Keywords: robust control, probabilistic robustness, first-passage reliability, parametric model uncertainty, stochastic simulation.

The existence of model uncertainty is important for modern control applications, as one of the main objectives is to establish optimum robustness over all possible operational conditions. Standard tools for robust control design, such as H_infinity, µ-synthesis and the many offshoots of these, consider only compact sets of possible models for the system. Information implying that some model parameters are more probable than others is not explicitly treated. However in most real engineering applications, there is considerable knowledge about the relative plausibility of the different model parameter values. A probability logic approach provides a rational and consistent framework for quantifying this knowledge. This is established by characterizing the relative plausibility of different properties of the system by probability models. A robust design may be then established by optimizing statistics of the objective function (probabilistic performance) under the statistically described plant uncertainty, rather than the objective function resulting from the nominal model (nominal performance).

The present paper discusses the robust-performance optimization of linear time invariant dynamical systems with probabilistically-described parametric model uncertainties and focuses on cases including a stochastic disturbance input. We consider H₂ and multi-objective H₂ control synthesis for quantification of the system nominal performance. The probabilistic measure of optimality is then defined either as the average (i.e. expectation) of the performance over the uncertain parameter space, or the probability that the performance will exceed acceptable bounds. We also examine robust stochastic design for minimal first-passage failure probability [1], i.e. maximal reliability of the dynamic response. In this case the definition of robust performance in presence of probabilistic model uncertainties follows directly from the axioms of probability logic. Analysis and synthesis methodologies are discussed, based on recently developed stochastic simulation techniques [2]. The influence of different probability models for describing plant uncertainty is also discussed.

The design approach is illustrated in a structural control application. Probabilistically-robust controllers are demonstrated to yield considerable different designs compared to controllers optimized using only a nominal model, or using the "worst-case" interpretation of system robustness. Also, differences are shown in the design characteristics between different probabilistic characterizations for the system uncertainty or for the performance objective.

1

A.A. Taflanidis, J.T. Scruggs, J.L. Beck, "Reliability-based performance objectives and probabilistic robustness in structural control applications", Journal of Engineering Mechanics, 134(4), 291-301, 2008. doi:10.1061/(ASCE)0733-9399(2008)134:4(291)

2

A.A. Taflanidis, J.L. Beck, "An efficient framework for optimal robust stochastic system design using stochastic simulation", Computer Methods in Applied Mechanics and Engineering, 198(1), 88-101, 2008. doi:10.1016/j.cma.2008.03.029

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