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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.

In this study, we focus on the controllability of fractional-order damped systems in linear and nonlinear cases with multiple time-varying delays in control. For the linear system based on the Mittag-Leffler matrix function, we define a controllability Gramian matrix, which is useful in judging whether the system is controllable or not. Furthermore, in two special cases, we present serval equivalent controllable conditions which are easy to verify. For the nonlinear system, under the controllability of its corresponding linear system, we obtain a sufficient condition on the nonlinear term to ensure that the system is controllable. Finally, two examples are given to illustrate the theory.

We address the output feedback stabilization for a Euler–Bernoulli beam equation with boundary moment control and disturbance. The stabilization of this system has been studied in Guo et al. (J Dyn Control Syst. 2014;20:539–58), where the controller is based on full state feedback. In order to derive the output feedback controller, we design a new disturbance estimator to estimate the total disturbance in the sense that the estimation error signal belongs $L^{2}(0,\infty )$ , and it decays exponentially if the initial state is smooth. Using the estimated total disturbance, we propose a control law to stabilize the system. Using admissibility theory, we show that the closed-loop system is exponentially stable and the signals in the disturbance estimator in the closed-loop are proved to be bounded.

This study aims to address the impulsive controllability of linear fractional impulse controlled systems. The authors establish the Kalman rank criterion to guarantee the impulsive controllability of the considered system. They also present the minimum number of impulsive controls for a realisation of such controllability. An example is finally presented to demonstrate the effectiveness of the authors' results.

This paper investigates performance output tracking for a boundary controlled multidimensional heat equation. It is assumed that the so-called total disturbance (which is composed of internal possibly nonlinear uncertainty and external disturbance) the equation is subject to is on one part of the boundary, and that the control is applied on the rest of the boundary. Using only partial boundary measurement, we first propose an extended state observer to estimate both system state and the total disturbance. This allows us to design a servomechanism and then an output feedback controller. Under the condition that both the reference signal and the disturbance vanish or belong to spaces $H^1(0,\infty ;L^1(\Gamma _1))$ and $L^2(0,\infty ;L^2(\Gamma _0))$ , respectively. We show that the over-all control strategy achieves three objectives on the system performance: first, exponentially output tracking for arbitrary given reference signal; second, uniformly boundedness of all internal signals; and, third, the internal asymptotic stability of the closed-loop system. In addition, the control strategy turns out to be robust to the measurement noise. We provide numerical experiments to illustrate the effectiveness of the proposed control strategy.

This paper considers boundary control of time fractional order diffusion–wave equation. Both boundary stabilization and disturbance rejection are considered. This paper, for the first time, has confirmed, via hybrid symbolic and numerical simulation studies, that the existing two schemes for boundary stabilization and disturbance rejection for (integer order) wave/beam equations are still valid for time fractional order diffusion–wave equations. The problem definition, the hybrid symbolic and numerical simulation techniques, outlines of the methods for boundary stabilization and disturbance rejection are presented together with extensive simulation results. Different dynamic behaviors are revealed for different fractional orders which may stimulate new future research opportunities.

This paper is concerned with the output feedback exponential stabilization for a one-dimensional wave equation with in-domain feedback/recirculation of a boundary velocity with a spatially constant coefficient, which is first studied in [ IEEE Trans. Autom. Control , vol. 62, no. 9, pp. 4760–4767, Sep. 2017]. When there are no boundary internal uncertainty and external disturbance, it is shown that by using one displacement measurement only, the output feedback makes the closed-loop system exponentially stable, which essentially improves the result of [ IEEE Trans. Autom. Control , vol. 62, no. 9, pp. 4760–4767, Sep. 2017]. When there are boundary internal uncertainty and external disturbance, using two displacement measurements only, we present an observer-based output feedback law that contains an infinite-dimensional disturbance estimator used to reject the boundary internal uncertainty and external disturbance. The resulting closed-loop system is shown to be exponentially stable and the state of all subsystem involved are uniformly stable. The Backstepping method for infinite dimensional system and active disturbance rejection control method play important roles in the design.

This paper is concerned with the adaptive error feedback regulator problem whereby the 1D Euler--Bernoulli beam equation with general unknown amplitude harmonic disturbance is rejected and general ...

In this study, the asymptotical stability for several classes of fractional order differential systems with time delay is investigated. The authors first present an integral inequality by which the Halanay inequality is extended to fractional order case. Based on the generalised Halanay inequality, the authors establish several asymptotical stability conditions under which the fractional order systems with time delay are asymptotically stable. It is worth to note that these stability conditions are easy to check without resorting to the solution expression of the systems.

The stability analysis of fractional-order systems with unbounded delay remains an open problem. In this paper, we firstly explore two new integral inequalities. Using these two integral inequalities obtained, the Halanay inequality with unbounded delay is extended to Caputo fractional-order case and Riemann–Liouville fractional-order case. Finally, several examples are presented to illustrate the effectiveness and applicability of the fractional Halanay inequalities in obtaining the asymptotic stability conditions of fractional-order systems with unbounded delay.

In this paper, we are concerned with the output feedback exponential stabilization problem for a system (plant) described by a one-dimensional Euler–Bernoulli beam equation. The measurements are only the displacement and the angular velocity at the right end. An infinite dimensional estimator is designed to estimate the disturbance. With the estimated disturbance, we propose a state observer that is exponentially convergent to the original system, then design two different kinds of stabilizing controllers: one is based on the velocity feedback, the other is based on the angular velocity feedback. In both cases, by adopting the Riesz basis approach, the exponential stability of the closed-loop systems is built with guaranteeing that all internal systems are uniformly bounded. The numerical experiments are carried out to illustrate the theoretical results.

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 ...

In this paper, we consider performance output tracking for a boundary controlled one-dimensional wave equation with possibly unknown internal nonlinear uncertainty and external disturbance. We first show that the open-loop system is well-posed and then propose a disturbance estimator. It is shown that the disturbance estimator can estimate successfully the total disturbance that consists of internal uncertainty and external disturbance. An servomechanism based on the estimated total disturbance is then designed. It is shown that the closed-loop system is well-posed. Three control objectives are achieved: (a) the output is tracking the reference signal; (b) all the internal signals are uniformly bounded; (c) the closed-loop system is internally asymptotically stable if both the reference signal and the disturbance vanish or belong to the space H2(0, ∞) and L2(0, ∞), respectively. The unmatched performance output tracking control is first time applied to a system described by the partial differential equation for complete general disturbance rejection and reference tracking purpose. Another key feature of this paper is that we do not use the high-gain to estimate total disturbance for unmatched system. The numerical experiments are carried out to illustrate effectiveness of the proposed control law.

In this paper, we consider boundary feedback stabilization for unstable time fractional reaction diffusion equations. New state feedback controls with actuation on one end are designed by the backstepping method for both Dirichlet and Neumann boundary controls. By the Riesz basis approach and the fractional Lyapunov method, we prove the existence and uniqueness and the Mittag--Leffler stability for the closed-loop systems. For both cases, the observers and the observer-based output feedback are designed to stabilize the systems.

In this paper, we consider boundary output feedback stabilization for a multi-dimensional wave equation with boundary control matched unknown nonlinear internal uncertainty and external disturbance. A new unknown input type extended state observer is proposed to recover both state and total disturbance which consists of internal uncertainty and external disturbance. A key feature of the proposed observer in this paper is that we do not use the high-gain to estimate the disturbance. By the active disturbance rejection control (ADRC) strategy, the total disturbance is compensated (canceled) in the feedback loop, which together with a collocated stabilizing controller without uncertainty, leads to an output feedback stabilizing feedback control. It is shown that the resulting closed-loop system is well-posed and asymptotically stable under weak assumption on internal uncertainty and external disturbance. The numerical experiments are carried out to show the effectiveness of the proposed scheme.

In this paper, we consider boundary output feedback stabilization for a multidimensional wave equation with boundary control matched unknown nonlinear internal uncertainty and external disturbance. A new unknown input type extended state observer is proposed to recover both state and total disturbance which consists of internal uncertainty and external disturbance by using the hidden regularity of the wave equation. By the active disturbance rejection control (ADRC) strategy, the total disturbance is compensated (canceled) in the feedback loop, which together with a collocated stabilizing controller without uncertainty, leads to an output feedback stabilizing feedback control. It is shown that the resulting closed-loop system is well-posed and asymptotically stable under weak assumption on internal uncertainty and external disturbance. The numerical experiments are carried out to show the effectiveness of the proposed scheme.

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 L2(0, ∞) 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.

We solve the output regulator problem for a system described by a Neumann boundary controlled anti-stable one-dimensional Schrodinger equation by the backstepping approach. The output to be controlled is not required to be measured and its observation operator is assumed to be admissible for a certain Co-semigroup that is related to the original system, while the measurements used for control are located at the boundary First we show that the open-loop system is well-posed. Then we find a state feedback control law to solve the output regulator problem by the backstepping method. We design a finite-dimensional reference observer and an infinite-dimensional disturbance observer. Based on this observer, we propose an output feedback controller to achieve output regulation.

In this paper, we propose a new output feedback control law to exponentially stabilize an one-dimensional wave equation with unknown anti-damping coefficient subject to general external disturbance entering the control end. With the proposed control law, although the anti-damping parameter is unknown and the disturbance is presented, it is shown that the resulting closed-loop system still decays exponentially. The numerical experiments are carried out to illustrate effectiveness of the proposed control law.

We investigate the exact controllability of a nonlinear plant described by the equation x˙(t)=Ax(t)+Bu(t)+BNN(x(t),t), where t ≥ 0. Here A is the infinitesimal generator of a strongly continuous group T on a Hilbert space X, B and BN, defined on Hilbert spaces U and UN, respectively, are admissible control operators for T and the function N:X×[0,∞)↦UN is continuous in t and Lipschitz in x, with Lipschitz constant LN independent of t. Thus, B and BN can be unbounded as operators from U and UN to X, in which case the nonlinear term BNN(x(t),t) in the plant is in general not Lipschitz in x. We assume that there exist linear operators F and Fb such that the triples (A,[BBN],F) and (-A,[BBN],Fb) are regular and A + BFΛ and −A + BFb, Λ are generators of operator semigroups Tf and Tb on X such that ∥Ttf∥·∥Ttb∥ decays to zero exponentially. We prove that if LN is sufficiently small, then the nonlinear plant is exactly controllable in some time τ > 0. Our proof is constructive, i.e. given an initial state ...

In this paper, we consider stabilisation for a cascade of ODE and first-order hyperbolic equation with external disturbance flowing to the control end. The active disturbance rejection control (ADR...

The active disturbance rejection control (ADRC) is now considered as a powerful control strategy in dealing with large uncertainty covering unknown dynamics, external disturbance, and unknown part in coefficient of the control. However, all theoretical works up to present are limited to deterministic uncertainty. In this technical note, we generalize the ADRC to uncertain nonlinear systems subject to external bounded stochastic disturbance described by an uncertain stochastic differential equation driven by white noise. We first design an extended state observer (ESO) that is used to estimate both state, and total disturbance which includes the internal uncertain nonlinear part and the external uncertain stochastic disturbance. It is shown that the resulting closed-loop system is practically stable in the mean-square topology. The numerical experiments are carried out to illustrate effectiveness of the proposed approach.

In this paper, we consider output feedback stabilisation for a wave PDE-ODE system with Dirichlet boundary interconnection and external disturbance flowing the control end. We first design a variable structure unknown input type state observer which is shown to be exponentially convergent. Then, we estimate the disturbance in terms of the estimated state, an idea from active disturbance rejection control. These enable us to design an observer-based output feedback stabilising control to this uncertain PDE-ODE system.

We consider boundary output feedback stabilization for a multi-dimensional Kirchhoff plate with boundary observation suffered from a general external disturbance. We adopt for the first time the active disturbance rejection control approach to stabilization of multi-dimensional partial differential equations under corrupted output feedback. In terms of this approach, the disturbance is estimated by a relatively independent estimator, based on (possibly) an infinite number of ordinary differential equations reduced from the original PDEs by infinitely many time-dependent test functions. This gives a state observer, an additional result via this approach. The disturbance is compensated in the feedback-loop. As a result, the control law can be designed almost as the same as that for the system without disturbance. We show that with a time varying gain properly designed, the observer driven by the disturbance estimator is convergent; and that all subsystems in the closed-loop are asymptotically stable. We also provide numerical simulations which demonstrate the convergence results and underline the effect of the time varying high gain estimator.

This paper investigates the controllability of fractional damped dynamical systems with control delay. Necessary and sufficient conditions for the controllability of fractional damped dynamical systems with control delay are established. The obtained conditions are easily checkable. An example is also given to illustrate the theory.

In this paper we solve the tracking and disturbance rejection problem for an unstable wave equation with reference and disturbance signals that are generated by a linear exosystem whose eigenvalues are located on the imaginary axis. We consider both unmatched disturbance at the boundary and also in the domain, and also matched disturbance at the control boundary. To make the control design more easy, the in-domain disturbance is first shifted to the control boundary by introducing a new variable transformation, and the anti-stable boundary term is then transformed into a dissipative boundary term by constructing a transport equation. We find an ordinary differential equation (ODE) with delay that describes the relationship between the output tracking error and the coupled signals consisting of the disturbance and reference signal, by using Riemann variables. Based on this ODE with delay, we apply the internal model principle to find the dynamic controller that achieves the output regulation while rejecting the disturbances, and uses only the tracking error as input.

In this research we are concerned with a switched system that switches very fast between two infinite-dimensional LTI subsystems, such that the overall system spends on average an equal amount of time in both possible subsystems. We ask if it is possible to approximate the switched system with an average system that is the limit of our switched system when the switching period tends to zero. We are mainly concerned with the situation when both subsystems are described by a contraction semigroup on a Hilbert space. We examine in particular the case when both subsystems are obtained from a basic subsystem via different output feedback operators.

We solve the output tracking and disturbance rejection problem for a plant described by a one-dimensional anti-stable wave equation, with reference and disturbance signals that belong to W1,∞ (0, ∞) and L∞ (0, ∞), respectively. Generally, these signals cannot be generated by an exogenous system. We explore an approach based on proportional control. We show that the proportional low gain controller can achieve exponential output tracking while rejecting the disturbance. We solve the problem in three steps. In the first step, we convert the original system without disturbance into the two transport equations with an ordinary differential equation by using Riemann variables, then we propose a proportional control law by making use of the properties of transport systems and time delay systems. In the second step, based on our recent results on disturbance estimators, we apply the estimation/cancellation strategy to cancel to the external disturbance and to track the reference asymptotically. In the third step, we design a controller for the state observer system. Since no explicit disturbance appears in this system, the disturbance is exactly compensated, and the output signal to be controlled is exponentially tracking the reference signal. As a byproduct, we obtain a new output feedback stabilizing control law by which the resulting closed-loop system is exponentially stable using only two displacement output signals. Numerical experiments demonstrate the effectiveness of the proposed control law.

Consider a nonlinear distributed parameter system (DPS) described by ẋ (t) = Ax(t)+Bu(t)+B N N(x(t), t) for all t ≥ 0. Here A is the infinitesimal generator of a strongly continuous group of operators 𝕋 on a Hilbert space X, B and B N , defined on Hilbert spaces U and U N , respectively, are admissible control operators for 𝕋 and N : X × [0,∞) ↦ U N is continuous in t and Lipschitz in x, with Lipschitz constant L N independent of t. Thus B and B N can be unbounded as operators from U and U N to X, in which case the nonlinear term B N N(x(t), t) in the DPS is in general not a Lipschitz map from X ×[0,∞) to the state space X. Our goal is to find conditions under which this DPS is exactly controllable in some time τ, which means that for any initial state x(0) ∈ X, we can steer the final state x(τ) of the DPS to any chosen point in X by using an appropriate input function u ∈ L2([0, τ];U). We suppose that there exist linear operators F and F b such that (A, [B B N ], F) and (−A; [B B N ], F b ) are regular triples and A + BF Λ and −A + BF b,Λ are generators of strongly continuous semigroups 𝕋f and 𝕋b on X such that ∥𝕋 t f∥ · ∥𝕋 t b∥ decays to zero exponentially. We prove that if L N is sufficiently small, then the nonlinear DPS is exactly controllable in some time τ > 0. Our proof is constructive and provides a numerical algorithm for approximating the required control signal. We illustrate our approach using a simple example.

In this paper we are concerned with the output regulation problem for a system (plant) described by a boundary controlled anti-stable one-dimensional Schrodinger equation by the backstepping approach. The output to be controlled is not required to be measurable and its output operator is assumed to be admissible for a certain C 0 -semigroup that is related to the original system whereas the measurement is located at the boundary. First we show that the open-loop system is well-posed. We construct a state feedback control law to solve the output regulation problem, using the backstepping method. We design a finite-dimensional reference observer and an infinite-dimensional disturbance observer. Based on these observers, we design an output feedback controller that achieves output regulation.

This paper investigates the controllability of fractional damped dynamical systems with distributed delays. A necessary and sufficient condition for the controllability of linear fractional damped dynamical system with distributed delay in control is established, and a sufficient condition for the nonlinear system is also established. Two examples are given to illustrate the theorems.

In this paper, we solve the output tracking and disturbance rejection problem for a system described by a one-dimensional anti-stable wave equation, with reference and disturbance signals that belong to W1,∞[0, ∞) and L∞[0, ∞), respectively. Generally, these signals cannot be generated from an exosystem. We explore an approach based on proportional control. It is shown that a proportional gain controller can achieve exponentially the output tracking while rejecting disturbance. Our method consists of three steps: first, we convert the original system without disturbance into two transport equations with an ordinary differential equation by using Riemann variables, then we propose a proportional control law by making use of the properties of transport systems and time delay systems. Second, based on our recent result on disturbance estimator, we apply the estimation/cancellion strategy to cancel to the external disturbance and to track the reference asymptotically. Third, we design a controller using a state observer. Since disturbance does not appear in the observer explicitly (the disturbance is exactly compensated), the controlled output signal is exponentially tracking the reference signal. As a byproduct, we obtain a new output feedback stabilizing control law by which the resulting closed-loop system is exponentially stable using only two displacement output signals.

In this comment, we point out that there are four errors leading to question of the validity of Theorem 3 and Proposition 4 of a recent paper Ran, Wang, and Dong (2016a).