Ying-Jen Chen

A New Sum-of-Squares Design Framework for Robust Control of Polynomial Fuzzy Systems With Uncertainties

This paper presents a new sum-of-squares (SOS, for brevity) design framework for robust control of polynomial fuzzy systems with uncertainties. Two kinds of robust stabilization conditions are derived in terms of SOS. One is global SOS robust stabilization conditions that guarantee the global and asymptotical stability of polynomial fuzzy control systems. The other is semiglobal SOS robust stabilization conditions. The latter is available for very complicated systems that are difficult to guarantee the global and asymptotical stability of polynomial fuzzy control systems. The main feature of all the SOS robust stabilization conditions derived in this paper are to be expressed as nonconvex formulations with respect to polynomial Lyapunov function parameters and polynomial feedback gains. Since a typical transformation from nonconvex SOS design conditions to convex SOS design conditions often results in some conservative issues, the new design framework presented in this paper gives key ideas to avoid the conservative issues. The first key idea is that we directly solve nonconvex SOS design conditions without applying the typical transformation. The second key idea is that we bring a so-called copositivity concept. These ideas provide some advantages in addition to relaxations. To solve our SOS robust stabilization conditions efficiently, we introduce a gradient algorithm formulated as a minimizing optimization problem of the upper bound of the time derivative of an SOS polynomial that can be regarded as a candidate of polynomial Lyapunov functions. Three design examples are provided to illustrate the validity and applicability of the proposed design framework. The examples demonstrate advantages of our new SOS design framework for the existing linear matrix inequality approaches and the existing convex SOS approach.

Stability Analysis and Region-of-Attraction Estimation Using Piecewise Polynomial Lyapunov Functions: Polynomial Fuzzy Model Approach

This paper proposes sum-of-squares (SOS) methodologies for stability analysis and region-of-attraction (ROA) estimation for nonlinear systems represented by polynomial fuzzy models via piecewise polynomial Lyapunov functions (PPLFs). At first, two SOS-based global stability criteria are proposed by applying maximum-type and minimum-type PPLFs, respectively. It is known that less-conservative results can be obtained by reducing global stability to local stability, since it is usually the case for nonlinear systems that the stability cannot be reached globally. Therefore, based on the two types of PPLFs, two local stability criteria are further proposed with the algorithms that enlarge the estimated ROA as much as possible. The constraints for checking (global and local) stability and enlarging the estimated ROA are represented in terms of bilinear SOS problems. Hence, the path-following method is applied to solve the bilinear SOS problems in the proposed methodologies. Finally, some examples are provided to illustrate the utility of the proposed methodologies.

An SOS-Based Control Lyapunov Function Design for Polynomial Fuzzy Control of Nonlinear Systems

This paper deals with a sum-of-squares (SOS)-based control Lyapunov function (CLF) design for polynomial fuzzy control of nonlinear systems. The design starts with exactly replacing (smooth) nonlinear systems dynamics with polynomial fuzzy models, which are known as universal approximators. Next, global stabilization conditions represented in terms of SOS are provided in the framework of the CLF design, i.e., a stabilizing controller with nonparallel distributed compensation form is explicitly designed by applying Sontags control law, once a CLF for a given nonlinear system is constructed. Furthermore, semiglobal stabilization conditions on operation domains are derived in the same fashion as in the global stabilization conditions. Both global and semiglobal stabilization problems are formulated as SOS optimization problems, which reduce to numerical feasibility problems. Five design examples are given to show the effectiveness of our proposed approach over the existing linear matrix inequality and SOS approaches.

Fuzzy Control Strategy for a Hexapod Robot Walking on an Incline

In this paper, the fuzzy control strategy for a hexapod robot walking on an incline is proposed. In order to maintain the vertical projection of the center of gravity (COG) remaining in the support pattern, the robot’s posture is adjusted by a fuzzy controller depending on the slope of incline. At first, Denavit–Hartenberg convention is applied to calculate the positions of motors and end points of legs. When the robot is walking on an incline, a rotation matrix, which can be acquired by an inertial measurement unit settled on the center of robot’s body, is required to obtain the vertical projection of COG. Then, the fuzzy controller is designed to adjust the angles of motors for supporting legs such that the vertical projection of COG approaches the COG of support polygon. Finally, several experiments are implemented by a hexapod robot to demonstrate the effectiveness of the proposed fuzzy control strategy.

Nonconvex stabilization criterion for polynomial fuzzy systems

This paper presents a nonconvex criterion represented in terms of bilinear sum of squares (SOS) conditions for the stabilization of polynomial fuzzy systems. In the existing literature dealing with the stabilization of polynomial fuzzy control systems, the construction of Lyapunov function candidates is restricted in order to transform the nonconvex SOS conditions into convex SOS conditions. Moreover, the transformation from nonconvex conditions into convex conditions itself also induces some conservativeness for the stabilization of polynomial fuzzy control systems. This study proposes a stabilization criterion for polynomial fuzzy control systems remaining in the nonconvex form of bilinear SOS conditions. Therefore, the proposed nonconvex SOS stabilization criterion, in some way, can provide more relaxed results than the existing convex SOS stabilization criteria. Moreover, the restriction on the construction of Lyapunov function candidates in literature does not exist in the proposed nonconvex stabilization criterion. An iterative algorithm is utilized to obtain solutions of the nonconvex stabilization criterion represented in terms of bilinear SOS conditions. The iterative algorithm does not guarantee either to obtain the global optimum of the nonconvex criterion or even to converge. However, this algorithm is easy to implement and can yield feasible solutions efficiently in the design example of this work.

Stabilization analysis of single-input polynomial fuzzy systems using control Lyapunov functions

This paper presents a novel method for stabilization analysis of the single-input polynomial fuzzy systems using a sum of squares (SOS) approach to construct control Lyapunov functions (CLFs) for the systems. First, we represent a nonlinear system as a polynomial fuzzy system. Then, sufficient conditions in SOS terms for a positive definite function to become a CLF is presented. We solve the sufficient conditions using SOS optimization technique to construct the CLF. Finally, after a CLF is constructed, the controller for the system is designed using Sontags formula. To illustrate the validity of the proposed approach, a design example is provided.

Polynomial fuzzy control design for synchronizing multi-scroll Chen chaotic systems

This paper proposes a polynomial fuzzy control design strategy for synchronizing multi-scroll Chen chaotic systems. At first, the master and slave multi-scroll Chen chaotic systems are transformed into the equivalent polynomial fuzzy models. Then a polynomial fuzzy control is designed to synchronize the master and slave polynomial fuzzy models. In the control design, the polynomial feedback gains are obtained by solving sum-of-squares (SOS) constraints. Finally, the simulation results show the validity of the proposed polynomial fuzzy control design.

A Simple Passive Attitude Stabilizer for Palm-Size Aerial Vehicles

This paper presents a simple passive attitude stabilizer (PAS) for vision-based stabilization of palm-size aerial vehicles. First, a mathematical dynamic model of a palm-size aerial vehicle with the proposed PAS is constructed. Stability analysis for the dynamics is carried out in terms of Lyapunov stability theory. The analysis results show that the proposed stabilizer guarantees passive stabilizing behavior, i.e., passive attitude recovering, of the aerial vehicle for small perturbations from a stability theory point of view. Experimental results demonstrate the utility of the proposed PAS for the aerial vehicle.

Stability region analysis for polynomial fuzzy systems by polynomial Lyapunov functions

This paper presents a sum-of-squares (SOS) based methodology to obtain inner bounds on the region-of-attraction (ROA) for nonlinear systems represented by polynomial fuzzy systems. The methodology searches a polynomial Lyapunov function to guarantee the local stability and the invariant subset of the ROA is presented as the level set of the polynomial Lyapunov function. At first the methodology checks whether the considered system can be guaranteed to be locally asymptotically stable. After confirming that the system is guaranteed to be locally asymptotically stable, the methodology enlarges the invariant subset of the ROA as much as possible. The constraints for both of checking stability and enlarging contractively invariant set are represented in terms of bilinear SOS optimization problems. The path-following method is applied to solve the bilinear SOS optimization problems in the methodology.

Piecewise polynomial lyapunov functions based stability analysis for polynomial fuzzy systems

This paper proposes two stability criteria for polynomial fuzzy systems by applying minimum-type and maximum-type piecewise polynomial Lyapunov functions (PPLFs) respectively. Piecewise Lyapunov functions and polynomial Lyapunov functions (PLFs) have been utilized to the stability analysis for fuzzy-model-based (FMB) control systems to obtain relaxed results in literature. However, the minimum-type and maximum-type PPLFs have not been employed to analyze the stability of FMB control systems. Therefore, this paper applies the minimum-type and maximum-type PPLFs to the stability analysis of polynomial FMB control systems. Two relaxed stability criteria represented in terms of bilinear sum-of-squares (SOS) conditions are proposed. The proposed stability criteria are represented in terms of bilinear SOS conditions that cannot be directly solved by the mathematical tools of solving SOS optimization problem (e.g. SOSTOOLS and SOSOPT). Therefore, the path-following method that has been shown to be effective for the nonconvex bilinear matrix inequality problem is employed for solving the bilinear SOS problem of the proposed stability criteria. A numerical example is provided to demonstrate the relaxation of the proposed stability criteria.