Optimal Noise Rejection in Structural Analysis by Means of Generalized Sampled-Data Hold Functions

In this paper, a technique for optimal noise rejection, based on generalized sampled-data hold functions is applied to the control of civil engineering structures. The technique consists in suitably modulating the sampled outputs of the system under control by periodically varying functions in order to attenuate the effect of the disturbances on the system states to an acceptable level, by minimizing a quadratic cost function. This minimization is performed by feeding back the outputs of the system, which are assumed to be corrupted by measurement noise. Moreover, in the present paper, the robustness properties of the GSHF based optimal regulator is analyzed and guaranteed stability margins, expressed in terms of elementary cost and system matrices, are proposed for such a type of optimal regulators. The effectiveness of the method is demonstrated by various simulation results. The results of the paper can be used to assess the detrimental effect of noise on the closed-loop system and the tradeoff involved in assuring good sampled-data performance and sufficient robustness.     

Optimal infinite-horizon feedback laws for a general class of constrained discrete-time systems: Stability and moving-horizon approximations

Stability results are given for a class of feedback systems arising from the regulation of time-varying discrete-time systems using optimal infinite-horizon and moving-horizon feedback laws. The class is characterized by joint constraints on the state and the control, a general nonlinear cost function and nonlinear equations of motion possessing two special properties. It is shown that weak conditions on the cost function and the constraints are sufficient to guarantee uniform asymptotic stability of both the optimal infinite-horizon and moving-horizon feedback systems. The infinite-horizon cost associated with the moving-horizon feedback law approaches the optimal infinite-horizon cost as the moving horizon is extended.     

SQP-methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and real-time control

Parametric nonlinear optimal control problems subject to control and state constraints are studied. Two discretization methods are discussed that transcribe optimal control problems into nonlinear programming problems for which SQP-methods provide efficient solution methods. It is shown that SQP-methods can be used also for a check of second-order sufficient conditions and for a postoptimal calculation of adjoint variables. In addition, SQP-methods lead to a robust computation of sensitivity differentials of optimal solutions with respect to perturbation parameters. Numerical sensitivity analysis is the basis for real-time control approximations of perturbed solutions which are obtained by evaluating a first-order Taylor expansion with respect to the parameter. The proposed numerical methods are illustrated by the optimal control of a low-thrust satellite transfer to geosynchronous orbit and a complex control problem from aquanautics. The examples illustrate the robustness, accuracy and efficiency of the proposed numerical algorithms.     

A stochastic approach for the optimization of open-loop engine control systems

The operation of sensors and actuators in engine control systems is always affected by errors, which are stochastic in nature. In this paper it is shown that, because of the non-linear interactions between engine performance and control laws in an open-loop engine control system, these errors can give rise to unexpected deviations of control variables, fuel consumption and emissions from the optimal values, which are not predictable in an elementary way.A model for vehicle performance evaluation on a driving cycle is presented, which provides the expected values of fuel consumption and emissions in the case of stochastic errors in sensors and actuators, utilizing only steady-state engine data.The stochastic model is utilized to obtain the optimal control laws; the resultant non-linear constrained minimization problem is solved by an Augmented Lagrangian approach, using a Quasi-Newton technique. The results of the stochastic optimization analysis indicate that significant reductions in performance degradation may be achieved with respect to the solutions provided by the classical deterministic approach.     

Computation of optimal controls for a nonlinear stochastic third-order system

This paper deals with the computation of optimal feedback control laws for a nonlinear stochastic third-order system in which the nonlinear element is not completely specified. It is shown that, due to the structure of the system, the optimal feedback control law, whenever it exists, is not unique. Also, it is shown that, in order to implement an optimal feedback control law, a nonlinear partial differential equation has to be solved. A finite-difference algorithm for the solution of this equation is suggested, and its efficiency and applicability are demonstrated with examples.     

Improved Optimal Control Methods Based Upon the Adjoining Cell Mapping Technique

In this paper, cell mapping methods are studied and refined for the optimal control of autonomous dynamical systems. First, the method proposed by Hsu (Ref. 1) is analyzed and some improvements are presented. Second, adjoining cell mapping (ACM), based on an adaptive time of integration (Refs. 2–3), is formulated as an alternative technique for computing optimal control laws of nonlinear systems, employing the cellular state-space approximation. This technique overcomes the problem of determining an appropriate duration of the integration time for the simple cell mapping method and provides a suitable mapping for the search procedures. Artificial intelligence techniques, together with some improvements on the original formulation lead to a very efficient algorithm for computing optimal control laws with ACM (CACM). Several examples illustrate the performance of the CACM algorithm.     

On ε-optimal continuous selectors and their application in discounted dynamic programming

A quadratic regulator problem for a class of nonlinear systems is considered in which the control cost is multiplied by a small parameter, which becomes a so-called cheap control problem. Conditions are found under which the minimum cost becomes zero (perfect regulation) and the linear part in the optimal control law becomes dominant as the small parameter goes to zero. Near optimality of control laws truncated from the optimal control law in series form is also found.     

Info-gap forecasting and the advantage of sub-optimal models

We consider forecasting in systems whose underlying laws are uncertain, while contextual information suggests that future system properties will differ from the past. We consider linear discrete-time systems, and use a non-probabilistic info-gap model to represent uncertainty in the future transition matrix. The forecaster desires the average forecast of a specific state variable to be within a specified interval around the correct value. Traditionally, forecasting uses a model with optimal fidelity to historical data. However, since structural changes are anticipated, this is a poor strategy. Our first theorem asserts the existence, and indicates the construction, of forecasting models with sub-optimal-fidelity to historical data which are more robust to model error than the historically optimal model. Our second theorem identifies conditions in which the probability of forecast success increases with increasing robustness to model error. The proposed methodology identifies reliable forecasting models for systems whose trajectories evolve with Knightian uncertainty for structural change over time. We consider various examples, including forecasting European Central Bank interest rates following 9/11.     

A new approach to discretization applied to a continuous, nonlinear, sliding-mode-like control using nonsmooth analysis

This paper introduces a new approach to discretization of nonlinearcontrol laws with a Lipschitz property. The sampling time isdefined as a parameter, which must be selected sufficientlysmall so that the closed-loop system is stable. In contrastto similar results, the stabilizing effect of the control istaken into account. This can result in less conservative constraintson the minimum sampling frequency. The discretization techniquesare explained on a general nonlinear model and applied to thediscretization of a novel nonlinear, robust sliding-mode-likecontrol law. Similar robustness features as for continuous controlare demonstrated. Nonsmooth Lyapunov functions are used forthe discretized sliding-mode-like control introducing cone shapedregions of the state space. One of these cone shaped regionscoincides with a cone shaped layer around the sliding mode definedby the continuous sliding-mode-like control. A stability theoremusing nonsmooth Lyapunov functions is provided.     

Synchronization of mechanical horizontal platform systems in finite time

The horizontal platform system (HPS) is a mechanical device that exhibits rich and chaotic dynamics. In this paper, the problem of finite-time synchronization of two non-autonomous chaotic HPSs is investigated. It is assumed that both drive and response systems are disturbed by model uncertainties, external disturbances and fully unknown parameters. Appropriate update laws are proposed to undertake the unknown parameters. Using the update laws and finite-time control theory, a robust adaptive controller is derived to synchronize the two uncertain HPSs in a given finite time. Subsequently, the effects of input nonlinearities are taken into account and a robust adaptive controller is introduced to synchronize the two uncertain HPSs within a finite time. The finite-time stability and convergence of the proposed schemes are analytically proved. Two illustrative examples are presented to show the robustness and applicability of the proposed adaptive finite-time control techniques.     

Optimal synchronization of Rössler system with complete uncertain parameters

The paper discusses the optimal control for the chaos synchronization of Rössler systems with complete uncertain parameters during finite and infinite time intervals. Based on the Liapunov–Bellman technique, optimal control laws are derived from the conditions that ensure asymptotic stability of the error dynamical system and minimizes the cost transfer of this system from arbitrary state to its equilibrium state. The derived control laws make the states of two identical Rössler systems asymptotically synchronized. Some special cases are introduced. Important numerical simulation is included to show the effectiveness of the optimal synchronization technique.     

Robust Output Tracking Control of an Uncertain Linear System via a Modified Optimal Linear-Quadratic Method

This study investigates the robust output tracking problem for a class of uncertain linear systems. The uncertainties are assumed to be time invariant and to satisfy the matching conditions. According to the selected nominal parameters, an optimal solution with a prescribed degree of stability is determined. Then, an auxiliary input via the use of an adapting factor, connected to the nominal optimal control, is introduced to guarantee the robustness and prescribed degree of stability for the output tracking control of the uncertain linear systems. This method is very simple and effective and can reject bounded uncertainties imposed on the states. A maglev vehicle model example is given to show its effectiveness.     

Guaranteed cost synchronization of complex networks with uncertainties and time‐Varying delays

This article focuses on the problem of Guaranteed cost synchronization of complex networks with uncertainties and time‐Varying delays. Sufficient conditions for the existence of the optimal guaranteed cost control laws are introduced in the light of linear matrix inequalities via the Lyapunov–Krasovskii stability theory. The time‐varying node delays and time‐varying coupling delays are simultaneously regarded in the complex network. The node uncertainties and coupling uncertainties are simultaneously considered as well. Numerical simulations are provided to account for the effectiveness and robustness of the proposed method. The results in this article generalize and improve the corresponding results of the recent works. © 2015 Wiley Periodicals, Inc. Complexity 21: 381–395, 2016     

Closed-form recursive formula for an optimal tracker with terminal constraints

Feedback control laws are derived for a class of optimal finite time tracking problems with terminal constraints. Analytical solutions are obtained for the feedback gain and the closed-loop response trajectory. Such formulations are expressed in recursive forms so that a real-time computer implementation becomes feasible. An example involving the feedback slewing of a flexible spacecraft is given to illustrate the validity and usefulness of the formulations.     

Optimality conditions for the control of a double-membrane complex system by a modal decomposition technique

The optimal of damping out the oscillations of an elastically rectangular double-membrane system by means of point-wise actuators is solved analytically. The membrane is clamped along the boundaries. The motion of the system is initiated by given initial displacement and velocity conditions. The basic control problem is to minimize the deflection and the velocity of displacements at a specified time with the minimum expenditure of actuation energy. A quadratic performance functional is chosen as the cost functional which comprises the functionals of the deflection, velocity and the point-wise actuators. Necessary and sufficient conditions of optimality are investigated. The necessary conditions of optimality are obtained from a variational approach and formulated in the form of degenerate integrals which lead to explicit optimal control laws for the actuators. Numerical results are given for various problem parameters and the efficiency of the control mechanism is investigated.     

Dynamics for controlled 2-D Boussinesq systems with distributed controls

The long-time behavior of solutions for an optimal distributed control problem associated with the Boussinesq equations is studied. First, a quasi-optimal solution for the Boussinesq equations is constructed; this quasi-optimal solution possesses the decay (in time) properties. Then, some preliminary estimates for the long-time behavior of all solutions of the Boussinesq equations are derived. Next, the existence of a solution for the optimal control problem is proved. Finally, the long-time decay properties for the optimal solutions is established.     

Dynamics for Controlled 2D Generalized MHD Systems with Distributed Controls

We study the dynamics of a piecewise (in time) distributed optimal control problem for Generalized MHD equations which model velocity tracking coupled to magnetic field over time. The long-time behavior of solutions for an optimal distributed control problem associated with the Generalized MHD equations is studied. First, a quasi-optimal solution for the Generalized MHD equations is constructed; this quasi-optimal solution possesses the decay (in time) properties. Then, some preliminary estimates for the long-time behavior of all solutions of Generalized MHD equations are derived. Next, the existence of a solution of optimal control problemis proved also optimality system is derived. Finally, the long-time decay properties for the optimal solutions is established.     

Asymptotic Behavior of the Value Functions of Discrete-Time Discounted Optimal Control

We study deterministic discounted optimal control problems associated with discrete-time systems. It is shown that, for small discount rates, the controllability properties of the underlying system can guarantee the convergence of the discounted value function to the value function of the average yield. An application in the theory of exponential growth rates of discrete inclusions is presented. This application motivates the analysis of infinite-horizon optimal control problems with running yields that are unbounded from below.     

Instantaneous control for traffic flow

The solution methods for optimal control problems with coupled partial differential equations as constraints are computationally costly and memory intensive; in particular for problems stated on networks, this prevents the methods from being relevant. We present instantaneous control problems for the optimization of traffic flow problems on road networks. We derive the optimality conditions, investigate the relation to the full optimal control problem and prove that certain properties of the optimal control problem carry over to the instantaneous one. We propose a solution algorithm and compare quality of the computed controls and run‐times. Copyright © 2006 John Wiley & Sons, Ltd.