Models linear discrete-time transfer

Following the MVC framework [102, 148[, consider a process described by a linear discrete-time transfer function model ... 

Bode diagram, 330-31, 334-37 frequency response, 323-24 interacting capacities, 197-200 noninteracting capacities, 194-96 pulse transfer function, 619 Multiple-input multiple-output system, 20 discrete-time model, 586 discrete transfer function, 612 input-output model, 83-85, 163-68 linearization, 121-26 transfer-function matrix, 164, 166 Multiple loop control systems, 394-409 Multiplexer, 560, 564 Multivariable control systems, 461-62 alternative configurations, 467-84 decoupling of loops, 503-8 design questions, 461-62 interaction of loops, 487-94 selection of loops, 494-503 Multivariable process (see Multiple-input multiple-output system)... 

Irrespective of the situation, process system identification is focused on determining the values for Gp and G/ as accurately as possible. Since most of the applications assume that the controller is digital, the system identification methods considered here will focus on the discrete time implementation of system identification. For this reason, the models for each of the blocks will be assumed to be linear, rational functions of the backshift operator Such models are most often referred to as transfer functions. The most general plant model is the prediction error model, which has the following form ... 

We begin with the single input, single output (SISO) case and assume that the process to be identified is stable, linear, time invariant and can be represented by the following discrete-time, finite impulse response (FIR) transfer function model ... 

A discrete-time impulse or step response model can be developed from a transfer function model or a linear differential (or difference) equation model. For example, consider a first-order transfer function with t = 1 min and K=l, and a unit step input change. The first-order difference equation corresponding to Eq. 7-34 with y(0) = 0 and Ar = 0.2 is... 

In earlier chapters, Simulink was used to simulate linear continuous-time control systems described by transfer function models. For digital control systems, Simulink can also be used to simulate open- and closed-loop responses of discrete-time systems. As shown in Fig. 17.3, a computer control system includes both continuous and discrete components. In order to carry out detailed analysis of such a hybrid system, it is necessary to convert all transfer functions to discrete time and then carry out analysis using z-transforms (Astrom and Wittenmark, 1997 Franklin et al., 1997). On the other hand, simulation can be carried out with Simulink using the control system components in their native forms, either discrete or continuous. This approach is beneficial for tuning digital controllers. 

A linear model predictive control law is retained in both cases because of its attracting characteristics such as its multivariable aspects and the possibility of taking into account hard constraints on inputs and inputs variations as well as soft constraints on outputs (constraint violation is authorized during a short period of time). To practise model predictive control, first a linear model of the process must be obtained off-line before applying the optimization strategy to calculate on-line the manipulated inputs. The model of the SMB is described in [8] with its parameters. It is based on the partial differential equation for the mass balance and a mass transfer equation between the liquid and the solid phase, plus an equilibrium law. The PDE equation is discretized as an equivalent system of mixers in series. A typical SMB is divided in four zones, each zone includes two columns and each column is composed of twenty mixers. A nonlinear Langmuir isotherm describes the binary equilibrium for each component between the adsorbent and the liquid phase. 

Our first task is to build a model where the complex vocal apparatus is broken down into a small number of independent components. One way of doing this is shown in Figure 11.1b, where we have modelled the lungs, glottis, pharynx cavity, mouth cavity, nasal cavity, nostrils and lips as a set of discrete, coimected systems. If we make the assumption that the entire system is linear (in the sense described in Section 10.4) we can then produce a model for each component separately, and determine the behaviour of the overall system fi om the appropriate combination of the components. While of course the shape of the vocal tract will be continuously varying in time when speaking, if we choose a sufficiently short time fi ame, we can consider the operation of the components to be constant over that short period time. This, coupled with the linear assumption then allows us to use the theory of linear time invariant (LTI) filters (Section 10.4) throughout. Hence we describe the pharynx cavity, mouth cavity and lip radiation as LTI filters, and so file speech production process can be stated as the operation of a series of z-domain transfer functions on the input. 

The present paper steps into this gap. In order to emphasize ideas rather than technicalities, the more complicated PDE situation is replaced here, for the time being, by the much simpler ODE situation. In Section 2 below, the splitting technique of Maas and Pope is revisited in mathematical terms of ODEs and associated DAEs. As implementation the linearly-implicit Euler discretization [4] is exemplified. In Section 3, a cheap estimation technique for the introduced QSSA error is analytically derived and its implementation discussed. This estimation technique permits the desired adaptive control of the QSSA error also dynamically. Finally, in Section 4, the thus developed dynamic dimension reduction method for ODE models is illustrated by three moderate size, but nevertheless quite challenging examples from chemical reaction kinetics. The positive effect of the new dimension monitor on the robustness and efficiency of the numerical simulation is well documented. The transfer of the herein presented techniques to the PDE situation will be published in a forthcoming paper. 

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