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In this paper, event-triggered active disturbance rejection control (ADRC) is first addressed for a class of uncertain random nonlinear systems driven by bounded noise and colored noise. The event-triggered extended state observer (ESO) and ADRC controller are designed, where two respective event-triggering mechanisms with a fixed positive lower bound for the inter-execution times are proposed. The random total disturbance representing the coupling of nonlinear unmodeled dynamics, external deterministic disturbance, bounded noise, and colored noise is estimated in real time by the event-triggered ESO and compensated in the event-triggered feedback loop. Both the mean square and almost surely practical convergence of the closed-loop systems is shown with rigorous theoretical analysis. Finally, some numerical simulations are implemented to validate the proposed control scheme and theoretical results., Comment: arXiv admin note: text overlap with arXiv:2302.02395

In this paper, linear and nonlinear event-triggered extended state observers are designed for a class of uncertain stochastic systems driven by bounded and colored noises. Two event-generators with an ensured positive minimum inter-event time for every sample path solution of the stochastic systems, are proposed for the designs of linear and nonlinear event-triggered extended state observers, respectively. The mean square and almost sure convergence of the estimation errors of unmeasured state and stochastic total disturbance including internal uncertainty and external stochastic noises is presented with rigorous theoretical proofs. Compared with the linear event-triggered extended state observer, the theoretical results show that the nonlinear one via homogeneity possesses higher estimation accuracy but is at the price of higher triggering frequency. Some numerical simulations are performed to authenticate the theoretical results.

In this paper, the convergence and noise-tolerant performance of a tracking differentiator in the presence of multiple stochastic disturbances are investigated for the first time. We consider a quite general case where the input signal is corrupted by additive colored noise, and the tracking differentiator itself is disturbed by additive colored noise and white noise. It is shown that the tracking differentiator tracks the input signal and its generalized derivatives in mean square and even in almost sure sense when the stochastic noise affecting the input signal is vanishing. Some numerical simulations are performed to validate the theoretical results.

In this paper, a novel control strategy namely disturbance observer-based control is first applied to stabilization and disturbance rejection for an anti-stable stochastic heat equation with Neumann boundary actuation and unknown boundary external disturbance generated by an exogenous system. A disturbance observer-based boundary control is designed based on the backstepping approach and estimation/cancellation strategy, where the unknown disturbance is estimated in real time by a disturbance observer and rejected in the closed-loop, while the in-domain multiplicative noise whose intensity is within a known finite interval is attenuated. It is shown that the resulting closed-loop system is exponentially stable in the sense of both mean square and almost surely. A numerical example is demonstrated to validate the effectiveness of the proposed control approach.

A linear control system with quadratic cost functional over infinite time horizon is considered without assuming controllability/stabilizability condition and the global integrability condition for the nonhomogeneous term of the state equation and the weight functions in the linear terms in the running cost rate function. Classical approaches do not apply for such kind of problems. Existence and non-existence of overtaking optimal controls in various cases are established. Some concrete examples are presented. These results show that the overtaking optimality approach can be used to solve some of the above-mentioned problems and at the same time, the limitation of this approach is also revealed.

We investigate the regulator problem (tracking and disturbance rejection) for a system (plant) described by a boundary controlled anti-stable linear one-dimensional Schrodinger equation, using the backstepping approach. The output to be controlled is not required to be measurable and its observation operator is assumed to be admissible for a certain operator semigroup that is related to the operator semigroup of the original plant. We consider both the state feedback and the output feedback regulator problem. In the latter case, the measurement from the Schrodinger equation is taken at the boundary. First we show that the open-loop system is well-posed. We design a state feedback control law that solves the regulator problem by the backstepping method. Then, a finite-dimensional reference observer and an infinite-dimensional disturbance observer are designed. Putting these together, we obtain an output feedback controller with internal loop that achieves output regulation., Comment: This appeared in "Pure and Applied Functional Analysis", published by Yokohama Publishers

We study the output feedback exponential stabilization of a one-dimensional unstable wave equation, where the boundary input, given by the Neumann trace at one end of the domain, is the sum of the control input and the total disturbance. The latter is composed of a nonlinear uncertain feedback term and an external bounded disturbance. Using the two boundary displacements as output signals, we design a disturbance estimator that does not use high gain. It is shown that the disturbance estimator can estimate the total disturbance in the sense that the estimation error signal is in $L^2[0,\infty)$. Using the estimated total disturbance, we design an observer whose state is exponentially convergent to the state of original system. Finally, we design an observer-based output feedback stabilizing controller. The total disturbance is approximately canceled in the feedback loop by its estimate. The closed-loop system is shown to be exponentially stable while guaranteeing that all the internal signals are uniformly bounded., Comment: Full Version

This paper develops systematically the output feedback exponential stabilization for a one-dimensional unstable/anti-stable wave equation where the control boundary suffers from both internal nonlinear uncertainty and external disturbance. Using only two displacement signals, we propose a disturbance estimator that not only can estimate successfully the disturbance in the sense that the error is in $L^2(0,\infty)$ but also is free high-gain. With the estimated disturbance, we design a state observer that is exponentially convergent to the state of original system. An observer-based output feedback stabilizing control law is proposed. The disturbance is then canceled in the feedback loop by its approximated value. The closed-loop system is shown to be exponentially stable and it can be guaranteed that all internal signals are uniformly bounded., Comment: Proc. of the IFAC World Congress, Toulouse,France, July 2017

In this paper, we prove that a sequence of generalized eigenvectors of a linear unbounded operator associated with an Euler-Bernoulli beam equation under bending moment boundary feedback forms a Riesz basis for the underlying state Hilbert space. As a consequence, the resulting closed-loop system is exponentially stable.