In Part I, a method is developed for determining the optimal control laws for a class of aerodynamical systems whose dynamics are linear in the thrust and nonlinear in the lift and thrust angle. Due to the presence of the linear thrust control, a singular subarc exists along which it is often possible to eliminate the Lagrange multipliers from the control equations. Conditions under which this elimination is possible are derived, and expressions for thrust and the rate of change of lift and thrust angle are obtained that depend only on state variables and a small number of time-invariant parameters. The optimal values of the unknown parameters are determined by a direct search in parameter space for that set which minimizes the system performance function. As a result, the proposed method is considerably simpler than standard numerical techniques that require a separate search in function space for each component of the control vector. Furthermore, since the control vector is generated by the direct solution of differential equations, the method appears suitable for use with in-flight guidance computers. Several numerical examples are presented consisting of one, two, and three dimensional control. In each case, it is shown that the search in multi-dimensional function space can be replaced by an equivalent search in the parameter space of initial conditions. In Part II, a three stage numerical algorithm is developed for a general class of optimal control problems. The technique is essentially a combination of the direct and indirect approaches. Like the indirect approach, the control law equations are used to eliminate the control vector from the system and adjoint equations. However, instead of trying to solve the two point boundary-value problem directly, the augmented performance function is first considered to be a function of the unknown initial conditions and is minimized by a gradient search in the initial condition space. It is shown that it is sufficient to search over the surface of any sphere for the intersection of the line μλ*₀ where λ*₀ is the classical solution of initial values. As a result, this first approach is not dependent on a good initial estimate of the optimal trajectory, and is therefore used in the first two stages of the proposed algorithm to provide the property of rapid initial convergence. The property of rapid final convergence is obtained by employing either a modified method of matching end points, or a method of determining the optimal step size for the gradient method of the first two stages. Either combination results in a three stage numerical algorithm that has good initial convergence, good final convergence, and which requires storage at terminal points only. Several examples are presented consisting of both bounded and unbounded control. / Applied Science, Faculty of / Electrical and Computer Engineering, Department of / Graduate

Methods for studying the behaviour of on-off feedback systems, with the emphasis on steady-state periodic phenomena, are presented in this thesis. The two main problems analyzed are (1) the determination of the periods of self and forced oscillations in single-, double-, and multiloop systems containing an arbitrary number of on-off elements; and (2) the investigation of the asymptotic stability in the small of single-loop systems containing one on-off element which may or may not have a linear region of operation. To study the periodic phenomena in on-off systems, methods of determining the steady-state response of a single on-r-off element are first described. Concepts pertaining to the steady-state behaviour are then introduced: in this respect it has been found that generalizations of the concepts of the Hamel and Tsypkin loci and also of the phase characteristic of Neimark are useful in the study of self and forced oscillations. Both the Tsypkin loci and the phase characteristic concepts are used to determine the possible periods of self and forced oscillations in single- and double-loop systems containing an arbitrary number of on-off elements; these concepts are also applied to multiloop systems. On-off elements containing a linear region of operation, called a proportional band, are then described: both the transient and periodic response are presented. An approximate method for determining the periodic response is given. The concept of the Tsypkin loci is used to determine the possible periods of self and forced oscillations in a single-loop system containing one on-off element with a proportional band. The asymptotic stability in the small, or local stability, of the periodic states of single-loop systems containing one ideal on-off element has been considered by Tsypkin. In this thesis, Tsypkin's results have been generalized to include the cases of on-off elements containing a proportional band. The stability of such systems is determined by the stability of equivalent sampled-data systems with samplers having finite pulse widths. Finally, this stability problem is solved by a direct approach, one that makes use of the physical definition of local stability; the results obtained by this method agree with those derived by the sampled-data approach. / Applied Science, Faculty of / Electrical and Computer Engineering, Department of / Graduate

M.Ing. / Ice storage systems are used to store thermal energy in the form of ice build-up inside storage tanks. During off peak hours, the chiller is used to charge up the storage tank until it is full. During on peak hours, the storage is discharged to meet a certain fraction of the building cooling load. The control strategy employed determines the extent to which the storage compensates the chiller and visa versa. Given the way in which electricity rates are structured, ice storage systems become an effective energy management strategy. The objective of the study is to compare energy savings derived by using conventional control strategies versus optimal control. Conventional control strategies can be classified as chiller priority control, constant proportion control and storage priority control. In chiller priority control, the chiller meets the cooling load subject to a pre-set limit not being exceeded. Should the limit be exceeded, the remaining cooling load (at each time horizon) is compensated for by the storage. In constant proportion control, both the storage and chiller meets a constant proportion of the cooling load. Storage priority control attempts to discharge as much of the storage as possible, such that at the end of the planning horizon, the ice build up in the storage tank is just depleted. Optimal control employs dynamic programming to ensure that the integrated cost of energy, during the entire planning horizon, is minimal. A steady state ice storage plant model for analysing the performance of the control strategies is presented. The model computes the inlet and outlet temperatures into the various components of the air-conditioning plant, being the air-handling units, heat exchanger, ice storage tanks and chiller. The maximal possible discharge and charging rate at each time period (for the different control strategies) is determined using the model. Given the state of charge of the ice storage tank at each time period, it is then possible to calculate chiller power consumption. The power consumed by fans, fan coil units and pumps (in the air-conditioning plant) has not been calculated in the present analysis, however, the model can easily be extended to include such calculations. The ice storage plant model, enabled simulations of the different control strategies to be carried out over building cooling load profiles for summer and winter. Based on a 24-hour planning horizon, optimal control is found to be optimal and the only consistently performing strategy for all seasons. For the 5000 kWh ice storage plant investigated, optimal control yielded 25% energy savings in June and 12% in January, amounting to a potential of R 11 000 per month. Chiller priority control was near optimal in January but consumed 25% more energy than the base case (without storage) in June. Constant proportion control was optimal in January but poorer in June. Storage priority control is found to be optimal in June but the lowest performer in January. The drawback of optimal control and storage priority control, however, is that they require prediction of future cooling loads. The variance when using auto-regressive neural network to predict the load is expected to be in the region of 2% and thus considered acceptable. Chiller priority control and constant proportion control are instantaneous and simple to implement hence their popularity.

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