Optimizing control plant, formulation

Formulating the Optimizing Control Problem for an Integrated Chemical Plant... 

For an optimal control problem, the complementary maxmin problem is equivalent to the controller having perfect knowledge of the plant parameters and disturbances, including future disturbances. This formulation could be termed the crystal ball approach to control. For many problems, the crystal ball control will be successful, but this says very little about whether any realizable control system exists that can meet the performance specification. 

It is common knowledge that for time invariant systems and cost criteria and an infinite horizon T —> 00 the resulting optimal control can be formulated as a statie state feedback control law k x) [26], i.e. the optimal control depends only on the eurrent state vector of the plant. In aecordanee with what has been said above, the eontroller nonlinearity is influeneed by (1) the plant dynamics Eq. (18), (2) the region of operation eharacterized by the set of initial eonditions B and (3) the performanee eriterion Eq. (19). 

This chapter has focused on the use of controller parametrization in the integration of design and control. The focus here has been on the parametrization of linear controllers, although there have been promising recent developments in the parametrization of nonlinear controllers. As with the IMC approach, we seek to provide a performance limit independent of controller type, and indeed for stable systems they are structurally equivalent. Its application to operability analysis has been posed within an optimization framework, and as such is able to account for the simultaneous presence of all plant-inherent performance-limiting characteristics. Both discrete and continuous-time formulations have been presented, and its implementation for both controllability analysis and controllable plant design shown. Its application has also been illustrated through two case studies. 

Clearly, the first six component types can be controlled during the initial formulation of the medium. It is these that have been the subject of optimization studies, which form a large part of the recent plant cell culture literature. The seventh category falls outside the capabilities of most investigators in the field. The concentrations of dissolved gases have also been neglected as components, possibly because they cannot be controlled in the same manner as dissolved solids. 

The controller receives the on-line composition measurement of the product outlets (extract and raffinate) as feedback data from the plant. These measurements are filtered through a periodic Kalman filter and used together with the simplified SMB model results to estimate the state of the system and to remove the possible moidel errors. The formulation of RMPC is based on the assumption that possible errors or disturbances are likely to repeat and will have a periodic effect on the output, which is the most likely correlation between disturbances and output in a SMB unit. The estimated future concentration profile in the SMB is used to optimize the future behaviour of the plant over a predefined prediction horizon. The controller implements the calculated optimal plant input by changing the external flow rates in order to control the internal flow rates, which are the manipulated variables. Time lags, e.g. between online concentration measurements and optimizer or between optimizer and SMB plant, are insignificant relative to the process dynamics and sampling time for the planned scheme. 

Bahri et al. [22] presented a backoff optimisation formulation to examine the disturbance rejection capability of the given design and find a backoff optimal design in order to reject the specified disturbance at steady state. One feature of the optimisation formulation of Bahri et al. [22] is the ability to include parameters characterising the design of the plant as decision variables without control design. In their later work, Bahri et al. [23] extended their work to dynamic systems. In this work dynamic performance was evaluated dependent on detailed control system design. 

Two broad approaches to dynamic operability analysis are the use of so-called open-loop indicators, and the solution of a suitably formulated optimization problem. Characteristics of the former are that they are based on steady-state or linear dynamic models, are relatively easy to compute and seek to provide indications of potential plant-inherent control problems independent of the choice of control system. Examples are the minimum singular value and the plant condition number which reflect sensitivity to input constraints and model uncertainty respectively. More detail may be found in [1] with a good overview of these methods given in [2]. 

While the main thrust of these analyses are to provide a plant that exhibits satisfactory closed-loop performance, the assumptions regarding the control system vary considerably across the various methods proposed. The open-loop indicators are largely based on factors that limit achievable closed-loop performance independent of controller type, whereas most of the optimization based integrated design formulations assume a specific controller type such as multiloop PI, LQG and so forth. While this is not considered to be a problem per se, it is important that the implications of these assumptions are clear so that appropriate deductions may be drawn. This chapter attempts to at least in part address this issue. 

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