Observer Design for Takagi-Sugeno Lipschitz Systems Affected by Disturbances using Quadratic Boundedness

Abstract

In this paper, a proportional observer design using quadratic boundedness is proposed in order to estimate the state of a system described by a Takagi-Sugeno model with a Lipschitz nonlinearity term, and affected by unknown disturbances. The conditions for ensuring that the error between the real and the estimated state converge within an ellipsoidal region about zero, are provided in the form of a linear matrix inequality (LMI) formulation. Then, the simulation results of this approach applied to a four-wheeled omni-directional mobile robot will be shown.

Observer‐based model reference control of Takagi–Sugeno–Lipschitz systems affected by disturbances using quadratic boundedness

“…In order to achieve this goal, both a reference model and an estimated‐feedback control law must be added to the control system. Due to the Lipschitz nonlinearity, the integration between the different components of the control system is not trivial, so the contributions of this paper are the following: Shows that the overall augmented system is also TSL, so that the analysis results developed in [26] can still be applied, although with some tweaks. Derivation of QB design conditions for the observer and the estimate‐feedback controller gains. In both cases, the design concerns a bilinear matrix inequalities (BMIs) feasibility problem due to the product of different decision variables. In the observer case, by prefixing some scalar decision variables, linear matrix inequalities (LMIs) [27] are obtained, which are much easier to solve using available solvers. Converting the controller design BMIs into LMIs is a more complex procedure, which involves constraining the corresponding Lyapunov function to have a specific structure, so that some conservativeness is introduced, albeit with the advantage of reducing the computational complexity. The overall design is illustrated by means of a numerical example, which demonstrates the effectiveness of the developed control strategy. …”

“…The work [26] considered the problem of designing a proportional observer using QB to estimate the state of a TS system with Lipschitz nonlinearity, in the following referred to as Takagi-Sugeno-Lipschitz (TSL) system, affected by bounded disturbances. It was shown that the QB approach allowed to tune the observer so as to increase the achieved performance (smaller estimation error when compared with a non-QB observer).…”

“…It was shown that the QB approach allowed to tune the observer so as to increase the achieved performance (smaller estimation error when compared with a non-QB observer). The present paper is built over the results previously obtained in [26] by considering that the estimated state is used to feed the controller in order to achieve QB of the tracking error between the real system and some desired reference trajectory. In order to achieve this goal, both a reference model and an estimated-feedback control law must be added to the control system.…”

“…• Shows that the overall augmented system is also TSL, so that the analysis results developed in [26] can still be applied, although with some tweaks. • Derivation of QB design conditions for the observer and the estimate-feedback controller gains.…”