Linear Feedback Law Research Articles

Uniform exponential stability approximations of semi‐discretization schemes for two hybrid systems

The uniform exponential stabilities (UESs) of two hybrid control systems comprised of a wave equation and a second‐order ordinary differential equation are investigated in this study. Linear feedback law and local viscosity are considered, as are nonlinear feedback law and internal anti‐damping. The hybrid system is first reduced to a first‐order port‐Hamiltonian system with dynamical boundary conditions, and the resulting system is discretized using the average central‐difference scheme. Second, the UES of the discrete system is obtained without prior knowledge of the exponential stability of the continuous system. The frequency domain characterization of UES for a family of contractive semigroups and the discrete multiplier approach are used to validate the main conclusions. Finally, the Trotter–Kato theorem is used to perform a convergence study on the numerical approximation approach. Most notably, the exponential stability of the continuous system is derived by the convergence of energy and UES, which is a novel approach to studying the exponential stability of some complex systems. Numerical simulation is used to validate the effectiveness of the numerical approximating strategy.

Global practical stabilization of the double integrator system with an imperfect sensor and subject to a bounded disturbance

This paper is concerned with global practical stabilization of the double integrator system with an imperfect sensor and subject to an additive bounded output disturbance. The imperfect sensor nonlinearity possesses the nonlinear characteristics of saturation and dead zone. Because of the presence of output dead zone and the additive disturbance, the states cannot be expected to driven into an arbitrarily small neighborhood of the origin. To solve the global practical stabilization problem, we proposes a low gain-based linear dynamic output feedback law, under which the first state enters and remains in a bounded set whose size is depended on the bound of disturbance and the range of dead zone and the second state enters and remains in a pre-specified arbitrarily small set, both in finite time. Simulation results illustrate the effectiveness of our proposed control method.

On the Feasibility of Self-Powered Linear Feedback Control

A control system is called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">self-powered</i> if the only energy it requires for operation is that which it absorbs from the plant. For a linear feedback law to be feasible for a self-powered control system, its feedback signal must be colocated with the control inputs, and its input-output mapping must satisfy an associated passivity constraint. The imposition of such a feedback law can be viewed equivalently as the imposition of a linear passive shunt admittance at the actuation ports of the plant. In this paper we consider the use of actively-controlled electronics to impose a self-powered linear feedback law. To be feasible, it is insufficient that the imposed admittance be passive, because parasitic losses must additionally be overcome. We derive sufficient feasibility conditions which explicitly account for these losses. In the finite-dimensional, time-invariant case, the feasibility condition distills to a more conservative version of the Positive Real Lemma, which is parametrized by various loss parameters. Three examples are given, in which this condition is used to determine the least-efficient loss parameters necessary to realize a desired feedback law.

Exact Linearization of Minimally Underactuated configuration Flat Lagrangian Control Systems by Quasi-Static Feedback of Classical States

We study the exact linearization of configuration flat Lagrangian control systems with p degrees of freedom and p - 1 inputs by quasi-static feedback of classical states. First, we present a detailed analysis of the structure of the parameterization of the system variables by the flat output. Based on that, we systematically construct a linearizing quasi-static feedback law of the classical state such that the closed-loop system shows the behavior of decoupled integrator chains. Our approach shows that the construction of a generalized Brunovsky state can be completely circumvented. Furthermore, we present a method for determining the lengths of the integrator chains achieved by quasi-static feedback laws that allow for rest-to-rest transitions.

Composite nonlinear feedback control of a DC-DC boost converter under input voltage and load variation

Voltage boost converters are one of the most important components of DC microgrids, since they are used to enhance the voltage of naturally intermittent energy sources such as solar panels in order to feed unknown demands. In this work, a novel tuning algorithm for Composite Nonlinear Feedback (CNF) is studied in depth to improve transient performance and address output voltage regulation for a DC-DC boost converter in the presence of DC input uncertainty. The proposed CNF controller comprises both linear and nonlinear feedback terms. The linear part contributes to the stability and output tracking with a small damping ratio and a quick response. The nonlinear part, i.e., damping term, reduces the overshoot stemming from the linear feedback law and increases the damping ratio of the overall closed-loop system. The nonlinear part is automatically tuned whereby the transient performance of the DC-DC boost converter improves significantly. To assess the performance of the proposed technique, a boost converter is simulated in MATLAB Simulink considering different scenarios such as changing load, DC input, and voltage reference. The numerical results demonstrate that the tuned CNF controller outperforms the linear controller in the DC boost converter. Additionally, several experiments are conducted to validate the efficacy of the suggested technique.

State feedback control law design for an age-dependent SIR model

An age-dependent SIR model is considered with the aim to develop a state-feedback vaccination law in order to eradicate a disease. A dynamical analysis of the system is performed using the principle of linearized stability and shows that, if the basic reproduction number is larger than 1, the disease free equilibrium is unstable. This result justifies the development of a vaccination law. Two approaches are used. The first one is based on a discretization of the partial integro-differential equations (PIDE) model according to the age. In this case a linearizing feedback law is found using Isidori’s theory. Conditions guaranteeing stability and positivity are established. The second approach yields a linearizing feedback law developed for the PIDE model. This law is deduced from the one obtained for the ODE case. Using semigroup theory, stability conditions are also obtained. Finally, numerical simulations are presented to reinforce the theoretical arguments.

Synchronization of stochastic fractional-order model of muscular blood vessels

This study deals with the synchronization problem of the stochastic fractional-order biomathematical model of muscular blood vessels (SFOMBVs) with nonlinear inputs using a state-feedback adaptive control law, in which the effects of uncertainties and external disturbances are considered in the modeling. The linear state feedback and adaptive laws are combined via an adaptive update law, leading to a simple and robust adaptive feedback scheme. The synchronization and tracking performances of the SFOMBV master and slave systems are investigated, and the stability of the proposed control system is proved based on stochastic analysis techniques and Lyapunov theory. The simulation results show that the proposed control method can lead an abnormal muscular blood vessel to a normal trajectory with good robustness by reducing the tracking error.

Adaptive Event-Triggered Model Predictive Load Frequency Control for Power Systems

This paper investigates the adaptive event-triggered (AET) output feedback model predictive control (MPC) method for the load frequency control (LFC) problem of hybrid power system involving the thermal power and the photovoltaic power. First, an AET communication mechanism, which possesses the ability to dynamically adjust the trigger threshold parameter according to an adaptive law, is addressed to reducing the communication overhead, while keeping the required control performance; Second, the output feedback model predictive controller which is composed of an off-line designed state observer and an on-line solved optimization problem, is proposed using the techniques of quadratic boundedness and S-procedure. The state observer gain is obtained by off-line maximizing the trace of a matrix which is closely related to minimal positively invariant set; While the output feedback MPC strategy is acquired by expressing the infinite horizon control moves as a free control move and a linear feedback law based on the estimated state; Third, the recursive feasibility of the on-line optimization problem and the stability of the closed-loop system are analyzed by the fact that the estimated state and estimation error will converge to the presented minimal positively invariant set. Finally, MATLAB and its linear matrix inequality (LMI) toolbox are used to solve the optimization problem, and the LFC system with load disturbance is simulated and analyzed. The simulation results show that the proposed method is feasible and effective.

Local and Global Stabilization of Switched Linear Systems With Actuator Saturation

Utilizing the min-composite Lyapunov function and the min-projection switching strategy, we solve the local and global stabilization problems for switched linear systems by saturated, not necessarily stabilizing individually, linear feedback laws. Sufficient conditions are derived in the form of matrix inequalities under which the closed-loop system is locally asymptotically stable with an estimate of the domain of attraction or is globally asymptotically stable. The derived conditions are shown to be less conservative than the existing ones and, when used to guide the design of feedback laws, enable local stabilization with a larger estimate of the domain of attraction or global stabilization for a larger class of systems. Numerical examples illustrate the theoretical conclusions.

On Optimal Linear Regulator with Polynomial Process of External Excitations

A linear control system over an infinite time-horizon is considered, where external excitations are defined as polynomials based on a time-varying Ornstein--Uhlenbeck process. An optimal control law with respect to long-run average type criteria is established. It is shown that the optimal control has the form of a linear feedback law, where the affine term satisfies a backward linear stochastic differential equation. The normalizing functions in the optimality criteria depend on the stability rate of the dynamic equation for the Ornstein--Uhlenbeck process.

Optimal Control of Systems with Distributed Parameters Under the Interval Uncertainty of an Object’s Characteristics

A constructive method is proposed for solving a linear-quadratic problem of the robust optimization of spatiotemporal control actions in systems with distributed parabolic parameters under conditions of interval uncertainty of the time-invariant parametric characteristics of an object. In accordance with the control strategy according to the principle of the best guaranteed result, the assessment of achievable target sets of the controlled system is carried out according to the accuracy of uniform approximation to the required final state on an extended set of arguments, which includes, in addition to spatial variables, the entire admissible set of uncertain factors. The developed approach is based on the previously developed alternance method for constructing parameterizable program control algorithms, which extends the results of the theory of nonlinear Chebyshev approximations to a wide range of optimization problems and uses the fundamental laws of the subject area. It is shown that the proposed equations of optimal controllers, which are reduced to linear feedback laws on the measured state of the object with nonstationary transmission coefficients, lead to achievable values of the optimality criteria that do not exceed their upper bounds under the considered uncertainty conditions

Global Practical Stabilization of the Double Integrator System In the Presence of Output Saturation and a Bounded Disturbance

In this paper we consider the problem of global practical stabilization of the double integrator system in the presence of output saturation and a bounded disturbance additive to the output. Because of the disturbance in the output, not all system states can be driven into an arbitrarily small set, as in the usual practical stabilization problem, the output can only be driven into a bounded set whose size is dependent on the bound of the disturbance. It is shown that there exist linear feedback laws that globally practically stabilize the double integrator system in the presence of output saturation and the bounded output disturbance, and we propose linear low gain based dynamic output feedback laws to solve the global practical stabilization problem. Simulation results illustrate the effectiveness of the proposed control design.

Cone-bounded feedback laws for linear second order systems

The strong stabilizability of linear second order equation is investigated in this paper. It is supposed that there exists a linear feedback law that makes the origin of the closed-loop system globally asymptotically stable. Then this control is subject to a cone-bounded nonlinearity. Well-posedness and stability results of the closed-loop system under such (nonlinear) control are stated. The first result is proven by using nonlinear semigroups techniques and the Schauder fixed-point theorem and the second one is based on the result of Marx et al. [12]. Our results are then applied to the particular damped and undamped wave equation. Simulation results are presented to validate the theoretical results. Note that some of the results of this paper apply for a large class systems.

Global control aspects for long waves in nonlinear dispersive media

A class of models of long waves in dispersive media with coupled quadratic nonlinearities on a periodic domain T are studied. We used two distributed controls, supported in ω ⊂ T and assumed to be generated by a linear feedback law conserving the “mass” (or “volume”), to prove global control results. The first result, using spectral analysis, guarantees that the system in consideration is locally controllable in Hs(T), for s ≥ 0. After that, by certain properties of Bourgain spaces, we show a property of global exponential stability. This property together with the local exact controllability ensures for the first time in the literature that long waves in nonlinear dispersive media are globally exactly controllable in large time. Precisely, our analysis relies strongly on the bilinear estimates using the Fourier restriction spaces in two different dispersions that will guarantee a global control result for coupled systems of the Korteweg—de Vries type. This result, of independent interest in the area of control of coupled dispersive systems, provides a necessary first step for the study of global control properties to the coupled dispersive systems in periodic domains.

A Semiglobal Approach for Stabilizing Nonlinear Systems With State Delays by Memoryless Linear Feedback

This article investigates the problem of how to control nonlinear systems with delays in the state by memoryless linear feedback. The notions of semiglobal asymptotic stabilization and sublevel sets are introduced in the context of time-delay systems. With the aid of Razumikhin theorem, we develop a semiglobal design method for the construction of Lyapunov functions, associated sublevel sets, and delay-free linear state feedback laws, step-by-step, achieving semiglobal asymptotic stabilization for time-delay nonlinear systems in a lower triangular form. In contrast to the global stabilization of nonlinear systems with delays in the state, which is usually achieved by <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dynamic state</i> feedback (Lin and Zhang, 2020), the significance of this work is to point out that a tradeoff of the control objectives, e.g., semiglobal versus global stabilization, makes it possible to control a class of time-delay nonlinear systems by <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">static linear</i> feedback. Extensions to nonlinear systems with globally asymptotically locally expoentially stable (GALES)-like inverse dynamics are also included in this article.

Constrained Controller and Observer Design by Inverse Optimality

Model predictive control (MPC) is often tuned by trial and error. When a baseline linear controller exists that is already well tuned in the absence of constraints and MPC is introduced to enforce them, one would like to avoid altering the original linear feedback law whenever they are not active. We formulate this problem as a controller matching similar to the works of Di Cairano and Bemporad (2009), Di Cairano and Bemporad (2010), and Tran et al. (2015), which we extend to a more general framework. We prove that a positive-definite stage-cost matrix yielding this matching property can be computed for all stabilizing linear controllers. In addition, we prove that the constrained estimation problem can also be solved similarly, by matching a linear observer with a moving horizon estimator. Finally, we discuss various aspects of the practical implementation of the proposed technique in some examples.

Improved Linear ADRC for Hybrid Energy Storage Microgrid Output-Side Converter

As an effective application form for large-scale and efficient use of distributed power, microgrid not only realizes the flexible control of distributed power but also provides reliable power supply for the load. However, the power quality on the load side will be severely deteriorated when the microgrid is disturbed. To cope with this issue, an improve linear active disturbance rejection control (LADRC) technique is proposed in this article. In this technology, the idea of series correction is introduced into the disturbance observation channel of linear extended state observer. This strategy can achieve more accurate tracking of disturbance items through the correction link, and handle the unknown modeling error and external disturbances of the system as an overall disturbance through a single structure. Then, according to the linear feedback law to achieve good control of the output-side converter. Moreover, the traceability, convergence, and stability of ILADRC are analyzed. More importantly, guidance on parameters configuration is given on the basis of performance analysis. Finally, the validity of the proposed improved LADRC technique is verified on a 40-kW experiment platform.

Spectral analysis of Euler–Bernoulli beam model with distributed damping and fully non-conservative boundary feedback matrix

The distribution of natural frequencies of the Euler–Bernoulli beam resting on elastic foundation and subject to an axial force in the presence of several damping mechanisms is investigated. The damping mechanisms are: ( i ) an external or viscous damping with damping coefficient ( − a 0 ( x )), ( ii ) a damping proportional to the bending rate with the damping coefficient a 1 ( x ). The beam is clamped at the left end and equipped with a four-parameter (α, β, κ 1 , κ 2 ) linear boundary feedback law at the right end. The 2 × 2 boundary feedback matrix relates the control input (a vector of velocity and its spacial derivative at the right end) to the output (a vector of shear and moment at the right end). The initial boundary value problem describing the dynamics of the beam has been reduced to the first order in time evolution equation in the state Hilbert space of the system. The dynamics generator has a purely discrete spectrum (the vibrational modes). Explicit asymptotic formula for the eigenvalues as the number of an eigenvalue tends to infinity have been obtained. It is shown that the boundary control parameters and the distributed damping play different roles in the asymptotical formulas for the eigenvalues of the dynamics generator. Namely, the damping coefficient a 1 and the boundary controls κ 1 and κ 2 enter the leading asymptotical term explicitly, while damping coefficient a 0 appears in the lower order terms.

A Discrete Approach to Feedback Linearization, Yaw Control of An Unmanned Helicopter

Nonlinear control laws often need to be implemented with digital hardware. Use of digital control systems leads to communication/processing delays which are widely neglected in control of mechanical systems. This paper proposes a discrete approach to feedback linearization that considers these commonly overlooked delays in design. The proposed approach is shown to both improve the performance and remove the need for continuous derivative terms. In feedback linearization control systems, designed in the continuous domain, derivative terms are required to speed up the control response of mechanical systems, but disadvantageously cause high sensitivity to noise. The proposed approach was used to design a feedback linearization control system for a common turning maneuver of an unmanned helicopter in yaw. At this maneuver, the helicopter centroid motion and pitch rotational speed are almost zero. Governing differential equations of the helicopter at this maneuver are nonlinear and coupled. A feedback linearization law was proposed to curb nonlinearity and, a discrete control system, considering the inevitable delay due to the use of digital control systems, was adopted to complete the control law. This innovative approach resulted in less sensitivity to noises and performance boost. Practical limits in terms of control input, rotor speed, sampling frequency and noises of the gyroscope, the tachometer and the acceleration sensor were taken into account in this research. The results show that the proposed control system leads to fast and smooth yaw turns even at a high pitch angle (close to vertical) or in the case of being hit by external objects.

Asymptotic Stabilization of a Flexible Beam with Attached Mass

A mathematical model of a simply supported Euler-Bernoulli beam with attached spring-mass system is considered. The model is controlled by distributed piezo actuators and lumped force. We address the issue of asymptotic behavior of solutions of this system driven by a linear feedback law. The precompactness of trajectories is established for the operator formulation of the closed-loop dynamics. Sufficient conditions for strong asymptotic stability of the trivial equilibrium are obtained.