Wiener structure based adaptive control for dynamic processes with approximate monotonic nonlinearities
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 modelingwhere v(t) denotes an intermediate variable, and 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 can be used to describe approximate monotonic nonlinearities of processes. In the meantime,
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 as
From Eq. (25), the nonlinear regression of is givenwhere
Tracking performance and stability analysis
Definition 1 Let and be two discrete scalar or vector signals, . We denote , if there exist positive constants , and such that , .
Definition 2 The discrete Nussbaum gain is proposed and defined aswhere is a discrete sequence with , and is the sign function.
Then, is defined in a following manner.The determination of the sign function at the next moment Begin 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 belowwhere and . After Eq. (53), the linear dynamic part can be written aswhere 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).
References (55)
- et al.
A reduced order soft sensor approach and its application to a continuous digester
J. Process Contr.
(2011) - et al.
“Assumed inherent sensor” inversion based ANN dynamic soft-sensing method and its application in erythromycin fermentation process
Comput. Chem. Eng.
(2006) - et al.
Monitoring the process of curing of epoxy/graphite fiber composites with a recurrent neural network as a soft sensor
Eng. Appl. Artif. Intel.
(1998) - et al.
A decoupled multiple model approach for soft sensors design
Control Eng. Pract.
(2011) - et al.
A novel identification method for Wiener systems with the limited information
Math. Comput. Model.
(2013) - et al.
Generalised Hammerstein-Wiener system estimation and a benchmark application
Control Eng. Pract.
(2012) - et al.
Modeling for soft sensor systems and parameters updating online
J. Process Contr.
(2014) - et al.
Modeling study of nonlinear dynamic soft sensors and robust parameter identification using swarm intelligent optimization CS-NLJ
J. Process Contr.
(2017) - et al.
On-line control of the heat exchanger network under fouling constraints
Energy
(2019) - et al.
H∞ non-fragile observer-based dynamic event-triggered sliding mode control for nonlinear networked systems with sensor saturation and dead-zone input
ISA Trans.
(2019)
Self-tuning regulator for a tractor with varying speed and hitch forces
Comput. Electron. Agric.
Design of industrial computer control system in grease production
Procedia Comput. Sci.
Adaptive neural network control for strict-feedback nonlinear systems using backstepping design
Automatica
Auxiliary model based recursive generalized least squares parameter estimation for Hammerstein OEAR systems
Math. Comput. Model.
Performance analysis of the recursive parameter estimation algorithms for multivariable Box-Jenkins systems
J. Franklin Inst.
Recursive least squares parameter identification for systems with colored noise using the filtering technique and the auxiliary model
Digit. Signal Process.
Auxiliary model-based multi-innovation least squares identification for multivariable OE-like systems with scarce measurements
J. Process Contr.
Nonlinear adaptive control using neural networks and multiple models
Automatica
Output feedback adaptive control of a class of nonlinear discrete-time systems with unknown control directions
Automatica
A self-tuning control method for Wiener nonlinear systems and its application to process control problems
Chin. J. Chem. Eng.
A novel parameter separation based identification algorithm for Hammerstein systems
Appl. Math. Lett.
Recursive parameter and state estimation for an input nonlinear state space system using the hierarchical identification principle
Signal Process.
States based iterative parameter estimation for a state space model with multi-state delays using decomposition
Signal Process.
Kalman state filtering based least squares iterative parameter estimation for observer canonical state space systems using decomposition
J. Comput. Appl. Math.
Development of a novel soft sensor using a local model network with an adaptive subtractive clustering approach
Ind. Eng. Chem. Res.
Developing dynamic soft sensors using multiple neural networks
Chin. J. Chem. Eng.
Discussion about dynamic soft-sensing modeling
Chin. J. Chem. Eng.
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