Robust control problem

The canonical robust control problem is shown in Figure 9.29. 

The MPC control problem illustrated in Eqs. (8-66) to (8-71) contains a variety of design parameters model horizon N, prediction horizon p, control horizon m, weighting factors Wj, move suppression factor 6, the constraint limits Bj, Q, and Dj, and the sampling period At. Some of these parameters can be used to tune the MPC strategy, notably the move suppression faclor 6, but details remain largely proprietary. One commercial controller, Honeywell s RMPCT (Robust Multivariable Predictive Control Technology), provides default tuning parameters based on the dynamic process model and the model uncertainty. 

This tutorial uses the MATLAB Control System Toolbox for linear quadratie regulator, linear quadratie estimator (Kalman filter) and linear quadratie Gaussian eontrol system design. The tutorial also employs the Robust Control Toolbox for multivariable robust eontrol system design. Problems in Chapter 9 are used as design examples. 

New research advances in control theory that are bringing it closer to practical problems are promising dramatic new developments and attracting widespread industrial interest. One of these advances is the development of "robust" systems. A robust control system is a stable, closed-loop system that can operate successfully even if the model on which it is based does not adequately describe the plant. A second advance is the use of powerful semiempirical formalisms in control problems, particularly where the range of possible process variables is constrained. 

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. 

Finally, the controller solving the robust regulation problem for linear system (1) takes the form... 

Consider here the nonlinear robust regulation problem (NRRP), which consists in finding, if possible, a dynamic controller of the form... 

Corollary 2. The Nonlinear Robust Regulation Problem is solvable by means of a linear controller if the pair Ao,Bq) is stabilizable, the pair (Co,j4o) is detectable, there exist mappings Xgs = (w, p), and Ugg = 7 (w, p), with... 

The methods for mercury analysis described earlier are based on continuous-flow technology and although this is easy to translate these procedures into an on-hne regine, several problems are associated with this. These are (a) presentation of a representative sample (b) conversion of all forms of mercury in the sample into the divalent form prior to reduction to mercury and (c) engineering the system to include sufficient robustness, control and flexibility. 

The formulation described above provides a useful framework for treating feedback control of combustion instability. However, direct application of the model to practical problems must be exercised with caution due to uncertainties associated with system parameters such as and Eni in Eq. (22.12), and time delays and spatial distribution parameters bk in Eq. (22.13). The intrinsic complexities in combustor flows prohibit precise estimates of those parameters without considerable errors, except for some simple well-defined configurations. Furthermore, the model may not accommodate all the essential processes involved because of the physical assumptions and mathematical approximations employed. These model and parameter uncertainties must be carefully treated in the development of a robust controller. To this end, the system dynamics equations, Eqs. (22.12)-(22.14), are extended to include uncertainties, and can be represented with the following state-space model ... 

Despite the optimistic overtones, robust control is not a solved problem. Some difficult theoretical questions remain in the synthesis area. The available software is, at best, experimental the controller is complex and its structure is not obvious. It generally uses all the measurements and all the manipulated variables in a centralized fashion. On-line tuning is difficult except when the IMC structure is employed [8], Fault tolerance, that is, continued satisfactory or at least stable performance in the event of an actuator or sensor failure, cannot be guaranteed. 

The optimal robust controller designed with one of the new synthesis techniques is generally not of a form that can be readily implemented. The main benefit of the new synthesis procedure is that it allows the designer to establish performance bounds that can be reached under ideal conditions. In practice, a decentralized (multiloop) control structure is preferred for ease of start-up, bumpless automatic to manual transfer, and fault tolerance in the event of actuator or sensor failures. Indeed, a practical design does not start with controller synthesis but with the selection of the variables that are to be manipulated and measured. It is well known that this choice can have more profound effects on the achievable control performance than the design of the controller itself. This was demonstrated in a distillation example [17, 18] in which a switch from reflux to distillate flow as the manipulated variable removes all robustness problems and makes the controller design trivial. 

A number of research issues in the areas of robust control, model predictive control, and control structure selection were mentioned previously. Unfortunately, even if all these problems were solved, a practical problem like the Shell Control Problem [23] could still not be tackled in a systematic fashion. All the research topics discussed so far in this paper are re-... 

Two main classes of optimization methods are available for handling uncertainty. The essential difference relates to whether or not measurements are used in the calculation of the optimal strategy. In the absence of measurements, a robust optimization approach is typically used, whereby conservatism is introduced to guarantee feasibility for the entire range of expected variations [18]. When measurements are available, adaptive optimization can help adjust to process changes and disturbances, thereby reducing conservatism [9]. It is interesting to note that the above classification is similar to that found in control problems with the robust and adaptive techniques. 

S. Kdrkel, E. Kostina, H. G. Bock, J.P. Schloder, 2004, Numerical methods for optimal control problems in design of robust optimal experiments for nonlinear dynamic processes. Opt. Methods and Software, 19, 327-338. 

Because of problem complexity, the separation system is usually decomposed into subsystems, as vapour, liquid and solid separations. For each sub-system, there are systematic procedures to generate design alternatives. For many separation operations, there are procedures, supported by industrial experience, that lead to economical optimal design with good controllability properties. However, there are also many cases where design procedures for robust controllability are still needed. 

Marrison, C. I. and Stengel, R. F. Stochastic robustness synthesis applied to a benchmark control problem. International Jourruil of Robust and Nonlinear Control 5(1) (1995), 13-31. 

Human modeling method has low sensitivity to noise. ALM is similar to human modeling methods and it is very robust to noise. Therefore it is very useful method for control problems. Up to now, several researches has been performed on the application of ALM in control problems. Some of these control researches are as below ... 

Instrumentation response times Sensor problems Time-delays Interactions between process states Interactions between process units Cascade strategies New sensors sensor location. Inferential measurement and control Predictive control. Robust controller designs Selection of control loop pairings. Decoupling control Feedforward strategies... 

Chapter 7 discusses robust control. This allows for the inclusion of uncertainty of process parameters in the control design. The concept of robustness refers to the preservation of closed-loop stability under allowable variations in system parameters. General stability results and integrity results are given for the LQR problem. 

Stein and Doyle [35] developed an expression to calculate ft for the Robust Performance Problem in the case where the plant is minimum phase and is controlled by an inverse-based decoupling controller. The modeled uncertainty is described by a complex unstructured input block with weighting function w, and performance requirement Wj measured by the closed-loop sensitivity function S. The decoupling controller K is based on the inverse of G in the form. (s) = G (s) (s), where k(s) is a scalar transfer function which makes K s) proper and gives a stable closed-loop system. Note that G s) is a linear stable system with stable inverse (i. e. G is minimum-phase).This compensator produces diagonal sensitivity and complementary sensitivity functions with identical diagonal elements, namely... 

In the field of polymer reactor engineering, the calorimetric estimation and control problems have been extensively studied with simulations and experiments [1, 33, 37,39]. EJCF [33,37] and L [39] observers have been employed to estimate the heat generation rate, on the basis of an off-line fitted heat transfer model [38, 39]. Various control techniques have been employed among them are adaptive, inferential, model predictive, and geometric control [1, 38, 39]. The robustness of the controller is shown by its successful implementations, regardless of the particular estimation and control techniques employed. Recently [15], it has been formally established, and experimentally demonstrated, the feasibility of jointly estimating the heat generation rate and the heat transfer coefficient in an exothermic reactor. 

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