Disturbance rejection

For the operating conditions, the set-points of EC and DMC compositions at the top and bottom are 0.01 and 0.2996, respectively, and the bottom temperature should not exceed 140°C to prevent the decomposition of reactants. From these plots, it can be concluded that the MPC outperforms the PI controller in terms of response speed in disturbance rejection, maintaining the variables at set points, and optimization capability. Especially, the PI controller failed to maintain the DMC composition set-point due to the slow long-term dynamics caused by the interaction between the RD column and azeotropic recovery column. 

Regulator problems Consider changes in disturbance with a fixed set point (R = 0) C = G oacjL. The goal is to reject disturbances, i.e., keep the system output at its desired value in spite of load changes. Ideally, we would like to have C = 0, i.e., perfect disturbance rejection. 

In Eq. (10-5), 1/Gp is the set point tracking controller. This is what we need if we install only a feedforward controller, which in reality, we seldom do.4 Under most circumstances, the change in set point is handled by a feedback control loop, and we only need to implement the second term of (10-5). The transfer function -GL/Gp is the feedforward controller (or the disturbance rejection... 

To illustrate the disturbance rejection effect, consider the distillation column reboiler shown in Fig. 8.2a. Suppose the steam supply pressure increases. The pressure drop over the control valve will be larger, so the steam flow rale will increase. With the single-loop temperature controller, no correction will be made until the higher steam flow rate increases the vapor boilup and the higher vapor rate begins to raise the temperature on tray 5. Thus the whole system is disturbed by a supply-steam pressure change. 

Centralized control can be also designed based on disturbance rejection or robustness requirements. In this case, the controller is not a static linear feedback law, as (45), but a dynamic feedback controller is obtained. Additionally, two degree of freedom controllers allow for a better control behavior in tracking and regulation. All these alternatives are beyond the scope of this introductory local control design treatment and are the subject of specialized references (see, for instance, [19]). 

Summary. In this chapter the control problem of output tracking with disturbance rejection of chemical reactors operating under forced oscillations subjected to load disturbances and parameter uncertainty is addressed. An error feedback nonlinear control law which relies on the existence of an internal model of the exosystem that generates all the possible steady state inputs for all the admissible values of the system parameters is proposed, to guarantee that the output tracking error is maintained within predefined bounds and ensures at the same time the stability of the closed-loop system. Key theoretical concepts and results are first reviewed with particular emphasis on the development of continuous and discrete control structures for the proposed robust regulator. The role of disturbances and model uncertainty is also discussed. Several numerical examples are presented to illustrate the results. 

Flo. 7.36. Block diagram for changes in load only (viz. load disturbance rejection case)... 

The control objective is different in the two cases. In disturbance rejection, the control objective is to reject the disturbance as quickly as possibly, i.e. to bring the process back to the original steady state by counteracting the effect of the disturbance (see Figure 5(a)). As an example, consider a person in a shower who wants to maintain the water temperature at a constant value. To achieve this, the person may have to turn the cold water tap up to increase the flow rate of the cold water if there is a sudden increase in temperature as a result of someone else in the house using cold water, i.e. to fill a kettle in the kitchen. (This is an example of a disturbance caused by a shared utility, here the cold water.)... 

Figure 5 Process responses (a) disturbance rejection and (b) set point tracking...

Figure 7 Closed loop responses of first-order process, (a) P only (set point tracking), (b) P only (disturbance rejection), (c) stable PI or PID (disturbance rejection) and (d) unstable PI or PID (disturbance rejection).

How do we choose the values of the controller parameters Kc, ii and td They must be chosen to ensure that the response of the controlled variable remains stable and returns to its steady-state value (disturbance rejection), or moves to a new desired value (set point tracking), quickly. However, the action of the controller tends to introduce oscillations. 

Using the methods presented in Chapter 2, the above formulation can be used to derive a state-space realization of the slow dynamics of the type in Equation (2.48). The resulting low-dimensional model should subsequently form the basis for formulating and solving the control problems associated with the slow time scale, i.e., stabilization, output tracking, and disturbance rejection at the process level. 

Energy-related control objectives are to be addressed in the fast time scale r, where w1 are available as manipulated inputs (Equations (7.3)). From a practical point of view, however, only a limited number of material flow rates may be available to address objectives related to temperature control thus, the set of manipulated inputs in the fast time scale could often consist solely of the large energy flows Qin and Q0ut- Simple, distributed controllers for the stabilization (and fast disturbance rejection) of unit temperatures are a typical choice at this level. 

In this study we identify an SMB process using the subspace identification method. The well-known input/output data-based prediction model is also used to obtain a prediction equation which is indispensable for the design of a predictive controller. The discrete variables such as the switching time are kept constant to construct the artificial continuous input-output mapping. With the proposed predictive controller we perform simulation studies for the control of the SMB process and demonstrate that the controller performs quite satisfactorily for both the disturbance rejection and the setpoint tracking. 

Here we shall treat two typical control problems of practical interest one is the disturbance rejection and the other is the setpoint tracking. First, we assume that the feed pump stops during 40 minutes after 40th switching. These may be considered as unmeasured disturbances introduced to the process. Figure 2 shows that the controller successfully rejects the unmeasured distu rbance. 

An SMB process is identified by using the subspace identification method. The input/output data-based prediction model is used to obtain the prediction model. The identified model exhibits an excellent prediction performance. The input/output data-based predictive controller based on the identified model is designed and applied to MIMO control problems for the SMB process under the presence of the input and output constraints. The simulation results demonstrate that the controller proposed in diis study shows an excellent control performance not only for the disturbance rejection but also for the setpoint tracking. 

The rest of this chapter is structured as follows. The next section considers general techniques for use in the integrated design approach proposed design with uncertainty, screening tools for disturbance rejection, and dynamic opti-... 

For example, optimizing control moves for disturbance rejection in a blending system subject to step disturbances of variable magnitude would give perfect disturbance rejection for a maxmin/perfect-knowledge formulation and a poor... 

A Screening Test for Disturbance Rejection in Nonlinear Processes Subject... 

These measures of delay all provide useful indicators of the effective delay, but they do not in themselves indicate whether the effect of the delay prevents a disturbance from being rejected before causing constraint violations. In Holt and Morari s analysis, the disturbances are assumed to appear as steps on the outputs, making the question of whether the disturbance causes constraint violation trivial. In practice, disturbances are often well approximated by steps, but the effect of the step usually propagates dynamically through part of the process before affecting the outputs. This means that the effect of the process dynamics in attenuating the disturbance should be included in order to assess disturbance rejection. 

A common performance estimation method in classical single-loop feedback controller design is to check the open-loop disturbance rejection of a system up to the point in time at which the controller action is assumed to take effect. If constraints are violated during this time, the controller cannot prevent the violation. The use of this test can be traced back to Velguth and Anderson (1954). 

This expression for the minimum delay from control action to the constrained output y, min, fj , . corresponds to Holt and Morari s lower bound on the minimum delay. Use of their upper-bound values is not appropriate in general, as they are based on a requirement for decoupling which may not be necessary or even desirable for disturbance rejection. If decoupling is assumed to be an additional performance requirement, the upper-bound delays could be substituted for those used. It should be noted that, in the absence of a decoupling requirement, tj and hence the predicted disturbance rejection, can never be improved by increasing any process delay. 

The difference between feedforward and feedback control can be conveniently represented by using an additional unconstrained measurement with the appropriate delay in relation to the disturbance to represent the feedforward measurement. The effective delay will usually be less for feedforward than for feedback control, giving an improved bound on disturbance rejection. 

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