observer gain matrices with much less conservatism than those recent methods reported in the literature. Furthermore, there are only two tuning parameters which are both located in a limited candidate set and thus this fact does benefit for giving the optimal solution of our developed multiinstant united switch‐type candidates. Finally, numerical comparisons are provided in order to verify the technical superiority of the results developed in this article.

Improved Results on Delay-Dependent Robust H ∞ Control of Uncertain Neutral Systems with Mixed Time-Varying Delays

In this paper, the problem of the delay-dependent robust H ∞ control for a class of uncertain neutral systems with mixed time-varying delays is studied. Firstly, a robust delay-dependent asymptotic stability criterion is shown by linear matrix inequalities (LMIs) after introducing a new Lyapunov–Krasovskii functional (LKF). The LKF including vital terms is expected to obtain results of less conservatism by employing the technique of various efficient convex optimization algorithms and free matrices. Then, based on the obtained criterion, analyses for uncertain systems and H ∞ controller design are presented. Moreover, on the analysis of the state-feedback controller, different from the traditional method which multiplies the matrix inequality left and right by some matrix and its transpose, respectively, we can obtain the state-feedback gain directly by calculating the LMIs through the toolbox of MATLAB in this paper. Finally, the feasibility and validity of the method are illustrated by examples.

Dependent delay stability characterization for a polynomial T-S Carbon Dioxide model

<p style='text-indent:20px;'>By extending some linear time delay systems stability techniques, this paper, focuses on continuous time delay nonlinear systems (TDNS) dependent delay stability conditions. First, by using the Takagi Sugeno Fuzzy Modeling, a novel relaxed dependent delay stability conditions involving uncommon free matrices, are addressed in Linear Matrix Inequalities (LMI). Then, as application a Nonlinear Carbon Dioxide Model is used and rewritten by a change of coordinate to the interior equilibrium point. Next, by using the non-linearity sector method the model is transformed to a corresponding Fuzzy Takagi Sugeno (TS) multi-model. Also, the maximum delay margin to which the model is stable, is identified. Finally, to prove the analytic results a numerical simulation is also performed and compared to other methods.</p>

Formulation and Thermomechanical Characterization of Functional Hydrogels Based on Gluten Free Matrices Enriched with Antioxidant Compounds

Native starch from potatoes and hybrid carrageenans from the red alga Mastocarpus stellatus have been used as gluten-free gelling matrices to obtain functional hydrogels. The enrichment of gelling matrices with antioxidant compounds from natural sources is an increasing market trend. In this context, this work is aimed at the formulation and thermo-rheological characterization of functional hydrogels using potato starch from agro-industrial waste and kappa–iota hybrid carrageenans extracted from the above seaweed, enriched with antioxidant compounds from different agro-industrial products, such as waste coming from the pruning of green tea and two varieties of hops used in the brewing industry. Environmentally friendly technologies such as microwave hydrodiffusion and gravity, microwave-assisted extraction, ultrasounds and autohydrolysis were used for the recovery of antioxidant compounds. The results point out that functional hydrogels based on potato starch and hybrid carrageenans with a wide range of viscoelastic features can be achieved, with the particularity of being suitable for people with celiac disease. The incorporation of selected antioxidant extracts from vegetable by-products involved the drop (about tenfold) of the viscous and elastic properties of the formulated gels. The sequential combination of the above treatments could even further expand the thermo-rheological properties of formulated hydrogels, with potential application in functional foodstuffs and novel gluten-free goods.

Bilaminate Synthetic Dermal Matrix versus Free Fascial Flaps: A Cost-Effectiveness Analysis for Full-Thickness Hand Reconstruction.

Full-thickness injuries to the hand require durable soft tissue coverage to preserve tendon gliding and hand motion. We aim to investigate the cost effectiveness of hand resurfacing comparing free fascial flap reconstruction versus bilaminate synthetic dermal matrices. Cost effectiveness was modeled using decision tree analysis with the rollback method. Total active range of motion was modeled as the common outcome variable based on systematic literature review. Costing was performed from a payer perspective using national Medicare reimbursements. The willingness to pay threshold was determined by average worker's compensation for hand disability. Probabilistic sensitivity analysis was conducted for range of motion outcomes and the costs using 10,000 Monte Carlo simulations. The average cost of free fascial flap reconstruction was $14,201.24 compared with $13,674.20 for Integra, yielding an incremental cost difference of $527.04. Incremental range of motion improvement was 18.0 degrees with free fascial flaps, yielding an incremental cost effectiveness ratio of $29.30/degree of motion. Assuming willingness to pay thresholds of $557.00/degree of motion, free-fascial flaps were highly cost effective. On probabilistic sensitivity analysis, free fascial flaps were dominant in 25.5% of simulations and cost effective in 32.1% of simulations. Thus, microsurgical reconstruction was the economically sound technique in 57.5% of scenarios. Free fascial flap reconstruction of complex hand wounds was marginally more expensive than synthetic dermal matrix and yielded incrementally better outcomes. Both dermal matrix and microsurgical techniques were cost effective in the base case and in sensitivity analysis. In choosing between dermal matrix and microsurgical reconstruction of complex hand wounds, neither technique has a clear economic advantage.

A new modified polynomial-based optimal control design approach

The main concern of this article is addressing a new modified approach to design a nonlinear optimal controller. The modification focuses on proposing a new approximate solution for the Hamilton–Jacobi–Bellman nonlinear partial differential equation. The introduced solution works based on the state-dependent power series expansion presentation of the involved functions in the Hamilton–Jacobi–Bellman partial differential equation. Applying this technique results in releasing a set of free state-dependent functions in the controller structure that can be adjusted to fulfill some special control missions in addition to the optimization objectives. They are formed based on the specific formulation of the candidate Lyapunov function. The proposed approach is exemplified for an intricate biological system, immunogenic tumor-immune cell interaction in the human body, to clarify the mechanism of designing the controller and adjusting the arrays of the free matrices. The closed-loop system by presented optimal state feedback controller meets the predefined optimization objectives without getting feedback from a hard-measurable state. It is achieved by adjusting the aforementioned released functions such that an optimal output feedback controller is obtained. To have some insights into the performance of the system and the effectiveness of the controller, the positiveness of the system’s states is proved and checked numerically by applying the differential transformation method to the system’s differential equations. Finally, to highlight the abilities of the proposed approach from different aspects, some simulations are carried out.

Smooth, time‐invariant regulation of nonholonomic systems via energy pumping‐and‐damping

SummaryIn this article we propose an energy pumping‐and‐damping technique to regulate nonholonomic systems described by kinematic models. The controller design follows the widely popular interconnection and damping assignment passivity‐based methodology, with the free matrices partially structured. Two asymptotic regulation objectives are considered: drive to zero the state or drive the systems total energy to a desired constant value. In both cases, the control laws are smooth, time‐invariant, state‐feedbacks. For the nonholonomic integrator we give an almost global solution for both problems, with the objectives ensured for all system initial conditions starting outside a set that has zero Lebesgue measure and is nowhere dense. For the general case of higher order nonholonomic systems in chained form, a local convergence result is given. Simulation results comparing the performance of the proposed controller with other existing designs are also provided.

Global synchronization of memristive hybrid neural networks via nonlinear coupling

This paper probes into the synchronization for memristor-based hybrid neural networks via nonlinear coupling. At first, a new condition is established to judge whether quadratic functions are negative or not on a closed interval regardless of their concavity or convexity. Then, by utilizing Legendre orthogonal polynomials, a recent extended integral inequality with free matrices is popularized to get tighter lower bound of some integral terms. Next, based on a novel Lyapunov functional, by applying our new integral inequality with free matrices, linear convex combination method and the new criterion, a new delay-dependent condition is gained to reach the global synchronization for the considered neural networks. At last, an example is presented to account for the validity of our results.

Finite-time dissipative control of uncertain singular T–S fuzzy time-varying delay systems subject to actuator saturation

This paper investigates the dissipative-based finite-time control for uncertain singular T–S fuzzy time-varying delay system affected by actuator saturation. First, the concept of dissipative stability and finite-time bound is presented. Then an appropriate Lyapunov–Krasovskii functional (LKF) is established and for the sake of reducing the conservatism of the results, some free matrices are introduced. Using the convexity property of the matrix inequality, some conditions are given to ensure the fuzzy system is finite-time bounded and dissipative. Moreover, by solving a series of linear matrix inequalities (LMIs), the controllers with the dissipative disturbance weakened level are derived. Finally, simulation examples are presented to show the feasibility and superiority of this method.