Polynomial Fuzzy Systems Research Articles

Polynomial Dynamic Output-feedback Controllers for Positive Polynomial Fuzzy Systems with Time Delay

Polynomial Dynamic Output-feedback Controllers for Positive Polynomial Fuzzy Systems with Time Delay

Design of output feedback proportional–integral controller for polynomial fuzzy systems

AbstractIn this paper, the design of dynamic output feedback control law for polynomial fuzzy systems is studied. The main purpose is to design a tracker control law for fuzzy polynomial systems that are subject to external bounded disturbances. The structure of the control law is considered as fuzzy proportional–integral control. The conditions for deriving the control law are formulated in the form of a sum of square (SOS) feasibility problem. The designed control law will be able to force the system state vector to follow the state vector of a stable reference model in addition to guaranteeing the performance measure. The simulation results show the efficiency of the proposed method exposed to external disturbance.

Fully distributed consensus for fuzzy multiagent systems with jointly connected topology

AbstractThis paper investigates the problem of fully distributed consensus for polynomial fuzzy multiagent systems (MASs) under jointly connected topologies. First, a polynomial fuzzy modeling method is presented to characterize the error dynamics that is constructed by one leader and multiple followers. Then, using the relative state information and the agents' dynamics, a distributed adaptive protocol is designed to guarantee that MASs under jointly connected topologies can achieve consensus in a fully distributed fashion. Utilizing the Lyapunov technique, a relaxed sufficient criterion is proposed to ensure consensus for fuzzy MASs under jointly connected topologies. Moreover, the adaptive coupling weights between neighboring agents can converge to certain values. The derived condition is transformed into a sum‐of‐squares form, which can be solved numerically. We provide an example to illustrate the proposed distributed adaptive consensus technique's validity.

Membership-Function-Dependent Design of $L_1$-Gain Output-Feedback Controller for Stabilization of Positive Polynomial Fuzzy Systems

This article presents the <inline-formula><tex-math notation="LaTeX">$L_1$</tex-math></inline-formula>-gain polynomial fuzzy output-feedback controller design and the stability analysis using the sum-of-squares (SOS) approach for positive polynomial fuzzy-model-based (PPFMB) control systems. The polynomials, positivity, and optimal <inline-formula><tex-math notation="LaTeX">$L_1$</tex-math></inline-formula> performance make some existing convex methods for general systems inapplicable. To overcome this problem, first, an augmented system of the PPFMB control system is constructed; then, by introducing some constraint conditions and mathematical techniques, nonconvex stability and positivity conditions are skillfully transformed into convex ones simultaneously. In addition, to control the systems flexibly and lower the implementation cost, the imperfect premise matching concept is taken into account for controller design. Besides, the high-degree-polynomial approximation method is adopted to conduct stability and positivity analysis by incorporating the information of membership functions and the boundary information of the state variables. On the basis of the Lyapunov stability theory, the relaxed stability and positivity conditions in terms of the SOS form are obtained. Finally, two simulation examples are presented to verify the feasibility of the theoretical results.

Robust H∞ Fuzzy Observer-Based Fault-Tolerant Tracking Control for Nonlinear Stochastic System: A Sum of Square Approach

In this study, the robust <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> fuzzy observer-based fault-tolerant tracking control strategy is proposed for the stochastic polynomial fuzzy system (SPFS) under the effect of external disturbance, measurement noise, system/sensor fault signals and continuous/discontinuous internal random fluctuations. At first, the smoothed models of fault signals are constructed to describe their dynamic behavior. Then, by integrating the SPFS with the smoothed models of fault signals as one augmented system, the state/fault signal estimation problem can be transformed to a state estimation problem of augmented system by the proposed polynomial fuzzy observer. With the utilization of estimated state/fault signals and reference trajectory, a fuzzy polynomial fault-tolerant tracking controller can be implemented. To attenuate the effect of undesired external disturbance and measurement noise on the state/fault signal estimation and tracking control performance, the robust <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> fuzzy observer-based fault-tolerant tracking control strategy is proposed in this study. By utilizing the homogeneous Lyapunov function and Lipschiz condition, the Itô-Lévy formula is reformulated to relax the compensation terms of stochastic processes during the design. Then, the design conditions are derived in terms of interpolation function-dependent matrix inequality and consequently transformed to a two-step sum of square (SOS) condition design problem for the robust <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> fuzzy observer-based fault-tolerant tracking control strategy. A simulation example of double lane maneuvering task for autonomous ground vehicle (AGV) is provided to illustrate the effectiveness of proposed method. Index terms: Observer-based tracking control, polynomial fuzzy system, stochastic control, fault-tolerant control, sum of square (SOS).

Synthesis of Polynomial Fuzzy Model-Based Designs with Synchronization and Secure Communications for Chaos Systems with H∞ Performance

This paper presents the sum of squares (SOS)-based fuzzy control with H∞ performance for a synchronized chaos system and secure communications. To diminish the influence of the extrinsic perturbation, SOS-based stability criteria of the polynomial fuzzy system are derived by using the polynomial Lyapunov function. The perturbation decreasing achievement is indexed in a H∞ criterion. The submitted SOS-based stability criteria are more relaxed than the existing linear matrix inequality (LMI)-based stability criteria. The cryptography scheme based on an n-shift cipher is combined with synchronization for secure communications. Finally, numerical simulations illustrate the perturbation decay accomplishment of the submitted polynomial fuzzy compensator.

Robust DOF control for uncertain polynomial fuzzy systems in finite frequency domain

This study addresses the issue of robust dynamic output feedback control (DOF) for polynomial Takagi–Sugeno (T–S) fuzzy systems in the Finite Frequency (FF) domain. Sufficient conditions for designing the robust DOF control are derived in terms of the sum of squares (SOS). The proposed strategy is built in the FF domain to reduce conservation-generated by the techniques established in the whole frequency domain. In addition, there are no transformation matrices or equality constraints under these conditions, which simplifies the numerical solution. To show the validity of the suggested technique, several numerical examples are presented.

Relaxed Sum-of-squares Approach to Stabilization of Polynomial Fuzzy Systems

Relaxed Sum-of-squares Approach to Stabilization of Polynomial Fuzzy Systems

Modeling and Control of Electrical Power and Energy Produced by a Synchronous Generator Using Polynomial Fuzzy Systems and Sum-Of-Squares Approach

The synchronous generator, as the main component of power systems, plays a key role in these system’s stability. Therefore, utilizing the most effective control strategy for modeling and control the synchronous generator results in the best outcomes in power systems’ performances. The advantage of using a powerful controller is to have the synchronous generator modeled and controlled as well as its main task i.e., stabilizing power systems. Since the synchronous generator is known as a complicated nonlinear system, modeling and control of it is a difficult task. This paper presents a sum of squares (SOS) approach to modeling and control the synchronous generator using polynomial fuzzy systems. This method as an efficacious control strategy has numerous superiorities to the well-known T-S fuzzy controller, due to the control framework is a polynomial fuzzy model, which is more general and effectual than the well-known T-S fuzzy model. In this case, a polynomial Lyapunov function is used for analyzing the stability of the polynomial fuzzy system. Then, the number of rules in a polynomial fuzzy model is less than in a T-S fuzzy model. Besides, derived stability conditions are represented in terms of the SOS approach, which can be numerically solved via the recently developed SOSTOOLS. This approach avoids the difficulty of solving LMI (Linear Matrix Inequality). The Effectiveness of the proposed control strategy is verified by using the third-part Matlab toolbox, SOSTOOLS.

A Novel Path-Following-Method-Based Polynomial Fuzzy Control Design.

This article presents a novel path-following-method-based polynomial fuzzy control design. By examining the stabilization problem, the nonconvex stabilization criterion represented in terms of bilinear sum-of-squares (SOS) constraints is proposed to complement the existing convex stabilization criteria. Based on the polynomial Lyapunov function and considering the operation domain, the stabilization control is designed with a systematic region of attraction (ROA) analysis method. Since the proposed stabilization criterion remains in nonconvex form, the conservativeness caused by the transformation from nonconvex (bilinear SOS) constraints into convex (SOS) constraints can be avoided. Moreover, the restriction on the Lyapunov function candidates for the convex transformation in the literature does not exist in the proposed nonconvex stabilization criterion. The stabilization analysis for polynomial fuzzy control systems is concerned with the double fuzzy summation problem that can be treated as the copositivity problem. Therefore, the SOS-based copositive relaxation technique is applied for the proposed stabilization criterion. Since the proposed nonconvex stabilization criterion is represented in terms of bilinear SOS constraints, the path-following method is employed for solving the bilinear SOS problem. Finally, design examples are provided to demonstrate that the proposed nonconvex stabilization criterion complements the existing convex stabilization criteria.

Relaxed Stabilization and Disturbance Attenuation Control Synthesis Conditions for Polynomial Fuzzy Systems.

This article presents both stabilization and disturbance attenuation control synthesis conditions for a class of polynomial fuzzy systems. The sum of squares conditions is relaxed by restricting the domain of the polynomial variables that represent the membership functions as well as by using the line integral polynomial fuzzy Lyapunov function. For the disturbance attenuation problem, we consider the transformation of the Hamilton-Jacobi-Isaacs equation into a set of inequalities and a policy iteration algorithm to approximate its value function. Furthermore, we propose an alternative path following to find the initial L2 gain stabilizing control policy. Five examples are provided to demonstrate the effectiveness of the proposed conditions.

More relaxed stability analysis and positivity analysis for positive polynomial fuzzy systems via membership functions dependent method

In this paper, the conservatism source of the positivity and stability analysis results of positive polynomial fuzzy-model-based (PPFMB) control system are studied. Also, in order to improve the flexibility of controller design, a fuzzy controller that does not depend on the membership functions of the fuzzy model is designed. In the existing literatures, it is proved that the LCLF can reduce the conservatism of stability results. However, the LCLF generally results in non-convex conditions which is still a conservatism source. To handle the non-convex conditions, the sector nonlinear concept is applied to handle non-convex terms in stability conditions, and the obstacles caused by mismatched membership functions can be eliminated by PLMF dependent method. In addition, to relax the conservatism caused by the lack of membership functions information, the PLMF dependent positivity analysis are performed for the first time. Meanwhile, PLMF dependent method is extended to stability conditions to obtained more relaxed conditions. Finally, a simulation example is presented to verify the feasibility of this method.

Admissibility and Admissibilization of Singular Polynomial Fuzzy Systems with Time-Varying Delay

This paper addresses the admissibility analysis and the polynomial fuzzy controller design for a class of singular polynomial fuzzy systems with time-varying delay. Firstly, according to Lyapunov stability theory, a sufficient condition is obtained by means of sum-of-squares (SOS) to ensure that the closed-loop singular polynomial fuzzy system is admissible. Secondly, by the help of the two equivalent sets, SOS-based admissibility conditions are proposed to avoid solving the equality conditions and the polynomial fuzzy controller is designed. Finally, three examples validate the usefulness and superiority of the proposed design procedure compared with existing results.

Path-Following-Based Design for Guaranteed Cost Control of Polynomial Fuzzy Systems

This paper presents a path-following-based design for guaranteed cost control of a class of nonlinear systems represented by polynomial fuzzy systems. First, this paper proposes a polynomial Lyapunov function approach to guaranteed cost control for the feedback system consisting of a polynomial fuzzy system and a polynomial fuzzy controller. In particular, we introduce a new type of polynomial fuzzy controllers based on an approximate solution for the Hamilton–Jacobi–Bellman inequality. To design a guaranteed cost polynomial fuzzy controller effectively, a path-following-based design algorithm is newly developed by formulating as a sum-of-squares (SOS) stabilization problem. Two new relaxations are provided by bringing a peculiar benefit of the SOS framework. One is an $${\mathcal{S}}$$ -procedure relaxation for the considered outmost Lyapunov function level set that is contractively invariant set. The other is an $${\mathcal{S}}$$ -procedure relaxation for design conditions obtained for polynomial membership functions redefined by variable replacements in considered ranges. Furthermore, this paper provides a practical and reasonable way for estimating lower upper-bounds of a given performance function by increasing the order of a considered polynomial function. Finally, a complicated nonlinear system design example is employed to illustrate the validity of the proposed design algorithm and the lower upper-bound estimation.

Stability Analysis and Estimation of Domain of Attraction for Positive Polynomial Fuzzy Systems With Input Saturation

In this paper, the stability and positivity of positive polynomial fuzzy model based (PPFMB) control system are investigated, in which the positive polynomial fuzzy model and positive polynomial fuzzy controller are allowed to have different premise membership functions from each other. These mismatched premise membership functions can increase the flexibility of controller design; however, it will lead to the conservative results when the stability is analyzed based on the Lyapunov stability theory. To relax the positivity/stability conditions, the improved Taylor-series-membership-functions-dependent (ITSMFD) method is introduced by introducing the sample points information of Taylor-series approximate membership functions, local error information and boundary information of substate space of premise variables into the stability/positivity conditions. Meanwhile, the ITSMFD method is extended to the PPFMB control system with input saturation to relax the estimation of domain of attraction. Finally, simulation examples are presented to verify the feasibility of this method.

Interval Observers Design for Polynomial Fuzzy Singular Systems by Utilizing Sum-of-Squares Program

This paper designs interval observers for polynomial fuzzy singular systems, in which the external disturbances and unknown parameters are included in the systems. The polynomial fuzzy modelings are proposed, which are more practical than Takagi–Sugeno fuzzy modelings. Then, the interval observers are designed for nonnegative and generic systems. The stability conditions are proposed by sum-of-squares (SOS) approach via SOSTOOLS. Compared to the linear matrix inequality-based approaches, the SOS approach is more relaxed and general. Estimation accuracy is analyzed via ${L_{1}/L_{2}}$ framework. The efficiency of the interval observers are demonstrated through two simulation examples.

Observer-based passive control for polynomial fuzzy singular systems with time-delay via sliding mode control

This paper is concerned with the problem of sliding mode observer design and observer-based sliding mode controller synthesis for polynomial fuzzy singular systems with time-delay. The polynomial fuzzy model is constructed for singular nonlinear system to eliminate the upper and lower bounds restriction of states in the system and reduce the calculation quantity. An integral-type sliding surface function, which has a stabilizing effect on the sliding variable, is proposed for the observer system. In terms of sum-of-squares approach, observer-based sliding mode passive controller is designed for the system with unknown membership functions, which drives the system trajectories onto the predefined sliding surface in a finite time, meanwhile, the stability and robust passivity of the closed-loop system are guaranteed. The information of membership functions is included in the stability conditions, which can reduce the conservatism. Finally, an example is performed to verify the merits of the method proposed.

A Polynomial Membership Function Approach for Stability Analysis of Fuzzy Systems

For the stability analysis of a polynomial fuzzy system, a new polynomial membership function approach is proposed to reduce conservatism. In this article, based on a state-feedback closed-loop system, a polynomials fitting method is utilized, and an improved membership function transformation technique is proposed to approximate the membership functions of the fuzzy system. Then, the membership-function-dependent polynomial-based stability conditions are derived. The obtained polynomial membership functions and approximation errors will be involved in the stability analysis process. Based on the sum-of-squares optimization technique, polynomial conditions can be directly solved. Finally, by several numerical and practical examples, conservatism reduction effects are shown by comparisons with existing methods.

Observer‐based tracking controller design for a class of polynomial fuzzy systems with disturbance

AbstractIn this paper, the tracking control law design problem for a class of polynomial fuzzy systems subjected to external bounded disturbance is studied. The tracking control design problem is solved using the observer‐based tracking controller design method in which the observer estimates the controlled system state vector and also the system states will follow the state vector of a stable reference model subject to an H∞ performance. The process of calculating controller parameters is formulated as an SOS feasibility program. Simulation results are presented to show the merits of the proposed control design approach.

Output Feedback and Stability Analysis of Positive Polynomial Fuzzy Systems

This article investigates the polynomial fuzzy output feedback (PFOF) control synthesis problem for positive polynomial fuzzy systems. When the states of a positive system cannot be fully obtained, the output feedback control strategy is a good method to stabilize the control system. From fuzzy control point of view, we employ the polynomial fuzzy model rather than the Takagi–Sugeno fuzzy model, and this kind of models can express a wider range of nonlinear positive systems. However, the polynomials in the system matrices and the feedback control gain matrices will make the nonconvex stability conditions more difficult to deal with. Thereby, in order to crack the hard nut, a nonzero transformation vector is introduced in this article to deal with the nonconvex problem skillfully. Furthermore, the imperfect premise matching technique is taken into account so that the implement of the controller is more simple and money-saving. In addition, the augmented dynamic of the positive polynomial fuzzy-model-based (PPFMB) control system is investigated to facilitate the stability and positivity analysis, and the basic conditions in the light of sum of squares (SOSs) are derived. Besides, the advanced membership function dependent (MFD) technique is used so that a great of useful information of MFs is extracted to improve the relaxation of the results. Finally, the effectiveness of the theoretical findings is illustrated by a simulation example.