Quasilinear Control of Systems with Time-Delays and Nonlinear Actuators and Sensors

This thesis investigates Quasilinear Control (QLC) of time-delay systems with nonlinear actuators and sensors and analyzes the accuracy of stochastic linearization for these systems. QLC leverages the method of stochastic linearization to replace each nonlinearity with an equivalent gain, which is obtained by solving a transcendental equation. The idea of QLC is to stochastically linearize the system in order to analyze and design controllers using classical linear control theory. In this thesis, the existence of the equivalent gain for a closed-loop time-delay system is discussed. To compute the equivalent gain, two methods are explored. The first method uses an explicit but complex algorithm based on delay Lyapunov equation to study the time-delay, while the second method uses Pade approximant. It is shown that, under a suitable criterion, Pade approximant can be effectively applied for QLC of time-delay systems. Furthermore, the method of Saturated-Root Locus (S-RL) is extended to nonlinear time-delay systems. It turns out that, in a time-delay system, S-RL always terminates prematurely as opposed to a delay-free system, which may or may not terminate prematurely. Statistical experiments are performed to investigate the accuracy of stochastic linearization compared to a system without time-delay. The impact of increasing the time-delay in the approach of stochastic linearization is also investigated. Results show that stochastic linearization effectively linearizes a nonlinear time-delay system, even though delays generally degrade accuracy. Overall, the accuracy remains relatively high over the selected parameters. Finally, this approach is applied to pitch control in a wind turbine system as a practical example of a nonlinear time-delay system, and its performance is analyzed to demonstrate the efficacy of the approach

Performance analysis of global local mean square error criterion of stochastic linearization for nonlinear oscillator

The paper presents a performance analysis of global-local mean square error criterion of stochastic linearization for some nonlinear oscillators. This criterion of stochastic linearization for nonlinear oscillators bases on dual conception to the local mean square error criterion (LOMSEC). The algorithm is generally built to multi-degree of freedom (MDOF) nonlinear oscillators. Then, the performance analysis is carried out for two applications which comprise a rolling ship oscillation and two-degree of freedom one. The improvement on accuracy of the proposed criterion has been shown in comparison with the conventional Gaussian equivalent linearization (GEL)

Sixty years of stochastic linearization technique

Stochastic linearization technique is a versatile method of solving nonlinear stochastic boundary value problems. It allows obtaining estimates of the response of the system when exact solution is unavailable; in contrast to the perturbation technique, its realization does not demand smallness of the parameter; on the other hand, unlike the Monte Carlo simulation it does not involve extensive computational cost. Although its accuracy may be not very high, this is remedied by the fact that the stochastic excitation itself need not be known quite precisely. Although it was advanced about six decades ago, during which several hundreds of papers were written, its foundations, as exposed in many monographs, appear to be still attracting investigators in stochastic dynamics. This study considers the methodological and pedagogical aspects of its exposition

Simultaneous Suspension Control and Energy Harvesting through Novel Design and Control of a New Nonlinear Energy Harvesting Shock Absorber

Simultaneous vibration control and energy harvesting of vehicle suspensions have attracted significant research attention over the past decades. However, existing energy harvesting shock absorbers (EHSAs) are mainly designed based on the principle of linear resonance, thereby compromising suspension performance for high-efficiency energy harvesting and being only responsive to narrow bandwidth vibrations. In this paper, we propose a new EHSA design -- inerter pendulum vibration absorber (IPVA) -- that integrates an electromagnetic rotary EHSA with a nonlinear pendulum vibration absorber. We show that this design simultaneously improves ride comfort and energy harvesting efficiency by exploiting the nonlinear effects of pendulum inertia. To further improve the performance, we develop a novel stochastic linearization model predictive control (SL-MPC) approach in which we employ stochastic linearization to approximate the nonlinear dynamics of EHSA that has superior accuracy compared to standard linearization. In particular, we develop a new stochastic linearization method with guaranteed stabilizability, which is a prerequisite for control designs. This leads to an MPC problem that is much more computationally efficient than the nonlinear MPC counterpart with no major performance degradation. Extensive simulations are performed to show the superiority of the proposed new nonlinear EHSA and to demonstrate the efficacy of the proposed SL-MPC

Peak response of non-linear oscillators under stationary white noise

The use of the Advanced Censored Closure (ACC) technique, recently proposed by the authors for predicting the peak response of linear structures vibrating under random processes, is extended to the case of non-linear oscillators driven by stationary white noise. The proposed approach requires the knowledge of mean upcrossing rate and spectral bandwidth of the response process, which in this paper are estimated through the Stochastic Averaging method. Numerical applications to oscillators with non-linear stiffness and damping are included, and the results are compared with those given by Monte Carlo Simulation and by other approximate formulations available in the literature

Improved stochastic linearization method using mixed distributions

A new procedure for the random vibration analysis of hysteretic structures using stochastic equivalent linearization is reported. Its aim is to improve the prediction of the response obtained by conventional Gaussian linearization technique. To this purpose, mixed discrete-continuous Gaussian distributions are used taking into account the bounded nature of the non-linear restoring force. The simple but important property of the mixed distribution is its linearity, which allows the use of the previous results obtained by the Gaussian hypothesis, avoiding the need of employing non-Gaussian continuous distributions or other time-consuming techniques such as local Monte Carlo simulations. Closed-form expressions of the new linearization coefficients for the Bouc-Wen-Baber model are then provided. The relative weights of the discrete and Gaussian distributions are calculated in dependence of the degree of non-linearity in each time step. The comparison of the results with previously published ones obtained by simulation shows a good agreement, providing a substantial improvement of the method with respect to the conventional Gaussian technique with the same calculation effort

A Fast Newton-Raphson Method in Stochastic Linearization

peer reviewedOwing to its accessible implementation and rapidity, the equivalent linearization has become a common probabilistic approach for the analysis of large-dimension nonlinear structures, as encountered in earthquake and wind engineering. It consists in replacing the nonlinear system by an equivalent linear one, by tuning the parameters of the equivalent system, in order to minimize some discrepancy error. Consequently classical analysis tools such as the spectral analysis may be reconditioned to approximate the solution of structures with slight to moderate nonlinearities. The tuning of the equivalent parameters requires the solution of a set of nonlinear algebraic equations involving integrals. It is typically performed with the fixed-point algorithm, which is known to behave poorly in terms of convergence. We therefore advocate for the use and implementation of a Newton-Raphson approach, which behaves much better, even in its dishonest formulation. Unfortunately, this latter option requires the costly construction of a Jacobian matrix. In the approach described in this paper, this issue is answered by introducing a series expansion method that provides a fast and accurate estimation of the residual function (whose solution provides the equivalent parameters) and a fast and approximate estimation of the Jacobian matrix. An illustration demonstrate the good accuracy obtained with the proposed method