On the Numerical Formulation of Parametric Linear Fractional Transformation (LFT) Uncertainty Mo for Multivariate Matrix Polynomial Problems

Robust control system analysis and design is based on an uncertainty description, called a linear fractional transformation (LFT), which separates the uncertain (or varying) part of the system from the nominal system. These models are also useful in the design of gain-scheduled control systems based on Linear Parameter Varying (LPV) methods. Low-order LFT models are difficult to form for problems involving nonlinear parameter variations. This paper presents a numerical computational method for constructing an LFT model from a given LPV model. The method is developed for multivariate polynomial problems, and uses simple matrix computations to obtain an exact low-order LFT representation of the given LPV system without the use of model reduction. Although the method is developed for multivariate polynomial problems, multivariate rational problems can also be solved using this method by reformulating the rational problem into a polynomial form. 1.0 Introduction Formulation of linear fractional transformation (LFF) models of systems involving nonlinear parameter variations is of interest for robust control system analysis and design, as well as for control of linear parameter varying (LPV) systems. Moreover, the LFF models should be of low order for efficient computation during analysis and design. A matrix singular value decomposition (svd) approach was presented in 1985 in references [1] and [2] for computing LFT's for problems involving linear parameter variations. However, construction of low-order LFF models for problems involving nonlinear parameter dependencies is very difficult, because it is equivalent to a multidimensional minimal state-space realization problem for which there is no general theory. The approach that has been taken to date for solving nonlinear parameterdependent problems is to successively decompose the system until all components are linear, and then to compute an LFF for each linear component based on the result presented in [1] and [2]. The LFT's associated with each system component are then combined using LFF properties to form the LFF model of the full system. Model reduction is usually required using this approach, because unnecessary repetitions of the varying parameters usually result. A decomposition method for LFF modeling of nonlinear parameter-dependent systems was first presented in reference [3], and later refined in reference [4]. This latter paper presented a special decomposition approach which reduces the number of unnecessary repetitions of the varying parameters, although model reduction is still employed to reduce the dimension of the resulting LFT model of the full system. The approach presented in this paper is an extension of the computational approach of references [1] and [2] for nonlinear parameter-dependent systems, and is based on reference [5]. Specifically, the computational approach is developed for multivariate matrix polynomial problems, although multivariate rational problems can be solved using this approach by reformulating the rational problem to be in a multivariate polynomial form. Reference [6] presents a method for doing this. The LFT modeling approach presented in this paper requires no matrix decompositions for multivariate polynomial problems, and achieves a low-order LFT model directly i.e., without the use of model reduction. Moreover, the computations are based on simple matrix operations, including the svd and solving linear matrix equations. 2.0 LFT Modeling Problem Definition The LFT modeling problem to be addressed in this paper is defined below. It is assumned that the problem to be solved is in a multivariate matrix polynomial form. However, as shown in reference [6], multivariate rational problems can be reformulated as multivariate polynomial problems and solved using this approach. The problem is stated as follows. Given: A linearparameter varying(LPV)modelof anonlinearparameter-dependent system,as represented by thefollowing equation (2.1a) = [_l,_2,...,_m] ER m (2.1b) where S(iS) has been separated into nominal and varying components, and the varying (or uncertain) component, SA(iS), has been formulated as an LFT problem given by the following equation S A (8) = P21A(I PllA)'I P12 = P21(I APll )'1AP12 (2.2) in which each element of SA(6) is a multivariate polynomial function of the varying parameters, 6 Find: A low-order state-space uncertainty model that satisfies equation (2.2) and is characterized by the constant matrices P21, P12, and P11 and the uncertainty matrix A(iS), as depicted below in Figure 1. WA zA