翻译论文

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Optimum Control Systems In recent years much attention has been focused upon optimizing the behavior of systems. A particular problem may concern maximizing the range of a rocket, maximizing the profit of a business, minimizing the error in estimation of position of an object,minimizing the energy or cost required to achieve some required terminal state, or any of a vast variety of similar statements. The search for the control which attains the desired objective ective while minimizing (or maximizing) a defined system criterion constitutes the fundamental problem of optimization theory. The design sequence for an optimal control system has five basic steps: 1. Modeling of plant 2. Establishment of constraints 3. Selection of the performance index 4. Minimization of the performance index 5. Determination of the controller configuration. For continuous, deterministic, and lumped parameter plant the end results of the first step are the state and output equations

Obviously, these equations must adequately describe the plant. Plant modeling is not a trivial task, nor is the selection of the best state, control, and output variables. Complete controllabillity in the mathematical sense is a necessary but not sufficient condition for the existence of an optimal control!'' In addition, if control is to be feedback, the plant must be completely observable. Remember that observability does not guarantee physical measurability. The constraints of the second step are physical constraints imposed on the state and control variables as well as any other physical constraints that might affect the performance of the plant. The lack of proper constraints leads to physically unrealistic and ridiculous solutions. State constraints may be equality constraints whereby the initial and/or final states are specified or inequality constraints restricting the range of permissible values of specific state variables. Control and other constraints are generally inequality constraints; e.g. the maximum acceleration of the plant or fuel used must be less than a specified value. State trajectories and controls that satisfy all the constraints are called admissible trajectories and admissible controls and are candidates for further investigation. Those trajectories and controls that do not satisfy the constraints are termed inadmissible and are rejected. The formulation of the performance index (P1) may well be the most critical and difficult step of all. The performance index is an attempt to express quantitatively the deviations in plant performance from an ideal express performance. The performance index is written as the functional:

whereandare the initial and final times. J, is evaluated at the final state and is not necessarily specified. , the cost or loss function, is evaluated over the entire control interval Weighting factors are used to assign relative importance to various terms in Jr that describe the deviations from the ideal performance. Each admissible control will yield a single value for a given performance index. This value of J can be used as a figure of merit in comparing competing controls; the smaller the value of J. the better the control, mathematically that is.t0t The admissible control that produces an admissible trajectory and minimizes the value of the performance index is called the optimal control and given the symbol . the trajectory is called the optimal trajectory and symbolized by If an optimal control, which is a function of time, is specified only for a particular initial state whereandare

Then the control is open-loop. If, however, the optimal control is a function of both time and the State

then the control is closed loop with state-variable feedback and we call the optimal control law. If,for example.

where E is a constant matrix, the optimal control law is a stationary linear feedback of state variables. An optimal control need not exist for a given performance index nor be unique if it does exist.in addition, changing the performance index will result in a different optimal control and trajectory.The designer must be capable of interpreting and choosing from several optimal controls in terms of physical and practical considerations. With the exception of some special cases, minimization of the performance index to obtain an optimal control does not yield analytical solutions and the computational efforts is high. The two basic methods for minimization arc dynamic programming and a calculus of variations approach known as Pontryagin's minimum principle. Dynamic programming is a multistage decision process that searches directly for minimum of the performance index, which is written as a recurrence equation. The distinguishing characteristic of dynamic programming is the use the principle of optimality to reduce the~of search sufficiently to make direct search feasible. The reduction of the search area by principle of optimality and the state and control constraints is illustrated in Fig. 2-6B-I. As applied to the optimal control problem the principle of optimality states that a control that is optimal over a complete interval must be optimal over every subinterval. The computational procedure is to start at a final state and work backward in stages to an initial state, finding the optimal control for each stage in turn. If the optimal control is found for N stages, it contains the optimal control for any lesser number of stages ending at the same final stage. This is known as the principle of embedding . Dynamic programming yields an optimal control law, but not in analytical form. If analytical approximation is not possible, the tabulated control values must be stored and be