System identification time invariance

As mentioned above, the backbone of the controller is the identified LTI part of Wiener model and the inverse of static nonlinear part just plays the role of converting the original output and reference of process to their linear counterpart. By doing so, the designed controller will try to make the linear counterpart of output follow that of reference. What should be advanced is, therefore, to obtain the linear input/output data-based prediction model, which is obtained by subspace identification. Let us consider the following state space model that can describe a general linear time invariant system ... 

Al-Haj Ali et al. [5,6] developed different types of linear time invariant models by system identification, which adequately represent the fluidized-bed drying dynamics. MBC techniques such as IMC and model predictive control (MPC) were used for the designing of the control system. Simulations with multivariable MPC strategy... 

For many years, there have basically been two distinct methods for finding the solution of the adaptive control problem [2]. These are direct and indirect control methods. When the controller parameters 6 k) are directly adjusted to reduce some norm of the output error between the reference model and the plant, this is called direct control or implicit identification. In indirect control, also referred to as explicit identification, the parameters of the plant are estimated as the elements of a vector p k) at each instant fc, and the parameter vector 0 k) of the controller is chosen assuming that p(fc) represents the true value of the plant parameter vector p. Figures 4.3 and 4.4 respectively show the direct and indirect model-reference adaptive control structures for a linear time invariant (LTI) plant. It is important to note that in both cases efforts have to be made to probe the system to determine its behaviour because control action is being taken based on the most recent in-... 

With the identification of the TS trajectory, we have taken the crucial step that enables us to carry over the constructions of the geometric TST into time-dependent settings. We now have at our disposal an invariant object that is analogous to the fixed point in an autonomous system in that it never leaves the barrier region. However, although this dynamical boundedness is characteristic of the saddle point and the NHIMs, what makes them important for TST are the invariant manifolds that are attached to them. It remains to be shown that the TS trajectory can take over their role in this respect. In doing so, we follow the two main steps of time-independent TST first describe the dynamics in the linear approximation, then verify that important features remain qualitatively intact in the full nonlinear system. 

One final point about closed-loop process control. Economic considerations dictate that to derive optimum benefits, processes must invariably be operated in the vicinity of constraints. A good control system must drive the process toward these constraints without actually violating them. In a polymerization reactor, the initiator feed rate may be manipulated to control monomer conversion or MW however, at times when the heat of polymerization exceeds the heat transfer capacity of the kettle, the initiator feed rate must be constrained in the interest of thermal stability. In some instances, there may be constraints on the controlled variables as well. Identification of constraints for optimized operation is an important consideration in control systems design. Operation in the vicinity of constraints poses problems because the process behavior in this region becomes increasingly nonlinear. 

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