State-space realization from time domain data of flight control systems

Abstract

During the past decades, modern control theory has become an effective analysis and design tool for various applications, especially in aerospace engineering. Many user-friendly software packages have become available. Probably due to the fact that the time-domain approach is more popular than the frequency domain approach among control engineers, software users are often required to input state space model data {A, B, C, D) to initiate their tasks. However, state-space data is not always available to software users. More specifically, the data which is made available to engineers might be in the form of time-series (e.g., unit step responses, impulse responses, and frequency responses). The objective of this paper is to present a technique which enables one to generate reduced-order state-space model data (A, B, C, D) from time-domain data (impulse and unit step response). Computational examples of flight control systems will be presented herein to demonstrate the merit of this proposed technique. Generation of State Space Models from the Impulse and Unlt Step Responses To identify a partially known system, test signals such as a unit step function or approximate impulsive function are frequently used to generate system responses. Based on these system responses, (i.e., unit step and impulse responses) a state space model (A, B. C, D) can be generated. This section is concerned with a approach for generating (A, B, C, D) from either an impulse or unit step response. The realization approach adopted here has a built-in capability for providing reduced order models. The impulse response h(t) specifies a linear SISO system which has the following state-space representation where the state vector x (t) is an n x 1 vector and the system matrix A is n x n. The dimension n is to be determined. From linear system theory (Chen, 19841, it is well known that any impulse response h(t) can be expressed in terms of A, b and namely where the state-transition matrix e*t denotes the exponential of the matrix At. In the sequel, focus will be placed on the problem of finding A, b and c from a known impulse response h(t). A numerical procedure will be developed along with computational results. A set of sampled data (time series) h ( k ~ ) is first obtained by sampling the impulse response h(t), where T is the sampling period. Recalling that h(t) = c eAt b then