Output Feedback Stabilization Research Articles

Invariant output feedback stabilisability: the scalar case

In this paper we prove that stabilisability is a static output feedback (SOF) invariant, for scalar and multivariable systems. Then we examine scalar stabilisability, from an invariant viewpoint. We prove that the signature of Hermite’s Bezoutian is constant within certain intervals that we call critical, and we give a very simple algebraic stabilisability criterion, consisted of a finite number of stability checks, one for each critical interval. We establish the validity of this criterion with Routh, Hurwitz and Lyapunov methods. We prove that the winding number of the Nyquist plot around points of the real axis, is constant within critical intervals. We correlate the stabilisability within critical intervals with the stability of corresponding critical polynomials defined at their limits. Finally, we discuss why our findings constitute a decisive step towards the analytical solution for the persisting problem of static output feedback stabilizability of linear time invariant multivariable systems.

Output feedback stabilization of type 2 fuzzy singular fractional-order systems with mismatched membership functions

Output feedback stabilization of type 2 fuzzy singular fractional-order systems with mismatched membership functions

Global Mittag-Leffler output feedback stabilization for nonlinear fractional-order system with known growth rate

Global Mittag-Leffler output feedback stabilization for nonlinear fractional-order system with known growth rate

Static output feedback stabilization of uncertain rational nonlinear systems with input saturation

In this paper, the notion of robust strict QSR-dissipativity is applied to solve the static output feedback control problem for a class of continuous-time nonlinear rational systems subject to input saturation and bounded parametric uncertainties. A local dissipativity condition is combined with generalized sector conditions to formulate the synthesis of a stabilizing controller in terms of linear matrix inequalities. Furthermore, an iterative algorithm based on linear matrix inequalities is proposed in order to compute the feedback gain matrix that maximizes the estimate of the closed-loop region of attraction. Numerical examples are provided to illustrate the applicability of this new approach in examples borrowed from the literature.

Uniform exponential stability of semi-discrete scheme for observer-based control of 1-D wave equation

In this paper, we consider the uniformly exponential stability of a semi-discretized coupled system generated by a 1-D wave equation under a stabilizing output feedback control. Since the control and observation are non-collocated, we first design an observer to recover the state. The closed-loop system under the observer-based feedback is composed of a two coupled wave equations. First, we transform the closed-loop system into an equivalent system by order reduction method, and the transformed system is shown to be exponentially stable by constructing a suitable Lyapunov function. As a consequence, two equivalent semi-discrete finite difference schemes are developed corresponding to the transformed system and original coupled system. Parallel to the proof of the continuous counterpart, the Lyapunov function is constructed to prove the uniformly exponential stability of the semi-discrete scheme for the transformed system, which leads to the uniformly exponential stability of the semi-discrete scheme for the original coupled system. The weak convergence of the discrete solution to the continuous counterpart is also briefly presented.

Asynchronous output feedback control for Markov jump systems under dynamic event‐triggered strategy

AbstractThis article addresses the synthesis of static output feedback stabilization via employing event‐based control, for discrete‐time Markov jump systems subject to limited bandwidth and mismatched modes. The event‐triggered communication strategy with an extra dynamic variable is considered in the design of controller to alleviate transmission burden. Then, for the purpose of further improving system performance, a special triggering threshold constructed as a diagonal matrix is introduced. Asynchronous behavior caused by mismatched modes between the controller and controlled systems is depicted by utilizing a hidden Markov model. With the assistance of output feedback theory, an asynchronous event‐based output feedback control law is sufficiently formulated to achieve the stochastic stabilization with desired performance. Ultimately, an illustrative example is implemented to verify the feasibility of theoretical results.

Quantized output feedback for continuous-time switched systems with time-delay

This paper studies the output feedback stabilization of the linear switched systems with quantization and time-delay. Under the coupled effect of time-delay and sampling, there is a complex mismatch between the system and the controller modes, which increases the difficulty of quantization rules design. Moreover, the switching of system modes brings challenges to the state reconstruction based on output signals. The purpose of this paper is designing a feedback controller based on a state observer to ensure the exponential convergence and Lyapunov stability of the switched systems. First, a virtual system is introduced to update the observer state to deal with the complex mismatch between the system and controller modes. Second, quantization rules are designed separately relying on the switching situations during the state reconstruction. Last, the upper bound of the system state is obtained by discussing the increasing/decreasing rate of Lyapunov function, and the system stability is guaranteed. Two-tank system is adopted to illustrate the effectiveness of the main results.

Global Finite-Time Stabilization for Uncertain Systems With Unknown Measurement Sensitivity.

This article focuses on global finite-time output feedback stabilization for uncertain nonlinear systems with unknown measurement sensitivity. The existence of the continuous measurement error resulting from limited accuracy of sensors invalidates the existing design strategies depending on the use of the precise output in the construction of an observer, which highlights the contribution of this article. Essentially, different from related works, we propose a new finite-time convergent observer by avoiding the use of the information on nonlinearities. By combining the homogeneous domination with the addition of a power integrator method, an output feedback controller composed of multiple nested sign functions is successfully developed. Finally, the effectiveness of the presented scheme is exhibited by a numerical example.

Nonlinear boundary output feedback stabilization of reaction–diffusion equations

This paper studies the design of a finite-dimensional output feedback controller for the stabilization of a reaction-diffusion equation in the presence of a sector nonlinearity in the boundary input. Due to the input nonlinearity, classical approaches relying on the transfer of the control from the boundary into the domain with explicit occurrence of the time-derivative of the control cannot be applied. In this context, we first demonstrate using Lyapunov direct method how a finite-dimensional observer-based controller can be designed, without using the time derivative of the boundary input as an auxiliary command, in order to achieve the boundary stabilization of general 1-D reaction-diffusion equations with Robin boundary conditions and a measurement selected as a Dirichlet trace. We extend this approach to the case of a control applying at the boundary through a sector nonlinearity. We show from the derived stability conditions the existence of a size of the sector (in which the nonlinearity is confined) so that the stability of the closed-loop system is achieved when selecting the dimension of the observer to be large enough.

Time space deformation approach to prescribed-time stabilization: Synergy of time-varying and non-Lipschitz feedback designs

A novel approach is proposed to the prescribed-time stabilizing design of feedback linearizable controllable systems regardless of their initial conditions. The basic design idea consists in scaling an appropriate finite-time stabilizing non-Lipschitz feedback by means of a certain time deformation which squeezes the infinite time interval [0,∞) to the prescribed one [0,T), while also changing the state variables. The resulting nonsmooth feedback design relies on time-varying gains, uniformly bounded over the infinite horizon, thereby yielding an attractive implementation opportunity which is opposed to the original design (Song et al., 2017) with time-varying gains, escaping to infinity as time goes to the prescribed time instant. To facilitate the exposition the approach is first developed for a perturbed double integrator, driven by a twisting-wise controller, which is fed by a supertwisting-wise observer. Both the controller and observer, chosen for their robustness properties, are properly scaled according to the time space deformation to constitute an innovative prescribed-time stabilizing output feedback design of the double integrator, operating in the presence of uniformly bounded matched disturbances. To test the versatility of the approach it is then extended to the prescribed-time state feedback design of multi-input multi-output systems representable in the normal form. Theoretical results are additionally supported by simulations.

Output feedback stabilization of stochastic planar nonlinear systems with output constraint

This paper is concerned with the problem of output feedback stabilization of stochastic planar nonlinear systems with output constraint. By introducing some new nonlinear mappings, adding a power integrator and homogeneous domination techniques, according to different value ranges of the homogeneous degree in the same form of assumptions, we establish two new design and analysis frames of output feedback stabilization to achieve respectively stochastic asymptotic stability (or asymptotical stability in probability) and stochastic finite-time stability of the trivial solution of the closed-loop system while guaranteeing the requirements of almost sure boundedness of the closed-loop signals and the symmetric output constraint.

Practical fixed-time ISS of neutral time-delay systems with application to stabilization by using delays

The concept of practical fixed-time input-to-state stability for neutral time-delay systems with exogenous perturbations is introduced. Lyapunov–Krasovskii theorems are formulated in explicit and implicit ways. Further, the problem of static nonlinear output-feedback stabilization of a linear system with parametric uncertainties, external bounded state and output disturbances by using artificial delays is considered. The constructive control design consists in solving linear matrix inequalities with only four tuning parameters to be chosen. It is shown both, theoretically and numerically, that the system governed by the proposed controller converges faster to the given invariant set than in the case of using its linear counterpart.

Output feedback stabilization of large-scale networked cyberphysical systems using cryptographic techniques

This paper aims to create a secure environment for state estimation and control design of a networked system composed of multiple dynamic entities and remote computational units, in the presence of disclosure attacks. In particular, both dynamic entities and computational units may be vulnerable to attacks and become malicious. The objective is to ensure that the input and output data of the benign entities are protected from the malicious parties as well as protected when they are transmitted over the network in a distributed environment. We propose a methodology integrating a novel double-layer cryptographic technique with an observer-based control algorithm to achieve the objective where the cryptographic technique addresses the security requirements and the control algorithm satisfies the performance requirements.

Boundary output feedback stabilisation of a class of reaction–diffusion PDEs with delayed boundary measurement

This paper addresses the boundary output feedback stabilisation of a general class of 1-D reaction–diffusion PDEs with delayed boundary measurement. The output takes the form of either a Dirichlet or Neumann trace. The output delay can be arbitrarily large. The control strategy is composed of a finite-dimensional observer that is used to observe a delayed version of the first modes of the PDE and a predictor component that is employed to obtain the control input to be applied at the current time. For any given value of the output delay, we assess the stability of the resulting closed-loop system provided the order of the observer is selected large enough. Taking advantage of this result, we discuss the extension of the control strategy to the case of simultaneous input and output delays.

Global stabilization via output feedback of stochastic nonlinear time‐delay systems with time‐varying measurement error: A Lyapunov–Razumikhin approach

AbstractThe output feedback stabilization for a class of stochastic nonlinear systems in lower‐triangular form is discussed. The key feature of the studied systems is that both of time‐delay factors and sensor measurement error are considered into system modeling. To handle these uncertainties in unmeasured states, a form of full‐order state observer is introduced to rebuild the states of the systems. By developing a simple but efficient control scheme, a linear output feedback controller coupled with two different static gains is designed to stabilize the whole system. In contrast to the conventional Lyapunov–Krasovskii functional approach, a framework of stability analysis based on Lyapunov–Razumikhin function is proposed for the closed‐loop stochastic system. The advantage of the proposed method lies in the superior capability of tackling various types of time delays. Specially, the time derivatives of time‐delay terms can be no less than 1, or even unbounded. Finally, two examples are shown to validate our theoretical results.

Global stabilization for a class of upper‐triangular stochastic nonlinear systems with input delay via sampled‐data output feedback

AbstractThe contribution of this investigation lies in solving a sampled‐data output feedback stabilization problem for a class of upper‐triangular stochastic nonlinear systems in ‐normal structure with unavailable states and uncertain time‐varying input delay. A reduced‐order observer with the available sampled output detection is developed for the focused system to estimate its unavailable states, and under such observer, a sampled‐data output feedback stabilizer is constructed by combining the backstepping and adding a power integral techniques. Through selecting the appropriate Lyapunov–Krasovskii functionals, we can demonstrate that the adverse influences caused by the uncertain time‐varying delays in the resulting closed‐loop system can be restricted effectively by the constructed stabilizer, and further, by following the stochastic differential theories, the global stability of the considered resulting closed‐loop system can be analyzed under the suitable design parameters and allowable sampling period. Finally, two examples are given to verify the effectiveness of the proposed control method.

A Characterization of approximate decentralized fixed modes of time‐delay systems

AbstractThe concept of decentralized fixed mode (DFM) is crucial in determining the output feedback stabilizability by linear time‐invariant (LTI) decentralized controllers. However, in some cases, although a system has no DFMs, some of its modes may be very close to being fixed. It is thus important to characterize these modes, which are called approximate DFMs (ADFMs). In this paper, the characterization of ADFMs is discussed for LTI neutral (possibly descriptor‐type) time‐delay systems under static and dynamic decentralized output feedback control laws, and a mode mobility measure is proposed to provide quantitative information about ADFMs. To analytically compare the proposed measure, some of the commonly used quantitative measures for the characterization of ADFMs of finite‐dimensional systems have also been extended to the considered type of systems. The herein proposed approach allows multiple time delays and does not require additional expansions or preliminaries for multi‐input multi‐output systems, like some other commonly used methods. Thus, characterization with this measure, in general, requires less computational load, while it can provide more explicit information about the controller effort than the other considered measures.

Robust implementable regulator design of linear systems with non-vanishing measurements

Robust implementable output regulator design approaches are studied for linear continuous-time systems with periodically sampled measurements, consisting of both the regulation errors and extra measurements that are generally non-vanishing in steady state. A digital regulator is first developed via the conventional emulation-based approach, rendering the regulation errors asymptotically bounded with a small sampling period. We then develop a hybrid design framework by incorporating a generalized hold device, which transforms the original problem into the problem of designing an output feedback controller fulfilling two conditions for a discrete-time system, given any (large) non-pathological sampling period. In our hybrid design framework, we show that such a controller can always be obtained by designing a discrete-time internal model, a discrete-time washout filter, and a discrete-time output feedback stabilizer. As a result, the regulation errors are shown to be globally exponentially convergent to zero. This design framework is further developed for a multi-rate digital regulator with a large sampling period of the measurements and a small control execution period.

Output feedback stabilization of reaction–diffusion PDEs with a non-collocated boundary condition

This paper addresses the control design problem of output feedback stabilization of a reaction–diffusion PDE with a non-collocated boundary condition. More precisely, we consider a reaction–diffusion equation with a boundary condition describing a proportional relationship between the left and right Dirichlet traces. Such a boundary condition naturally emerges, e.g., in the context of reaction–diffusion partial differential equations presenting a transport term and with a periodic Dirichlet boundary condition. The control input takes the form of the left Neumann trace. Finally, the measurement is selected as a pointwise Dirichlet measurement located either in the domain or at the boundary. The adopted control strategy takes the form of a finite-dimensional controller coupling a state feedback and a finite-dimensional observer. The stability of the closed-loop system is obtained provided the order of the observer is selected to be large enough. Finally, we extend this result to the establishment of an input-to-state stability estimate with respect to an additive perturbation in the application of the boundary control.

Robust Output Feedback Stabilization of MIMO Invertible Nonlinear Systems With Output-Dependent Multipliers

This note studies the robust output feedback stabilization problem of multi-input multi-output invertible nonlinear systems with output-dependent multipliers. An "ideal" state feedback is first designed under certain mild assumptions. Then, a set of extended low-power high-gain observers is systematically designed, providing a complete estimation of the "ideal" feedback law. This yields a robust output feedback stabilizer such that the origin of the closed-loop system is semiglobally asymptotically stable, while improving the numerical implementation with the power of high-gain parameters up to 2.