On the moment dynamics of stochastically delayed linear control systems

Abstract

In this article, the dynamics and stability of a linear system with stochastic delay and additive noise are investigated. It is assumed that the delay value is sampled periodically from a stationary distribution. A semi-discretization technique is used to time-discretize the system and derive the mean and second-moment dynamics. These dynamics are used to obtain the stationary moments and the corresponding necessary and sufficient stability conditions. The application of the proposed method is illustrated through the analysis of the Hayes equation with stochastic delay and additive noise. The method is also applied to the control design of a connected automated vehicle. These examples illuminate the effects of stochastic delays on the robustness of dynamical systems.

Critical parameters for the robust stabilization of the inverted pendulum with reaction delay: State feedback versus predictor feedback

“…It shall be noted that although the presented analysis used the concept of static uncertainties according to References 17‐20,39, the results can also be interpreted to stochastic parameter uncertainties, for example, noise in the sensory perception or in the motor control. As shown in References 40‐42, stochastic perturbation has a similar effect on the performance of the control process: the stable parameter region in the presence of noise is typically smaller than the stable region for the nominal (noise‐free) system. In this sense, the stability radii can be used to demonstrate the robustness of the system against noise, too.…”

Adaptive Boundary Control of Reaction-Diffusion PDEs with Unknown Input Delay

“…where p 1n = 2 1 0 sin(nπξ)p(1, ξ)dξ, which is the Fourier coefficient of p(1, y), i.e., the function p(1, y) can be represented in the form of Fourier series as p(1, y) = ∞ n=1 p 1n sin(nπy). Similarly, one can derive the kernels of inverse transformations (22) and (23) given by q(x, y) = −λy…”

“…In [18], stability conditions of a nonlinear plant undergoing stochastic delays are derived from the mean and the second moment dynamics (see [21] and references therein). In the context of the control of autonomous vehicles, [22] performed a stability analysis of a class of linear systems with lag time, assuming a periodic sampling of the stochastic delay value from a stationary distribution. Employing a vehicleto-vehicle communication control, [23] considered delays of stochastic nature induced by packet loss and stabilize both inter-vehicle distance and velocity dynamics by constructing robust nonlinear control laws.…”

Prediction-based controller for linear systems with stochastic input delay

“…Indeed, prediction-based control laws have been applied to linear SDDEs in [6], but the delay itself is assumed to be constant. Up to our knowledge, ones of the few studies to consider the delay as a stochastic variable are [21,22,27,40]. While [40] studies a piecewise constant process and [27] analyzes a deterministic delay term multiplied by a random variable, [21,22] consider stochastic state delays modeled as a Markov process with a finite number of states.…”

“…Up to our knowledge, ones of the few studies to consider the delay as a stochastic variable are [21,22,27,40]. While [40] studies a piecewise constant process and [27] analyzes a deterministic delay term multiplied by a random variable, [21,22] consider stochastic state delays modeled as a Markov process with a finite number of states. The authors then consider each delay value separately, following the so-called technique of probabilistic delay averaging.…”