Showing papers in "Optimal Control Applications & Methods in 1994"

A gradient flow approach to computing lq optimal output feedback gains

Abstract: This short communication considers the linear quadratic problem with static output feedback. It is shown that an optimal solution can be successfully computed by finding the limiting solution of an ordinary differential equation which is given in terms of the gradient flow associated with the cost function. Several properties are obtained concerning the gradient flow. For example, it is shown that the flow contains a subsequence convergent to a locally optimal output feedback gain. In the special case of state feedback the flow is guaranteed to converge to the optimal gain. The effectiveness of the method is demonstrated by an example.

Multivariable non‐linear controller for a synchronous generator

Abstract: The problem of multivariable controller synthesis for a turbogenerator is considered. Using the feedback linearization approach for multi-input non-linear systems, the existence of a linearizing state feedback for the improved reduced-order model of a turbogenerator is proved and the simplest form of this feedback is derived. Application of the obtained non-linear controller to the multivariable control of a turbogenerator provides very good results not only for the reduced-order model but also for the ‘exact’ (Park's) model of a turbogenerator. Simulations of fault and post-fault conditions in the obtained nonlinear control system confirm its superiority over a system with a voltage regulator and power system stabilizer.

Optimal control of the penicillin G fed-batch fermentation: An analysis of the model of heijnen et al.

Abstract: This paper presents the application of optimal control theory in determining the optimal feed rate profile for the penicillin G fed-batch fermentation, using a mathematical model based on balancing methods. Since this model does not fulfil all requisites for standard optimal control, we propose a sequence of new models — that converges to the original one in a smooth way — to which the standard techniques are applicable. The unusual optimization of some initial conditions is included. We then state the conjecture that allows us to obtain the optimal control for the original model. The enormous gains in production and the vanishing of the characteristic biphasic behaviour through feed rate profile optimization raise some questions concerning the validity of this model. In this way this optimal control study can prove to be very useful for model discrimination purposes. Furthermore, mathematical and microbial insights lead to the construction of a suboptimal heuristic strategy — which we show to be a limiting case of the optimal scheme — that can serve as a basis for the development of robust, model-independent, optimal adaptive control schemes.

Necessary optimality conditions for a generalized problem of production scheduling

Abstract: This communication concerns a derivation and study of necessary optimality conditions for production scheduling. The continuous-time optimal control problem is stated for the analysis of extremal behaviour of a flexible flow shop. This is accomplished by means of the maximum principle, which is applied to the problem with irregular constraints and three typical forms of objective function.

H∞ control for a singularly perturbed aircraft model

Abstract: State space solutions for H∞ control are formulated for composite state feedback control of singularly perturbed systems. Scaled frequency domain realizations of dynamic systems are developed and used to generate H∞ solutions that approach the H2 solution as an H∞-norm design parameter is allowed to approach infinity. The realizations also account for non-orthogonal cost function terms that occur naturally in the slow subsystem design for singularly perturbed systems. Traditional H2 results are also provided. The formulation is applied to compute reduced, composite and full-order controller gains for the longitudinal dynamics of an F-8 aircraft.

Mixed H2/H∞ control with pole placement in a class of regions

Abstract: The problem of mixed H2/H∞ control with pole placement is considered for linear time-invariant systems. This is the problem of determining a controller for linear time-invariant systems which minimizes the H2-norm of a certain closed-loop transfor function subject to an H∞-norm constraint on another closed-loop transfer function and an additional constraint on the location of the closed-loop poles in the complex plane. An optimization problem is posed for the pole-constrained H2/H∞, problem in such a way that the objective function is expressed as a weighted sum of the actual H2 cost and its upper bound. A necessary condition for the optimization problem is derived via the Lagrange multiplier technique. The condition involves a set of highly coupled equations. By sacrificing the H2 performance, an alternative optimization problem is posed in order to simplify the necessary condition. An iterative algorithm for solving the coupled equations arising in the necessary conditions is proposed and numerical examples are presented.

A numerical algorithm for singular optimal control synthesis using continuation methods

Abstract: This paper presents an efficient computational method for the synthesis of singular optimal control problems. The proposed numerical procedure consists of two phases. In the first phase the original singular optimal control problem is converted into a non-singular one by adding to the performance index a perturbed (or weighted) energy term. The resultant boundary value problem can easily be solved for an appropriately large value of the perturbation parameter. In the second phase the solution obtained from the first phase is refined in a systematic manner based on continuation methods (imbedding methods or homotopy methods) until the optimal (or suboptimal) solution to the original problem is achieved. One of the major advantages of the proposed algorithm is that the resultant two-point boundary value problem need be solved just once for a properly large perturbation parameter and the refinement of the solution is accomplished by solving a set of initial value problems sequentially and/or in parallel as the perturbation parameter goes to zero. The proposed algorithm is therefore computationally efficient and applicable to a large class of optimal control problems with various boundary conditions (e.g. fixed and free terminal time). The practicability of the method is demonstrated by computer simulations on an example problem.

Discrete time lq design from the viewpoint of the inverse optimal regulator

Abstract: In this communication a new design method for the discrete-time LQ regulator based on the solution of the inverse optimal regulator is presented. The main feature of the method is that it requires no Riccati solution in addition to having a number of tuning parameters which help to shape a desirable response of the closed-loop system.

Differential dynamic programming technique for optimal control

Abstract: Continuous- and discrete-time differential dynamic programming (DDP) approaches to solve general optimal control problems are described and analysed. A comparison of the two approaches shows the continuous-time approach to be more general and flexible compared with the discrete-time approach, since it is not tied to any discretization scheme. A comparison of DDP with the non-linear programming (NLP) approach is also given. Three structural control problems — a linear model of a space structure, a single degree of freedom non-linear impact absorber and a non-linear flexible beam subjected to an impulsive load — are used to numerically evaluate the continuous- and discrete-time DDP approaches. Several grid sizes are used to show that the continuous-time approach with a reasonable number of grid points is more accurate and efficient (in most cases) than the discrete-time approach. It is therefore recommended to fully develop and evaluate the technique for the optimal control of large-scale systems.

A decomposition approach to suboptimal control of discrete-time systems

Abstract: A suboptimal controller for a class of discrete-time systems is presented. The controller is derived by first solving ‘off-line’ a simplified optimal control problem obtained by neglecting part of the system state and by considering a larger time step, then by solving ‘on-line’ at each time step an optimization problem based on the results of the previously solved ‘off-line’ problem. A simple numerical example is presented to illustrate the control scheme.

Optimization and control of a fedbatch fermentation process

Abstract: This paper describes the optimization problem of a fedbatch fermentation process for alcohol production which initially consists of minimizing a time-optimal global criterion. Dynamic programming methods have previously allowed one to solve this global optimization problem and to deduce a local criterion to be maximized at each sampling step by using non-linear programming techniques. It is then stated that under a few realistic constraints the optimization problem can be translated into the regulation problem of the substrate concentration in the fermentor. The optimal value is a function of a certain growth model. A control law based on the quotient between measurement and optimal value is then implemented. Simulation results and real-life experiments validate the problem transformation.

An application of optimal control theory to the trajectory tracking of rigid robot manipulators

Abstract: In this paper optimal control theory is applied to the trajectory tracking problem for a rigid robot manipulator. First a stability result is developed showing the relationship between optimal control theory and the global exponential stability of a class of second-order non-linear systems. Next this class of nonlinear systems is shown to contain the trajectory tracking error dynamics for a rigid robot manipulator when using a modified computed torque controller with an auxiliary input. Finally the stability result is applied to the tracking error dynamics to determine the auxiliary input to the robot controller. The result is a non-linear feedback control law which provides exponentially stable position and velocity tracking for a rigid robot manipulator without complete feedback linearization of the robot dynamics.

Optimization of the protrusion shape for a Couette-type flow

Abstract: A study is made of the longitudinal 2D viscous steady flow and heat flux between two plates. Optimal shape design problems are solved in explicit form and shown to have unique global extrema. Conformal mappings are used to bring the problems into a fixed domain and solve them as Dirichlet boundary value problems in the form of Cauchy integrals and series expansions. For the simplest problem statement the optimum is shown to coincide with the well-known concrete dam outline of constant hydraulic gradient.

Identification of rudder—yaw and rudder—roll steering models by using recursive prediction error techniques

Abstract: Linear and non-linear recursive prediction error (RPE) methods have been employed for the purpose of identification of the dynamics of ship rudder—yaw and —roll motions. The evaluations of the goodness of model structures have been carried out by comparing simulated outputs from the identified model with real measurements. In the linear case Nomoto's first-order model has been initially considered for the transfer functions between rudder—yaw and —sway, with a second-order model for rudder—roll with coupling effects considered only between sway—yaw and —roll. Identification results have shown that such a model is not appropriate. Changing the model to second-order between rudder—sway and —yaw and further considering the coupling effects between roll angle—sway and —yaw shows that a significant improvement is achieved, while the coupling effect between yaw and roll appears negligible. In the nonlinear case effects due to various non-linear terms have been studied with the non-linear RPE algorithm.

Optimal operation of solar heat storage with off-peak energy price incentive

Abstract: The problem of minimum cost control of a solar-assisted heat pump/heat storage system is considered The system is configured to have separate storage for collected solar heat and high-temperature heat, so that night-time (‘off-peak’) operation of the heat pump is possible The problem of finding the optimal operating strategy when electrical energy is priced at a discount during off-peak hours is considered The solution of the periodic optimization problem is obtained through the use of state increment dynamic programming, simplified through the use of state feedback linearization/decoupling and augmented with a label-tracing procedure to identify the periodic but unknown initial/terminal state The algorithm is applied for a typical case study The results lead to several observations concerning economical operating strategies

Optimal pulsing in an advertising diffusion model

Abstract: Empirical evidence indicates that a given moderate number of ads per year may achieve higher effect when concentrated in flights than when spread equally. In the control-theoretic literature a few approaches have been developed for which the optimal policy is a pulsing schedule. The present communication analyses a two-state diffusion advertising model for repeat purchasing. Recognizing the interaction between customers and potential buyers, the optimal advertising policy turns out to be a persistent periodic oscillation. This provides a further interesting example in marketing for the existence of a stable limit cycle.

A multi-input/multi-output optimal PI controller for redundant robots in the presence of flexible disturbances

Abstract: The two-stage synthesis of a multi-input/multi-output optimal proportional-integral (PI) controller is described for linear, time-invariant systems. In the first stage the PI controller is designed by solving a steady state algebraic Riccati equation. As a result, the optimal cost is expressed in terms of the system's constant output set-points. In the second stage the cost is further reduced by optimally selecting the output set-points to minimize a static quadratic performance index subject to linear algebraic constraints. The design framework is applied to a planar redundant robotic manipulator equipped with four joints and mounted at the tip of a long flexible arm. We then address the problem of self-motion control in the presence of vibratory disturbances.

Terrain tracking based on optimal aim strategies

Abstract: This short communication discusses two recently proposed terrain-tracking schemes, reformulates both schemes as special cases of a previously proposed optimal aim strategy, outlines a more precise and explicit continuous-time control law based on a fixed-point neural network, describes a more rapid and effective discrete-time control law based on a fixed-point iteration process, and derives criteria for matching reference models to the dynamical characteristics of individual systems.

Discrete time optimal control of linear time-delay systems

Abstract: A discrete time optimal control for linear time-delay systems is developed to ensure that all closed-loop eigenvalues will lie inside a circular region centred at (β;, 0) with radius α. It is shown that by suitable manipulations the problem can be reduced to a standard discrete time quadratic regulator problem. An illustrative example is included to demonstrate the applicability of the proposed method.

New performance bounds of continuous‐time LQ systems with uncertain parameters

Abstract: In this short communication we present new upper and lower performance bounds of LQ (linear—quadratic) systems with time-varying parametric uncertainties. These new performance bounds improve on those proposed by Wang and Kuo by giving more accurate measures of performance robustness. We use the same example as that used by Wang and Kuo to show that the improvements are substantial.

Further development of an approximate loop transfer recovery methodology for fixed order dynamic compensator design

Abstract: This paper further develops a methodology for designing fixed order compensators. The formulation uses an approximate loop transfer recovery (LTR) method to achieve performance and stability subject to a disturbance and unstructured uncertainty at either the plant input or the plant output. The approximate LTR methodology has the advantage that the quadratic performance index weighting matrices are uniquely defined. To reduce sensitivity to real parameter uncertainty, the performance index is modified to include a penalty on the closed-loop sensitivity dynamics. The formulation avoids the explicit introduction of sensitivity states into the performance index and thus does not increase the overall dimension of the problem. An example design for the coupled mass benchmark control problem is provided to demonstrate this methodology.