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Contemporary stochastic optimal control by synergy of the probability density evolution method (PDEM) and conventional optimal controller exhibits less capability to guarantee economical energy consumption versus control efficacy when non-stationary stochastic excitations drive hysteretic structures. In this regard, a novel multiscale stochastic optimal controller is invented based on the wavelet transform and the PDEM.

For a representative point, a conventional control law is decomposed into sub-control laws by deploying the multiresolution analysis. Then, the sub-control laws are classified into two generic control laws using resonant and non-resonant bands. Both frequency bands are established by employing actual natural frequency(ies) of structure, making computed efforts depend on actual structural properties and time-frequency effect of non-stationary stochastic excitations. Gain matrices in both bands are then acquired by a probabilistic criterion pertaining to system second-order statistics assessment. A multi-degree-of-freedom hysteretic structure driven by non-stationary and non-Gaussian stochastic ground accelerations is numerically studied, in which three distortion scenarios describing uncertainties in structural properties are considered.

Time-frequency-dependent gain matrices sophisticatedly address non-stationary stochastic excitations, providing efficient ways to independently suppress vibrations between resonant and non-resonant bands. Wavelet level, natural frequency(ies), and ratio of control forces in both bands influence the scheme’s outcomes. Presented approach outperforms existing approach in ensuring trade-off under uncertainty and randomness in system and excitations.

Presented control law generates control efforts relying upon resonant and non-resonant bands, and deploys actual structural properties. Cost-function weights and probabilistic criterion are promisingly developed, achieving cost-effectiveness of energy demand versus controlled structural performance.

In real life, excitations are highly non-stationary in frequency and amplitude, which easily induces resonant vibration to structural responses. Conventional control algorithms in this case cannot guarantee cost-effective control effort and efficient structural response alleviation. To this end, this paper proposes a novel adaptive linear quadratic regulator (LQR) by integrating wavelet transform and genetic algorithm (GA).

In each time interval, multiresolution analysis of real-time structural responses returns filtered time signals dominated by different frequency bands. Minimization of cost function in each frequency band obtains control law and gain matrix that depend on temporal-frequency band, so suppressing resonance-induced filtered response signal can be directly achieved by regulating gain matrix in the temporal-frequency band, leading to emphasizing cost-function weights on control and state. To efficiently subdivide gain matrices in resonant and normal frequency bands, the cost-function weights are optimized by a developed procedure associated to genetic algorithm. Single-degree-of-freedom (SDOF) and multi-degree-of-freedom (MDOF) structures subjected to near- and far-fault ground motions are studied.

Resonant band requires a larger control force than non-resonant band to decay resonance-induced peak responses. The time-varying cost-function weights generate control force more cost-effective than time-invariant ones. The scheme outperforms existing control algorithms and attains the trade-off between response suppression and control force under non-stationary excitations.

Proposed control law allocates control force amounts depending upon resonant or non-resonant band in each time interval. Cost-function weights and wavelet decomposition level are formulated in an elegant manner. Genetic algorithm-based optimization cost-efficiently results in minimizing structural responses.