D.V. Iourtchenko

Optimal bounded control of steady-state random vibrations

Abstract A SDOF system is considered, which is excited by a white-noise random force. The systems response is controlled by a force of bounded magnitude, with the aim of minimizing integral of the expected response energy over a given period of time. The integral to be minimized satisfies the Hamilton–Jacobi–Bellman (HJB) equation. An analytical solution of this PDE is obtained within a certain outer part of the phase plane. This solution is analyzed for large time intervals, which correspond to the limiting steady-state random vibration. The analysis shows the outer domain expanding onto the whole phase plane in the limit, implying that the simple dry-friction control law is the optimal one for steady-state response. The resulting value of the (unconditional) expected response energy, for the case of a stationary excitation, is also obtained. It matches with the corresponding result of energy balance analysis, as obtained by direct application of the SDE Calculus, as well as that of stochastic averaging for the case where the magnitude of dry friction force and intensity of excitation are both small. A general expression for mean absolute value of the response velocity is also obtained using the SDE calculus. Certain reliability predictions both for first-passage and fatigue-type failures are also derived for the optimally controlled system using the stochastic averaging method. These predictions are compared with their counterparts for the system with a linear velocity feedback and same r.m.s. response, thereby illustrating the price to be paid for the bounds on control force in terms of the reduced reliability of the system.

Hybrid solution method for dynamic programming equations for MDOF stochastic systems

An optimal control problem is considered for a multi-degree-of-freedom (MDOF) system, excited by a white-noise random force. The problem is to minimize the expected response energy by a given time instantT by applying a vector control force with given bounds on magnitudes of its components. This problem is governed by the Hamilton-Jacobi-Bellman, or HJB, partial differential equation. This equation has been studied previously [1] for the case of a single-degree-of-freedom system by developing a “hybrid” solution. Specifically, an exact analitycal solution has been obtained within a certain outer domain of the phase plane, which provides necessary boundary conditions for numerical solution within a bounded in velocity inner domain, thereby alleviating problem of numerical analysis for an unbounded domain. This hybrid approach is extended here to MDOF systems using common transformation to modal coordinates. The multidimensional HJB equation is solved explicitly for the corresponding outer domain, thereby reducing the problem to a set of numerical solutions within bounded inner domains. Thus, the problem of bounded optimal control is solved completely as long as the necessary modal control forces can be implemented in the actuators. If, however, the control forces can be applied to the original generalized coordinates only, the resulting optimal control law may become unfeasible. The reason is the nonlinearity in maximization operation for modal control forces, which may lead to violation of some constraints after inverse transformation to original coordinates. A semioptimal control law is illustrated for this case, based on projecting boundary points of the domain of the admissible transformed control forces onto boundaries of the domain of the original control forces. Case of a single control force is considered also, and similar solution to the HJB equation is derived.

Reliability of Strongly Nonlinear Single Degree of Freedom Dynamic Systems by the Path Integration Method

This paper presents a first passage type reliability analysis of strongly nonlinear stochastic single-degree-of-freedom systems. Specifically, the systems considered are a dry friction system, a stiffness controlled system, an inertia controlled system, and a swing. These systems appear as a result of implementation of the quasioptimal bounded in magnitude control law. The path integration method is used to obtain the reliability function and the first passage time.

Response spectral density of linear systems with external and parametric non-Gaussian, delta-correlated excitation

An exact, closed form, analytical expressions for a response spectral density of a certain type of systems, subjected to non-Gaussian, stationary, delta-correlated noise are derived. A new, extended mean square stability conditions are derived for such systems.

Solution to a class of stochastic LQ problems with bounded control

Abstract A new approach for finding an exact analytical solution to the modified Hamilton–Jacobi–Bellman equation is proposed. Together with the recently developed hybrid solution method, the proposed strategy allows to find a solution to a whole class of stochastic optimal control problems with bounded in magnitude control force.

Transition from Planar to Whirling Oscillations in a Certain Nonlinear System

A single-mass two-degrees-of-freedom system is considered, witha radially oriented nonlinear restoring force. The latter is smooth andbecomes infinite at a certain value of a radial displacement. Stabilityanalysis is made for planar oscillation, or motion along a givendirection. As long as this motion is periodic, the nonlinearity in therestoring force provides a periodic parametric excitation in thetransverse direction. The linearized stability analysis is reduced tostudy of the Mathieu equation for the (infinitesimal) motions in thetransverse direction. For the case of free oscillations in the givendirection an exact solution is obtained, since a specific analyticalform is used for the (strongly nonlinear) restoring force, which permitsexplicit integration of the equation of motion. Stability of the planarmotion in this case is shown to be very sensitive to even slightdeviations from polar symmetry in the restoring force (as well as to theamplitude of oscillations in the given direction). Numerical integrationof the original equations of motion shows the resulting motion to be awhirling type indeed in case of the transversal instability. For thecase of a sinusoidal forcing in the given direction solution for the(periodic) response is obtained by Krylov–Bogoliubov averaging. Thisresults in the ‘transmitted’ Ince–Strutt chart – namely, stabilitychart for transverse direction on the amplitude-frequency plane of theexcitation in the original direction.

Bounded Control of Random Vibration: Hybrid Solution to HJB Equations

Problems of optimal bounded control for randomly excited systems are studied by the Dynamic Programming approach. An effective hybrid solution method has been developed for the corresponding Hamilton–Jacobi–Bellman (HJB) equations for the optimized functional of the response energy. These PDEs are shown to be amenable to an exact analytical solution within a certain ‘outer’ domain of the phase space. The solution provides boundary conditions for numerical study within the remaining ‘inner’ domain. The simple ‘dry-friction’ law had been shown previously to be optimal for a SDOF system within the outer domain and to become asymptotically optimal within the whole phase plane for the important case of so-called ‘long-term’ control, whereby steady-state response is to be optimized according to the integral cost functional. These results are extended in this paper to MDOF systems by using modal transformation. Thus, the multidimensional dry-friction law is shown to be the optimal one for the case of long-term control. The expected response energy is predicted for this case by a direct energy balance based on application of the SDE Calculus to the energy equation. Stochastic averaging is also used in order to obtain certain reliability estimates. In particular, numerical results for the expected time to first-passage failure are presented, illustrating reduction of reliability due to the imposed bound on the control force. Monte-Carlo simulation results are presented, which demonstrate adequate accuracy of the predictions far beyond the expected applicability range of the asymptotic approaches.

Brief paper: Solution to a class of stochastic LQ problems with bounded control

Abstract A new approach for finding an exact analytical solution to the modified Hamilton–Jacobi–Bellman equation is proposed. Together with the recently developed hybrid solution method, the proposed strategy allows to find a solution to a whole class of stochastic optimal control problems with bounded in magnitude control force.