Non-linear model approximation and reduction by new input-state Hammerstein block structure

https://doi.org/10.1016/j.compchemeng.2011.01.037Get rights and content

Abstract

In this paper, the focus will be on approximating nonlinear large scale mathematical model of process systems using full order block-structured model. Further, the objective is to achieve a reduced order model for the nonlinear large model with reduced computational complexity, while at the same time being the good approximation of nonlinear model. The modeling approach used for this purpose is block structure models. Input–output Hammerstein structure referred to the classical Hammerstein model has been extended to new Hammerstein structure making use of states and inputs, hence called as input-state (IS) Hammerstein structure. In this paper it is shown that expansion of Taylor series leads to IS-Hammerstein structure. The accuracy of the approximation is improved by including higher order (second order) approximation. The input-state Hammerstein structure provides opportunities for model reduction in context of reducing the computational load by order reduction of states and Jacobians. IS-full order Hammerstein model has been implemented on a case study from the process industry namely the high purity distillation column. Within the operational domain of a process, the IS-Hammerstein structure provides a reduced order mode that can be used for online application purposes (i.e., optimization, model predictive control, etc.).

Introduction

Typically, first order principle models (rigorous models) are stiff and large, thus are computationally inefficient. Modeling of complex systems based on first principles often results in large scale dynamic models that typically comprise nonlinear (NL) coupled differential algebraic equations. First principle models (rigorous models) are stiff and large, thus are computationally inefficient. Since the (rigorous) NL models are not always exact match of real processes and there is mismatch at some point between the two, reduced models can be very useful if they match the rigorous NL model over a certain operation window. Advantage of reduced mathematical models for NL processes include low computational effort, better approximation of process within the operating window and beneficial for the real-time applications (e.g., control and optimization purposes). Furthermore, these models cannot usually be employed directly for design and implementation of estimation and control schemes as developed in systems and control theory. The approximation of NL systems to be controlled by simpler, yet accurate mathematical models is common practice in process control as has been discussed by Follinger (1982), Loffler and Marquardt (1991) and Deutscher (2005). The common approach to control system design for NL systems is to use Jacobian linearization of the plant about an operating point in order to apply linear control theory. However, in many applications the resulting performance of the closed loop system is not satisfactory, since the linear approximation model does not take the nonlinear character of the plant into account. The solution to this problem is the application of nonlinear control design strategies such as feedback linearization yielding a nonlinear controller (e.g., Isidori, 1995). A nonlinear controller in this study (Isidori, 1995) for the closed loop system achieves the desired properties on a wider range of operating system. But in general, for many process systems, the mathematical models for nonlinear controllers are complex and inefficient in the context of computational load and hence the simulation time. Hence, in order to facilitate the design of nonlinear controllers, it is reasonable to use a simplified model of the nonlinear plant or process. Definitely it is required that reduced order model reproduces and approximates the behavior of the nonlinear system to be controlled in an operational domain (in the vicinity of the operating point).

A viable approach to NL model simplification is to derive a reduced order model that approximates the significant behavior of the full order rigorous model. Generally, model reduction is associated with a reduction in the number of the differential equations that describe the process behavior with acceptable accuracy. Similarly modification of the online algebraic computations to offline mappings and calculations can be regarded as a major model simplification step. The advantage of reduced mathematical models for NL processes are low computational effort, better approximation of process within the operating window and facilitation of real-time applications (e.g., control and optimization purposes).

The rigorous models available for large scale industrial processes can be characterized as a set of differential and algebraic equations (DAE). DAE class of models is capable to express the majority of processes. Thus the methodology to achieve a reduced model of a nonlinear process should be capable of handling DAE models. The transformation from DAE to ordinary differential equation (ODE) format is regarded as a major model reduction step since the algebraic computations are then eliminated. A methodology which involves this step is advantageous for the process models of DAE class.

There is not a lot of literature available on model reduction in the context of computational load. Balasubramhanya and Doyle (2000) developed a reduced order model of batch distillation column using traveling waves. The closed loop simulation of this reduced model was six times faster than the original model with high accuracy. Chimowitz and Lee (1985) reported increase of computational efficiency of a factor of order three by the use of local thermodynamic models. According to Chimowitz, up to 90% of computational time was used for thermodynamic computations during simulation, motivating their approach of model approximation. The local thermodynamic models are integrated with the numerical solver where an updating mechanism of the parameters of the local models was included. This approach is not easy to use since model reduction and numerical algorithm are integrated. Ledent and Heyen (1994) attempted to use local models within dynamic simulations but were not to the acceptable mark largely due to discontinuities introduced by updating the local models. Kumar and Daoutidis (1999) applied a nonlinear input–output linearizing feedback controller to a high purity distillation column that was non-robust using the original model, but had better robustness properties using a singularly perturbed reduced model. No details were presented on the effect of the model reduction on the computational load.

Model reduction by projection has also been a widely used approach, specifically for chemical processes. In most papers, projection based model reduction is applied to ordinary differential equations except in Loffler and Marquardt (1991) where the projection method is applied to DAE equations to achieve a low order model. Aling et al. (1997) used POD to get reduced model for the rapid thermal processing system. The number of differential equations was reduced from one hundred and ten to less than forty. Reduction of computational load for simulation was up to a factor of ten. Hahn and Edgar (2000) elaborated on model reduction by balancing empirical Gramians and showed model order reduction, but reduction in computational effort and time was limited.

Perregaard (1993) simplified and reduced chemical processes models for simulation and optimization purposes. He simplified the calculation of algebraic equations, which resulted in computational effort reduction. Gani, Perregaard, and Johansen (1990) distinguished between differential, explicit algebraic and implicit algebraic equations. The key observation is that for Newton-like methods, the Jacobian can be approximated during intermediate iterations. They replaced the true Jacobian by a an approximate that is cheap to compute. This approximate Jacobian information is derived from local thermodynamic models with analytical derivatives. They present in their paper several cases and report reductions of overall computational times of the order 20–60% without loss of accuracy and no side affects on the convergence of the numerical method. Empirical modeling has been one of the major approaches for achieving low computational complexity (which allows fast simulations). Ling and Rivera (1998) used a Hammerstein structure for model reduction, but did not report reduction in computational load (on polymerization benchmark). Berg (2005) reported that, if computational load has to be reduced, not only model order reduction is to be targeted but also the complexity (and stiffness) of reduced model has to be lower; as Gani et al. (1990) achieved the computational load reduction by reducing the complexity. Additionally, the block structure models have been identified for the chemical processes and systems by Eskinat, Johnson, and Luyben (1991), Billings and Fakhouri (1977), Norquay, Palazoglu, and Romangnoli (1999), Harnischmacher and Marquardt (2007), but the block structures have not been developed for the model reduction purposes, which is the objective for our research.

The important observation of this literature overview is that there is hardly any reduction technique available for the reduction of computational load or simulation time (for online applications). All techniques have different focuses, and the effect on computational load can only be evaluated by implementation. Not every model reduction methodology works for every process, but it is desired to have a model reduction methodology which is generic and applicable to a wide class of processes (represented by DAE class of models). Moreover the literature review shows the lack of research material on this subject (to recent time); whatever is available, mostly addresses the model order reduction rather than the reduction in complexity of reduced model (which is a major source of computational load). Not many model reduction methodologies have addressed the problem of simplification of complexity in reduced model. The field is open for research to achieve reduced models, which are not only simple and reduced ordered but also accurate and also leading to computational load reduction.

The block structure models (also known as model-based schemes) have been used for the control of nonlinear models Zhu and Seborg (1994), Eskinat et al. (1991) and Wigren (1993). These models have an advantage over other model approximation techniques and reduction methodologies: they exploit the nonlinear, complex and large dynamic models and this gives insight and understanding to the complexity of the process. This provides opportunities to feel for the complexity of the nonlinear process (or rigorous model) and consequently proceed to achieve the reduced order simplified model. The use of block structure, hence enhances the chances to get a reduced (simplified) model that is uncomplicated and is computationally efficient (e.g., Chang & Luus, 1971; Eskinat et al., 1991; Haist, Chang, & Luus, 1973; Pearson, 1995; Pottmann & Pearson, 1998; Sadegh, Melgaard, Madsen, & Holst, 1994).

In this paper, a block structure scheme (Hammerstein) has been used to achieve full order and reduced order models for nonlinear chemical processes. In the subsequent section, the theoretical background and the derivation of IS-Hammerstein structure are discussed. In Section 2.4, the new reduction methodology is developed. In Section 3, implementation on high purity distillation column and its results are considered. The last section concludes the paper with key points.

Section snippets

Block structure models

Empirical models, such as block schemes generally have low computational complexity because of straightforward structure and simple computations, hence facilitating the fast simulations. Model-based structures, that can exploit nonlinear dynamic models for the simplified model for online application purposes (e.g., control, optimization, etc.) are becoming increasingly popular (e.g., Doyle, Ogunnaike, & Pearson, 1995; Meadows & Rawlings, 1997; Pajunen, 1984; Zhu & Seborg, 1994). Significantly,

Implementation of IS-Hammerstein model

This section presents a process example to illustrate both the applicability and mechanics of the full order as well as reduced order IS-Hammerstein block structure models. The case studies considered are CSTR and a high purity distillation column (based on a first-principles model of Lévine & Rouchon, 1991).

Conclusions

In this paper, a new methodology has been developed for the nonlinear approximation of nonlinear process systems. The approximation model developed is based on block structures and it is shown that the new model structure (known as input-state Hammerstein structure) is derived from a Taylor expansion. The prediction accuracy of the model structure is improved by including higher order terms of Taylor series and hence approximating nonlinearly at steady-state (equilibrium) and dynamic

Notations

    A

    state matrix

    C

    output matrix

    g(u)

    nonlinear mapping

    H(z)

    scalar, linear, dynamic model

    J

    Jacobian matrix

    Jb

    Jacobian basis

    L( · )

    scalar, linear, dynamic model

    N( · )

    vector-valued, nonlinear, static model

    n

    number of measurements

    Rai

    air intake

    Rrc

    catalyst recycle

    x

    process state, x(t) in continuous time, xk in discrete time

    x

    steady-state vector

    x˙=dx/dt

    differentiation of state

    Tra

    temperature of the reactor

    t

    time

    u

    input, u(t) in continuous time, uk in discrete time

    U1

    transformation matrix

    v

    intermediate variable of a

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