Computation of probabilistic stability measures for a controlled distributed parameter system

Abstract

Parameter uncertainty can degrade the performance of an otherwise well-designed control system, sometimes leading to system instability. In the context of structural control, performance degradation and instability imply excessive vibration and even structural failure. The ability of a controller to maintain the stability of a system in spite of parameter uncertainty is measured by its robustness, which can be viewed as a probability measure, wherein the joint distribution is of dimension equal to the number of uncertain parameters and the failure hypersurface is defined by the onset of instability in the eigenspace. This observation has led to some recent analyses employing FORM/SORM methods and Monte Carlo simulation.The extension of these concepts to distributed parameter systems is, unfortunately, not immediate. The mere fact that these systems are infinite dimensional precludes the use of much of the machinery available for discrete systems, unless the distributed system is first discretized, which itself introduces error into the analysis, or is represented by an eigenfunction expansion, which requires truncation after some finite number of modes, also a potential source of error. In fact, the system will behave as one with an infinite number of subsystems with highly dependent failure modes in series.In Bergman and Hall, Effect of controller uncertainty on the stability of a distributed parameter system, Structural Safety and Reliability, eds. Schuëller, Shinozuka and Yao, Balkema, Rotterdam, 1993, pp. 210–220, root locus analysis was employed to assess the reliability of the system, requiring the repetitive solution of a transcendental characteristic equation over a range of the parameter under investigation. The loci then provide a mapping from the probability distribution of the random parameter to the probability distribution of the system eigenvalues. This approach was utilized over 30 years ago by Boyce, Random vibration of strings and bars, Proc. of the Fourth US National Congress of Applied Mechanics, Berkeley, 18–21 June, 1962, pp. 77–85, who examined eigenvalue distributions for undamped taut strings and Euler-Bernoulli beams, each subjected to the action of a single point actuator. He demonstrated that, for the case of uncertainty in the actuator gain alone, a simple, closed form mapping leading to the distributions of the eigenvalues of the system could be determined directly from the distribution of the actuator gain, and for uncertainty in the remaining parameters, approximate distributions could be obtained through the application of perturbation methods.In the current paper, the FORM/SORM approach is applied to the taut string problem, where the distributed nature of the system is maintained throughout the analysis. Uncertain parameters, in this case the proportional gain and time delay, are characterized by probability distributions with known mean and variance. Each is transformed to a standard normal variate via Rosenblatt transformations, and the most likely failure point in the parameter space is found using a constrained optimization procedure. The effect of distribution is shown through parameter studies, and verification is provided by Monte Carlo simulation. As expected, time delay is shown to have a pronounced effect upon system robustness.