Expected Response Energy Research Articles

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.

ENERGY BALANCE FOR RANDOM VIBRATIONS OF PIECEWISE-CONSERVATIVE SYSTEMS

Vibrations of systems with instantaneous or stepwise energy losses, e.g., due to impacts with imperfect rebounds, dry friction forces(s) (in which case the losses may be treated as instantaneous ones by appropriate introduction of the response energy) and/or active feedback “bang–bang” control of the systems' response are considered. Response of such (non-linear) systems to a white-noise random excitation is considered for the case where there are no other response energy losses. Thus, a simple linear energy growth with time between “jumps” is observed. Explicit expressions for the expected response energy are derived by direct application of the stochastic differential equations calculus, which contains the expected time interval between two consecutive jumps. The latter may be predicted as a solution to the relevant first-passage problem. Perturbational analysis of the relevant PDE for this problem for a certain vibroimpact system demonstrated the possibility for using the solution to the corresponding free vibration problem as a zero order approximation. The method is applied to an s.d.o.f. system with a feedback inertia control, designed according to a certain previously introduced “generalized reversed swings law”. Extensive Monte-Carlo simulation results are presented for this system as well as for several previously analyzed ones: system with impacts; system with dry friction; system with stiffness control; pendulum with controlled length. The results are compared with those due to the asymptotic stochastic averaging approach. Both methods are shown to provide adequate accuracy far beyond the expected applicability range of the asymptotic approach (which requires both excitation intensity and losses to be small), with direct energy balance being generally superior.

Stochastic Optimal Bounded Control for a System With the Boltz Cost Function

A single-degree-of-freedom mass-spring system is considered under white-noise excitation. To reduce expected response energy of the system, a bounded control force can be applied and optimal control law is sought for. An exact explicit analytical solution, for the Lagrange cost function, is derived for the Hamilton-Jacobi-Bellman (HJB) equation within a certain "outer" domain. This solution provides boundary conditions for numerical simulation of the HJB equation within the remaining "inner" domain. Comparison of optimal control law, with a previously obtained one for the Mayer cost function, is made. The analytical solution to the HJB equation for the "combined" Boltz cost function is derived for the outer domain as a linear combination of the solutions for the Mayer and Lagrange cost functions. Sensitivity of the Bellman functional to the above optimal control laws is analyzed.

Bounded parametric control of random vibrations

The dynamic programming approach is used to study a feedback control problem for a randomly excited single–degree–of–freedom system. The available actuator for control can provide temporal stiffness variations for the system, which are of a bounded magnitude. An analytical solution is obtained for the corresponding Hamilton–Jacobi–Bellman equation for the expected response energy, which should be minimized according to the integral cost criterion. While this solution is valid within some parts of the phase plane only, it extends to the whole phase plane with increasing time–interval of control. Thus the exact explicit expression for the optimal control law is obtained for the case where steady–state response is to be controlled. This law requires feedback–controlled switching stiffness between the given bounds. Stationary random vibration of the system with this control law is studied then, by a direct energy balance approach and by a stochastic averaging method. The latter of these provides a more extensive description of the response, which may provide reliability estimates for the controlled system, but this asymptotic approach is valid for the limiting case of a weak control only. The direct energy balance predicts only the expected response energy, or the mean square displacement. However, its range of applicability is expected to be less restrictive with respect to the maximal magnitude of the system9s stiffness variations: the non–dimensional variations may not be much smaller than unity. The analytical studies are extended to the case of a controlled system with a nonlinear restoring force.

Optimal bounded control of steady-state random vibrations

A SDOF system is considered, which is excited by a white-noise random force. The system's 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.

Optimal Bounded Response Control for a Second-Order System Under a White-Noise Excitation

A single-degree-of-freedom system is excited by a white-noise random force. The system's re sponse can be reduced by a control force of limited magnitude R, and the problem is to minimize the expected response energy at a given time instant T under this constraint. A "hybrid" solution to the corresponding Hamilton-Jacobi-Bellman (or HJB) equation is obtained for the case of a linear controlled system. Specif ically, an exact analytical solution is obtained within a certain outer domain with respect to a "strip" with switching lines, indicating optimality of a "dry-friction," or the simplest version of the "bang-bang" control law within this domain. This explicit solution is matched by a numerical solution within an inner domain, where switching lines are illustrated. In the limiting case of a weak control, or small R, the hybrid solution leads to a simple asymptotically suboptimal "dry-friction" control law, which is well-known for deterministic optimal control problems; more precisely, the difference in expected response energies between cases of opti mal and suboptimal control is shown to be proportional to a small parameter. Numerical results are presented, which illustrate the optimal control law and evolution of the minimized functional. They are used in particular to evaluate convergence rate to the derived analytical results for the suboptimal weak control case. A special case of a nonlinear controlled system is considered also, one with a rigid barrier at the system's equilibrium position. The resulting vibroimpact system is studied for the case of perfectly elastic impacts/rebounds by us ing special piecewise-linear transformation of state variables, which reduces the system to the nonimpacting one. The solution to the HJB equation is shown to be valid for the transformed system as well, resulting in the optimal control law for the vibroimpact 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.

Towards incorporating impact losses into random vibration analyses: a model problem

A benchmark model random vibration problem is considered for an SDOF system with a one-sided inelastic barrier at its equilibrium position under Gaussian white-noise excitation. An analytical solution is obtained for an expected response energy by cycle-to-cycle application of the method of moments with successive conditional and unconditional averaging. The solution, valid both for small and `not very small' impact losses, leads to a formula for the equivalent viscous damping factor, which may be used to account for impact losses. Monte-Carlo simulation results show that this formula provides generally better accuracy than one obtained by stochastic averaging. They also illustrate applicability of the formula for certain more complicated random vibration problems.