Design of a nonlinear controller based on a piecewise-linear Hammerstein model

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Abstract

This paper presents a nonlinear controller, which can be used to control single-input single-output nonlinear systems that can be approximated in terms of Hammerstein models. The proposed nonlinear controller is based on the principles of a classic linear pole placement controller and the piecewise-linear Hammerstein model. The controller can be used to control processes with highly nonlinear or even discontinuous static functions, while keeping simple controller structure and a very low computational burden.

Introduction

Many industrial processes are characterized by nonlinear, rather than linear, dynamic behaviour. The efficient control of such processes still represents an important motivation for both scientists and engineers in the field of automatic control. The particular challenge nowadays is the transfer of the existing control methods and theoretical ideas from theory to operation under realistic process conditions.

Existing nonlinear control algorithms are typically based on nonlinear process models of different forms. For control purposes nonlinear models with a simple structure that require a low computational effort are preferred. At the same time, the models should be supported with simple and efficient underlying parameter identification algorithms accommodated for “realistic” excitation signals, i.e., signals that are likely to be allowed to be applied to processes. This is extremely important since the development of an appropriate model is a critical step that has a direct impact on the success of any model-based control algorithm.

The problem of efficient nonlinear model identification was addressed in our recent paper [5]. The discussion was focused on the class of nonlinear Hammerstein models, which can be used to represent a wide range of industrial processes. After discussing the main weaknesses of the classic Hammerstein model we proposed a new model, referred to as the piecewise-linear Hammerstein model, which addresses all the above mentioned requirements for efficient implementation. In [5] the properties of the modified model were studied and the identification algorithm was proposed together with the desired properties of the excitation signals.

This paper represents a logical continuation of our work by proposing the utilization of the piecewise-linear Hammerstein model in a control algorithm for single-input single-output processes.

In recent years the Hammerstein model was widely used, especially within predictive control laws, e.g., [1], [6], which gained a lot of popularity, not only in research, but also in industrial implementation. Other kinds of control laws based on the Hammerstein model are also discussed in the literature, e.g., dead-beat control [14], adaptive dead-beat feedforward compensation of measurable disturbances [4], an indirect adaptive controller based on a linear quadratic control and an approximation of the nonlinear static function using neural networks [11], and a nonlinear dynamic compensator for Hammerstein systems with passive nonlinear dynamics [7].

In our paper the principles of linear pole placement controller design [3], [10] were selected as the starting point. This classic principle was then modified in a manner as to be compatible with processes described in terms of the piecewise-linear Hammerstein model. A similar approach can be found in [2], [16], [17], where the authors proposed a nonlinear control structure based on linear pole placement controller design, extended for the classic single polynomial based Hammerstein model or in [13], where a generalized minimum variance controller is accommodated for the single polynomial based Hammerstein model. The main drawback of these control algorithms is the significant computational effort, since an embedded polynomial has to be numerically inverted in each sampling interval of the controller. Similarly, in [8], [9] the identification and the utilization of the Hammerstein model are proposed, where the nonlinear static function is approximated by the Bezier function. Also in this case the resulting pole placement controller requires the inversion of the nonlinear static function to be repeated in each sampling interval. The inversion is in this case performed by the numerical algorithm (the inverse of the de Casteljau algorithm). Although several solutions have been studied to overcome the problem of the inversion of the nonlinear static function (one possibility is, e.g., via internal feedback linearization[15]), the inherent problem remains unsolved.

In our approach the problem of the computational burden of the inversion of the nonlinear static function is completely circumvented, since the controller is based on the piecewise-linear static function, which has an analytical inverse, and thus requires only a minimum computational effort during each sampling interval of the controller. This gives the possibility of implementing the algorithm within the programmable logic controllers (PLCs). There are also other important advantages of the proposed controller, i.e., the possibility for model identification in the presence of a signal with a temporarily bounded amplitude and the possibility to control processes with a highly nonlinear or even a discontinuous static function. These advantages become extremely important when practical applications of the control algorithm are considered.

The paper is organized as follows. In Section 2 the background of the piecewise-linear Hammerstein model is described and its properties are briefly discussed. In Section 3 the linear controller design is summarized. The linear controller is then integrated with the piecewise-linear Hammerstein model. In this section some important facts about the resulting control algorithm are also discussed. The testing of the controller operation by simulation is documented in Section 4.

Section snippets

The piecewise-linear Hammerstein model

In [5] the piecewise-linear Hammerstein model was proposed and its properties were discussed in detail. Here, the approach is briefly summarized.

The proposed model follows the known structure of the classic Hammerstein model, which contains a nonlinear static function with input u and output x, followed by a linear dynamic block with input x and output y. The idea was to use a piecewise-linear representation [12] of the nonlinear static function x(u) instead of the usual polynomial

Controller design

The primary goal of this paper is the utilization of the model, i.e., the development of a control algorithm for processes whose models can be expressed in terms of a piecewise-linear Hammerstein model. For the sake of generality the design of a general linear controller is adopted as the starting point. This approach is then modified to be integrated with the piecewise-linear Hammerstein model.

Simulation results

In this section the operation of the proposed controller is demonstrated by the simulation. The process was simulated by means of a continuous-time classic Hammerstein model with the following nonlinear static function (Fig. 3), which is discontinuous at u=0.6:x(u)=-0.3738u+0.9350u1/2if 0.0u<0.6,8.4343+13.6069uif 0.6u1.0.-21.0412u1/2The linear part of the process was simulated by the following transfer function:GP(s)=y(s)u(s)=(20s+1)(40s+1)(10s+1).To design the controller a set of linear

Conclusion

In this paper a control algorithm for single-input single-output nonlinear processes that can be approximated in terms of a piecewise-linear Hammerstein model is proposed. The piecewise-linear Hammerstein model is an alternative to the classic, i.e., single polynomial based Hammerstein model, offering significant simplification and many advantages. Some of them become obvious when the model is used in model-based control algorithms, as proposed in this paper. It was shown that the proposed

Acknowledgement

The authors are grateful to the Ministry of Education, Science and Sports of the Republic of Slovenia for financial support of this work.

References (17)

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