Nonautonomous regulator problem in Hilbert spaces

In this paper, we study the quadratic optimal control problem on the half linett _o, for nonautonomous control processes in Hilbert spaces. We prove that the quadratic optimal control problem has a solution if, and only if, an associated Riccati equation has a positive solution fortt _o. The optimal control is given in feedback form. If a detectability assumption holds, then we prove that the optimal control is a stabilizing feedback control when the associated Riccati equation has a positive solution which is bounded fortt _o.This work was performed under the auspices of the National Research Council of Italy (CNR).     

Optimal control of distributed parabolic systems with multiple time delays given in the integral form

Various optimal control problems for linear parabolic systemswith multiple time delays given in the integral form are considered.Necessary and sufficient conditions of optimality are derivedfor the Neumann problem. The optimal control is obtained inthe feedback form. Making use of the results of Schwartz, therepresentation of the optimal feedback control is given. A simpleexample of application is also provided.     

Optimal Feedback Control for Linear Systems with Input Delays Revisited

The design problem of optimal feedback control for linear systems with input delays is very important in many engineering applications. Usually, the linear systems with input delays are firstly converted into linear systems without delays, and then all the design procedures are based on the delay-free linear systems. In this way, the feedback controllers are not designed in terms of the original states. This paper presents some new closed-form formula in terms of the original states for the delayed optimal feedback control of linear systems with input delays. We firstly reveal the essential role of the input delay in the optimal control design of the linear system with a single input delay: the input delay postpones the action of the optimal control only. Based on this fact, we calculate the delayed optimal control and find that the optimal state can be represented by a simple closed-form formula, so that the delayed optimal feedback control can be obtained in a simple way. We show that the delayed feedback gain matrix can be “smaller” than that for the controlled system with zero input delay, which implies that the input delay can be considered as a positive factor. In addition, we give a general formula for the delayed optimal feedback control of time-variant linear systems with multiple input delays. To show the effectiveness and advantages of the main results, we present five illustrative examples with detailed numerical simulation and comparison.     

Stability analysis of dynamic collaboration model with control signals on two lanes

In this paper, the influence of control signals on the stability of two-lane traffic flow is mainly studied by applying control theory with lane changing behaviors. We present the two-lane dynamic collaboration model with lateral friction and the expressions of feedback control signals. What is more, utilizing the delayed feedback control theory to the two-lane dynamic collaboration model with control signals, we investigate the stability of traffic flow theoretically and the stability conditions for both lanes are derived with finding that the forward and lateral feedback signals can improve the stability of traffic flow while the backward feedback signals cannot achieve it. Besides, direct simulations are conducted to verify the results of theoretical analysis, which shows that the feedback signals have a significant effect on the running state of two vehicle groups, and the results are same with the theoretical analysis.     

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.     

INFINITE DIMENSIONAL TRACKING OPTIMAL CONTROL VIA DYNAMIC OUTPUT FEEDBACK~*

This paper explores implemantation problems of infinite dimensional linear-quadratictracking optimal control.Based on the closed-loop result,a new formula of optimal controlexpressed by past-time state feedback is proved.From this,on the conditions of observa-bility,expressions of optimal control via dynamic output feedback are derived.The mainfeedback operator functions are given by solution of linear integral equations.     

Realtime control of robots with initial value perturbations via nonlinear programming methods

The mathematical model of an industrial robot with initial value perturbations is considered as a parametric nonlinear control problem subject to control and state constraints. Based on recent stability results for parametric control problems, a robust nonlinear programming method is proposed for computing the sensitivity derivatives of optimal solutions. Real-time control approximations of perturbed optimal solutions are obtained by evaluating a first order Taylor expansion of the perturbed solution. The efficiency of the real-time approximation is demonstrated for the robot model     

Feedback Control of a Distributed Parabolic System with Boundary Condition Involving a Time-Varying Lag

In this paper, a feedback control problem for a distributed-parametersystem with boundary condition involving a time-varying lagis considered. Neccessary and sufficient conditions for optimalityare derived. The optimal control is obtained in feedback form.Estimates for the solutions of parabolic systems with specifiedforms of feedback control are established.     

Nonlinear optimal feedback control for lunar module soft landing

In this paper, the task of achieving the soft landing of a lunar module such that the fuel consumption and the flight time are minimized is formulated as an optimal control problem. The motion of the lunar module is described in a three dimensional coordinate system. We obtain the form of the optimal closed loop control law, where a feedback gain matrix is involved. It is then shown that this feedback gain matrix satisfies a Riccati-like matrix differential equation. The optimal control problem is first solved as an open loop optimal control problem by using a time scaling transform and the control parameterization method. Then, by virtue of the relationship between the optimal open loop control and the optimal closed loop control along the optimal trajectory, we present a practical method to calculate an approximate optimal feedback gain matrix, without having to solve an optimal control problem involving the complex Riccati-like matrix differential equation coupled with the original system dynamics. Simulation results show that the proposed approach is highly effective.     

Determination of the optimal constant output feedback gains for linear discrete systems

The optimal constant gain output feedback control of linear time-invariant discrete systems with respect to an averaged performance criterion is discussed. The algebraic conditions necessary for minimizing the averaged performance cost are derived and a method for computing the constant optimal feedback gain is presented.     

Stochastic optimal time-delay control of quasi-integrable Hamiltonian systems

A nonlinear stochastic optimal time-delay control strategy for quasi-integrable Hamiltonian systems is proposed. First, a stochastic optimal control problem of quasi-integrable Hamiltonian system with time-delay in feedback control subjected to Gaussian white noise is formulated. Then, the time-delayed feedback control forces are approximated by the control forces without time-delay and the original problem is converted into a stochastic optimal control problem without time-delay. After that, the converted stochastic optimal control problem is solved by applying the stochastic averaging method and the stochastic dynamical programming principle. As an example, the stochastic time-delay optimal control of two coupled van der Pol oscillators under stochastic excitation is worked out in detail to illustrate the procedure and effectiveness of the proposed control strategy.     

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.     

最速反馈控制的不变性

变结构控制对系统模型和扰动具有一定的不变性是众所周知的事实。最速反馈控制是以其开关曲线为滑动曲线的变结构控制。本文用变结构控制理论来讨论修正了的最速反馈控制对一定范围的系统扰动具有完全的不变性,即完全能够抑制一定范围的扰动作用,而且闭环系统的所有轨线,在理论上,都以有限时间到达原点。这就为设计高效非线性反馈提供了一条有效途径,还给出了避免高频颤震来实现最速反馈控制的数字化办法。     

Modeling and nonlinear control of a flexible-link manipulator

The problem of modeling and controlling the tip position of a one-link flexible manipulator is considered. The proposed model has been used to investigate the effect of the open-loop control torque profile, and the payload. The control strategy is based on the nonlinear State Dependent Riccati Equation (SDRE) design method in the context of application to robotics and manufacturing systems. In this paper, an experimental test-bed was developed to demonstrate the concept of end-point position feedback on a single-link elastic manipulator, and the control strategy for a single-link flexible manipulator. The controller is designed based on the nonlinear SDRE developed by the authors and applied to a flexible manipulator. The experimental results are compared with conventional PD controller strategy. The results reveal that the nonlinear SDRE controller is near optimal and robustly; and its performance is improved comparing to the PD control scheme.     

Optimal control of parabolic systems with time-varying lags

Various optimal control problems for linear parabolic systemswith time-varying lags are considered. Necessary and sufficientconditions of optimality are derived for the Neumann problem.The optimal control is obtained in the feedback form. Makinguse of the results of Schwart's, the representation of the optimalfeedback control is given.     

The sufficient conditions for continuous ɛ-optimal feedbacks in control problems with a terminal cost

The existence of continuous positional strategies of ?-optimal feedback is proved for linear optimal control problems with a convex terminal cost. These continuous feedbacks are determined from Bellman's equation in ?-perturbed control problems with an integral-terminal cost and a smooth value function. An example is given in which an ?-optimal continuous feedback does not exist. It is shown that the point limit of the ?-optimal feedbacks when ?→0 determines the optimal feedback, that is, a positional strategy and, possibly, a discontinuous strategy.     

Approximation design of optimal controllers for nonlinear systems with sinusoidal disturbances

This paper considers an infinite-time optimal damping control problem for a class of nonlinear systems with sinusoidal disturbances. A successive approximation approach (SAA) is applied to design feedforward and feedback optimal controllers. By using the SAA, the original optimal control problem is transformed into a sequence of nonhomogeneous linear two-point boundary value (TPBV) problems. The existence and uniqueness of the optimal control law are proved. The optimal control law is derived from a Riccati equation, matrix equations and an adjoint vector sequence, which consists of accurate linear feedforward and feedback terms and a nonlinear compensation term. And the nonlinear compensation term is the limit of the adjoint vector sequence. By using a finite term of the adjoint vector sequence, we can get an approximate optimal control law. A numerical example shows that the algorithm is effective and robust with respect to sinusoidal disturbances.     

Implementation of dynamic programming for chaos control in discrete systems

In this paper the control of discrete chaotic systems by designing linear feedback controllers is presented. The linear feedback control problem for nonlinear systems has been formulated under the viewpoint of dynamic programming. For suppressing chaos with minimum control effort, the system is stabilized on its first order unstable fixed point (UFP). The presented method also could be employed to make any desired nth order fixed point of the system, stable. Two different methods for higher order UFPs stabilization are suggested. Afterwards, these methods are applied to two well-known chaotic discrete systems: the Logistic and the Henon Maps. For each of them, the first and second UFPs in their chaotic regions are stabilized and simulation results are provided for the demonstration of performance.     

A general view to relaxation methods in control theory1

A general method of construction of relaxed optimal control problems is proposed. Two standard relaxed problem in optimal control of ordinary differential equation and relation between them are thus obtained as special cases, and besides these relaxed problems are justified in accord with their extremal properties.     

Boundary optimal feedback controller for time-periodic Stokes–Oseen flows

This work considered the Dirichlet boundary optimal control of time-periodic Stokes–Oseen equations. The existence of optimal solution and maximum principle are obtained without assuming that the normal component of the control is equal to zero. Moreover, we get the regularity result of the optimal solution via the Euler–Lagrange system. The existence of solution to the HJB equation is proved. The feedback form of the optimal controller is given, and with this feedback controller, we can get for the solution to the periodic Navier–Stokes equations the property of continuous dependence on the outer force term.