Linear Output Feedback Control Research Articles

On the Optimization Landscape of Dynamic Output Feedback Linear Quadratic Control

The convergence of policy gradient algorithms hinges on the optimization landscape of the underlying optimal control problem. Theoretical insights into these algorithms can often be acquired from analyzing those of linear quadratic control. However, most of the existing literature only considers the optimization landscape for static full-state or output feedback policies (controllers). We investigate the more challenging case of dynamic output-feedback policies for linear quadratic regulation (abbreviated as dLQR), which is prevalent in practice but has a rather complicated optimization landscape. We first show how the dLQR cost varies with the coordinate transformation of the dynamic controller and then derive the optimal transformation for a given observable stabilizing controller. One of our core results is the uniqueness of the stationary point of dLQR when it is observable, which provides an optimality certificate for solving dynamic controllers using policy gradient methods. Moreover, we establish conditions under which dLQR and linear quadratic Gaussian control are equivalent, thus providing a unified viewpoint of optimal control of both deterministic and stochastic linear systems. These results further shed light on designing policy gradient algorithms for more general decision-making problems with partially observed information.

Finite-time stabilization of discrete-time conewise linear systems

The finite-time stabilizing control design problem for discrete-time conewise linear systems is tackled in this paper. Such a class of systems consists of the union of ordinary linear time-invariant subsystems, whose dynamics are defined in prescribed conical regions, constituting a conical partition of the state space. By imposing some cone-copositivity properties to a suitable piecewise quadratic function, two sufficient conditions are preliminarily derived concerning the system’s finite-time stability. By building on them, novel results are then presented for the system’s finite-time stabilization through a piecewise linear output feedback controller. Such results are based on the solution of feasibility problems involving sets of Linear Matrix Inequalities (LMIs). A numerical example illustrates the effectiveness of the proposed approach.

Criteria for Stability and Stabilization of Variable Fractional-Order Uncertain Neutral Systems With Time-Varying Delay: Delay-Dependent Analysis

This work focuses on the stability and stabilization of variable fractional-order neutral systems (VFONS) with time-varying delays and time-varying structured uncertainties. Via the FO Lyapunov function, a new set of delay-and order-dependent criterion is proposed for both nominal and uncertain VFONS. The resultant criteria are formulated by designing a linear matrix inequality-based static output feedback control strategy in a set of linear matrix inequalities. The numerical simulations verify the theoretical formulation.

Suboptimal Linear Output Feedback Control of Discrete-Time Systems With Multiplicative Noises

This brief considers the fundamental problem of output feedback control for a class of discrete-time systems with multiplicative noises. It is well-known that the classical separation principle fails due to the existence of multiplicative noises in state and control variables. Thus, the traditional optimal solutions for the controller and estimator can not be obtained. Different from the previous methods of enforced separation principle, the explicit expression of the suboptimal linear controller is derived by using the solution of the forward and backward stochastic difference equations (FBSDEs) and the linear optimal estimator. However, it leads to the other mathematical difficult problem that the control gain and the estimator gain are coupled with each other. Specifically, the control gain obeys the backward Riccati equation and the estimator gain satisfies the forward Riccati equation such that the solutions can not be acquired simultaneously. A novel method of iterative solutions to the coupled backward and forward Riccati equations (CBFREs) is proposed. Numerical examples are illustrated to show that the proposed algorithms have better performance than the previous methods.

Disturbance rejection control of multiple integrators system by bounded linear extended observer based output feedback

SummaryThis paper is concerned with the disturbance rejection control problem of multiple integrators systems subject to actuator saturation and external disturbance. By using the solutions to a class of parametric Lyapunov equations (PLEs), extended observer based bounded linear output feedback controller is designed to solve the considered problem. By constructing a novel Lyapunov function and using the properties of the PLE, the stability of the closed‐loop system is proved and the domain of attraction is estimated. In order to improve control performance, bounded linear time‐varying output feedback controller is also proposed such that the state of the closed‐loop system tends to zero as the time approaches to the prescribed setting time which is independent of the initial conditions and can be assigned in advance. In addition, some comparisons of the proposed methods to existing results are given. Finally, numerical examples verify the effectiveness of the proposed methods.

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.

Experimental study on vibration control of spur geared rotor system with active magnetic bearings

The gearbox vibrations and noise caused by variation of contact forces often causes failure in the components of gearbox, which are then transmitted to the surrounding structures. The concept of active control of gearbox vibrations with piezoelectric actuators at the mounting points has been analysed earlier in several studies, then with certain limitations. There has been little research on how Active Magnetic Bearings (AMBs) might mitigate gear vibrations. AMBs are used in this study to investigate active internal shaft vibration control caused by rotor unbalance, gear runout, transmission error, which usually resides within the gearbox. Hence, minimizing the effect of vibrational amplitude at gear mesh frequency and its harmonics to have quieter gear operation. A mathematical model is developed for the torsional-lateral dynamic analysis of a spur geared rotor-AMB system. Responses are computed numerically with different gear-rotor faults. Based on the mathematical model, setted up an experimental test rig in the laboratory, and the effectiveness of the proposed model is compared with and without the application of AMBs. The approach is based on active control of the shaft transverse vibration with an electromagnetic actuator. The control forces are applied to the rotor shafts supported on conventional rolling element bearings by an eight-pole radial AMB, as an auxiliary component and a closed-loop linear output feedback control is employed for stable, reliable, and robust operation. A linear PID controller working on differential mode is used to generate the appropriate control signals and the experimental results are presented. It was found that there is a considerable amount of reduction in the geared rotor vibration levels and correspondingly in overall measured gear noise levels.

Input–Output Feedback Linearization Control of a Linear Induction Motor Taking Into Consideration Its Dynamic End-Effects and Iron Losses

This article proposes a new input–output feedback linearization control (FLC) technique of linear induction motors (LIMs), taking into consideration both the dynamic end-effects and the iron losses. Starting from a previously conceived dynamic model, including the dynamic end-effects and the iron losses, all the theoretical framework of the FLC has been developed. The proposed FLC improves a previous version of FLC in accounting also the iron losses, which in LIMs with fixed-secondary sheet play a pivotal role more than in rotating induction motors (RIMs). The proposed FLC has been experimentally tested on a suitably developed test setup, and experimental comparisons between the proposed FLC, the classic field-oriented control and a previously developed FLC, not accounting for the iron losses, have been shown in variable flux working conditions.

On the Stabilization through Linear Output Feedback of a Class of Linear Hybrid Time-Varying Systems with Coupled Continuous/Discrete and Delayed Dynamics with Eventually Unbounded Delay

This research studies a class of linear, hybrid, time-varying, continuous time-systems with time-varying delayed dynamics and non-necessarily bounded, time-varying, time-differentiable delay. The considered class of systems also involves a contribution to the whole delayed dynamics with respect to the last preceding sampled values of the solution according to a prefixed constant sampling period. Such systems are also subject to linear output-feedback time-varying control, which picks-up combined information on the output at the current time instant, the delayed one, and its discretized value at the preceding sampling instant. Closed-loop asymptotic stabilization is addressed through the analysis of two “ad hoc” Krasovskii–Lyapunov-type functional candidates, which involve quadratic forms of the state solution at the current time instant together with an integral-type contribution of the state solution along a time-varying previous time interval associated with the time-varying delay. An analytic method is proposed to synthesize the stabilizing output-feedback time-varying controller from the solution of an associated algebraic system, which has the objective of tracking prescribed suited reference closed-loop dynamics. If this is not possible—in the event that the mentioned algebraic system is not compatible—then a best approximation of such targeted closed-loop dynamics is made in an error-norm sense minimization. Sufficiency-type conditions for asymptotic stability of the closed-loop system are also derived based on the two mentioned Krasovskii–Lyapunov functional candidates, which involve evaluations of the contributions of the delay-free and delayed dynamics.

Deep Reinforcement Learning and Simultaneous Stabilization-Based Flight Controller for Nano Aerial Vehicle

The plants of nano aerial vehicles (NAVs) are inherently unstable. Hence, a NAV needs a flight controller to accomplish a mission. Furthermore, the sensing and computational capabilities of NAV's autopilot hardware are limited. Hence, the implementation of the full state feedback controllers with gain scheduling is difficult. This paper proposes a flight controller scheme that consists of two parts: a Simultaneously Stabilizing Output Feedback Linear (SSOFL) controller and a Proximal Policy Optimization (PPO) deep reinforcement learning agent, which is connected in parallel to the SSOFL controller. In this scheme, the single SSOFL controller provides stabilization and nominal tracking performance to the NAV throughout its flight envelope by accomplishing simultaneous stabilization (SS). Additionally, the PPO agent is trained using the closed-loop (CL) nonlinear plant with this SSOFL controller to enhance the tracking performance. The effectiveness of the proposed flight controller scheme is verified using the six-degree-of-freedom nonlinear simulations of the fixed-wing nano aerial vehicle.

Observer-based gust load alleviation via reduced-order models*

In this paper, we investigate the design of linear active output feedback control for aircraft gust load alleviation, involving standard sensors and control surfaces. The control architecture is composed of (i) a robust observer, allowing for estimating some wing bending moments despite the wind perturbation, and (ii) a structured H∞-norm oriented control law exploiting this additional information. To cope with the complexity of the aeroelastic model and make the control and observer synthesis computationally tractable, a model reduction step, based on Petrov-Galerkin projections, is first applied. One specificity of this work is that the obtained projector is exploited in the observer design, linking the reduction and the estimation. The effectiveness of the methodology is illustrated through numerical simulation involving an open access high complexity aeroelastic model.

Complete model-free sliding mode control (CMFSMC)

This study presents a complete model-free sliding mode control (CMFSMC) framework for the control of continuous-time non-affine nonlinear dynamic systems with unknown models. The novelty lies in the introduction of two equalities to assign the derivative of the sliding functions, which generally bridges the designs of those model-based SMC and model-free SMC. The study includes a double SMC (DSMC) design, state observer design, and desired reference state vector design (whole system performance), which all do not require plant nominal models. The preconditions required in the CMFSMC are the plant dynamic order and the boundedness of plant and disturbances. U-model based control (U-control) is incorporated to configure the whole control system, that is (1) taking model-free double SMC as a robust dynamic inverter to cancel simultaneously both nonlinearity and dynamics of the underlying plants, (2) taking a model-free state observer to estimate the state vector, (3) taking invariant controller to specify the whole control system performance in a linear output feedback control and to provide desired reference state vector. The related properties are studied to support the concept/configuration development and the analytical formulations. Simulated case studies demonstrate the developed framework and show off the transparent design procedure for applications and expansions.

Emerging approaches for nonlinear parameter varying systems

Emerging approaches for nonlinear parameter varying systems

Model‐free linear active disturbance rejection output feedback control for electro‐hydraulic proportional system with unknown dead‐zone

Here, a model-free linear active disturbance rejection controller (LADRC) integrating dead-zone inverse compensation for an electro-hydraulic proportional (EHP) system with an unknown dead-zone is proposed. Unlike the existing LADRC design idea, the presented method uses a new hydraulic system block diagram to select the order of LADRC instead of the highest degree of the system. Furthermore, the rationality of order selection is illustrated by the frequency domain analysis theory. To solve the dead-zone of the proportional valve, a novel structural transformation is proposed by constructing a new dead-zone, which has a definite physical meaning and can be obtained by the experimental measurement. Then, the dead-zone disturbance is reduced, and the burden of the observer is relieved. At the same time, the controller parameters are decreased from three to two equivalent PI controller. Besides, the stability of the closed-loop system is analysed by the describing function method. The performance of the proposed control solution has been investigated through extensive comparative experiments under different working conditions. The experimental results successfully demonstrate the effectiveness and practicality of the presented method.

A generalized exponential stabilization for a class of semilinear parabolic equations: Linear boundary feedback control approach

This paper investigates the generalized exponential stabilization (GES) of a class of semilinear parabolic equations, which is a generalization of the result of exponential stabilization and has the character of accelerating decay process locally. From the point of admissibility of unbounded input operators of view, the well‐posedness of the open‐loop system is analyzed. Then, under some sufficient conditions expressed by linear matrix inequality, a linear boundary output feedback controller only using boundary measurement is proposed such that the closed‐loop system satisfies GES in absence of external disturbance and satisfies an H∞ performance index when disturbance exists. The well‐posedness of the resulting closed‐loop system is also given via semi‐group theory. Finally, we give numerical examples to verify the performance of the designed boundary controller.

Active disturbance rejection adaptive control for uncertain nonlinear systems with unknown time‐varying dead‐zone input

AbstractThis paper rigorously investigates the problem of active disturbance rejection control for nonlinear systems subject to uncertainties and dead‐zone input. One of the features of the considered systems is that the dead‐zone parameters are allowed to be time varying and asymmetrical. The main idea of this paper is to utilize an extended state observer (ESO) to estimate all the external disturbance, the uncertain nonlinearity, the unknown dead‐zone model, and the unmeasurable states. A switched dynamic parameter estimation law is established to separate the dead‐zone model from the total uncertainty and estimate the parameters of dead‐zone online. Based on the proposed estimation law and the ESO, an output feedback linearization control law is designed to cancel the total disturbance and compensate the dead zone. The theoretical analysis of stability shows that the closed‐loop system has multi‐time scale convergence properties. Finally, simulation results are provided to illustrate the efficiency of the proposed method.

Output feedback control for quadratic systems: A Lyapunov function approach

AbstractThis article deals with the design of quadratic and linear dynamic output feedback controllers for quadratic systems to ensure, on the one hand, local exponential stability of the origin in the closed‐loop form and, on the other hand, to increase an estimate of the basin of attraction of the origin as large as possible. The introduction of a quadratic term in the controller allows us to consider a controller in the same class of dynamics as the studied system. Here, our approach is to consider a Lyapunov function with respect to the extended state gathering the states of the system and the controller. The induced nonlinear inequalities are treated thanks to an auxiliary vector repeating the extended state adequately and by combining distinct linearization techniques to finally obtain linear matrix inequalities. For comparison purposes, we also provide another approach characterizing the extended state as belonging to a polytope in the state space. Numerical examples illustrate the results.

An Input–Output Feedback Linearization Control Method Synthesized by Artificial Neural Network for Grid-Tied Packed E-Cell Inverter

In this article, a trained artificial neural network (ANN) input-output feedback linearization (IOFL) control strategy is proposed for a grid-connected nine-level packed E-cell (PEC9) converter under various operating variations. For the first time, a second-order dynamic model in d-q reference frame is developed for PEC9 converter. This model enforces the converter to employ IOFL-based proportional-resonance compensators for accurately contributing to current tracking enhancement, mitigation at the current ripple, and consequently, the capacitors dc voltages ripple. As another contribution, the current tracking ability of the proposed controller is much more promoted by decoupling the d-q dynamic components. Indeed, the effect of each proposed control component is fortified through decoupling feedback coefficients. In addition, as the third contribution, ANN as the complementary controller adapts the proposed IOFL-based controller coefficients to improve efficiency and stability of the whole control loop system, including the PEC9 capacitor voltages in unstable operation of load, grid, and dc source as well as in the presence of parameters mismatch and disturbances. To support the aforementioned contributions with more details, the closed-loop description and the dynamic model of the proposed controller-based PEC9 converter are utilized to present several stability evaluation processes in both time and frequency domains. This assessment process is accurately executed to confirm the stable performance of PEC9 when the errors of the PEC9 converter currents are changed as well as the decoupling feedback coefficients are varied. Finally, comparative dSPACE setup-based experimental and simulation results are attained to further verify the validity of the proposed control strategy under various operating conditions.

Novel Master-Slave Synchronization Conditions for Chaotic Fractional-Order Lur’e Systems Based on Small Gain Theorem

The master-slave synchronization problem for chaotic fractional-order (FO) Lur'e systems is investigated in this brief. Firstly, order-dependent stability criteria for FO Lur'e systems are established using small gain theorem and the results are transformed into linear matrix inequalities (LMIs). Then, by employing the derived results, linear output feedback controllers that asymptotically synchronize chaotic FO Lur'e systems can be obtained. Finally, numerical examples are provided to illustrate that the proposed results are valid and less conservative than existing ones.

Observer-based finite-time output feedback control of heat equation with Neumann boundary condition

This paper is concerned with the problem of boundary observer-based finite-time output feedback control for a heat system with Neumann boundary condition and Dirichlet boundary actuator. In this system, it is endowed a single sensor at the left boundary and actuator at the right boundary. First, a backstepping finite-time observer is designed. Based on linear switching observer gains with state-dependent switching, the proposed observer estimates the system state variables in a finite-time only using one displacement boundary measurement. Then, an observer-based linear switching output feedback control with state-dependent switching is presented, which steers any solution of the output feedback closed-loop system to zero in a finite-time. Finally, the simulation results are provided to support the theoretical results.