Showing papers on "Optimal control published in 1984"

Optimal Power Flow By Newton Approach

Abstract: The classical optimal power flow problem with a nonseparable objective function can be solved by an explicit Newton approach. Efficient, robust solutions can be obtained for problems of any practical size or kind. Solution effort is approximately proportional to network size, and is relatively independent of the number of controls or binding inequalities. The key idea is a direct simultaneous solution for all of the unknowns in the Lagrangian function on each iteration. Each iteration minimizes a quadratic approximation of the Lagrangian. For any given set of binding constraints the process converges to the Kuhn-Tucker conditions in a few iterations. The challenge in algorithm development is to efficiently identify the binding inequalities.

Applications of Singular Perturbation Techniques to Control Problems

Abstract: This paper discusses typical applications of singular perturbation techniques to control problems in the last fifteen years. The first three sections are devoted to the standard model and its time-scale, stability and controllability properties. The next two sections deal with linear-quadratic optimal control and one with cheap (near-singular) control. Then the composite control and trajectory optimization are considered in two sections, and stochastic control in one section. The last section returns to the problem of modeling, this time in the context of large scale systems. The bibliography contains more than 250 titles.

Hierarchical Economic Dispatch for Piecewise Quadratic Cost Functions

Abstract: This paper presents a method to solve the economic power dispatch problem with piecewise quadratic cost functions. The solution approach is hierarchical, which allows for decentral i zed computations. An advantage of this approach is the capability to optimize over a greater variety of operating conditions. Traditionally, one cost function for each generator is assumed. In this formulation multiple intersecting cost functions are assumed. This method has appl ication to fossil generation units capable of burning gas and oil , as well as other problems which result in multiple intersecting cost curves for a particular unit. The results show that the solution method is practical and valid for real-time application. The motivation for this research stems from the actual operational and planning problems of a large Southwestern Utility.

Quadratically Convergent Optimal Power Flow

Abstract: A newly developed sparse implementation of an optimization method using exact second derivatives is applied to the optimal power flow problem. Four utility systems are studied using a variety of objective functions, including fuel costs, active and reactive losses, and new shunt capacitors. Systems solved range from 350 buses to 2000 buses. Comparisons are made with an older algorithm which uses an Augmented Lagrangian to demonstrate the advantages of run time and robustness of the new method. The algorithm and accompanying software represent a technological breakthrough, since they are suitable for solving systems on the order of 2000 buses and demonstrate solution speeds of 5 minutes on large mainframe computers. The method is particularly well suited to infeasible, or even divergent starting points. An option to add shunt capacitors in the event of hopeless infeasibility guarantees an optimal solution for many difficult to solve systems. An automatic scaling feature is added to correct numerical ill-conditioning resulting from series compensation or poor R/X ratios.

Optimal control of two interacting service stations

Abstract: Optimal controls described by switching curves in the two-dimensional state space are shown to exist for the optimal control of a Markov network with two service stations and linear cost. The controls govern routing and service priorities. Finite horizon and long run average cost problems are considered and value iteration is a key tool. Nonconvex value functions are shown to exist for slightly more general networks. Nonconvex value functions are also shown to arise for a simple single station control problem in which the instantaneous cost is convex but not monotone. Nevertheless, optimality of threshold policies is established for the single station problem. The proof is based on a novel use of stochastic coupling and policy iteration.

The Tracking Index: A Generalized Parameter for α-β and α-β-γ Target Trackers

Abstract: A generalized, optimal filtering solution is presented for the target tracking problem Applying optimal filtering theory to the target tracking problem, the tracking index, a generalized parameter proportional to the ratio of the position uncertainty due to the target maneuverability to that due to the sensor measurement, is found to have a fundamental role not only in the optimal steady-state solution of the stochastic regulation tracking problem, but also in the track initiation process Depending on the order of the tracking model, the tracking index solution yields a closed form, consistent set of generalized tracking gains, relationships, and performances Using the tracking index parameter, an initializing and tracking procedure in recursive form, realizes the accuracy of the Kalman filter with an algorithm as simple as the well-known α-β filter or α-β-γ filter depending on the tracking order

Fuel-cost minimisation for both real-and reactive-power dispatches

Abstract: The fuel cost formula is developed for optimal real-and reactive-power dispatch for the economic operation of power systems The problem is decomposed into a P-optimisation and a Q-optimisation module, where both modules use the same fuel cost objective function resulting in the optimal load flow The control variables are generator real-power outputs for the real-power module; and generator reactive-power outputs, shunt capacitors/reactors and transformer tap settings for the reactive-power module The constraints are the operating limits of the control variables, power-line flows and busbar voltages The optimisation problem is solved using the gradient projection method (GPM) for the quadratic objective function and linear constraints The GPM allows the use of functional constraints without the need of penalty functions or Lagrange multipliers among other advantages Mathematical models are developed to represent the sensitivity relationships between dependent and control variables for both real- and reactive-power optimisation modules; and thus eliminate the use of B-coefficients Results of two test systems are presented and compared with conventional methods

H ∞ -optimal feedback controllers for linear multivariable systems

Abstract: This paper treats the problem of designing, for a linear multivariable plant, a feedback controller which minimizes the H^{\infty} -norm of a weighted sensitivity matrix. There exists a family of optimal improper feedbacks. This family is determined by application of a theory of Ball and Helton. A method for computing optimal feedbacks is described in detail and a numerical example is included. It is shown that an optimal improper feedback can be approximated by a proper one under certain conditions on the weighting matrices.

Connections between Optimal Stopping and Singular Stochastic Control I. Monotone Follower Problems

Abstract: The stochastic control problem of tracking a Brownian motion by a nondecreasing process (Monotone Follower) is related to a question of Optimal Stopping. Direct probabilistic arguments are employed to show that the two problems are equivalent, and that both admit optimal solutions.

The linear regulator problem for parabolic systems

Abstract: An approximation framework is presented for computation (in finite imensional spaces) of Riccati operators that can be guaranteed to converge to the Riccati operator in feedback controls for abstract evolution systems in a Hilbert space. It is shown how these results may be used in the linear optimal regulator problem for a large class of parabolic systems.

Optimal Control in Some Variational Inequalities

Abstract: We study a nonconvex, nondifferentiable problem of optimal control where the state of the system is defined by an elliptic variational inequality with obstacle, and where the cost function is quadr...

Approximate solutions of the Bellman equation of deterministic control theory

Abstract: We consider an infinite horizon discounted optimal control problem and its time discretized approximation, and study the rate of convergence of the approximate solutions to the value function of the original problem. In particular we prove the rate is of order 1 as the discretization step tends to zero, provided a semiconcavity assumption is satisfied. We also characterize the limit of the optimal controls for the approximate problems within the framework of the theory of relaxed controls.

Mathematical Theory of Production Planning.

Abstract: The book proposes a unified mathematical treatment of production planning and production smoothing problems, in the framework of optimal control theory. General concave and convex cost models which relate most closely to real life applications are considered. Planning horizon results are always central to the discussion and developments. This allows the treatment of finite horizon problems only, and guarantees that the production plan implemented over the first periods is optimal with regard to any demand pattern beyond the planning horizon. Algorithms are proposed to compute the optimal production policy, together with the corresponding software designed to be implemented on any microcomputer. The book is organized in seven chapters and a mathematical appendix, which provides the reader with all the necessary background to render the volume self-contained.

Non-Linear Var Optimization Using Decomposition And Coordination

Abstract: This paper introduces a new approach to solve the var optimization problem. It computes the desired optimal solution on-line and is applicable to large system deviations, without the need of an OPF. Problem non-linearities are retained. The overall problem is decomposed into subproblems which are solved separately using a suitable NLP Method. These results are then coordinated. This process is repeated until it converges to an overall optimum point.

Differential dynamic programming and Newton's method for discrete optimal control problems

Abstract: The purpose of this paper is to draw a detailed comparison between Newton's method, as applied to discrete-time, unconstrained optimal control problems, and the second-order method known as differential dynamic programming (DDP). The main outcomes of the comparison are: (i) DDP does not coincide with Newton's method, but (ii) the methods are close enough that they have the same convergence rate, namely, quadratic. The comparison also reveals some other facts of theoretical and computational interest. For example, the methods differ only in that Newton's method operates on a linear approximation of the state at a certain point at which DDP operates on the exact value. This would suggest that DDP ought to be more accurate, an anticipation borne out in our computational example. Also, the positive definiteness of the Hessian of the objective function is easy to check within the framework of DDP. This enables one to propose a modification of DDP, so that a descent direction is produced at each iteration, regardless of the Hessian.

A golden ratio control policy for a multiple-access channel

Abstract: Consider n stations sharing a single communications channel. Each station has a buffer of length one. If the arrival rate of station i is r i , then 1-\Pi_{i}(1- r_{i}) is shown to be an upper bound (over all policies) on the throughput of the channel. Moreover, an optimal policy always exists and is stationary and periodic. The throughput of two policies, the random-policy, and the golden-ratio policy, are analyzed for a finite and infinite number of stations. The latter is shown to approach a limit which is within at least 98.4 percent of the upper bound.

Optimal dynamic routing in communication networks with continuous traffic

Abstract: New characterizations of optimal state-dependent routing strategies are obtained for the continuous traffic network model proposed by Segall for linear cost with unity weighting at each node and for constant inputs. The concept of flow relaxation is introduced and is used to transform the optimal routing problem into an initial flow optimization problem with convex cost and linear constraints. Three algorithms are given for open-loop computation of the optimal initial flow. The first is a simple iterative algorithm based on gradient descent with bending and it is well suited for decentralized computation. The second algorithm reduces the problem to a series of max-flow problems and it computes the exact optimal flow in O (mod N to the fourth) computations, where mod N is the number of nodes in the network. The third algorithm is based on a search for successive bottlenecks in the network. (Author/TRRL)

Competition Versus Optimal Control in Groundwater Pumping When Demand is Nonlinear

Abstract: This article considers the issue of optimal control versus no control in groundwater pumping under the assumption of a nonlinear demand function for water use. We confirm for the case of the nonlinear demand function what had been demonstrated for the case of a linear demand function: namely, if water rights are properly defined and if the storage capacity of the aquifer is relatively large, the difference between a strategy of no control and a strategy of optimal control is small and thus can be ignored for practical policy considerations. Furthermore, we demonstrate that even if simulated optimal control yields slightly better results than no control, a strategy of no control is likely to yield better results than optimal control, unless we can be sure that the estimated demand for groundwater is very close to the true demand.

Ritz–Galerkin Approximation of the Time Optimal Boundary Control Problem for Parabolic Systems with Dirichlet Boundary Conditions

Abstract: This paper deals with Ritz–Galerkin approximations of the following two problems: (i) boundary-value problems with $L_2 $-boundary data given in the form of Dirichlet boundary conditions; (ii) time...

Necessary Conditions in Nonsmooth Optimization

Abstract: The paper contains four theorems concerning first order necessary conditions for a minimum in nonsmooth optimization problems in Banach spaces: a Lagrange multiplier rule for a mathematical programming problem in which an infinite dimensional equality constraint is included in the constraints, a general maximum principle for nonsmooth optimal control problems with state constraints, and a kind of multiplier rule for mathematical programming problems which applies when only finitely many equality constraints are present but when the Lipschitz continuity assumptions are removed. A summary of relevant background results from analysis is provided.

On the Number and Placement of Actuators for Independent Modal Space Control

Abstract: A new formulation of independent modal space control is developed to handle the attitude and shape control problem for large flexible spacecraft. The main advantage of this method is that one can obtain an analytical solution for the optimal control law for very high dimensional systems. The fundamental limitation of previous work—the requirement of one actuator for each mode to be controlled—is relaxed in the new formulation. The closed-loop design is obtained while independently assuring stability and the design may be iterated to improve closed-loop performance. The process is shown to be simple and efficient in a realistic numerical example of spacecraft shape and attitude control. The ease of control law generation by this approach is seen to be obtained at the expense of the ability to adjust directly the penalties on the actuator effort. Actuator placement is seen to be of fundamental importance, and methods are developed which are comparatively simple to use and which can determine optimal actuator locations.

Approximation procedures for the optimal control of bilinear and nonlinear systems

Abstract: Optimal control problems for bilinear systems are studied and solved with a view to approximating analogous problems for general nonlinear systems. For a given bilinear optimal control problem, a sequence of linear problems is constructed, and their solutions are shown to converge to the desired solution. Also, the direct solution to the Hamilton-Jacobi equation is analyzed. A power-series approach is presented which requires offline calculations as in the linear case (Riccati equation). The methods are compared and illustrated. Relations to classical linear systems theory are discussed.