Control of nonlinear processes using functional expansion models

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Abstract

Functional expansion (FEx) models are a subclass of the general block-oriented model structure for nonlinear process systems. Controller design in this context uses the internal model control (IMC) paradigm, and one can show that the resulting controllers are easily implementable. The primary advantage arises from the fact that inverting the nonlinear dynamic operator is avoided by taking advantage of the partitioned model inverse due to the special structure of FEx models. The robust stability and performance of the closed-loop system can be analyzed by expressing the FEx model as a linear uncertain system and using the structured singular value framework. We present a case study of a polymerization reactor and, for this SISO system, analyze nominal and robust stability and performance conditions as a function of the closed-loop filter constants for a given range of the input variable.

Introduction

The performance benefits of nonlinear control are generally well established in the literature (Allgöwer & Doyle, 1997, Bequette, 1991, Henson & Seborg, 1997). Demands on product quality, safe operation, and stringent environmental constraints provide motivation to explore nonlinear controller designs, seeking this improved process performance (Ogunnaike & Wright, 1997). Nonlinear designs can provide a wider range of process operation, superior setpoint and disturbance rejection performance, and, in theory, a more stable closed-loop process. While the former two properties are a direct result of a more accurate process model, the stability properties of nonlinear control systems remain difficult to quantify.

Despite the presence of successful nonlinear design methodologies, often the analysis associated with such designs are formidable. The stability and performance analyses often yield conservative designs, are difficult to apply in practice, or do not provide guarantees of performance and stability properties. Consequently, there is a need for attractive stability and performance analyses to address these issues. Ideally, these stability frameworks would provide insight into the tuning and the stability and performance levels of the nonlinear closed-loop process.

For LTI systems, the solution to the stability problem is well known, providing global stability and performance conditions. Most commonly, the tools of robust control are utilized (Morari & Zafiriou, 1989, Skogestad & Postlethwaite, 1996). In this case, the uncertain plant is parameterized by a set of perturbations. These perturbations are commonly of two forms:

  • 1

    Structured perturbations resulting from real parameter variations. These uncertainties result from uncertain parameters in the system, such as heat transfer coefficients or reaction rate constants.

  • 2

    Unstructured perturbations due to unknown modeling errors. These uncertainties can account for unmodeled dynamics, such as using a low order model for a higher order process.

These perturbations are then used to parameterize a family of plants, to which the real plant belongs. The structured singular value (SSV) calculation is used to guarantee robust stability or performance for the family of plants. Consequently, stability or performance is guaranteed for any plant in this family. The SSV is a generalized form of the singular value, and accounts for particular perturbation directions in the standard singular value computation. The general concepts of this theory are provided in several references in a comprehensive fashion (Skogestad & Postlethwaite, 1996, Packard & Doyle, 1993).

While the theory of robust control has been successful for linear systems, very rarely can global stability results be achieved for nonlinear systems. Consequently the focus is typically on local conditions. Several methods have been developed to address the local stability problems for nonlinear control systems based on the block-oriented models and input/output (I/O) operators. A local form of the small-gain theorem has been derived that guarantees stability of a control law using inverse Volterra systems (Zheng & Zafiriou, 1999). The small-gain theorem is based on requiring the product of the gains of the elements in the closed-loop to be less than unity. If this condition is met, then stability of the system in guaranteed as all signals in the loop will be attenuated. The local form of the small gain theorem is used, since the gain of nonlinear operators can vary over the range of operation. However, since small-gain arguments only take into account magnitude, and not directionality of the elements in the loop, the results are often overly conservative. Indeed, by the addition of higher-order Volterra terms, it is possible to constrain the system to an arbitrarily small region of operation in which stability can be guaranteed.

Additionally, contraction mapping arguments have been applied to assess the local stability of nonlinear control laws (Economou, 1985, Sistu & Bequette, 1996). The contraction mapping principle ‘tracks’ a particular norm of the closed-loop operator over time. If this norm can be shown to be always decreasing over time, then stability of the system is achieved. The difficulty in application of these concepts lies in the selection of an appropriate operator norm. Furthermore, this norm is a function of the operating point of the system. Previously, the maximum singular value has been used for this norm, although the results are conservative. Additionally, the spectral radius (maximum absolute eigenvalue) can be used to estimate the value of this norm, however, this does not result in a guarantee of stability.

Lyapunov's methods have also been used to estimate a region of local stability. O'Shea (1964) extended Zubov's method to address the stability of nonlinear difference equations. This allows the use of numerically computed Lyapunov surfaces to estimate the region of stability for model-based controllers Sistu and Bequette (1996). These surfaces are contours of constant ‘energy’ of the system. By determining the regions in the state-space which this energy is decreasing, stability can be estimated. However, mapping these surfaces in the state-space requires a significant number of numerical simulations, which may not be feasible for large systems.

The stability of polynomial nonlinear, auto-regressive, moving average, with exogenous inputs (NARMAX) models and their inverse is analyzed by Hernández and Arkun (1996). This approach was based on expressing the nonlinear model and its inverse as uncertain linear systems. The uncertainty description bounded the nonlinearities using sector conditions. From this structure, linear robust control tools were utilized to determine a local region of stability of the model and its inverse. A similar approach was previously used in a robustness analysis of a nonlinear CSTR (Doyle, Packard & Morari, 1989), while subsequent applications of these concepts have been applied in the stability analysis of a nonlinear model predictive control law (Maner & Doyle, 1997).

In this work, the approach of Hernández and Arkun (1996) is expanded to address the stability of control systems based on functional expansion (FEx) models. The FEx control framework has been demonstrated as a flexible model-based design, applicable to nonminimum-phase and multivariable systems Harris & Palazoğlu, 1997a, Harris & Palazoğlu, 1997b. The present work focuses on determining local conditions for the nominal stability (NS), robust stability, and robust performance (RP) problems. The next section will define the stability problems of interest, followed by a brief review of the FEx controller design procedure.

Section snippets

Definition of the stability problems

In this section, the stability problems to be addressed will be defined. We will deal with causal operators representing a particular dynamic I/O behavior. For example, in this operator framework, a model of the process will be denoted simply as y=Y[u]. This operator then maps inputs u, taken from an input space U, to outputs y in an output space Y. Throughout the development, the following will be assumed:

  • Stability is defined as I/O stability. In other words, every bounded input in an input

Controller design with FEx models

FEx models are a subclass of the general block-oriented model structure (Pearson, 1999). These general block models consist of static nonlinear and LTI elements and their use in model based control has been previously demonstrated (Harris & Palazoğlu, 1998). Other types of block-oriented models are Hammerstein and Wiener models, as well as the Volterra model (Henson & Seborg, 1997, Pearson, 1999). To develop the FEx model in this spirit, consider the following state-space system:ẋ=Ax+bu+ηx(x,u)

Stability of FEx systems

The work of Hernández and Arkun (1996) addresses the stability of NARMAX models and their inverse. The idea is to express nonlinear elements in the system as linear elements with the nonlinearities represented as uncertainties. The resulting uncertain linear system can then be analyzed using the methods of robust control theory. A similar development will be followed in this work, expanding these results to address the MS, NS, RS and RP problems for FEx systems. The presentation will begin by

Case study: control of a polymerization reactor

We shall focus on the control of a polymerization reaction, particularly the polymerization of methyl methacrylate in a jacketed CSTR. The CSTR is operated isothermally and assumed to be perfectly mixed. This process has been studied previously (Congalidis, Richards & Ray, 1989; Doyle et al., 1995). Here we will address the stability of a particular discrete FEx control law, whose design and details appear elsewhere (Harris & Palazoğlu, 1998).

The model is represented by the following state

Conclusions

Practical stability conditions are essential in the use of advanced nonlinear control schemes. This work presented a method to address the robust stability and performance of FEx control structures. By expressing nonlinearities in the closed-loop as linear systems with an associated uncertainty, the stability problem was converted to a linear stability problem. Robust control theory then provided conditions of local stability, that is, regions of operation where robust stability and performance

Acknowledgements

The support of the National Science Foundation is gratefully acknowledged (CTS-9800073). We also thank the anonymous reviewers for their valuable comments.

References (36)

  • J.C. Doyle et al.

    Feedback control theory

    (1992)
  • J.C. Doyle

    Analysis of feedback systems with structured uncertainties

    IEE Proceedings

    (1982)
  • Economou, C. G. (1985). An operator theory approach to nonlinear controller design. Ph.D. dissertation. Pasadena, CA:...
  • C.G. Economou et al.

    Internal model control: 5. Extension to nonlinear systems

    Industrial and Engineering and Chemical Process in Design and Development

    (1986)
  • Harris, K. R. (1998). Control and analysis of nonlinear chemical processes using functional expansions. Ph.D....
  • Harris, K. R., & Palazoğlu, A. (1997). Control of MIMO nonlinear systems via functional expansions. Proceedings of...
  • Harris, K.R., & Palazoğlu, A. (1997b). Model-based control of nonlinear processes via functional expansions....
  • Harris, K. R., & Palazoğlu, A. (1998). Studies on the dynamics of nonlinear processes via functional expansions: III....
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