Linear Closed-loop System Research Articles

Robust H∞ consensus for uncertain linear multi-agent systems on directed topologies: A dynamic event-triggered strategy

Considering the asymmetry of the system matrix caused by directed communication networks, a dynamic event-triggered control protocol for the robust [Formula: see text] consensus of uncertain linear multi-agent systems accompanied by external disturbances on directed topologies is proposed. Through the introduction of the controlled output and generalized algebraic connectivity for directed graphs, the consensus problem is converted to the event-triggered stabilization of the uncertain linear closed-loop system with disturbance. On the basis of the relative state of neighbor agents at event-times and the introduced dynamic variables, a dynamic event-triggered control protocol is put forward, and the linear matrix inequality sufficient conditions for the system to achieve robust [Formula: see text] consensus are obtained after a rigorous proof. Then, linear matrix inequalities are further decoupled to greatly reduce the computational complexity, and the parameters of the control protocol are determined by solving the decoupled linear matrix inequalities. Finally, simulations are applied for the verification of conclusions.

Fully actuated system approach for spacecraft rendezvous system with actuator saturation

This paper explores the application of a fully actuated system (FAS) approach to a spacecraft rendezvous system with actuator saturation. A FAS model for the spacecraft rendezvous system is developed, followed by the design of an extended state observer (ESO) to estimate unmeasurable states and the system's nonlinear terms. The saturation issue is addressed using the convex hull method. By the designed ESO, a state feedback controller for the system is designed using the FAS approach, resulting in a constant linear closed-loop system whose eigenvalues can be arbitrarily assigned. This method simplifies the controller design process and provides a more accurate system model without the linearisation of nonlinear terms, thereby improving control accuracy. Simulation results show the effectiveness and usefulness of the proposed method.

Minimum Data-Rate for Emulating a Linear Feedback System

In this work, we ask what the minimum data-rate is for a possibly nonlinear control law acting on a linear system to emulate the closed-loop behavior of a desired linear control law. This result gives an implicit estimate for the information flow in the feedback path of the closed-loop linear system. We formally introduce the notion of emulation and control law, and we present a data-rate theorem. Remarkably, we note that the minimum data-rate varies discontinuously with the parameters of the open and closed-loop systems. This feature is new since the usual data-rate theorem, which only requires stabilization instead of emulation, gives a continuously varying minimum data-rate.

Observer-based robust high-order fully actuated attitude autopilot design for spinning glide-guided projectiles

This paper investigates the design of an attitude autopilot for a dual-channel controlled spinning glide-guided projectile (SGGP), addressing model uncertainties and external disturbances. Based on fixed-time stable theory, a disturbance observer with integral sliding mode and adaptive techniques is proposed to mitigate total disturbance effects, irrespective of initial conditions. By introducing an error integral signal, the dynamics of the SGGP are transformed into two separate second-order fully actuated systems. Subsequently, employing the high-order fully actuated approach and a parametric approach, the nonlinear dynamics of the SGGP are recast into a constant linear closed-loop system, ensuring that the projectile's attitude asymptotically tracks the given goal with the desired eigenstructure. Under the proposed composite control framework, the ultimately uniformly bounded stability of the closed-loop system is rigorously demonstrated via the Lyapunov method. Validation of the effectiveness of the proposed attitude autopilot design is provided through extensive numerical simulations.

Robust estimation of state vector coordinates in the controlled helicopter motion problem

The problem of finding H∞ – a observer of the state vector of a linear continuous non-stationary dynamical system with a semi-infinite time of functioning is considered. It is assumed that a mathematical model of a closed-loop linear continuous deterministic dynamical system with an optimal linear regulator, found as a result of minimization of the quadratic quality criterion, is known. For solving the state observer synthesis problem the reduction of the problem to a min-max optimal control problem is used. In this problem, the minimum of the quality criterion is sought by the observer’s gain matrix, and the maximum – by the external influence, measurement noise, and initial conditions. To solve this problem, the extension principle is applied and sufficient optimality conditions are obtained that requires the choice of auxiliary functions of the Krotov–Bellman type. As a result of the implementation of the procedure for choosing an auxiliary function and using the rules of matrix differentiation, relations for the synthesis of the observer and formulas for finding the best matrix of observer gains, as well as the laws for choosing the worst external influences and noise, were obtained. We find a solution to the problem of state vector coordinates estimation in the presence of limited external influences and disturbances in a linear model of the measuring system. As an example, the equations of motion of the Raptor-type helicopter are used.

Model-Free Fault Detection Based on Performance Residual for Feedback Control Systems

This brief proposes a control performance-oriented fault detection (PFD) strategy for closed-loop linear time-invariant systems. Firstly, the performance index with the summation of quadratic form over the infinite time interval is investigated in virtue of a data-driven residual-centered closed-loop model, which can be constructed utilizing input and output data instead of system models. Then the adaptive residual generator is presented to update the parameters of the data-driven model in case of system changes caused by anomalies, serving the information of the faulty plant for PFD. Further, the performance residual, which represents the performance degradation, is defined based on the Bellman equation. By embedding the predesigned controller parameters in the performance residual generator, the performance degradation caused by system faults and controller mismatches can be detected. Additionally, the proposed scheme can be applied in various operating points without revising the detector parameters. Finally, a randomized algorithm-aided threshold design is developed to handle unknown input. Case studies on a circuit model are conducted to show the superiority of PFD.

Accurate stabilization for linear stochastic systems based on region pole assignment and its applications

On the premise of ensuring the stability of the closed-loop linear stochastic system, this paper uses region stability to further explore the precise control of its dynamic performance, which is an essential consideration in practical control. First, based on the relationship between the poles (eigenvalues) of the system state matrix and system performance, the sufficient conditions are presented for the distribution of system eigenvalues in the specific convex regions, i.e., a novel criterion for the region stability of the linear stochastic systems. Second, the region stabilization conditions are given by LMIs, which can guarantee that the system eigenvalues distribute in the connected region. Third, a new algorithm is designed to solve the stabilization condition of the eigenvalues of the state matrix located in the unconnected region. Accordingly, some dynamic performance indexes of linear stochastic systems can be controlled more accurately. Finally, two cases are provided to show the precise control for the performance indexes of the new stabilization method. • A new criterion of region stability for linear stochastic systems is presented, which can describe the system performance more accurately. • According to the region stability, a more accurate controller design method is provided, which can not only guarantee the stability of the systems, but also control the dynamic performance of the system accurately. • The convergent speed and damping response of the systems can be controlled effectively by using the region stabilization method.

High-order fully actuated system approaches: Part X. Basics of discrete-time systems

A basic introduction to high-order fully actuated (HOFA) approaches for discrete-time systems is given. Firstly, it is shown that, different from the continuous-time systems, general dynamical discrete-time systems can be represented by two types of discrete-time HOFA models, namely, the step forward HOFA models and the step backward HOFA models. Secondly, controllers for the step backward HOFA models are designed, which result in constant linear closed-loop systems with arbitrarily assignable eigenstructures. The related problem of feasibility is also discussed. The well-known discrete-time feedback linearisable systems and strict-feedback systems are shown to be equivalent to both the step forward and step backward HOFA models. Finally, a generalised step backward HOFA model containing control vectors of different time instants is proposed and investigated and demonstrated with a type of proposed pseudo feed-forward systems. The contribution in this paper has laid a fundamental basis for discrete-time HOFA approaches. Further analysis and design problems can be naturally established parallel to the continuous-time system case.

High-order fully-actuated system approaches: Part VI. Disturbance attenuation and decoupling

Two types of high-order fully actuated (HOFA) system models subject to external disturbances are firstly introduced. For the type of HOFA systems with deterministic disturbances, the problem of disturbance attenuation via state feedback is treated. While for the type of HOFA systems with dynamical disturbances, the problem of asymptotic disturbance decoupling via output feedback is considered. Utilising the full-actuation feature of the HOFA systems, disturbance attenuation and decoupling controllers for the corresponding systems are conveniently designed such that constant linear closed-loop systems with designed disturbance rejection properties are resulted in. Parametric designs for both controllers are provided, and disturbance attenuation is achieved by establishing a parametric form of the closed-loop transfer function from the disturbance to the output, while the parametric form of the disturbance decoupling controller is derived based on a complete parametric solution to a type of generalised Sylvester equations (GSEs). As a consequence of the parameter approaches, additional performance requirements on the closed-loop systems can be also easily handled. An illustrative example demonstrates the effect of the proposed approach.

High-order fully actuated system approaches: Part II. Generalized strict-feedback systems

An advantage of a high-order fully-actuated (HOFA) system is that there exists a controller such that a constant linear closed-loop system with an arbitrarily assignable eigenstructure can be obtained. In this paper, a generalised form of the conventional first-order strict-feedback systems (SFSs) is firstly proposed, and a recursive solution is proposed to convert equivalently the generalised SFS into a HOFA model. Then the second- and high-order SFSs are defined and their equivalent HOFA models are also derived. It is further shown that, under certain common conditions, the recursive solutions for converting the generalised SFSs into HOFA models can be rearranged into direct analytical explicit solutions. Such a high-order system approach is more direct and simpler than the first-order system approach since it avoids the process of converting firstly these second- and high-order SFSs into first-order ones for control, and can finally produce a constant linear closed-loop system. Particularly, it is more effective than the well-known method of backstepping since, for the generalised complicated SFSs with more subsystems, the method of backstepping may simply be not applicable due to more serious ‘differential explosion’ problem. Two examples are worked out to demonstrate the effect of the approach.

Enhanced Gradient Tracking Algorithms for Distributed Quadratic Optimization via Sparse Gain Design

In this paper we propose a new control-oriented design technique to enhance the algorithmic performance of the distributed gradient tracking algorithm. We focus on a scenario in which agents in a network aim to cooperatively minimize the sum of convex, quadratic cost functions depending on a common decision variable. By leveraging a recent system-theoretical reinterpretation of the considered algorithmic framework as a closed-loop linear dynamical system, the proposed approach generalizes the diagonal gain structure associated to the existing gradient tracking algorithms. Specifically, we look for closed-loop gain matrices that satisfy the sparsity constraints imposed by the network topology, without however being necessarily diagonal, as in existing gradient tracking schemes. We propose a novel procedure to compute stabilizing sparse gain matrices by solving a set of nonlinear matrix inequalities, based on the solution of a sequence of approximate linear versions of such inequalities. Numerical simulations are presented showing the enhanced performance of the proposed design compared to existing gradient tracking algorithms.

Experimental verification of a hybrid control scheme with chaotic whale optimization algorithm for nonlinear gantry crane: A comparative study

This paper proposes an experimental verification of a hybrid partial feedback linearization (PFL) and deadbeat (DB) control scheme as in Hamdy et al. (2018) with chaotic whale optimization algorithm (CWOA) for a nonlinear gantry crane (GC) system. The PFL linearizes the nonlinear model to end up with a linear closed-loop system. The DB controller is utilized for the desirable accelerated response without any oscillation or undesirable effects on the internal dynamics stability. The CWOA is used to tune the controller parameters. A sliding-mode observer (SMO) is utilized to estimate the unmeasured states. Using this hybrid scheme, a better payload sway elimination can be obtained. Finally, the experimental results are presented to illustrate the efficiency and the effectiveness of the proposed scheme with a comparative study.

Recursive Identification Algorithms for a Class of Linear Closed-loop Systems

This paper focuses on the identification problems for a class of linear closed-loop systems. On one hand, the identifiability condition is investigated for the case where the controller is in series with the plant on the forward channel. On the other hand, the identification model is derived after parametrization, in which the parameter vector only contains the parameters of the controlled plant instead of the whole closed-loop system, and a recursive least squares algorithm and a stochastic gradient algorithm are proposed for closed-loop systems. In order to improve the parameter estimation accuracy, a forgetting factor and a convergence index are introduced into the proposed stochastic gradient algorithm. The simulation results demonstrate the effectiveness of the proposed algorithms.

Explicit Reference Governor for the Constrained Control of Linear Time-Delay Systems

This paper introduces an explicit reference governor to supervise closed-loop linear time-delay systems. The proposed scheme enforces state and input constraints by modifying the reference of the supervised system so that the state vector always belongs to admissible sublevel sets of a suitably defined Lyapunov–Krasovskii functional. To accomplish this, this paper extends the existing definition of “dynamic safety margin” to a time-delay setting and illustrates how to employ classic Lyapunov–Krasovskii functionals even though the reference is time varying. Constraint enforcement for arbitrary reference signals and asymptotic convergence to any strictly steady-state admissible set point is rigorously proven. Experimental results are reported to demonstrate the simplicity, practicality, and robustness of the proposed method.

Nonstandard use of anti-windup loop for systems with input backlash

Control systems with backlash at the input are considered in this paper. The goal of this work is to characterize the attractor of such nonlinear dynamical systems, and to design anti-windup inspired loops such that the system is globally asymptotically stable with respect to this attractor. The anti-windup loops affect the dynamics of the controllers, and allow to increase the performance of the closed-loop systems. Different performance issues are considered throughout the paper such as robustness with respect to uncertainty in the backlash, and L2 gain when external disturbances affect the dynamics. Numerically tractable algorithms with feasibility guarantee are provided, as soon as the linear closed-loop system, obtained by neglecting the backlash effect, is asymptotically stable. The results are illustrated on an academic example and an open-loop unstable aircraft system.

Optimal control of large-scale singular linear systems via hierarchical strategy

This paper states a hierarchical strategy of large-scale singular linear system in which the system is composed of J singular linear subsystems with interconnections. Among hierarchical strategies, the two-level optimization conditions based on Interaction Prediction Method (IPM) are derived such that whole large-scale singular system is optimized. Based on the two-level coordination method, the optimization analysis and controller design of large-scale singular linear system is discussed. For this purpose, two decentralized nonlinear state feedback controllers are designed for each subsystem, such that the whole closed-loop linear system is optimized. These conditions are used to extract two IPM algorithms. Then, by using both algorithms, two numerical examples are given to confirm the analytical results and illustrate the effectiveness of the proposed method.

A Predictor-Based Framework for Delay Compensation in Networked Closed-Loop Systems

One of the main challenges in networked closed-loop systems is that communication delays can significantly deteriorate system performance and even cause instability. A predictor-based framework was proposed previously to improve the networked closed-loop system performance by compensating constant network delays. The key feature of this prediction scheme is that it does not require a dynamic model of the remote subsystem and has only one design parameter to tune for each predictor to mitigate the communication delays. In this paper, the predictor-based framework is developed further. Specifically, previously the predictor was developed and its stability was analyzed for constant delays only. In this paper, the predictor structure is extended to the varying delay case and its stability is established for the first time for varying delays in a sufficient but not very conservative range by leveraging a Markov chain based approach from the literature. To that end, the key assumption of the approach that varying delays can be modeled using a two-time-scale Markov chain is tested using actual network delay data. Next, to test the performance of the framework on a networked closed-loop system, it is applied to a simulated networked motor-shaft-geartrain system with different constant and varying delay models, as well as a real network. Predictors are shown to be effective in significantly reducing the negative effects caused by delays and greatly improve the fidelity of the networked integration. Finally, stability of the closed-loop system with predictor-based framework is studied with constant delays using a “mixed” small gain and passivity theorem from the literature and it is demonstrated that the predictors can even help stabilize a closed-loop linear time-invariant system when the nonpredictor-based networked system is unstable due to constant delays.

Harnessing the Kelvin–Helmholtz instability: feedback stabilization of an inviscid vortex sheet

In this investigation, we use a simple model of the dynamics of an inviscid vortex sheet given by the Birkhoff–Rott equation to obtain fundamental insights about the potential for stabilization of shear layers using feedback control. As actuation, we consider two arrays of point sinks/sources located a certain distance above and below the vortex sheet and subject to the constraint that their mass fluxes separately add up to zero. First, we demonstrate using analytical computations that the Birkhoff–Rott equation linearized around the flat-sheet configuration is in fact controllable when the number of actuator pairs is sufficiently large relative to the number of discrete degrees of freedom present in the system, a result valid for generic actuator locations. Next, we design a state-based linear-quadratic regulator stabilization strategy, where the key difficulty is the numerical solution of the Riccati equation in the presence of severe ill-conditioning resulting from the properties of the Birkhoff–Rott equation and the chosen form of actuation, an issue that is overcome by performing computations with a suitably increased arithmetic precision. Analysis of the linear closed-loop system reveals exponential decay of the perturbation energy and the corresponding actuation energy in all cases. Computations performed for the nonlinear closed-loop system demonstrate that initial perturbations of non-negligible amplitude can be effectively stabilized when a sufficient number of actuators is used. We also thoroughly analyse the sensitivity of the closed-loop stabilization strategies to the variation of a number of key parameters. Subject to the known limitations of inviscid vortex models, our findings indicate that, in principle, it may be possible to stabilize shear layers for relatively large initial perturbations, provided that the actuation has sufficiently many degrees of freedom.

Gain Scheduled Attitude Control of Fixed-Wing UAV With Automatic Controller Tuning

Fixed-wing unmanned aerial vehicles (UAVs) have become increasingly important in military, civil, and scientific sectors. Because of the existing nonlinearities, effective control this type of UAV remains a challenge. This paper proposes a gain scheduled proportional-integral derivative (PID) control system for fixed-wing UAVs where a family of PID cascade control systems is designed for several operating conditions of airspeed. This is done using an automatic tuning algorithm, where the controllers are automatically selected by deploying an airspeed sensor positioned ahead of the aircraft. Furthermore, the actual gain scheduling is carried out by forming an interpolation between the family members of the linear closed-loop system, which ensures a smooth transition from one operating point to another. Experimental results are conducted in a wind tunnel to show the successful design and implementation of the gain scheduled control system for the fixed-wing UAV and the significant performance improvement over a linear control system without controller adaptation.

Sum-of-Squares-Based Region of Attraction Analysis for Gain-Scheduled Three-Loop Autopilot

A conventional method of designing a missile autopilot is to linearize the original nonlinear dynamics at several trim points, then to determine linear controllers for each linearized model, and finally implement gain-scheduling technique. The validation of such a controller is often based on linear system analysis for the linear closed-loop system at the trim conditions. Although this type of gain-scheduled linear autopilot works well in practice, validation based solely on linear analysis may not be sufficient to fully characterize the closed-loop system especially when the aerodynamic coefficients exhibit substantial nonlinearity with respect to the flight condition. The purpose of this paper is to present a methodology for analyzing the stability of a gain-scheduled controller in a setting close to the original nonlinear setting. The method is based on sum-of-squares (SOS) optimization that can be used to characterize the region of attraction of a polynomial system by solving convex optimization problems. The applicability of the proposed SOS-based methodology is verified on a short-period autopilot of a skid-to-turn missile.