A new approach for modeling and control of nonlinear systems via norm-bounded linear differential inclusions

Abstract

A systematic approach to model nonlinear systems using norm-bounded linear differential inclusions (NLDIs) is proposed in this paper. The resulting NLDI model is suitable for the application of linear control design techniques and, therefore, it is possible to fulfill certain specifications for the underlying nonlinear system, within an operating region of interest in the state-space, using a linear controller designed for this NLDI model. Hence, a procedure to design a dynamic output feedback controller for the NLDI model is also proposed in this paper. One of the main contributions of the proposed modeling and control approach is the use of the mean-value theorem to represent the nonlinear system by a linear parameter-varying model, which is then mapped into a polytopic linear differential inclusion (PLDI) within the region of interest. To avoid the combinatorial problem that is inherent of polytopic models for medium- and large-sized systems, the PLDI is transformed into an NLDI, and the whole process is carried out ensuring that all trajectories of the underlying nonlinear system are also trajectories of the resulting NLDI within the operating region of interest. Furthermore, it is also possible to choose a particular structure for the NLDI parameters to reduce the conservatism in the representation of the nonlinear system by the NLDI model, and this feature is also one important contribution of this paper. Once the NLDI representation of the nonlinear system is obtained, the paper proposes the application of a linear control design method to this representation. The design is based on quadratic Lyapunov functions and formulated as search problem over a set of bilinear matrix inequalities (BMIs), which is solved using a two-step separation procedure that maps the BMIs into a set of corresponding linear matrix inequalities. Two numerical examples are given to demonstrate the effectiveness of the proposed approach.

Design of a linear quadratic regulator for nonlinear systems modeled via norm-bounded linear differential inclusions

“…is a nonlinear function of class 1 C . By using a two-step modeling procedure proposed in (Kuiava, Ramos and Pota, 2010) this nonlinear system can be represented by an NLDI model written in the form of the following linear system:…”

“…This paper adopts an approximation method formulated as an LMI optimization problem (Boyd et al, 1994;Kuiava, Ramos and Pota, 2010) that calculates matrices F and G in such a way that the obtained NLDI overbounds the following PLDI:…”

“…Boyd et al, 1994;Xie, Fu & de Souza, 1992). This paper adopts the two-step modeling procedure proposed in (Kuiava, Ramos and Pota, 2010) to describe the nonlinear system as an NLDI model. Basically, the first step of the modeling procedure consists of using the mean value theorem to describe the nonlinear system via a linear parametervarying (LPV) system.…”

“…So, for some applications the use of descriptions of nonlinear systems via NLDI models may be preferable, once that its quadratic stability conditions involve only two LMI constraints. In this case, the second step of the modeling procedure proposed in (Kuiava, Ramos and Pota, 2010) consists of an extension of a method proposed in (Boyd et al, 1994) that provides an efficient outer approximation (or overbounding) of a PLDI by an NLDI, which means that every trajectory of the former is also a trajectory of the latter. The major benefit of this extension method in comparison to the standard one (Boyd et al, 1994) is to allow us to choose a particular structure for the NLDI parameters in order to reduce the conservatism in the representation of the underlying nonlinear system by the NLDI model.…”

“…The main contribution of this work is to extend (Kuiava, Ramos and Pota, 2010) to the LQR framework and to provide a design procedure of LQR controller for nonlinear systems modeled via NLDIs. The adopted controller is a dynamic output feedback controller, once this type of control is widely used in many practical problems (see the examples in Ramos et al, 2004;de Oliveira et al, 2009;Hossain et al, 2009).…”