Optimal Periodic Control Problems Research Articles

Periodic control of continuous stirred tank reactors—I: The pi criterion and its applications to isothermal cases

The pi criterion which is useful for examining if the optimal periodic control is proper is briefly introduced in a somewhat generalized form and then it is applied to optimal periodic control problems of CSTR's in isothermal condition.

A Dynamic Example of the Intertemporal Price Discrimination Problem

Recently in this Journal, Gately (1976) presented a model of a retailer's one-day sale with two groups of customers having different demand elasticities. Using a simple example he showed that in some cases the retailer can make extra profit by temporarily reducing prices for a short time. The resulting price policy exemplifies a type of price discrimination, which L6fgren (1971) termed intertemporal price discrimination. The solution is obtained by using a static analysis. In this note we should like to point out, however, that the intertemporal price discrimination problem can more generally be considered to be an optimal periodic control problem. Gately's example resembles in many respects the one formulated by Guardabassi & Locatelli (1974), which includes a dynamic model for the recovery of customers' demand after a sales period. We report here on the results of a more general dynamic analysis of a corresponding pricing problem; see Timonen & HIamiliinen (1979). We derived the optimal periodic pricing policies for an example with three consumer groups. In this model the length of the time interval between two consecutive sales affects the retailer's profit and it can thus be taken as an optimization variable. The results show that the time pattern of the optimal pricing policy need not be of the one-day sale type even in this kind of simple problem. Consider an example where a retailer has three groups of consumers for a certain product. The dynamic demand behavior of the groups is described by the consumption rates of the groups and by the desired levels of storage that each group is going to maintain for different prices. Storage can be increased by buying a certain amount of the product at the current price at discrete points in time and the storage is decreased at the rate of the group's consumption. Fig. 1 a shows the desired level of storage Ah that each group of consumers is going to maintain for different prices. The dependence of the groups' con-

Optimal Periodic Control: A Solution Set Theory of Necessary and Sufficient Conditions

The paper reviews, in a simplified form, recent research [8] concerning the role of first order necessary conditions in a general optimal periodic control problem (OPC). Three notions of optimality are considered: solutions of OPC; solutions of OPC which are steady-state (constant); solutions of OSS, a steady-state optimization problem subsumed in OPC. The three types of optima and the corresponding necessary conditions are studied by introducing a theory of solution sets. This puts earlier research into a comprehensive framework and establishes certain limits to what can be proved. A principal topic is tests for proper (time-dependent control is better than steady-state control). Some sufficient conditions for global optima of OPC are given and related to the necessary conditions. These results are new and generate an interesting class of OPC problems which are not proper.

Optimal Periodic Control: A General Theory of Necessary Conditions

Does time-dependent periodic control yield better process performance than optimal steady-state control? This paper examines exhaustively the role of first order necessary conditions in answering this question. For processes described by autonomous, ordinary differential equations, a very general optimal periodic control problem (OPC) is formulated. By considering control and state functions which are constant, a finite-dimensional optimal steady-state problem (OSS) is obtained from OPC. Three solution sets are introduced: $\mathcal{S}$(OSS)—the solutions of OSS, $\mathcal{S}$(OPC)—the solutions of OPC, $\mathcal{S}$(NCOSS)—the solutions of OPC which are constant. Necessary conditions for elements of each of these sets are derived; their solution sets are denoted, respectively, by $\mathcal{S}$(NCOSS), $\mathcal{S}$(NCOPC), and $\mathcal{S}$(NCSSOPC). The relationship between these six solutions sets is a central issue. Under various hypotheses certain pair-wise inclusions of the six sets are determined a...

Periodic optimization of a chemical reactor system using perturbation methods

A perturbation approach is presented to the periodic optimization problem of certain classes of nonlinear dynamical systems, for small-amplitude forcing functions, and within specific frequency ranges derived from Guardabassi'sπ-criterion. The procedure is illustrated by means of a classical example in chemical engineering, involving the optimal periodic operation of a continuous stirred tank reactor, in which two parallel reactions take place. The analysis is performed at cycling frequencies slightly above the minimum frequency, where, according to theπ-criterion, performance improvement by cycling becomes possible. The per cent production gains over the optimal stationary operation are evaluated, as a function of the amplitude and frequency of the oscillations allowed to exist in the system, and as a function of the process parameters. Also the characteristics of the control and state waveforms are analysed. Thus the existence and practical applicability are demonstrated of a mathematical relationship between the optimal periodic control problem, theπ-criterion, and the theory of perturbations.