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The Explicit Robust Model - Based Control Law via Parametric Programming

The optimal operating point of the vast majority of chemical, mechanical and electrical systems resides usually at the intersection of environmental, safety and operational restrictions, the so-called constraints. Hence, the inevitable presence of varying reference conditions (e.g. cyclic operation) and uncertainties (e.g. fluctuations in feed conditions or capacitor's properties) tend to result in severe infeasibilities, including off-spec production, excessive waste - disposal or even hazardous situations such as mechanical failure of equipment. Contrary to conventional control design techniques, robust model based control (rMBC) has been particularly effective for a wide class of realistic constrained systems subject to the incessant presence of uncertainty. Most types of robust model based controllers determine the future control policy according to a prediction of the system behaviour over a receding time horizon. The computation of the control sequence is obtained implicitly by solving iteratively an on-line robust optimal control problem. The solution of this problem relies on minimizing usually the worst-case quadratic or $-1/infty$ norm of the input output deviations while accounting explicitly for the process characteristics via the mathematical model of the system. The capabilities of robust MBC are limited mainly due to (i) the rigorous on-line calculations that grow exponentially with the number of possible uncertain scenarios and (ii) the resulting conservative control action that jeopardizes the system performance. An approach for moving off-line the rigorous calculations involved in the nominal MBC has been developed recently. It is based on well-cited state of the art parametric programming algorithms, with which the explicit mapping of the optimal control actions in the space of the state measurements is derived. This technique however, has not fully been extented to the design of robust parametric controllers that have as a performance index the general quadratic or $-1/infty$ cost. Thus, in the presence of uncertainties they may underperform causing operational difficulties. This technique has been adopted by citeasnoun{bem01} for robust optimal control but their approach merely minimizes the worst case $-infty$ norm. Thus, this method may cause dead-beat or conservative control and additionally, its computational complexity increases exponentially with the number of bounded uncertain parameters. In this talk a novel methodology is presented for the design of robust model based parametric controllers for linear dynamic systems subject to additive input bounded uncertainty. Here, feasibility analysis is utilized for constructing a set of infinite dimensional constraints that when satisfied ensure that all the process requirements are met for every possible uncertainty scenario. These constraints are transformed to an equivalent set of tangible finite dimensional constraints by using multiparametric linear programming. Then an optimal control problem is formulated that minimizes the more general input/output quadratic or l$-1 / infty$ norm cost, subject to the process model and the feasibility constraints. The control policy is derived off-line via solving this optimization problem parametrically in terms of the states. Thus, the solution of this parametric program directly portrays an explicit feedback control law for the system. This controller succeeds in providing a single control sequence that steers the plant into the feasible operating region for every possible uncertainty scenario. The control action minimizes the nominal or the average and not the worst case cost implying in some case less conservative performance. Furthermore, in the previous approaches the complexity of the optimization problem is strongly dependent on the size of the uncertainty vector and the number of output constraints; whereas in our approach, the complexity of the controller structure relies only on the number of non-redundant output constraints. At the end of the talk, the significant characteristics of the method are demonstrated via typical process engineering examples. Additionally, the similarity of these type of parametric controllers to the anti-windup design is discussed, thus clearly demonstrating their simple structure characteristics and broad applicability.

Type of Seminar:
Public Seminar
Vassilis Sakizlis
Centre for Process Systems Engineering, Department of Chemical Engineering, Imperial College London SW7 2BY, U.K.
Jan 22, 2003   17:15

ETH Zentrum, Gloriastr. 35, 8006 Zurich, Building ETZ, Room E6
Contact Person:

Prof. M. Morari
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