Design Of Feedback Control Laws Research Articles

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.

Isolation‐oriented Lyapunov‐based design of simultaneous feedback strategies for disease control of SIQR

AbstractThis article pursues a new approach to the design of feedback control laws mitigating the spread of human infectious diseases. The control inputs of vaccination, isolation, and outing/gathering regulation are derived as the gradient of a positive definite function of all the population variables of the SIQR model. Lyapunov functions in which susceptible and infected populations are decoupled are known to be non‐self‐sufficient for proving stability of the SIQR model as well as other variants of the SIR model. This fact has been hampering Lyapunov‐based control design of infectious disease models. This article demonstrates that a popular decoupled function can still serve as a control Lyapunov function as long as isolation is made state‐dependent. Techniques realizing this novel idea are presented and shown to allow the Lyapunov function to establish a robustness guarantee in terms of input‐to‐state stability with respect to immigration and newborn perturbation on an arbitrarily large domain. The effectiveness of the proposed simultaneous controller is illustrated through numerical simulations performed with a parameter set of COVID‐19.

A distributed model predictive control with neighborhood state feedback invariant set for reconfigurable networked systems

AbstractFor a class of networked systems, this article proposed a distributed model predictive control (DMPC) design where the actual control law is composed of the solution of an independent optimization problem and a neighborhood states feedback part. By the proposed method, each subsystem‐based MPC is designed in a distributed way and the neighborhood feedback provides more potential for the designed DMPC to be applied to a wider class of systems. First, an LMI optimization problem is designed for determining the parameters of the neighborhood feedback matrix and corresponding invariant sets where the information of neighboring subsystems is explicitly involved. The designed feedback control law deals with the affection of the interacted subsystems and limits the deviation between the nominal model and the real system within the robust invariant set. Then, the independent suboptimal problem is developed using a predictive model without considering interconnections among subsystems, where the state is restricted by a tightened constraint which is related to the designed robust invariant set. The feasibility and stability of the closed‐loop system are analyzed. The proposed DMPC enables the direct removal or plugging‐in of subsystems. Simulation results demonstrated the effectiveness of the method.

Optimum Path Tracking Control for the Inverse Problem of Vehicle Handling Dynamics Based on the hp-Adaptive Gaussian Pseudospectral Method

We propose a vehicle path-tracking method based on the hp-adaptive Gaussian pseudospectral method (GPM), which tackles the problem of the slow convergence speed of the optimal control of vehicle path tracking. First, we establish a kinematic vehicle model by considering the path constraints and boundary constraints during the process of tracking the described path of a vehicle. Subsequently, finding the minimum error of the lateral distance between the prescribed path and the expected trajectory is set as the performance objective function. Finally, the vehicle path tracking problem is transformed into an optimal control problem. The optimization algorithm is combined with the sequential quadratic programming algorithm to optimize problems related to the control and state variables and the boundary and path constraints. The simulated results show that the hp-adaptive GPM can improve the convergence rate of the optimal control problem for vehicle path tracking. The proposed hp-adaptive pseudospectral method has a higher solving efficiency compared with traditional approaches for the vehicle path-tracking problem. Concurrently, the verification results of a real vehicle test indicate the feasibility of the proposed algorithm for solving the vehicle path tracking problem. This research provides valuable insight into the design work of lane changes and is an important step towards the design of feedback control laws for path tracking.

Stability/Instability Study and Control of Autonomous Dynamical Systems: Divergence Method

A novel method of instability and stability study of equilibrium points of autonomous dynamical systems using a flow and divergence of the vector field is proposed. A relation between the Lyapunov, Gauss (Ostrogradsky) and Chetaev theorems with the divergence ones is established. The generalizations of Bendixon and Bendixon-Dulac theorems about a lack of periodic solutions in arbitrary order systems are considered. The state feedback control law design based on new divergence conditions is proposed. Examples illustrate the efficiency of the proposed method and comparisons with some existing ones.

Predictive Descriptor Observer Design for a Class of Linear Time-Invariant Systems With Applications to Quadrotor Trajectory Tracking

This article deals with sensor fault tolerant control design and large delay compensation for a class of linear time-invariant systems. Based on predictor-based state transformation and predictive descriptor observer design, both the state and output feedback control laws are constructed. Within the framework of Lyapunov–Krasovskii functionals, a set of stability conditions are identified. The closed-loop systems is globally asymptotically stable at the origin under the state feedback and is ultimately bounded under the output feedback. Finally, the proposed design is validated through numerical simulations and then implemented on a quadrotor trajectory tracking test. Both simulation and experimental results demonstrate the effectiveness of the proposed design.

Robust adaptive anti-windup wheel slip tracking control for intelligent vehicle with fast terminal sliding mode observer

Aiming at the requirement of the intelligent vehicle for the fast and stable tracking control of the wheel slip, a novel robust adaptive anti-windup wheel slip tracking control method with fast terminal sliding mode observer is proposed. First, a fast terminal sliding mode observer based on equivalent control on the sliding surface is proposed to estimate the states of the wheel slip dynamic system to lay the foundation for the full state feedback control law design. Second, a robust adaptive anti-windup wheel slip tracking control law with lumped uncertainty observer and additional anti-windup dynamics is derived based on Lyapunov-based method. The lumped uncertainty observer utilizes the nonlinear mapping ability of the radius basis function neural network to estimate and compensate the lumped uncertainty of the system, and the unknown optimal weight vector of the radius basis function neural network is updated by adaptive law. The additional anti-windup dynamics is used to suppress the effect of the input saturation on the stability of the system. Finally, the performance of the proposed method is verified through simulations of various maneuvers on vehicle dynamics simulation software.

Exponential Stabilization of a Star-Shaped Thermoelastic Network System Based on the Extended State Observer With Time-Varying Gains

The boundary stabilization problem of a star-shaped type II thermoelastic system is considered, where there is a nonuniform bounded disturbance at the control end. A time-varying extended state observer is proposed to estimate and cancel the disturbance, thereby largely removing the uncertainties and simplifying the feedback control law design. The exponential stability is shown for the extended state observer, so are the well-posedness properties of the closed-loop system, all based on the linear operator semigroup theory. Finally, it is shown that the disturbances can be attenuated and the resulting closed-loop system can exponentially approach zero. Numerical simulations illustrate the effectiveness of the proposed control strategy.

Continuous dynamic sliding mode control strategy of PWM based voltage source inverter under load variations

For closed-loop controlled DC-AC inverter system, the performance is highly influenced by load variations and online current measurement. Any variation in the load will introduce unwanted periodic error at the inverter output voltage. In addition, when the current sensor is in faulty condition, the current measurement will be imprecise and the designed feedback control law will be ineffective. In this paper, a sensorless continuous sliding mode control (SMC) scheme has been proposed to address these issues. The chattering effect due to the discontinuous switching nature of SMC has been attenuated by designing a novel boundary-based saturation function where the selection of the thickness of boundary is dependent to the PWM signal generation of the inverter. In order to remove the dependency on the current sensor, a particle swarm optimization(PSO) based modified observer is proposed to estimate the inductor current in which the observer gains are optimized using PSO by reducing the estimation errors cost function. The proposed dynamic smooth SMC algorithm has been simulated in MATLAB Simulink environment for 0.2-kVA DC-AC inverter and the results exhibit rapid dynamic response with a steady-state error of 0.4V peak-to-peak voltage under linear and nonlinear load perturbations. The total harmonic distortion (THD) is also reduced to 0.20% and 1.14% for linear and non-linear loads, respectively.

A new three-dimensional chaotic system: its adaptive control and circuit design

A new three-dimensional chaotic system with two nonlinearities is proposed in this research work. Specifically, the two nonlinearities considered in the construction of the new chaotic system are a quadratic nonlinearity and a quartic nonlinearity. A systematic study of the three-dimensional chaotic system has been made including phase portraits, dissipativity, rest points and their stability, Lyapunov chaos exponents, Kaplan-Yorke fractal dimension, etc. As main control results, we design feedback control laws using adaptive control theory to achieve global stabilisation of the new chaotic system and also global synchronisation of identical chaotic systems with unknown parameters. Finally, an electronic circuit design of the new chaotic system using electronic work bench (EWB) is described in detail to confirm the feasibility of the theoretical chaotic model.

Principle and experimental research on vibration reduction of flexible solar array using reaction flywheel

PurposeThe rest-to-rest movements for a spacecraft, such as attitude adjustment and orbital manoeuver, are likely to excite residual vibration of flexible appendages, which may affect the attitude accuracy and even result in severe structural damage. The purpose of this paper is to present an approach to attenuating the vibration of flexible solar array by using reaction flywheel.Design/methodology/approachThe reaction flywheel installed on solar array served as an actuator to provide reaction torque to a structure according to a designed feedback control law. This torque can be considered as an artificial damping. Experiment on a scale model of the solar array is first performed to verify the effectiveness of this method. Numerical simulation on finite element model of a full-scale solar array is subsequently carried out to confirm the validity of this method for practical engineering application.FindingsThe vibration suppression effect on the structure using a reaction flywheel is deduced by theoretical analysis. Results from both experiment and numerical simulation reveal that the efficiency of vibration attenuation is promoted.Research limitations/implicationsImprovements on control law are left for further study. Additionally, only the first-order bending vibration of the flexible solar array is attenuated, and further study is required for other types of vibration suppression.Practical implicationsAn effective method is proposed for spacecraft designers to actively suppress the vibration of the flexible solar array.Originality/valueA novel active vibration reduction scheme is proposed using a reaction flywheel to suppress vibration of the flexible solar array. This paper fulfils a source of theoretical analysis and experimental studies for vibration reduction measure design and provides practical help for the spacecraft designers.

Design of Feedback Control Laws for Information Transfer in Spintronics Networks

Information encoded in networks of stationary, interacting spin-1/2 particles is central for many applications ranging from quantum spintronics to quantum information processing. Without control, however, information transfer through such networks is generally inefficient. Currently available control methods to maximize the transfer fidelities and speeds mainly rely on dynamic control using time-varying fields and often assume instantaneous readout. We present an alternative approach to achieving efficient, high-fidelity transfer of excitations by shaping the energy landscape via the design of time-invariant feedback control laws without recourse to dynamic control. Both instantaneous readout and the more realistic case of finite readout windows are considered. The technique can also be used to freeze information by designing energy landscapes that achieve Anderson localization. Perfect state or super-optimal transfer and localization are enabled by conditions on the eigenstructure of the system and signature properties for the eigenvectors. Given the eigenstructure enabled by super-optimality, it is shown that feedback controllers that achieve perfect state transfer are, surprisingly, also the most robust with regard to uncertainties in the system and control parameters.

Stochastic Stability of Some Classes of Nonlinear 2D Systems

The paper considers nonlinear discrete and differential stochastic repetitive processes using the state-space model setting. These processes are a special case of 2D systems that originate from the modeling of physical processes. Using the vector Lyapunov function method, sufficient conditions for stability in the mean square are obtained in the stochastic setting, where the vast majority of the currently known results are for deterministic dynamics. Based on these results, the property of stochastic exponential passivity in the second moment is used, together with the vector storage function, to develop a new method for output feedback control law design. An example of a system with nonlinear actuator dynamics and state-dependent noise is given to demonstrate the effectiveness of the new results.

State-feedback control of Markov jump linear systems with hidden-Markov mode observation

In this paper, we study state-feedback control of Markov jump linear systems with partial information about the mode signal responsible for switching between dynamic modes. We assume that the controller can only access random samples of the mode signal according to a hidden-Markov observation process. Our formulation provides a novel framework to analyze and design feedback control laws for various Markov jump linear systems previously studied in the literature, such as the cases of (i) clustered observations, (ii) detector-based observations, and (iii) periodic observations. We present a procedure to transform the closed-loop system with hidden-Markov observations into a standard Markov jump linear system while preserving stability, H2 norm, and H∞ norm. Furthermore, based on this transformation, we propose a set of Linear Matrix Inequalities (LMI) to design feedback control laws for stabilization, H2 suboptimal control, and H∞ suboptimal control of discrete-time Markov jump linear systems under hidden-Markov observations of the mode signals. We conclude by illustrating our results with some numerical examples.

Stabilization of a second order ODE–heat system coupling at intermediate point

This paper considers the stabilization of a second order ODE–heat system coupling at an intermediate point under natural and checkable assumptions, which is motivated by the thermoelastic coupling physics arising in microelectromechanical systems (MEMS). A novel backstepping transformation form is proposed in this work and the feedback gain for the lumped-parameter component is constructed using the eigenvector of coefficient matrix in the ODE system. Then, we prove the existence of smooth kernels with second order continuously derivative for the forward and inverse transformations in the backstepping feedback control law design. At the same time, we show that the forward and inverse transformations are mutually invertible transformation pair. Finally, the effectiveness of the stabilization feedback controller design is shown with some numerical examples.

Swarm Coordination Based on Smoothed Particle Hydrodynamics Technique

The focus of this study is on the design of feedback control laws for swarms of robots that are based on models from fluid dynamics. We apply an incompressible fluid model to solve a pattern generation task. Possible applications of an efficient solution to this task are surveillance and the cordoning off of hazardous areas. More specifically, we use the smoothed-particle hydrodynamics (SPH) technique to devise decentralized controllers that force the robots to behave in a similar manner to fluid particles. Our approach deals with static and dynamic obstacles. Considerations such as finite size and nonholonomic constraints are also addressed. In the absence of obstacles, we prove the stability and convergence of controllers that are based on the SPH method. Computer simulations and actual robot experiments are shown to validate the proposed approach.

Robust Constraint Satisfaction for Continuous-Time Nonlinear Systems in Strict Feedback Form

This work is concerned with the design of feedback control laws that guarantee satisfaction of hard output constraints. A recursive procedure is used to derive sufficient conditions for robust constraint satisfaction in input affine nonlinear systems with bounded disturbances or parametric uncertainties. The control objective is achieved by a suitable switching between two control modes, one responsible for stabilization and one for constraint satisfaction.

A Multiple Model Adaptive Wave Filter for Dynamic Ship Positioning

This paper addresses a filtering problem that arises in the design of dynamic positioning systems for ships subjected to the influence of sea waves. Its key contribution is the use of a multiple model adaptive wave filter that relies on measurements of the ship's position and heading only. To this effect, a bank of Kalman filters is designed for a finite number of parameter values, each corresponding to a different peak frequency of the assumed wave spectrum model. Tools from multiple model adaptive estimation (MMAE) theory are exploited to blend the information provided by the different observers, yielding position and velocity estimates of the ship. These estimates are then be used in an appropriately designed feedback control law. Simulations illustrate the efficacy of the MMAE techniques proposed.

New approach for swinging up the Furuta pendulum: Theory and experiments

The problem of swinging up inverted pendulums has often been solved by stabilizing a particular class of homoclinic structures present in the dynamics of a physical pendulum. Here, new arguments are suggested to show how other homoclinic curves can be preplanned for dynamics of the passive-link of the robot. This is done by reparameterizing the motions according to geometrical relations among the generalized coordinates, which are known as virtual holonomic constraints. After that, conditions that guarantee the existence of periodic solutions surrounding the planned homoclinic orbits are derived. The corresponding trajectories, in contrast to homoclinic curves, admit efficient design of feedback control laws ensuring exponential orbital stabilization. The method is illustrated by simulations and supported by experimental studies on the Furuta pendulum. The implementation issues are discussed in detail.

Control of frictional dynamics of a one-dimensional particle array

Control of frictional forces is required in many applications of tribology. While the problem is approached by chemical means traditionally, a recent approach was proposed to control the system mechanically to tune frictional responses. We design feedback control laws for a one-dimensional particle array sliding on a surface subject to friction. The Frenkel–Kontorova model describing the dynamics is a nonlinear interconnected system and the accessible control elements are average quantities only. We prove local stability of equilibrium points of the un-controlled system in the presence of linear and nonlinear particle interactions, respectively. We then formulate a tracking control problem, whose control objective is for the average system to reach a designated targeted velocity using accessible elements. Sufficient stabilization conditions are explicitly derived for the closed-loop error systems using the Lyapunov theory based methods. Simulation results show satisfactory performances. The results can be applied to other physical systems whose dynamics is described by the Frenkel–Kontorova model.