Wiener structure based adaptive control for dynamic processes with approximate monotonic nonlinearities

https://doi.org/10.1016/j.jfranklin.2020.10.006Get rights and content

Abstract

The modeling of industrial processes requires to consider the complex features of systems, such as nonlinearities, dynamics and uncertainties, etc. In this paper, a simplified Wiener structure (SWS) is proposed for the modeling of dynamic processes with approximate monotonic nonlinearities. The nominal SWS not only considers the dynamic characteristics in processes, but also takes full advantages of the process nonlinear properties. Then, an adaptive control method for the SWS is proposed, in order to achieve exact output tracking of reference signals in the servo control mode. The recursive estimation is implemented before the adaptive control law is ready. In the recursive computation, the ideas of both the discrete Nussbaum gain and the dead-zone factor are introduced. The tracking theorem verifies the stability of adaptive control under the suitable assumption. Finally, two illustrative examples demonstrate the effectiveness of both the SWS and the adaptive control.

Introduction

Since most industrial processes have both nonlinear and dynamic properties, it is necessary to build full dynamic models. Although there are many methods for modeling nonlinear dynamic processes [1], [2], [3], [4], [5], [6], they meet the same problem: the mutual fusion relationship between dynamics and nonlinearities. This leads to inadequate and inaccurate modeling of processes [7,8]. Further, the separated modeling of dynamics and nonlinearities is recommended. That is, the nonlinear block-oriented models are adopted for process modeling. There are three types of block-oriented structures: Hammerstein structure [9], Wiener structure [10], and their combinations [11]. Hammerstein structure, i.e., the input-nonlinear form, can describe systems like power amplifiers [12], excavator arms [13], etc. Wiener structure, i.e., the output-nonlinear form, can describe processes like distillation columns [14], pH processes [15], biological systems [16], etc. Besides, Wiener structure can be used to deal with the dynamic differences [17,18] of transition processes at different time scales. In this paper, a simplified Wiener structure is founded as a general modeling method for describing dynamic processes with approximate monotonic nonlinearities. It should be noted that the approximate monotonic nonlinearities usually exist in process industries. For example, there is a monotonic relationship between fluid velocity and differential pressure. For another example, cold fluid temperature increases monotonically with the increase of hot fluid flow [19]. What's more, the dead-zone or saturation phenomenon [20], which is approximate monotonic, often appears in actuators or sensors.

On the other hand, for industrial nonlinear dynamic processes, the control target is to design a suitable controller, in order to make the controlled variable track the designed reference signals in the servo mode. Facing this situation, the idea of adaptive control can deal with uncertainties both inside and outside of the controlled processes [21], [22], [23], and then enhances the control performances. It means that adaptability makes output responses quickly and timely. Specifically, adaptive closed-loop systems consist of the controlled processes and the controllers. Here, the controlled processes contain actuators, real processes and sensors. Meanwhile, the adaptive controllers usually change all the time, including structures and parameters. Among these controllers, the self-tuning regulators [24] involve the changes of control parameters, and the sliding mode controllers [25], [26], [27] involve the variable structure control. Because of both complexity and diversity of the adaptive control, the related digital control algorithms are usually embedded into computer control systems (CCS) [28].

The adaptive control of continuous-time nonlinear systems has been studied extensively. The backstepping design method is generalized to the nonlinear continuous-time systems, which can be transformed into output feedback forms or parametric strict-feedback forms [29], [30], [31], [32]. The results have also been extended into MIMO systems [33], [34], [35]. In contrast to the above results of continuous systems, their discrete counterparts remain largely unexplored, and the Lyapunov design for stability analysis of discrete models becomes much more intractable.

In seminal works [36,37], the adaptive control schemes of linear discrete-time models have been developed successively. It should be mentioned that, in the previous literature for adaptive control, the signs of control gains are required to be known as a priori knowledge. Without a priori knowledge of control directions, it is difficult to determine the updating direction of recursive parameter estimation [38], [39], [40], [41], [42]. To overcome the theoretical limitation, further in [43,44], the discrete Nussbaum gain is firstly proposed to present a global stable adaptive control with unknown control directions. Later, the discrete Nussbaum gain has been developed successively for the adaptive control of nonlinear discrete systems in the forms of NARMAX (nonlinear autoregressive moving average with exogenous inputs), output-feedback and strict-feedback [45], [46], [47], [48], [49]. Unfortunately, Nussbaum gain has not been applied into the adaptive control of the Wiener-type processes so far.

In the view of above statements, the motivation of the paper contains two aspects: i) It is meaningful to find a general modeling method for dynamic processes with approximate monotonic nonlinearities; ii) For these processes, it is necessary to exploit an adaptive control method to achieve exact output tracking of reference signals in the servo control systems. The innovation also includes three points: i) a simplified Wiener structure is extracted to describe such kind of processes; ii) an adaptive control scheme is exploited to guarantee the stability of control systems; iii) Both the discrete Nussbaum gain and the dead-zone factor [50] are introduced into recursive parameter updating [51], [52], [53], [54], [55].

The rest of the paper is organized as follows. A unified Wiener structure is analyzed for process modeling in Section 2. In Section 3, a nominal simplified Wiener structure is proposed for modeling of the dynamic processes with approximate monotonic nonlinearities. Next, the adaptive control design scheme is exploited in Section 4. Section 5 gives the stability analysis of the proposed control method. Illustrative examples are shown in Section 6. Finally, conclusions are drawn in Section 7.

Section snippets

A unified Wiener structure in process modeling

For lots of univariate nonlinear dynamic processes, a following unified Wiener structure (UWS) is a good choice for process modelingv(t)=i=1ngiu(ti),y(t)=f(v(t)),where v(t) denotes an intermediate variable, and f(·) is a continuous function. The unified modeling based on Eq. (1) has two reasons: (a) the process dynamics can be replaced by adequate input dynamics in v(t); (b) Wiener structure that separates the static part from the dynamic one can describe nonlinear system characteristics

A simplified Wiener structure for modeling of the nonlinear dynamic processes

In this paper, both modeling and adaptive control are considered for the dynamic processes with approximate monotonic nonlinearities.

However, if the UWS in Eq. (1) is used for modeling of these processes, the unknown intermediate variable v(t) makes system identification difficult. Thus, a simplified Wiener structure (SWS) is exploited in this paper.

Assume that a continuous monotonic function fW(·) can be used to describe approximate monotonic nonlinearities of processes. In the meantime, fW(·)

Adaptive control design and parameter estimation

In this section, an adaptive control scheme is designed for the proposed SWS in Eq. (20), and corresponding recursive parameter estimation is exploited to solve the problem of updating direction. From Eq. (20), the following model is obtained asy(t+1)=i=1paij=1qbjgj[y(t+1i)]+c1u(t)+k=2mcku(t+1k)+υ(t+1).

From Eq. (25), the nonlinear regression of y(t+1) is giveny(t+1)=φT(t)·θ+c1u(t)+υ(t+1),whereφjT(t)=[gj(y(t)),gj(y(t1)),...,gj(y(t+1p))],j=1,...,q,θjT=[a1bj,a2bj,...,apbj],j=1,...,q,φuT(

Tracking performance and stability analysis

Definition 1

Let q1(t) and q2(t) be two discrete scalar or vector signals, tN+. We denote q1(t)=O[q2(t)], if there exist positive constants m1, m2 and t0 such that q1(t)m1maxttq2(t)+m2, t>t0.

Definition 2

The discrete Nussbaum gain N(p(t)) is proposed and defined asN(p(t))=ps(t)·sN(p(t)),ps(t)=suptt{p(t)},where p(t) is a discrete sequence with p(0)=0, and sN(p(t)) is the sign function.

Then, sN(p(t)) is defined in a following manner.

The determination of the sign function at the next moment
Begin
 1. sN(p(0))=+1

Numerical examples

Example 1

In order to test the proposed SWS and the proposed adaptive control, an input-nonlinear process is considered, i.e., a Hammerstein process. This process is designed to include both a monotonic quadratic nonlinearity of input and a linear dynamic part. The expression of input-nonlinearity is shown as belowifulowuuupv=ulow+1/(uupulow)·(uulow)2elsev=uendwhere ulow=5 and uup=27. After Eq. (53), the linear dynamic part can be written asy¯(t)+i=16a¯iy¯(ti)=k=16c¯kv(tk),where the linear

Conclusions

Unified modeling is a challenge for nonlinear dynamic processes. In this paper, the main contributions are shown as follows: i) the proposed SWS provides a simple but effective modeling method for dynamic processes with approximate monotonic nonlinearities; ii) the proposed adaptive control method, which contains both the discrete Nussbaum gain and the dead-zone factor, provides a reliable online control scheme to achieve exact output tracking; iii) the stability of proposed adaptive control is

Future recommendation

However, we don't know whether the proposed SWS is appropriate for dynamic processes with non-invertible nonlinearities. In process industries, the generalization ability of Wiener structure should be further discussed. The following three aspects should be considered: i) the transformation ability of complex block-oriented nonlinear systems; ii) the descriptive ability of hysteresis or bifurcation nonlinearities; iii) the descriptive ability of non-invertible or non-monotonic nonlinearities.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work is supported by National Natural Science Foundation of China (No. 61703434).

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