On-board System Identification of Systems with Unknown Input Nonlinearity and System Parameters

Input nonlinearities, or actuator nonlinearities, can be seen in a lot of systems and have significant effects on the system performance. From the controller design point of view, accurate yet simple model of input nonlinearities is essential to compensate their effects and to achieve high level control performance. Unfortunately, most input nonlinearities are neither known nor easy to characterize, especially when the input nonlinearities and unknown system parameters are present simultaneously in the system dynamics. Off-board calibration may be possible yet it is very time consuming and requires additional calibration systems. This paper focuses on a class of systems with unknown input nonlinearities and system parameters and proposes an on-board system identification process to model the unknown nonlinearities. The input nonlinearities are decomposed into localized orthogonal basis and then estimated together with the system parameters. The proposed method is applied to model the nonlinear flow mapping of cartridge valves. Simulation and experimental results are obtained to illustrate the effectiveness and practicality of the proposed method. INTRODUCTION Input nonlinearities can be seen in a lot of systems, such as electrical motion control systems with nonlinear amplifiers, valve controlled electro-hydraulic systems, room temperature ∗Address all correspondence to this author. 1 control systems, chemical and biological processes, etc., and have significant effects on the system performance [1]. Some simple input/actuator nonlinearities, such as deadband, have been well addressed in many researches [1, 2]. When the input nonlinearities are known, they can be compensated by properly designed controllers, as the work done in [3, 4]. When the input nonlinearities are not exactly known but their structures are known, adaptive control technique can be applied to solve the problem, such as [1, 5]. In order to achieve precision control performance, the system control law must take into account the input nonlinearity and compensate it instead of neglecting or linearizing the input nonlinearity. Therefore it is essential to have an accurate yet simple model of the input nonlinearity for controller design purpose. Most input nonlinearities are due to imperfect manufacturing processes or input devices’ internal structures. There does not exist a common model or even common model structure for a same type of input devices. In a word, the input nonlinearity in mass produced systems is usually non-repeatable and has to be identified one by one. Though off-board calibration may be a solution, in addition to the need of a calibration system that increases cost, it is a very time consuming task, which is not suitable for industrial mass production. This paper focuses on a class of nonlinear systems described in (1). ẋ = φT (x) ·θ+ f (u,x) (1) Copyright c © 2005 by ASME where x and u are the system state and input signal, respectively; φ(x) = [φ1(x), φ2(x), ..., φr(x)] is a set of known regressors, and θ = [θ1, θ2, ..., θr] is a set of unknown system parameters; f (u,x) is the unknown input nonlinearity, which is a function of the input signal u as well as the system state x. An automated on-board identification algorithm to model the input nonlinearity, which utilize only the available system information, such as system output or state, is proposed in the paper. Difficulties of on-board identification include: 1) Available information is very limited. Even though the system has full state feedback, there is usually no measurement of the value of the input nonlinearity f (u,x) and the derivatives of the system states ẋ. 2) The system dynamics is also subject to parametric uncertainties. The presence of unknown system parameters makes the system identification even harder. 3) Experimental condition is very difficult to satisfied due to the inability to arbitrarily set system states. In this paper, the unknown input nonlinearity and other unknown system parameters will be determined simultaneously from the system dynamics via certain intelligent integration of nonlinear function decomposition and on-line parameter estimation algorithms. Experimental conditions for accurate parameter estimations are carefully examined to obtain practical on-board tests that can run for accurate modelling of the input nonlinearity. The proposed technique is applied to model the nonlinear flow mapping of cartridge valves. Simulation and experimental results are presented to illustrate the effectiveness of the proposed method. The objective of the proposed technique is to build a model of the input nonlinearity for the controller design purpose. The model obtained by the proposed technique is not necessary to be a ’perfect’ model for analysis purpose. As long as the model is accurate enough in the important working range and improves control performance significantly, the objective is achieved. PROBLEM FORMULATION The following assumptions are made in this paper: Assumption 1. The system has full state feedback, hence u and x are known or measurable. However, the value of f (u,x) and the derivative of the system state, i.e., ẋ, are not measurable. Assumption 2. f (u,x) is bounded though the value of f (u,x) is not measurable. Assumption 3. The system is controlled by a well designed controller and stable, therefore both x and u have bounded value, which implies that the unknown input nonlinearity f (u,x) is practically defined on a compact set on u− x surface. 2 Assumption 4. The system parameters θ are unknown but bounded with known bounds. The regressor φ(x) is known. There are many ways to approximate an unknown function, such as the neural network [6], Fourier decomposition [7], Wavelet decomposition [8] and so on. The basic idea of all approximation methods is to decompose an unknown function into a set of basis functions with their own weighting factors, as shown in (2). f (u,x) = f̄ (u,x)+∆, f̄ = φf (u,x) ·θ f (2) where f̄ (u,x) is the approximation of f (u,x) and ∆ represents the approximation or modelling error, φf = [φ f 1, φ f 2, ...,φ f n] is a finite dimension vector of basis functions, and θf = [θ f 1, θ f 2, ...,θ f n] is a vector of unknown parameters or weighting factors. However, straightforward application of the above approximation seldom leads to satisfactory approximation due to the following practical limitation. Namely, to have a reasonably small approximation error ∆, a huge number of basis functions (i.e., large n) have to be used. This is especially true for the input nonlinearities having non-smooth components such as discontinuous frictions or deadband. Consequently, to obtain the approximation of the input nonlinearity f̄ (x,u), huge number of parameters θ f have to be adapted or estimated simultaneously from the limited experimental data sets, which makes it impossible to use the well-known parameter estimation algorithms having better converging properties due to the difficulty of running on-board experiments to satisfy the experimental conditions needed by those algorithms. For example, the least square estimation (LSE) algorithm needs the persistent excitation condition for parameter convergence, which, loosely speaking, can be satisfied only if rich enough data sets covering the entire working ranges of the control input u and the system state x are available. However, for on-board identification where the input component cannot be removed from the actual system, x is not an experimental variable that can be freely controlled and there is usually no way to conduct experiments that cover the entire range of u− x. Another problem is the Gibbs phenomena, i.e., when a discontinuous function is approximated by a finite number of basis functions, such as Fourier decomposition, severe oscillations exist around the discontinuous points. If the input u and state x do not span the entire region, the nonlinearities that are not adequately exercised can not be observed in the experimental data, hence it is impossible to estimate the nonlinearities accurately in the unaccessible area. However, from controller design point of view, the unaccessible area are trivial because if a well designed experiment can not access the Copyright c © 2005 by ASME area the closed loop system is unlikely to operate in the area. Instead of a global model for the unknown nonlinearity, only local properties on the area where the system operates are essential to improve control performance and therefore the main concern of the paper. The paper uses localized basis functions to bypass the problems associated with the global basis functions. Substituting (2) into (1), one can obtain: ẋ = φT (x) ·θ+φf (u,x) ·θ f +∆ (3) Defining φnew = [φ(x)T , φ f (u,x)T ] and θnew = [θT , θf ], (3) can be written in a compact form as y = φnew ·θnew +∆ (4) (4) is in the standard linear regression form with respect to the unknown parameters θnew with y = ẋ being the model output and ∆ as the model error. Thus, the original problem of automated on-board modelling of input nonlinearities in the presence of unknown system parameters is transformed into the tractable problem of accurate parameter estimation based on the linear regression model (4). The rest of the paper thus focuses on the selection of suitable basis functions φ f (u,x), the design of experiments, and the use of the least square algorithm (LSE) to minimize the effect of model error ∆ for accurate parameter estimation. IDENTIFICATION OF INPUT NONLINEARITIES Based on Assumption 3, the input nonlinearity is practically defined on a compact support on u−x surface. It is reasonable to cut the support on u−x surface into small blocks, where uNu and xNx represent the maximal values of u and x respectively. Each block is named after the indices of u and x, e.g. Ii j, as shown in Fig. 1. The distances between ui and ui+1 or x j and x j+1 do not have to be equally spaced. As the function approximation will be done on each small block instead of on the entire region, a priori knowledge about the input nonlinearity can be used to help choose the spacing to have a reasonable good model approximation accuracy while minimizing the number of blocks needed. For example, if it is known that the deadband may happen around some input value though the exact value is not known, relatively small spacing should be used in the region near that input value. On the other hand, at the region where one knows the input nonlinearity may not change drastically, relatively larger spacings can be used to reduce the computation load. On each small block Ii j, it is assumed that the value of the input nonlinearity does not change drastically and thus the input nonlinearity can be approximated as a Taylor series with respect