Time-delay stochastic optimal control and stabilization of quasi-integrable Hamiltonian systems

Innovative procedures for the time-delay stochastic optimal control and stabilization of quasi-integrable Hamiltonian systems subject to Gaussian white noise excitations are proposed. First, the problem of time-delay stochastic optimal control of quasi-integrable Hamiltonian systems is formulated and converted into the problem of stochastic optimal control without time delay. Then the converted control problem is solved by applying the stochastic averaging method for quasi-integrable Hamiltonian systems and the stochastic dynamical programming principle. The time-delay feedback stabilization of quasi-integrable Hamiltonian systems is formulated as an ergodic control problem with an un-determined cost function which is determined later by minimizing the largest Lyapunov exponent of the controlled system. As an example, a two-degree-of-freedom quasi-integrable Hamiltonian system with time-delay feedback control forces is investigated in detail to illustrate the procedures and their effectiveness.     

Stochastic averaging of quasi-integrable Hamiltonian systems under bounded noise excitations

A stochastic averaging method is proposed to predict approximately the response of quasi-integrable Hamiltonian systems to bounded noise excitations. Two cases of external resonance only and both internal and external resonances are considered separately. The reduced averaged Fokker–Planck–Kolmogorov equation is solved by using the combination of the finite difference method and the successive over relaxation method. An example of two coupled Duffing oscillators under bounded noise excitations is given to illustrate the application of the proposed procedure. It is shown that the analytical results agree well with those from digital simulation of original system.     

Stochastic minimax optimal time-delay state feedback control of uncertain quasi-integrable Hamiltonian systems

In this paper, free vibrations of symmetric angle-ply laminated cylindrical shells are presented under different thickness variations. The thickness variations of the shells are linear, exponential, and sinusoidal. Spline function techniques are adopted to approximate the displacement functions, and the point collocation method is used to obtain the generalized eigenvalue problem with appropriate boundary conditions. The effects of the geometric parameters, thickness parameters, and material parameters on the frequencies of symmetric angle-ply laminated cylindrical shells with two kinds of material properties under different boundary conditions are analyzed.     

Nonlinear stochastic optimal control of Preisach hysteretic systems

The nonlinear stochastic optimal control of Preisach hysteretic systems is studied, and the control procedure is illustrated with an example of the single-degree-of-freedom Preisach system. The Preisach hysteretic system subjected to a stochastic excitation is first replaced by an equivalent non-hysteretic nonlinear stochastic system with displacement-amplitude-dependent damping and stiffness, by using the generalized harmonic balance technique. Then, the relationship between the displacement amplitude and total system energy is established, and the equivalent damping and stiffness coefficients are expressed as functions of the system energy. The averaged Itô stochastic differential equation for the system energy as one-dimensional controlled diffusion process, is derived by using the stochastic averaging method of energy envelope. For the semi-infinite time-interval ergodic control, the dynamical programming equation is obtained based on the stochastic dynamical programming principle, and is solved to yield the optimal control force. Finally, the Fokker–Planck–Kolmogorov equation associated with the averaged Itô equation is established, and the stationary probability density of the system energy is obtained, from which the variances of the controlled system response and the optimal control force are predicted and the control efficacy is evaluated. Numerical results show that the proposed control strategy for Preisach hysteretic systems is very effective and efficient.     

Parametric optimal control of uncertain systems under an optimistic value criterion

It is well known that the optimal control of a linear quadratic model is characterized by the solution of a Riccati differential equation. In many cases, the corresponding Riccati differential equation cannot be solved exactly such that the optimal feedback control may be a complex time-oriented function. In this article, a parametric optimal control problem of an uncertain linear quadratic model under an optimistic value criterion is considered for simplifying the expression of optimal control. Based on the equation of optimality for the uncertain optimal control problem, an approximation method is presented to solve it. As an application, a two-spool turbofan engine optimal control problem is given to show the utility of the proposed model and the efficiency of the presented approximation method.     

Instantaneous optimal control with acceleration and velocity feedback

In the experimental demonstration of aseismic control systems, difficulties were encountered in the measurement of the displacement response of the structure. During earthquake ground motions, both the building and the ground are moving so that there is no absolute reference for the determination of the displacement response. An optimal control algorithm is proposed herein, which utilizes the measurements of acceleration and velocity responses rather than the displacement and velocity measurements. Such an optimal algorithm is developed based on the instantaneous optimal control theories, and it is evaluated and compared with other available optimal control laws. Numerical results indicate that the performance of the proposed optimal algorithm is as good as that of other optimal control laws currently available. However, the contribution of such an optimal control law to the practical implementation of active control systems for seismic hazard mitigations may be quite significant.     

First-passage failure of quasi linear systems subject to multi-time-delayed feedback control and wide-band random excitation

A procedure for studying the first-passage failure of quasi-linear systems subject to multi-time-delayed feedback control and wide-band random excitation is proposed. The stochastic averaging method for quasi-integrable Hamiltonian systems is first introduced. The backward Kolmogorov equation governing the conditional reliability function and a set of generalized Pontryagin equations governing the conditional moments of first-passage time are then established. The conditional reliability function, the conditional probability density and moments of first-passage time are obtained by solving the backward Kolmogorov equation and generalized Pontryagin equations with suitable initial and boundary conditions. An example is given to illustrate the proposed procedure and the results from digital simulation are obtained to verify the effectiveness of the proposed procedure. The effects of time delay in feedback control forces on the conditional reliability function, conditional probability density and moments of first-passage time are analyzed.     

含控制时滞的海洋平台前馈反馈最优跟踪控制

通过引入状态变换,将含控制时滞的海洋平台系统转化为无时滞系统.然后,通过引入渐近稳定的期望系统,基于波浪力线性外系统近似模型,给出了海洋平台系统基于二次型性能指标泛函的前馈反馈最优跟踪控制器设计方法及其控制器的存在唯一性条件.控制器的前馈增益矩阵和反馈增益矩阵可分别经由求解代数Riccati方程和Sylvester方程得到.仿真算例表明了方法的有效性,且在减小平台的振动幅值和控制力方面,前馈反馈最优跟踪控制方法优于前馈反馈最优控制方法.     

Optimal bounded control of harmonically and stochastically excited strongly nonlinear oscillators

A procedure for designing optimal bounded control to minimize the response of harmonically and stochastically excited strongly nonlinear oscillators is proposed. First, the stochastic averaging method for controlled strongly nonlinear oscillators under combined harmonic and white noise excitations using generalized harmonic functions is introduced. Then, the dynamical programming equation for the control problem of minimizing response of the systems is formulated from the partially completed averaged Itô equations by using the dynamical programming principle. The optimal control law is derived from the dynamical programming equation and control constraint without solving the dynamical programming equation. Finally, the stationary probability density of the amplitude and mean amplitude of the optimally controlled systems are obtained from solving the reduced Fokker–Planck–Kolmogorov equation associated with fully completed averaged Itô equations. An example is given to illustrate the proposed procedure and the results obtained are verified by using those from digital simulation.     

Stabilisation of a class of stochastic bilinear interconnected systems by suboptimal decentralised feedback controls

A computationally simple method has been suggested for decentralised stabilisation of a class of stochastic bilinear interconnected systems by optimal feedback controls at subsystem levels, appropriately designated “Suboptimal Decentralised Feedback Control”. The proposed method makes use of an aggregation-decomposition technique based on algebraic Riccati equations. The method has been illustrated through a numerical example of the stochastic bilinear interconnected system consisting of three subsystems, each of second order.     

Stochastic control of hysteretic structural systems

A method of stochastic optimal control of hysteretic structural systems under earthquake excitations is presented. Stochastic estimation and control problems are formulated in the form of Itô stochastic differential equations on the basis of the theory of continuous Markov processes. The conditional moment equations given observation data are derived for nonlinear filtering, and are closed by introducing appropriate analytical form of the conditional probability density functions of the state variables. Under the assumption that the admissible controls are expressed as functions of the conditional moment functions the Bellman equation is derived. If the spatial variables of the Bellman equation are defined by a part of the full set of conditional moment functions appearing in the closed moment equations, the resulting Bellman equation is coupled with conditional moment equations both for filtering and for prediction. The Gaussian and non-Gaussian stochastic linearization techniques combined with simple solution techniques to the Bellman equation are examined to solve the Bellman equation or extended Riccati equations without prediction procedures.     

Direct control method for improving stability and reliability of nonlinear stochastic dynamical systems

Optimal control for improving the stability and reliability of nonlinear stochastic dynamical systems is of great significance for enhancing system performances. However, it has not been adequately investigated because the evaluation indicators for stability (e.g. maximal Lyapunov exponent) and for reliability (e.g. mean first-passage time) cannot be explicitly expressed as the functions of system states. Here, a unified procedure is established to derive optimal control strategies for improving system stability and reliability, in which a physical intuition-inspired separation technique is adopted to split feedback control forces into conservative components and dissipative components, the stochastic averaging is then utilized to express the evaluation indicators of performances of controlled system, the optimal control strategies are finally derived by minimizing the performance indexes constituted by the sigmoid function of maximal Lyapunov exponent (for stability-based control)/the reciprocal of mean first-passage time (for reliability-based control), and the mean value of quadratic form of control force. The unified procedure converts the original functional extreme problem of optimal control into an extremum value problem of multivariable function which can be solved by optimization algorithms. A numerical example is worked out to illustrate the efficacy of the optimal control strategies for enhancing system performance.     

受需求扰动的供应链系统最优跟踪控制

针对动态供应链系统中市场需求扰动引起的牛鞭效应问题,提出一种基于前馈补偿的最优跟踪控制设计方法.首先,针对需求扰动可模型化为线性外系统的情形,引入了一个渐近稳定的期望系统;然后,基于线性二次型性能指标,给出了受需求扰动的供应链系统的前馈反馈最优跟踪控制器设计方法.控制器的反馈增益和前馈增益可分别通过求解Riccati方程和Stein方程得到.仿真结果表明,提出的前馈反馈最优跟踪控制方法能有效抑制需求扰动对供应链系统的影响,改善供应链系统的动态性能,且该方法明显优于经典的最优跟踪控制.     

Semi-active control of wind excited building structures using MR/ER dampers

A semi-active control strategy for building structures subject to wind loading and controlled by MR/ER dampers is proposed. The power spectral density (PSD) matrix of the fluctuating part of wind velocity vector is diagonalized in the eigenvector space. Each element of the diagonalized PSD matrix is modeled as a set of second-order linear filter driven by white noise. A Bingham model for MR/ER dampers is used. The forces produced by MR/ER dampers are split into passive and active parts and the passive part is combined with structural damping forces. A set of partially averaged Itô equations for controlled modal energies are derived by applying the stochastic averaging method for quasi-integrable-Hamiltonian systems. The optimal control law is then determined by using the stochastic dynamical programming principle and the cost function is so selected that the optimal control law can be implemented by the MR/ER dampers. The response of semi-active controlled structures is predicted by using the reduced Fokker–Planck–Kolmogorov equation associated with fully averaged Itô equations of the controlled structures. A comparison with clipped linear quadratic Gaussian (LQG) control strategy, for an example, shows that the proposed semi-active control strategy for MR/ER dampers is superior to clipped LQG control strategy.     

最优有界控制下色噪声驱动多时滞拟线性系统瞬态响应

基于Fokker-Planck-Kolmogorov方程瞬态求解研究了受最优有界控制的色噪声驱动的多时滞拟线性系统的瞬态响应。利用等价变换将时滞系统转化为非时滞系统。在弱扰动假设下应用标准随机平均法得到振幅过程的部分平均It?随机微分方程。由动态规划原理和控制力界值条件得到最优有界控制率从而得到完全平均的Fokker-Planck-Kolmogorov方程。通过原系统的退化线性系统导出一组正交基并在该基空间内进行Galerkin变分得到近似瞬态响应。最后将该方法应用到受最优有界控制率和色噪声共同作用的时滞Duffing-Van Der Pol振子进行理论求解并综合讨论了色噪声、时滞、控制力和共振对系统瞬态响应的影响,采用Monte-Carlo模拟验证了所有理论和计算结果的正确性。     

Stochastic optimal control of wind-excited tall buildings using semi-active MR-TLCDs

A new semi-active control device, magneto-rheological tuned liquid column damper (MR-TLCD), has been devised recently by the authors for mitigation of wind-induced vibration response of tall building structures. The developed device combines the benefits of magneto-rheological smart materials and tuned liquid column dampers. In this paper, real-time semi-active vibration control of tall building structures incorporating nonlinear MR-TLCDs under random wind excitation is studied by means of the statistical linearization method and the optimal linear quadratic (LQ) control strategy. The equations of motion of a tall building structure subjected to random wind loading and controlled by using MR-TLCDs at the top floor are first derived and represented in modal coordinate. After linearizing the uncontrollable part of MR-TLCD damping force and incorporating it with structural components, the classical linear quadratic (LQ) control strategy is applied to the linearized structural system to determine optimal control force of the MR-TLCDs. Clipping treatment is performed to ensure the commanded control force implementable by the MR-TLCDs. Wind-excited response of the semi-actively controlled structural system is evaluated by using the frequency-response function and then compared with that of the passively controlled structure to determine the control efficacy. A case study of a 50-story building structure is conducted to illustrate excellent control efficacy of the proposed semi-active MR-TLCD control system.     

Implementation and acceleration of optimal control for systems biology

Optimal control theory provides insight into complex resource allocation decisions. The forward–backward sweep method (FBSM) is an iterative technique commonly implemented to solve two-point boundary value problems arising from the application of Pontryagin’s maximum principle (PMP) in optimal control. The FBSM is popular in systems biology as it scales well with system size and is straightforward to implement. In this review, we discuss the PMP approach to optimal control and the implementation of the FBSM. By conceptualizing the FBSM as a fixed point iteration process, we leverage and adapt existing acceleration techniques to improve its rate of convergence. We show that convergence improvement is attainable without prohibitively costly tuning of the acceleration techniques. Furthermore, we demonstrate that these methods can induce convergence where the underlying FBSM fails to converge. All code used in this work to implement the FBSM and acceleration techniques is available on GitHub at https://github.com/Jesse-Sharp/Sharp2021.     

负刚度时滞反馈控制动力吸振器的反共振优化

提出一种含接地负刚度弹簧的时滞动力吸振器的反共振峰最小优化方法。首先,研究的模型不考虑时滞位移控制和主系统的阻尼,利用固定点理论得到接地负刚度系统的最优结构参数。其次,考虑优化调节负刚度系数,利用反共振峰最小准则,把反共振点幅值控制在给定的足够小的范围内,得到一组最优结构参数,使调控的反共振频率频带最宽。接着,调节主系统的阻尼系数,得到最终的一组最优结构参数。最后,通过幅频响应和时间历程响应曲线证明了反共振峰最小准则的正确性,并且将该研究结果与其他两种等峰优化的吸振器模型的减振效果进行对比,证明了接地负刚度时滞动力吸振器减振效果的优越性。     

Adaptive optimal tuning of a general class of stable LTI systems with restricted inputs

The problem addressed is one of model reference adaptive control (MRAC) of asymptotically stable plants of unknown order with zeros located anywhere in thes-plane except at the origin. The reference model is also asymptotically stable and lacking zero(s) ats = 0. The control law is to be specified only in terms of the inputs to and outputs of the plant and the reference model. For inputs from a class of functions that approach a non-zero constant, the problem is formulated in an optimal control framework. By successive refinements of the sub-optimal laws proposed here, two schemes are finally designed. These schemes are characterized by boundedness, convergence and optimality. Simplicity and total time-domain implementation are the additional striking features. Simulations to demonstrate the efficacy of the control schemes are presented. This paper is dedicated to Dr R M Umesh.     

具有类反斜线回滞系统的自适应控制

考虑一类具有类反斜线回滞的不确定非线性系统自适应控制问题.利用神经网络提出了变结构自适应控制方法和近似自适应控制方法,变结构自适应控制方法能保证跟踪误差趋于零,近似自适应控制方法的控制律是光滑的,能保证跟踪误差最终小于任意给定的正数.文中举例说明了结果的有效性.