6. Nash game based mixed H2/H∞ MPC is a linear controller design technique whereas the Laguerre-Wavelet Network is inherently a nonlinear modeling tech- nique. Hence it is desired in the present thesis to suggest a suitable linearising technique that would retain the local information of LWN model at the region of operation and also would serve the linearity requirement of MPC design technique.
7. It is also aimed to apply the above techniques for a benchmark chemical en- gineering process viz., continuous bioreactor. Bioreactor is inherently a non- linear process with input multiplicity nonlinearity i.e., different inputs to the process may give the same output. This nonlinear gain of the process cannot be properly approximated by a simple linear model and cannot be efficiently controlled by a linear controller as well. Thus the bioreactor example chosen in the present thesis posses enough control challenge in terms of nonlinearity.
The change in the process gain also has a serious effect on the closed loop sta- bility when regular PID type controllers are used to regulate the process. Thus bioreactor process which shows change in its gain over its operating region is a potential problem with enough control challenges is chosen to demonstrate the developments made in the present thesis both on system identification and control.
8. Finally, it is also desired to explore and present a cautionary note on the fragility of output feedbackH2/H∞ MPC design for the closed loop sensitivity to controller’s design parameter(s) that may lead to instability.
1.6 Outline of the Thesis
This thesis is conceptually divided into two parts viz., system identification and robust model predictive control design. A brief summary of each of the following chapters is furnished below.
1.6. Outline of the Thesis 29
Chapter 2 deals with the identification of a class of nonlinear systems using a new species of Wiener type model, namely Laguerre-Wavelet Network (LWN) model. In this model, the linear dynamic part of the Wiener structure is formed by orthogonal Laguerre filters and the static nonlinear portion is formed by wavelet network structure. The efficacy of the proposed model is demonstrated with two case studies: (i) simulation of continuous bioreactor - which exhibits input-multiplicity nonlinearity, and (ii) real time operation of a lab scale pasteurization process.
A newer kind of model predictive control is proposed in Chapter 3 which is based on the combination of the celebrated control methods, namely, mixedH2/H∞control and Nash game theory. From the problem formulation to the numerical approach of solving the desired control problem, the design of a robust controller has been dealt in detail in this chapter. The performance of the proposed robust MPC design is compared with the mixed H2/H∞ based MPC design developed in [121].
In Chapter 4, a complete and thorough theoretical analysis of the closed-loop stability and robustness of the proposed Nash game based H2/H∞ MPC is given.
The concepts of set theory have been used to provide a theoretical analysis of the robustness of this newer kind of feedback model predictive control. For the sake of completeness, the analysis is also carried out for the mixed H2/H∞ MPC design given in [121]. Some simulation results are also furnished in support of the theoretical claims made in this chapter.
Chapter 5 deals with the extension of the mixed H2/H∞ MPC controller for the output feedback case. In order to make use of the LWN model furnished in chapter 2 for control application, a suitable linearising technique is advocated that would retain the local information of LWN model at the region of operation and also would serve the linearity requirement of controller design. The linearised LWN model is used with the output feedback mixed H2/H∞ MPC controller design and has been demonstrated for a benchmark chemical engineering process viz., continuous bioreactor.
Finally, based on the works furnished in the chapters 2 through 5, some con- cluding remarks are drawn in Chapter 6. Also, based on experience and knowledge
1.6. Outline of the Thesis 30
gained by the author while working for this thesis, few recommendations for future research directions are provided.
Chapter 2
Wiener type Laguerre-Wavelet Network model
2.1 Introduction
There has been tremendous progress in the field of process model development (also called System Identification) over the years, though they were mainly limited to linear models. However, nonlinear system identification enjoyed much attention in the research community in the recent years [19, 42, 134]. There are many or- thonormal basis filters used in system identification like Finite Impulse Response (FIR) model, Laguerre filter, Kautz filter and Generalized Orthonormal Basis Filter (GOBF) [123, 134, 159]. With these orthonormal basis filters it is possible to obtain efficient mathematical model of linear systems even with time delays, with some a priori information about the process dynamics. Among the recent model develop- ments, the models developed by [134], namely Laguerre polynomial and Laguerre ANN, are found to be interesting. These models are developed in Wiener model structure, wherein the polynomial and ANN components form the static nonlinear part of the Wiener model and Laguerre filters form the linear dynamic part. Thus, the models do not require any external feedback for representing dynamic systems.
2.1. Introduction 32
Both polynomials and ANN provide static global nonlinear approximations. How- ever, they have limitations for local approximations. Such local approximations are highly required for representing processes with severe nonlinearities. Wavelets are found to provide such local approximations parsimoniously to an appreciable level of accuracy.
Wavelets, which basically find applications in the field of signal processing, are getting attention into the system modeling in the recent years [19, 42] for their property of multi-scale decomposition of signals. Representation of real time signals at multiple scales serves as a very efficient and useful way to analyze them. This can be achieved by expressing the data as a weighted sum of orthonormal basis functions, which are defined in both time and frequency, such as wavelets. The weights become the wavelet coefficients. Wavelets are a computationally efficient family of multi-scale basis functions. A signal can be represented at multiple resolutions by decomposing the signal on a family of wavelets and scaling functions.
The same properties of wavelets that are exploited in signal processing such as orthogonality and multi-scale resolution, make the wavelets a suitable system approximation tool too. The feature extraction abilities of multi-scale representation of data are utilized to construct multi-scale nonlinear models that are less affected by the presence of noise in the data. The main idea is to decompose the input-output data at multiple scales and construct a nonlinear model using them. It is in this sense wavelets are being used in the present work, exploiting the structural property of discrete-time wavelet transform, which is called Wavelet Network.
Wavelet network, a blend of idea of neural networks and wavelets, is reported by [166] to give a good static nonlinear mapping. In the present work, utilising the wavelets’ static nonlinear mapping into the Wiener model proposed by [134]
a newer kind of nonlinear model is developed, namely, Laguerre-Wavelet Network model for representing a class of nonlinear systems. In the proposed model the static nonlinear part, basically formed in terms of wavelets, is expected to give an efficient local and global nonlinear approximation parsimoniously. Thereby, the model can give a better performance than that of Saha et al [134].