An SOS-based observer design for polynomial fuzzy systems

Abstract

Abstract-This paper presents a sum of squares (SOS, for brevity) based observer design for a more general class of polynomial fuzzy systems with the polynomial matrices Ai(x(t)) and Bi(x(t)) that are permitted to be dependent of the states x(t). First, we briefly summarize previous works on SOS-based observer designs for two limited classes of polynomial fuzzy systems. To overcome the difficulty of the fact that does not realize the so-called separation principle design for the more general class, this paper provides a practical design procedure of a polynomial fuzzy controller and a polynomial fuzzy observer without lack of guaranteeing the stability of the overall control system in addition to converging state estimation error (via the observer) to zero. The design approach discussed in this paper is more general than the existing LMI approaches (to T-S fuzzy controller and observer designs) and also than the previous SOS-based observer designs. To illustrate the validity of the design approach, a design example is provided. The example shows the utility of our SOS approach to the polynomial fuzzy observer-based control for the more general class of polynomial fuzzy systems.

Polynomial observer design for unknown inputs polynomial fuzzy systems: A Sum of Squares approach

“…Now we propose to give more relaxed conditions to design unknown inputs polynomial observers. Hence, a transformation based on the introduction of a slack variable is used in order to separate the term P N i (y) in equality (27) [34]. The following result deduced from theorem 1, gives more relaxed SOS conditions with equality constraints that aims to design a polynomial observer for polynomial fuzzy systems subject to unknown inputs.…”

“…Indeed, [25] examined the case of PFS such that the consequent local model is independent of unmeasurable states, i.e., A i and B i only depend on y, and sufficient design conditions for a polynomial fuzzy observer are given in SOS based SDP program. In [26], [27] and [28], this result is generalized for a wider class of PFS in which one of matrices A i and B i or both are permitted to depend on x. However, all the afore-mentioned works concerned PFS driven only by known inputs.…”

An SOS-Based Observer Design for Discrete-Time Polynomial Fuzzy Systems

“…The paper uses polynomial fuzzy system to model and control the nonlinear systems via an SOS approach. The authors also study the SOS-based polynomial fuzzy observer design for three classes of continuous polynomial fuzzy systems: The polynomial matrices A i and B i are independent of the states x to be estimated (shortly name it as Class I) [15], [18], the polynomial matrices A i are permitted to be dependent of the states x to be estimated (shortly name it as Class II) [16], [18], the polynomial matrices A i and B i are permitted to be dependent of the states x to be estimated (shortly name it as Class III) [17,18]. And some extensive results have been obtained, e.g.…”

“…another paper on relaxation for T-S systems' stability analysis via SOS method is addressed in [19]. The SOS approach [14][15][16][17][18][19] presents that it is an extensive representation of LMIs. Obviously, the problems in them cannot be solved by interior point algorithms, e.g.…”

“…However, for polynomial fuzzy systems, there are a few results. Polynomial fuzzy observer designs for continuous polynomial fuzzy systems were provided in [15][16][17][18] via a sum of squares (SOS) approach. Observer designs for discrete-time polynomial fuzzy systems have not been addressed in the literature.…”