A minimax optimal control strategy for partially observable uncertain quasi-Hamiltonian systems

A stochastic minimax optimal control strategy for partially observable uncertain quasi-Hamiltonian systems is proposed. First, the stochastic optimal control problem of a partially observable nonlinear uncertain quasi-Hamiltonian system is converted into that of a completely observable linear uncertain system based on a theorem due to Charalambous and Elliot. Then, the converted stochastic optimal control problem is solved by a minimax optimal control strategy based on stochastic averaging method and stochastic differential game. The worst-case disturbances and the optimal controls are obtained by solving a Hamilton-Jacobi-Isaacs (HJI) equation. As an example, the stochastic minimax optimal control of a partially observable Duffing–van der Pol oscillator with uncertain disturbances is worked out in detail to illustrate the procedure and effectiveness of the proposed control strategy.     

Optimal bounded control for nonlinear stochastic smart structure systems based on extended Kalman filter

An optimal bounded control strategy for smart structure systems as controlled Hamiltonian systems with random excitations and noised observations is proposed. The basic dynamic equations for a smart structure system with smart sensors and actuators are firstly given. The nonlinear stochastic control system with noised observations is then obtained from the simplified smart structure system, and the system is expressed by generalized Hamiltonian equations with control, random excitation and dissipative forces. The optimal control problem for nonlinear stochastic systems with noised observations includes two parts: optimal state estimation and optimal response control based on estimated states, which are coupled each other. The probability density of optimally estimated systems has generally infinite dimensions based on the separation theorem. The proposed optimal control strategy gives an approximate separate solution. First, the optimally estimated system state is determined by the observations based on the extended Kalman filter, and the estimated nonlinear system with controls and stochastic excitations is obtained which has finite-dimensional probability density. Second, the dynamical programming equation for the estimated system is determined based on the stochastic dynamical programming principle. The control boundedness due to actuator saturation is considered, and the optimal bounded control law is obtained by the programming equation with the bounded control constraint. The optimal control depends on the estimated system state which is determined by noised observations. The proposed optimal bounded control strategy is finally applied to a single-degree-of-freedom nonlinear stochastic system with control and noised observation. The remarkable vibration control effectiveness is illustrated with numerical results. Thus the proposed optimal bounded control strategy is promising for application to nonlinear stochastic smart structure systems with noised observations.     

随机激励的耗散的Hamilton系统理论的研究进展

近几年中,利用Ｈａｍｉｌｔｏｎ系统的可积性与共振性概念及Ｐｏｉｓｓｏｎ括号性质等,提出了高斯白噪声激励下多自由度非线性随机系统的精确平稳解的泛函构造与求解方法,并在此基础上提出了等效非线性系统法,提出了拟Ｈａｍｉｌｔｏｎ系统的随机平均法,并在该法基础上研究了拟Ｈａｍｉｌｔｏｎ系统随机稳定性、随机分岔、可靠性及最优非线性随机控制,从而基本上形成了一个非线性随机动力学与控制的Ｈａｍｉｌｔｏｎ理论框架．本文简要介绍了这方面的进展．     

拟哈密顿系统非线性随机最优控制

主要介绍近十几年来拟哈密顿系统非线性随机最优控制理论方法及其应用的研究成果, 包括基于拟哈密顿系统随机平均法与随机动态规划原理的非线性随机最优控制基本策略, 即响应极小化控制、随机稳定化、首次穿越损坏最小化控制、以概率密度为目标的控制, 为将它们应用于工程实际而作的部分可观测系统最优控制、有界控制、时滞控制、半主动控制、极小极大控制的进一步研究, 以及综合考虑这些实际问题的非线性随机最优控制的综合策略, 非线性随机最优控制在滞迟系统、分数维系统等中的若干应用, 介绍与这些研究有关的背景, 并指出今后有待进一步研究的问题.     

A NEW STOCHASTIC OPTIMAL CONTROL STRATEGY FOR HYSTERETIC MR DAMPERS

I. INTRODUCTION Magneto-rheological (MR) ?uid as a smart material possesses fairly good essential characteristics suchas reversible change between liquid and semi-solid in milliseconds with a controllable yield strengthwhen exposed to a magnetic ?eld. A…     

Stochastic minimax semi-active control for MDOF nonlinear uncertain systems under combined harmonic and wide-band noise excitations using MR dampers

A stochastic minimax semi-active control strategy for multi-degrees-of-freedom (MDOF) strongly nonlinear systems under combined harmonic and wide-band noise excitations is proposed. First, a stochastic averaging procedure is introduced for controlled uncertain strongly nonlinear systems using generalized harmonic functions and the control forces produced by Magneto-rheological (MR) dampers are split into the passive part and the active part. Then, a worst-case optimal control strategy is derived by solving a stochastic differential game problem. The worst-case disturbances and the optimal semi-active controls are obtained by solving the Hamilton–Jacobi–Isaacs (HJI) equations with the constraints of disturbance bounds and MR damper dynamics. Finally, the responses of optimally controlled MDOF nonlinear systems are predicted by solving the Fokker–Planck–Kolmogorov (FPK) equation associated with the fully averaged Itô equations. Two examples are worked out in detail to illustrate the proposed control strategy. The effectiveness of the proposed control strategy is verified by using the results from Monte Carlo simulation.     

STOCHASTIC OPTIMAL VIBRATION CONTROL OF PARTIALLY OBSERVABLE NONLINEAR QUASI HAMILTONIAN SYSTEMS WITH ACTUATOR SATURATION

An optimal vibration control strategy for partially observable nonlinear quasi Hamiltonian systems with actuator saturation is proposed. First,a controlled partially observable non-linear system is converted into a completely observable linear control system of finite dimension based on the theorem due to Charalambous and Elliott. Then the partially averaged It stochastic differential equations and dynamical programming equation associated with the completely observable linear system are derived by using the stochastic averaging method and stochastic dynamical programming principle,respectively. The optimal control law is obtained from solving the final dynamical programming equation. The results show that the proposed control strategy has high control effectiveness and control effciency.     

Optimal bounded control of stochastically excited MDOF nonlinear viscoelastic systems

A new procedure for designing optimal bounded control of stochastically excited multi-degree-of-freedom (MDOF) nonlinear viscoelastic systems is proposed based on the stochastic averaging method and the stochastic maximum principle. First, the system is formulated as a quasi-integrable Hamiltonian system with viscoelastic terms and each viscoelastic term is replaced approximately by an elastically restoring force and a visco-damping force based on the randomly periodic behavior of the motion of quasi-integrable Hamiltonian system. Thus, a stochastically excited MDOF nonlinear viscoelastic system is converted to an equivalent quasi-integrable Hamiltonian system without viscoelastic terms. Then, by applying stochastic averaging, the system is further reduced to a partially averaged system of less dimension. The adjoint equation and maximum condition for the optimal control problem of the partially averaged system are derived by using the stochastic maximum principle, and the optimal bounded control force is determined from the maximum condition. Finally, the probability and statistics of the stationary response of optimally controlled system are obtained by solving the Fokker–Plank–Kolmogorov equation (FPK) associated with the fully averaged Itô equation of the controlled system. An example is worked out to illustrate the proposed procedure and its effectiveness.     

STOCHASTIC OPTIMAL CONTROL FOR THE RESPONSE OF QUASI NON-INTEGRABLE HAMILTONIAN SYSTEMS~

A strategy is proposed based on the stochastic averaging method for quasi nonintegrable Hamiltonian systems and the stochastic dynamical programming principle. The proposed strategy can be used to design nonlinear stochastic optimal control to minimize the response of quasi non-integrable Hamiltonian systems subject to Gaussian white noise excitation. By using the stochastic averaging method for quasi non-integrable Hamiltonian systems the equations of motion of a controlled quasi non-integrable Hamiltonian system is reduced to a one-dimensional averaged Ito stochastic differential equation. By using the stochastic dynamical programming principle the dynamical programming equation for minimizing the response of the system is formulated.The optimal control law is derived from the dynamical programming equation and the bounded control constraints. The response of optimally controlled systems is predicted through solving the FPK equation associated with It5 stochastic differential equation. An example is worked out in detail to illustrate the application of the control strategy proposed.     

A note on polynomial chaos expansions for designing a linear feedback control for nonlinear systems

This paper presents a polynomial chaos-based framework for designing optimal linear feedback control laws for nonlinear systems with stochastic parametric uncertainty. The spectral decomposition of the original stochastic dynamical model in an orthogonal polynomial basis, prescribed by the Wiener–Askey scheme, provides a deterministic model from which the optimal linear control law is designed. Optimality of the proposed control law is proved by solving the Hamilton–Jacobi–Bellman equation, and asymptotic stability of the controlled nonlinear systems is guaranteed in the Lyapunov sense. We are especially interested in synchronization of chaotic systems. For this reason, the control strategy is applied in the trajectory tracking of periodic orbits for the Duffing oscillator and the Rössler system with uncertain stochastic parameters and initial conditions. The results are verified with Monte Carlo simulations.     

Analysis of the structure of generalized forces in the Lagrange equations

Stability of mechanical systems in the sense of Thomson and Tait [1] can be judged from the type of forces applied to them. The forces are usually divided into potential (conservative), circular, dissipative, accelerating, gyroscopic, etc. The decomposition itself of generalized positional forces into conservative and properly nonconservative forces is well known for the case in which these forces linearly depend on the generalized coordinates (e.g., see [1–5]). Such a decomposition is associated with the unique representation of an arbitrary matrix of these forces as the sum of symmetric and skew-symmetric parts. Generalized forces linearly depending on the velocities can in a similar way be divided into dissipative and gyroscopic parts. In the present paper, we show how the same decomposition can be performed in the general nonlinear case.     

Optimal bounded control of quasi-nonintegrable Hamiltonian systems using stochastic maximum principle

A new procedure for designing optimal bounded control of quasi-nonintegrable Hamiltonian systems with actuator saturation is proposed based on the stochastic averaging method for quasi-nonintegrable Hamiltonian systems and the stochastic maximum principle. First, the stochastic averaging method for controlled quasi-nonintegrable Hamiltonian systems is introduced. The original control problem is converted into one for a partially averaged equation of system energy together with a partially averaged performance index. Then, the adjoint equation and the maximum condition of the partially averaged control problem are derived based on the stochastic maximum principle. The bounded optimal control forces are obtained from the maximum condition and solving the forward–backward stochastic differential equations (FBSDE). For infinite time-interval ergodic control, the adjoint variable is stationary process, and the FBSDE is reduced to an ordinary differential equation. Finally, the stationary probability density of the Hamiltonian and other response statistics of optimally controlled system are obtained by solving the Fokker–Plank–Kolmogorov equation associated with the fully averaged Itô equation of the controlled system. For comparison, the bang–bang control is also presented. An example of two degree-of-freedom quasi-nonintegrable Hamiltonian system is worked out to illustrate the proposed procedure and its effectiveness. Numerical results show that the proposed control strategy has higher control efficiency and less discontinuous control force than the corresponding bang–bang control at the price of slightly less control effectiveness.     

Feedback minimization of first-passage failure of quasi integrable Hamiltonian systems

A nonlinear stochastic optimal control strategy for minimizing the first-passage failure of quasi integrable Hamiltonian systems (multi-degree-of-freedom integrable Hamiltonian systems subject to light dampings and weakly random excitations) is proposed. The equations of motion for a controlled quasi integrable Hamiltonian system are reduced to a set of averaged Itô stochastic differential equations by using the stochastic averaging method. Then, the dynamical programming equations and their associated boundary and final time conditions for the control problems of maximization of reliability and mean first-passage time are formulated. The optimal control law is derived from the dynamical programming equations and the control constraints. The final dynamical programming equations for these control problems are determined and their relationships to the backward Kolmogorov equation governing the conditional reliability function and the Pontryagin equation governing the mean first-passage time are separately established. The conditional reliability function and the mean first-passage time of the controlled system are obtained by solving the final dynamical programming equations or their equivalent Kolmogorov and Pontryagin equations. An example is presented to illustrate the application and effectiveness of the proposed control strategy.     

Stochastic optimal control of cable vibration in plane by using axial support motion

A stochastic optimal control strategy for a slightly sagged cable using support motion in the cable axial direction is proposed.The nonlinear equation of cable motion in plane is derived and reduced to the equations for the first two modes of cable vibration by using the Galerkin method.The partially averaged Ito equation for controlled system energy is further derived by applying the stochastic averaging method for quasi-non-integrable Hamiltonian systems.The dynamical programming equation for the controlled system energy with a performance index is established by applying the stochastic dynamical programming principle and a stochastic optimal control law is obtained through solving the dynamical programming equation.A bilinear controller by using the direct method of Lyapunov is introduced.The comparison between the two controllers shows that the proposed stochastic optimal control strategy is superior to the bilinear control strategy in terms of higher control effectiveness and efficiency.     

STOCHASTIC OPTIMAL CONTROL OF HYSTERETIC SYSTEMS UNDER EXTERNALLY AND PARAMETRICALLY RANDOM EXCITATIONS

A stochastic optimal control method for nonlinear hysteretic systems under exter-nally and/or parametrically random excitations is presented and illustrated with an example ofhysteretic column system.A hysteretic system subject to random excitation is first replaced bya nonlinear non-hysteretic stochastic system.An It stochastic differential equation for the to-tal energy of the system as a one-dimensional controlled diffusion process is derived by usingthe stochastic averaging method of energy envelope.A dynamical programming equation is thenestablished based on the stochastic dynamical programming principle and solved to yield the op-timal control force.Finally,the responses of uncontrolled and controlled systems are evaluatedto determine the control efficacy.It is shown by numerical results that the proposed stochasticoptimal control method is more effective and efficient than other optimal control methods.     

On the stability of a circular system subjected to nonlinear dissipative forces

A circular system is a mechanical system subjected to potential forces and positional nonconservative forces (circular forces). The latter linearly depend on the coordinates and are characterized by a skew-symmetric matrix. The influence of linear dissipative forces on the stability of a circular system is ambiguous: on the one hand, they can stabilize a stable circular system (making it asymptotically stable); on the other hand, they can destabilize it [1–4]. The action of linear dissipative forces on a circular system results in the so-called destabilization paradox: the stability threshold decreases by a finite value.A detailed survey of this phenomenon can be found in [5]. The destabilization effect is also preserved under the action of nonlinear dissipative forces. The influence of these forces on the stability of the Ziegler pendulum with a tracking force was studied in [6]. It was shown that the critical value of the tracking force decreases by a finite value. A similar effect was discovered in the analysis of a continual system in [7].In the present paper, we study how nonlinear dissipative forces affect the stability of the equilibrium of a circular mechanical system with two degrees of freedom. The stability problem is solved without any references to specific mechanical systems. The results are used to analyze the stability of a gimbal gyro with allowance for dry friction in the rotor bearings.     

On the stability of equilibria of nonholonomic systems with nonlinear constraints

Lyapunov＇s first method, extended by V. V. Kozlov to nonlinear mechani- cal systems, is applied to the study of the instability of the position of equilibrium of a mechanical system moving in the field of conservative and dissipative forces. The mo- tion of the system is limited by ideal nonlinear nonholonomic constraints. Five cases determined by the relationship between the degree of the first nontrivial polynomials in Maclaurin＇s series for the potential energy and the functions that can be generated from the equations of nonlinear nonholonomic constraints are analyzed. In the three eases, the theorem on the instability of the position of equilibrium of nonholonomic systems with linear homogeneous constraints （V. V. Kozlov （1986）） is generalized to the case of nonlin- ear nonhomogeneous constraints. In the other two cases, new theorems are set extending the result from V. V. Kozlov （1994） to nonholonomic systems with nonlinear constraints.     

非线性随机动力学与控制的哈密顿理论框架

 近几年来，笔者提出与发展了随机激励的耗散的哈密顿系统理 论，包括精确平稳解、等效非线性系统法、拟哈密顿系统随机平均法、 拟哈密顿系统的随机稳定性与随机分岔、首次穿越损坏分析方法及非 线性随机最优控制策略，从而构成了一个非线性随机动力学与控制的 哈密顿理论框架.本文简要介绍这一理论框架.     

Time-delayed stochastic optimal control of strongly non-linear systems with actuator saturation by using stochastic maximum principle

A time-delayed stochastic optimal bounded control strategy for strongly non-linear systems under wide-band random excitations with actuator saturation is proposed based on the stochastic averaging method and the stochastic maximum principle. First, the partially averaged Itô equation for the system amplitude is derived by using the stochastic averaging method for strongly non-linear systems. The time-delayed feedback control force is approximated by a control force without time delay based on the periodically random behavior of the displacement and velocity of the system. The partially averaged Itô equation for the system energy is derived from that for the system amplitude by using Itô formula and the relation between system amplitude and system energy. Then, the adjoint equation and maximum condition of the partially averaged control problem are derived based on the stochastic maximum principle. The saturated optimal control force is determined from maximum condition and solving the forward–backward stochastic differential equations (FBSDEs). For infinite time-interval ergodic control, the adjoint variable is stationary process and the FBSDE is reduced to a ordinary differential equation. Finally, the stationary probability density of the Hamiltonian and other response statistics of optimally controlled system are obtained from solving the Fokker–Plank–Kolmogorov (FPK) equation associated with the fully averaged Itô equation of the controlled system. For comparison, the optimal control forces obtained from the time-delayed bang–bang control and the control without considering time delay are also presented. An example is worked out to illustrate the proposed procedure and its advantages.     

结构振动的滑模变结构半主动控制

研究应用磁流变阻尼器(MRD)对结构振动半主动控制的算法和原理。研制并对磁流变阻尼器进行了阻尼特性实验，采用非线性滞回双粘性模型描述磁流变阻尼器的阻尼特性，模型结果与实验结果非常一致。采用滑模控制算法和趋近律方法设计了半主动控制器。利用滑模控制方法所建立的控制器，本文给出了地震激励下结构振动半主动控制算例。计算分析表明，半主动滑模控制具有控制效果明显、鲁棒性好等优点，是一种非常有发展前途的控制方法。