Feedback Exponential Stabilization Research Articles

On a new method for the state and output feedback exponential stabilization of a cascaded wave PDE-ODE system with nonlinear boundary condition

Abstract This study presents a novel approach for the state and output feedback exponential control of cascaded unstable ODE-wave equations with nonlinear boundary condition. The new approach can be also used to investigate the state and output feedback stabilization of other cascaded ODE-partial differential equation (PDE) systems with nonlinear boundary condition. The interconnection between the ODE and wave PDE with the nonlinear boundary is bi-directional at two points. First, an exponential controller is planned and the well-posed and exponential stability of the closed-loop system is derived. Next, an exponential observer is designed, whereby the exponential stability of the overall error system is shown. Then, an observer-based output feedback control that causes the overall real system to be exponentially stable is built. The existence and exponential stability of the solution to the overall closed-loop system are further deduced. Finally, simulation results are given to verify the effectiveness, feasibility and validity of the proposed technique.

Infinite‐dimensional extended state observer based stabilization for an ODE‐wave cascaded system with control matched disturbance

AbstractThis paper focuses on the output feedback exponential stabilization for an ODE‐wave cascaded system with control matched disturbance at one end of the wave equation. We construct an infinite‐dimensional extended state observer to trace the state and estimate the total disturbance simultaneously. An estimated state and estimated total disturbance based output feedback control law is then designed. It is proved that the original system is exponentially stabilized and the whole closed‐loop system is bounded.

Feedback Exponential Stabilization of GHZ States of Multiqubit Systems

In this article, we consider stochastic master equations describing the evolution of a multiqubit system interacting with electromagnetic fields undergoing continuous-time measurements. By considering multiple <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$z$</tex-math></inline-formula> -type (Pauli <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$z$</tex-math></inline-formula> matrix on different qubits) and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$x$</tex-math></inline-formula> -type (Pauli <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$x$</tex-math></inline-formula> matrix on all qubits) measurements and one control Hamiltonian, we provide general conditions on the feedback controller and the control Hamiltonian ensuring almost sure exponential convergence to a predetermined Greenberger–Horne-Zeilinger (GHZ) state, which is assumed to be a common eigenstate of the measurement operators. We provide explicit expressions of feedback controllers satisfying such conditions. We also consider the case of only <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$z$</tex-math></inline-formula> -type measurements and multiple control Hamiltonians. We show that local stability in probability holds true, however due to the absence of random displacements generated by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$x$</tex-math></inline-formula> -type measurements, the reachability of a neighborhood of a predetermined GHZ state is not clear. In this case, we provide a heuristic discussion on some conditions which may ensure asymptotic convergence toward the target state. Finally, we demonstrate the effectiveness of our methodology for a three-qubit system through numerical simulations.

Feedback exponential stabilization of stochastic quantum systems based on state space division

In this paper, the global exponential stabilization of two-level stochastic quantum systems is achieved by using measurement-based feedback strategies, including state feedback and noise-assisted feedback, and the state convergence rate is improved by dividing the state space into multiple subspaces and using different feedback controls in different state subspaces. For both the combination of two continuous noise-assisted feedback and the combination of state feedback and noise-assisted feedback, the state space is divided based on the comparison of real-time state convergence rate under different measurement-based feedback strategies, respectively. The global exponential convergence and the effect of dividing state space in improving state convergence rate for two-level quantum systems are proved in theory and verified in numerical simulations.

Output feedback exponential stabilization for a 1-d wave PDE with dynamic boundary

We study the output feedback exponential stabilization for a 1-d wave PDE with dynamic boundary. When there is no disturbance, with only one measurement, we construct an infinite-dimensional state observer to trace the state and design an estimated state based controller to exponentially stabilize the original system. This is an essentially important improvement for the existence literature Morgül et al. (1994) [32] where two measurements including the high order angular velocity feedback were adopted. When a control matched nonlinear internal uncertainty and external disturbance are taken into consideration, we construct an infinite-dimensional extended state observer (ESO) to estimate the total disturbance and state simultaneously. By compensating the total disturbance, an estimated state based controller is designed to exponentially stabilize the original system while making the closed-loop system bounded. Riesz basis approach is crucial to the verifications of the exponential stabilities of two coupled systems of the closed-loop systems. Some numerical simulations are presented to illustrate the effectiveness.

Feedback exponential stabilization of the semilinear heat equation with nonlocal initial conditions

The present paper is devoted to the problem of stabilization of the one-dimensional semilinear heat equation with nonlocal initial conditions. The control is with boundary actuation. It is linear, of finite-dimensional structure, given in an explicit form. It allows to write the corresponding solution of the closed-loop equation in a mild formulation via a kernel, then to apply a fixed point argument in a convenient space.

Oblique projection output-based feedback exponential stabilization of nonautonomous parabolic equations

A new output injection operator and a new feedback control operator are proposed to estimate and stabilize the state of linear nonautonomous parabolic equations. The proposed operators are time-independent and involve appropriately constructed oblique projections. Firstly, it is shown the estimating property of the dynamical Luenberger observer and the stabilizing property of the feedback controller. Then, it is shown the stability of the observer-controller coupled system, where the control input is dynamically constructed from the measured output. The numbers of actuators and sensors are finite. Finally, results of simulations are presented showing the stability of the closed-loop observer-controller coupled system.

Exponential stability of nonlinear systems with event-triggered schemes and its application

In this paper, we discuss exponential stability for nonlinear systems with sampled-data-based event-triggered schemes. First, a framework is proposed to analyze exponential stability for nonlinear systems under some different triggering conditions. Based on these results, output feedback exponential stabilization is investigated for a class of inherently nonlinear systems under a kind of event-triggered strategies. Finally, the rationality of the theoretical work is verified by numerical simulations.

Output Feedback Exponential Stabilization for a One-Dimensional Wave Equation With Control Matched Nonlinear Disturbance

In this article, we study the output feedback exponential stabilization of a one-dimensional wave equation with control matched nonlinear disturbance. The well posedness of the open-loop system, which is essentially important for the designs of observer and controller, is verified. By designing an infinite-dimensional unknown input state observer in place of the conventional extended state observer, we estimate the total disturbance in real time. Moreover, we prove the convergence of the disturbance estimator. Based on the disturbance estimator, we design a state observer by compensating the total disturbance, and an estimated state feedback controller to exponentially stabilize the original system by means of the Riesz basis approach. Some numerical simulations are presented.

Boundary state feedback exponential stabilization for a one-dimensional wave equation with velocity recirculation

In this paper, we consider boundary state feedback stabilization of a one-dimensional wave equation with in-domain feedback/recirculation of an intermediate point velocity. We firstly construct an auxiliary control system which has a nonlocal term of the displacement at the same intermediate point. Then by choosing a well-known exponentially stable wave equation as its target system, we find one backstepping transformation from which a state feedback law for this auxiliary system is proposed. Finally, taking the resulting closed-loop of the auxiliary system as a new target system, we obtain another backstepping transformation from which a boundary state feedback controller for the original system is designed. By the equivalence of three systems, the closed-loop of original system is proved to be well-posed and exponentially stable. Some numerical simulations are presented to validate the theoretical results.

Output Feedback Exponential Stabilization of One-Dimensional Wave Equation With Velocity Recirculation

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.

Output Feedback Stabilization of an ODE-Schrödinger Cascade System Subject to Boundary Control Matched Unknown Disturbance

In this paper, we consider the output feedback exponential stabilization problem of ODE-Schrodinger cascade systems with the external disturbance. We propose a new extended state observer (ESO) that estimates both state and disturbance by the three output signals, then design a stabilizing control law by utilizing the backstepping technique. The resulting closed-loop system is shown to be exponentially stable guaranteeing that all internal systems involved are uniformly bounded. Finally, some numerical experiments are carried out to verify the effectiveness of the proposed control law.

Output feedback stabilization for a cascaded heat PDE‐ODE system subject to uncertain disturbance

SummaryIn this paper, we are concerned with the output feedback exponential stabilization of a cascaded parabolic partial differential equation–ordinary differential equation system where the control end suffers from the external uncertain disturbance. We design an unknown input–type state observer with the disturbance estimator term, which is used to stabilize this cascaded system and compensate the external disturbance simultaneously. The stabilizing state feedback control is designed for the observer system by the three‐step backstepping transformation, which is the corresponding observer‐based output feedback stabilizing control for the original system. Finally, the closed‐loop system is shown to be exponentially stable.

Robust H∞ control for uncertain switched nonlinear polynomial systems: Parameterization of controller approach

SummaryThis paper considers the global feedback exponential stabilization and L2 gain disturbance attenuation problems of the switched nonlinear polynomial systems with passive and nonpassive subsystems for any given average dwell time. In the existing result, it needs that there exists at least one open loop passive subsystem in the switched nonlinear system, which is unnecessary for the switched nonlinear polynomial system in this paper because the passivity of the subsystem can be obtained according to the passivity‐based feedback dissipative Hamiltonian realization. In addition, a parameterization of controller approach–based feedback exponential stabilization method of nonlinear polynomial system is described and utilized such that the Lyapunov function of the passive subsystem possesses the required decay rate to exponentially stabilize the switched nonlinear polynomial system. Comparing with the controller designed with the existing method, the obtained controller in this paper has much simpler structure and better performance. An example simulation is provided to illustrate the effectiveness of the proposed approach.

Stabilization to trajectories for parabolic equations

The feedback exponential stabilization to trajectories for semilinear parabolic equations in a given bounded domain is addressed. The controls take values in a finite-dimensional space and are supported in a small region. Both internal and boundary controls are considered.

Disturbance estimator based output feedback exponential stabilization for Euler–Bernoulli beam equation with boundary control

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.

Output Feedback Exponential Stabilization for One-Dimensional Unstable Wave Equations with Boundary Control Matched Disturbance

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 ...

Output feedback exponential stabilization of a nonlinear 1-D wave equation with boundary input

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.

Output Feedback Stabilization of Linear Systems with Unknown Additive Output Sinusoidal Disturbances

The output feedback exponential stabilization problem is addressed for known linear stabilizable and detectable systems when the measured output is affected by sinusoidal disturbances generated by an unknown exosystem and only an upper bound on the exosystem order is supposed to be known.Necessary and sufficient conditions are given: in particular a solution to the problem exists if and only if the set of exosystem eigenvalues is disjoint from the set of system eigenvalues. Two adaptively stabilizing control algorithms are proposed: the first one drives the state of the given system exponentially to zero when the actual disturbances are exactly modelled by the exosystem. When the exosystem overmodels the actual disturbances an on-line detector of the number of excited frequencies is included in the second more complex algorithm: the exponentially converging estimates of the system state variables are then used to drive the state of the given system exponentially to zero. An illustrative example with a disturbance containing a variable number of frequencies is worked out in details and simulated.

Output feedback exponential stabilization of uncertain chained systems

This paper deals with chained form systems with strongly nonlinear disturbances and drift terms. The objective is to design robust nonlinear output feedback laws such that the closed-loop systems are globally exponentially stable. The systematic strategy combines the input-state-scaling technique with the so-called backstepping procedure. A dynamic output feedback controller for general case of uncertain chained system is developed with a filter of observer gain. Furthermore, two special cases are considered which do not use the observer gain filter. In particular, a switching control strategy is employed to get around the smooth stabilization issue (difficulty) associated with nonholonomic systems when the initial state of system is known.