Model Reduction Techniques

In the case of model reduction for linear elastic, actively controlled structures, a comprehensive survey is given by Craig and Su [6]. It is often advantageous to transform such systems into modal space before reduction, control design, simulation and analysis are carried out. Reduction is then performed by selection of natural modes which 

Numerical analysis and simulation of adaptronic systems can be performed in the time or in the frequency domain depending on the representation of the system in the state space or as a matrix of transfer functions. In addition to performance criteria, important goals are stability and robustness of an adaptronic system. In the case of adaptronic structures, performance criteria are often given in terms of allowable static and dynamic errors relating to structural shape if subjected to specified disturbances. Many applications also involve limits in energy consmnption and actuator stroke or force, which must be checked in time-history simulations. A comprehensive introduction on the different aspects and their interaction can be found in [14]. Current research in the field is for instance presented in [15] and [16]. 

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In order to solve the first principles model, finite difference method or finite element method can be used but the number of states increases exponentially when these methods are used to solve the problem. Lee et u/.[8] used the model reduction technique to reslove the size problem. However, the information on the concentration distribution is scarce and the physical meaning of the reduced state is hard to be interpreted. Therefore, we intend to construct the input/output data mapping. Because the conventional linear identification method cannot be applied to a hybrid SMB process, we construct the artificial continuous input/output mapping by keeping the discrete inputs such as the switching time constant. The averaged concentrations of rich component in raffinate and extract are selected as the output variables while the flow rate ratios in sections 2 and 3 are selected as the input variables. Since these output variables are directly correlated with the product purities, the control of product purities is also accomplished. 

Model Reduction Techniques for Dynamic Optimization of Chemical Plants Operation... 

The application of model reduction techniques in the context of dynamic optimization of chemical plants operation is investigated. The focus is on the derivation and use of reduced models for the design and implementation of optimal dynamic operation in large-scale chemical plants. The recommended procedure is to apply the model reduction to individual units or groups of units, followed by the coupling of these reduced models, to obtain the reduced model of the plant. The procedure is flexible and accurate and leads to a major reduction of the simulation time. 

As seen in the previous chapter, all the approaches used to solve the dynamie optimization problem integrate, at some point, the dynamical system of the ehemieal proeess. In order to obtain more effieiently the values of the optimum profile of the control variable, a suitable model of the system should be developed. That means that the complexity of the model should be limited, but, in the same time, the model should represent the plant behaviour with good accuracy. The best way to obtain such a model is by using the model reduction techniques. However, the use of a classical model reduction approach is not always able to lead to a solution [6]. And very often, the physical structure of the problem is destroyed. Thus, the procedure has to be performed taking into account the process knowledge (units, components, species etc.). 

Unit Model reduction technique Full nonlinear model Reduced model... 

In this paper we present a Model Reduction technique incorporated with multi-parametric programming and control, namely Balanced Truncation (T3T). The use of Balanced Truncation eliminates a number of states of dynamic linear systems, while a bound on the maximum error obtained for the output vector can be established. This then allows for the derivation of (approximate) linear parametric controllers, which can be tested and validated (against the original high-fidelity model) off-line. These theoretical developments are presented next. 

Balanced tnmcation is one model reduction technique, which is particularly suitable in the context of state-space dynamic models, linear Model Predictive Control and Multi-parametric controller design, as discussed in the following. 

A novel gradient-based optimisation framework for large-scale steady-state input/output simulators is presented. The method uses only low-dimensional Jacobian and reduced Hessian matrices calculated through on-line model-reduction techniques. The typically low-dimensional dominant system subspaces are adaptively computed using efficient subspace iterations. The corresponding low-dimensional Jacobians are constructed through a few numerical perturbations. Reduced Hessian matrices are computed numerically from a 2-step projection, firstly onto the dominant system subspace and secondly onto the subspace of the (few) degrees of freedom. The tubular reactor which is known to exhibit a rich parametric behaviour is used as an illustrative example. 

Chapter 2 presents three model reduction techniques based on the consideration of energy flows in a bond graph model. One approach ranks energy stores and dis-sipators on the basis of a power norm called activity in order to reduce the model complexity by eliminating the least active elements. 

This chapter considers the reduction approach to proper modeling and describes a set of model reduction techniques that are particularly amenable to bond graph models in the sense that the techniques take advantage of the explicit energetic nature of bonds in a bond graph model and yield reductions not only at the equation level but also directly at the graph level. 

For static and (structural) dynamic analysis, for determination of eigenfre-quencies and eigenmodes, several different commercial tools exist such as NASTRAN, ABAQUS or ANSYS. Some of them are also able to handle actuators and piezoelectric materials, and also to carry out some types of model reduction techniques. Nevertheless, specific techniques might have to be established by the user via accessing the modal data base. These data are then also used to set up a modal or otherwise condensed state-space representation possibly including specific actuator and sensor models. A description of the transformation of finite-element models from ANSYS to dynamic models in state space form in MATLAB can be found in [20]. 

Further, we used the Linearize routine of gPROMS to obtain a linear model. Starting with the OCFE discretization, the linear model has 48 states. This might be too much for the purposes of controllability analysis and control system design. Therefore, we applied different model-reduction techniques (Skogestad and Postlethwaite, 1996). 

For dynamic simulation, pure OC is unsuitable. OCFE is found to give realistic representation of column s behaviour, together with a small-size model. This presents a good option to FD scheme. Linear model reduction techniques are further applied to reduce the model for control design purpose. Balanced residualization with 15 states approximates satisfactorily the column dynamics. 

In this chapter several model reduction techniques will be discussed. The first method is based on firequency response matching, other methods make use oficonversion ofi the model structure to a state space model and subsequently truncating the states that have a minimum impact on the input-output relationship. The main indicator used fior this purpose is the so-called Hankel singular value. In addition, the model structure is converted to a balanced realization, afiter which the reduction techniques can be applied. Several examples are given on how to apply the dififierent methods. 

In this section, MATLAB will be used to illustrate the differences between the different model reduction techniques that have been discussed in the previous section. Data is obtained for a process with two inputs and two outputs and was taken from Zhu (2001). The file is called glassdata2.mat and can be downloaded from Zhu (2001). In this exercise only output 7i will be modeled. 

File modred25.m is used to compare the various model reduction techniques and show their results. First the system is balanced and the Hankel singular values or Gramian diagonal elements are calculated for a 25th order state space model. Figure 26.5 shows the result. 

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