Z.G. Ying

An Optimal Nonlinear Feedback Control Strategy for Randomly Excited Structural Systems

A strategy for optimal nonlinear feedback control of randomlyexcited structural systems is proposed based on the stochastic averagingmethod for quasi-Hamiltonian systems and the stochastic dynamicprogramming principle. A randomly excited structural system isformulated as a quasi-Hamiltonian system and the control forces aredivided into conservative and dissipative parts. The conservative partsare designed to change the integrability and resonance of the associatedHamiltonian system and the energy distribution among the controlledsystem. After the conservative parts are determined, the system responseis reduced to a controlled diffusion process by using the stochasticaveraging method. The dissipative parts of control forces are thenobtained from solving the stochastic dynamic programming equation. Boththe responses of uncontrolled and controlled structural systems can bepredicted analytically. Numerical results for a controlled andstochastically excited Duffing oscillator and a two-degree-of-freedomsystem with linear springs and linear and nonlinear dampings, show thatthe proposed control strategy is very effective and efficient.

Nonlinear stochastic optimal control of partially observable linear structures

Abstract A strategy for nonlinear stochastic optimal control of partially observable linear structures is proposed and illustrated with linear building structures equipped with control devices and sensors under horizontal ground acceleration excitation. The control problem of a partially observable structure is first converted into that of a completely observable structure based on the separation principle. Then, a partially averaged control system of Ito equations is obtained from the completely observable structure by using the stochastic averaging method for quasi-Hamiltonian systems. Dynamical programming equations for finite and infinite time-interval controls are established based on the stochastic dynamical programming principle and solved to obtain the optimal control law and value function. Finally, the response of controlled structure is obtained from solving the Fokker–Planck–Kolmogorov equation associated with the fully averaged system of Ito equations. The numerical results for a five-story building structure model are obtained by using the proposed control strategy and compared with those by using linear quadratic Gaussian control strategy to show the effectiveness and efficiency of the proposed strategy.

Micro-vibration response of a stochastically excited sandwich beam with a magnetorheological elastomer core and mass

Magnetorheological (MR) elastomers are used to construct a smart sandwich beam for micro-vibration control. The micro-vibration response of a clamped–free sandwich beam with an MR elastomer core and a supplemental mass under stochastic support micro-motion excitation is studied. The dynamic behavior of MR elastomer as a smart viscoelastic material is described by a complex modulus which is controllable by external magnetic field. The sixth-order partial differential equation of motion of the sandwich beam is derived from the dynamic equilibrium, constitutive and geometric relations. A frequency-domain solution method for the stochastic micro-vibration response of the sandwich beam is developed by using the frequency-response function, power spectral density function and spatial eigensolution. The root-mean-square velocity response in terms of the one-third octave frequency band is calculated, and then the response reduction capacity through optimizing the complex modulus of the core is analyzed. Numerical results illustrate the influences of the MR elastomer core parameters on the root-mean-square velocity response and the high response reduction capacity of the sandwich beam. The developed analysis method is applicable to sandwich beams with arbitrary cores described by complex shear moduli under arbitrary stochastic excitations described by power spectral density functions.

Brief paper: A stochastically averaged optimal control strategy for quasi-Hamiltonian systems with actuator saturation

A stochastic optimal control strategy for quasi-Hamiltonian systems with actuator saturation is proposed based on the stochastic averaging method and stochastic dynamical programming principle. First, the partially completed averaged Ito stochastic differential equations for the energy processes of individual degree of freedom are derived by using the stochastic averaging method for quasi-Hamiltonian systems. Then, the dynamical programming equation is established by applying the stochastic dynamical programming principle to the partially completed averaged Ito equations with a performance index. The saturated optimal control consisting of unbounded optimal control and bounded bang-bang control is determined by solving the dynamical programming equation. Numerical results show that the proposed control strategy significantly improves the control efficiency and chattering attenuation of the corresponding bang-bang control.

Stochastic optimal coupling-control of adjacent building structures

A stochastic optimal coupling-control method for adjacent building structures is proposed. The coupled structures with control devices under random seismic excitation are condensed to form a reduced-order model for the control analysis. The stochastic averaging method is applied to the reduced model to obtain Ito stochastic differential equations with respect to structural modal vibration energies. Then the stochastic dynamical programming principle is applied to the energy processes to establish a dynamical programming equation, by which the optimal coupling-control law is determined. The seismic response mitigation is achieved through the structural energy control and the dimension of optimal control problem is reduced. The seismic excitation spectrum is taken into account according to the stochastic dynamical programming principle. The random response of the non-linear controlled structures is predicted by using the stochastic averaging method and is compared with that of the uncontrolled structures to evaluate the control efficacy. A numerical study is conducted to demonstrate the response reduction capacity of the proposed stochastic optimal coupling-control method for adjacent building structures.

Stochastic optimal semi-active control of hysteretic systems by using a magneto-rheological damper

A stochastic optimal semi-active control strategy for stochastically excited hysteretic systems by using a magneto-rheological (MR) damper is proposed. The dynamics of both the hysteretic system and the MR damper is characterized by using the Bouc?Wen hysteretic model. The control force produced by the damper is split into a passive part and a semi-active part. The passive part is combined with the uncontrolled system to form a passively controlled system. Then the system is converted into an equivalent nonlinear non-hysteretic stochastic system, from which a partially averaged It? stochastic differential equation is derived by using the stochastic averaging method of the energy envelope. For the ergodic control problem, a dynamical programming equation is established based on the stochastic dynamical programming principle and solved to yield the optimal semi-active control law. The fully averaged It? equation is obtained by substituting the optimal semi-active control force into the partially averaged It? equation and completing the averaging. Finally, the response of the semi-actively controlled system is obtained from solving the Fokker?Planck?Kolmogorov equation associated with the fully averaged It? equation. The efficacy of the proposed control strategy is illustrated by the numerical results and comparison with clipped LQG control for an example.

Non-linear stochastic optimal control for coupled-structures system of multi-degree-of-freedom

Coupled structures under random excitation are modelled as a quasi-integrable Hamiltonian system of multi-degree-of-freedom and the reduced-order model in structural mode space is formulated. A non-linear stochastic optimal control method for the system is presented. The non-linear optimal control of adjacent tall building structures coupled with supplemental control devices and under random seismic excitation is performed by using the proposed method. First, applying the stochastic averaging method to the system yields Ito stochastic differential equations for modal vibration energy processes, so that the system energy control is conducted generally instead of the system state control and the dimension of the control problem is reduced. Then applying the stochastic dynamical programming principle to the controlled diffusion processes yields a dynamical programming equation, taking into account random excitation spectra. An explicit polynomial solution to the equation is proposed to determine the non-linear optimal control forces. Furthermore, the response statistics of the controlled non-linear coupled structures under random seismic excitation are evaluated by using the stochastic averaging method, and are compared with those of the uncontrolled structures to determine the control efficacy. Numerical results illustrate the high control effectiveness and efficiency of the proposed non-linear stochastic optimal control method for coupled structures as a quasi-integrable Hamiltonian system.

Random response of integrable Duhem hysteretic systems under non-white excitation

Abstract In this study, an integrable Duhem hysteresis model is derived from the mathematical Duhem operator. This model can represent a wide category of hysteretic systems. The stochastic averaging method of energy envelope is then adapted for response analysis of the integrable Duhem hysteretic system subjected to non-white random excitation. Using the integrability of the proposed model, potential energy and dissipated energy of the hysteretic system can be represented in an integration form so that the hysteretic restoring force is separable into conservative and dissipative parts. Based on the equivalence of dissipated energy, a non-hysteretic non-linear system is obtained to substitute the original system, and the averaged Ito stochastic differential equation of total energy is derived with the drift and diffusion coefficients being expressed as Fourier series expansions in space averaging. The stationary probability density of total energy and response statistics are obtained by solving the Fokker–Planck–Kolmogorov (FPK) equation associated with the Ito equation. Verification is given by comparing the computational results with Monte Carlo simulations.

Brief paper: Stochastic minimax control for stabilizing uncertain quasi- integrable Hamiltonian systems

A procedure for designing feedback control to asymptotically stabilize, with probability one, quasi-integrable Hamiltonian systems with bounded uncertain parametric disturbances is proposed. First, the partially averaged Ito stochastic differential equations are derived from given system by using the stochastic averaging method for quasi-integrable Hamiltonian systems. Second, the Hamilton-Jacobi-Issacs (HJI) equation for the ergodic control problem of the averaged system and a performance index with undetermined cost function is established based on the principle of optimality. This equation is then solved to yield the worst disturbances and the associated optimal controls. Third, the asymptotic Lyapunov stability with probability one of the optimally controlled system with worst disturbances is analyzed by evaluating the maximal Lyapunov exponent of the fully averaged Ito equations. Finally, the cost function and feedback control are determined by the requirement of stabilizing the worst-disturbed system. A simple example is worked out to illustrate the application of the proposed procedure and the effects of optimal control on stabilizing the uncertain system.

Electric potential response analysis of a piezoelectric shell under random micro-vibration excitations

The response characteristics of a spherically symmetric piezoelectric shell under random boundary micro-vibration excitations are analyzed and calculated. The equation for electric potential is integrated radially to obtain the electric potential as a function of displacement, so that the differential equations for the piezoelectric shell with electrical and mechanical coupling are converted into an equation only for the displacement. The displacement transformation is constructed to convert the random boundary conditions into homogeneous ones, and the transformed displacement is expanded in space to further convert the partial differential equation for the displacement into ordinary differential equations using the Galerkin method. The equations represent a multi-degree-of-freedom dynamic system with an asymmetric stiffness matrix under random micro-vibration excitations. The frequency-response function matrix, power spectral density matrix and correlation function matrix of the system response are derived from these equations based on the theory of random vibration. The expressions of mean-square displacement, stress and electric potential of the piezoelectric shell are finally obtained and illustrated by numerical results for random micro-vibration excitations. The random electrical and mechanical coupling properties, in particular the relations between boundary electric potential responses and micro-displacement excitations, are explored.