Continuous-Time Bilinear System Identification

The objective of this paper is to describe a new method for identification of a continuous-time multiinput and multi-output bilinear system. The approach is to make judicious use of the linear-model properties of the bilinear system when subjected to a constant input. Two steps are required in the identification process. The first step is to use a set of pulse responses resulting from a constant input of one sample period to identify the state matrix, the output matrix, and the direct transmission matrix. The second step is to use another set of pulse responses with the same constant input over multiple sample periods to identify the input matrix and the coefficient matrices associated with the coupling terms between the state and the inputs. Numerical examples are given to illustrate the concept and the computational algorithm for the identification method. Introduction System identification is a methodology used to characterize a dynamical or other engineering system with measurements of the input-output signals. Mathematicians and engineers have developed a number of approaches to address the identification problem. The identification of a linear timeinvariant system is relatively vel1 understood and theoretically well developed [1,2]. This is not true for the identification of a nonlinear system, although some progress has been made in the identification of nonlinear systems over the past few decades [3-191. There is a class of nonlinear systems called bilinear systems whose dynamics are jointly linear in the state a d the force variables. It is a simple nonlinear extension of a linear system. The concept of bilinear systems was introduced in the 1960’s (see the surveys of Refs. [5] and [6]). References [7] and [SI provide a survey of bilinear-related systemtheory methods and their contributions to problems such as stabilization, controllability, and obsembility. Bilinear systems have been studied extensively and applied successfully to several problems [ 151. Recently, research activities in identification of bilinear systems have been focused on the so-called “discrete-time” model identification p 91. The discrete-time model is an approximation obtained by linearizing the continuous one with a method such as the finite difference. In contrast, we focus on the identification of a continuous-time bilinear system without any approximation. A new method is introduced in this paper for identification of a continuous-time multkinput and multkoutput bilinear system. When the input of a bilinear system is a constant, the bilinear system becomes a linear system. This special characteristic is the basis for the identification method. Two steps are required for the identification process. The first step begins with generating a set of pulse responses with a constant input applied OE at a time over one sample period. The pulse responses are then used to form a Hankel matrix consisting of system Markov parameters to identify the state matrix, the output matrix, and the direct transmission matrix. The identification step is quite similar, if not identical, to the identification of a linear system [1,2]. This step establishes a specific set of coordinates for the whole identification process. This set of coordinates is not unique, depending mainly on the size of Hankel matrix and the resulting choice of matrix that represents the observability matrix. The second step starts by generating another set of pulse responses with the same constant input as the frst step but for multiple sample periods. These multiple-pulse responses are used to define another set of Markov parameters to form a Hankellike matrix for each input. The observability matrix obtained in the first step is then applied to the Hankel-like matrix to compute the corresponding controllability matrix of the input to identify the input vector and the coefficient matrix associated with the coupling terms between the state and the input. Simple examples are given to demonstrate how to apply the method to identify a continuous-time bilinear system and how to transfer the identified model from one set of