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

Abstract

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 semi-global 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 non-convex formulations with respect to polynomial Lyapunov function parameters and polynomial feedback gains. Since a typical transformation from non-convex 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 non-convex 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 LMI approaches and the existing convex SOS approach.

Finite-Time Stability and Controller Design of Continuous-Time Polynomial Fuzzy Systems

“…Three illustrative examples have been provided to demonstrate the validity and effectiveness of the proposed results. In the future, we will consider the similar problem for the discrete-time polynomial fuzzy system and the polynomial fuzzy model with time delays or (and) uncertainties [17].…”

Feedback-linearization-based control of discrete-time quadratic TS fuzzy systems with disturbances

“…. (16) Note that the signals needed in controller (16) are all measurable. Therefore, the controller can be implemented.…”

“…Therefore, the controller can be implemented. (14), then the output of fuzzy systems (2) can asymptotically track the step reference signal with zero steadystate error under controller (16)…”

“…Quadratic rule consequent, and more generally polynomial rule consequent, has attracted attention of fuzzy control researchers in the recent years because they offer better modeling capabilities than the linear rule consequent. For instance, system analysis and controller design has been conducted using the Sum-of-Squares (SOS) optimization approach for the TS fuzzy systems with polynomial rule consequent [11][12][13][14][15][16][17][18][19][20]. It is important to point out that various non-feedback-linearization control schemes, including the SOS-based approach, have been developed in the literature to attain output tracking control (but not perfect output tracking control).…”

A review on stability analysis of continuous-time fuzzy-model-based control systems: From membership-function-independent to membership-function-dependent analysis

“…Along the line of PDC design concept, stability conditions in terms of SOS were obtained in [60]. Since then, a lot of research on stability analysis of PFMB control systems have been carried on and variations of SOS-based stability conditions have been obtained for different control problems, just to name a few, such as observer-based control problems [63,64,65,66,67], output-feedback control problems [68], positive control problems [69], regulation control problems [70], sampled-data control problems [71], stabilization control problems [60,72,61,73,74,75,76,77,78,79,80,81,82,83,84,85,86,11,87,88], switching control problems [79], tracking control problems [89] and etc.…”