Linear State Feedback Control Research Articles

On multi-class automated vehicles: Car-following behavior and its implications for traffic dynamics

This paper develops a unifying framework to unveil the physical car-following (CF) behaviors of automated vehicles (AVs) under different control paradigms and parameter settings. The proposed framework adopts the flexible asymmetric behavior (AB) model to reveal the control mechanisms and their manifestation in the physical CF behavior, particularly their response to traffic disturbances. A mapping relationship between the AB model parameters and control parameters is then obtained to understand the range of CF behavior possible. Finally, a predictive modeling approach based on a logistic classifier coupled with a convoluted Multivariate Gaussian Process (MGP) is designed to predict the CF behavior of an AV. Analysis of two well-known controllers, linear state-feedback and Model Predictive Control (MPC), show how the proposed framework can uncover the CF mechanisms and provide insights into traffic-level disturbance evolution. The proposed analysis framework remains scalable and can be applied to a variety of controllers. Ultimately, it can guide AV control design that is not myopic, but considers traffic-level performance.

A 4D hyperchaotic Sprott S system with multistability and hidden attractors

This paper derived a new simple hyperchaotic system from the famous Sprott, S system via the linear state feedback control. Compared with the available systems, the new system has eight terms, one constant, two parameters control, and a single quadratic nonlinear term. So this system is considered a simple relying on the number of terms and nonlinearities. The proposed system without equilibrium points and exhibits chaotic hidden attractors. Also, multistability or coexisting attractors are found through experimental simulations using phase portraits and the Lyapunov spectrum. Finally, anti-synchronization is implemented in the new system.

Tensor product‐based model transformation approach to cart position modeling and control in pendulum‐cart systems

AbstractThe paper presents the application of the tensor product (TP)‐based model transformation technique to model and control the cart position of single‐input multi‐output pendulum‐cart systems (PCSs). The modeling is first carried out. The derived TP model, the nonlinear model of PCS, and the laboratory equipment are tested in the same open‐loop scenario, and their corresponding outputs are collected, measured, and compared. The parallel distributed compensation (PDC) technique is next applied to TP‐based cart position controller design. A linear state feedback control design is also designed in order to conduct a comparative analysis. The two proposed control system structures are experimentally validated and tested in the same scenarios on the laboratory equipment, and their performance indices are analyzed and compared.

3-Dimensional Modeling and Attitude Control of Multi-Joint Autonomous Underwater Vehicles

To achieve rapid and flexible vertical profile exploration of deep-sea hybrid structures, a multi-joint autonomous underwater vehicle (MJ-AUV) with orthogonal joints was designed. This paper focuses on the 3-dimensional (3D) modeling and attitude control of the designed vehicle. Considering the situation of gravity and buoyancy imbalance, a 3D model of the MJ-AUV was established according to Newton’s second law and torque balance principle. And then the numerical simulation was carried out to verify the credibility of the model. To solve the problems that the pitch and yaw attitude of the MJ-AUV are coupled and the disturbance is unknown, a linear quadratic regulator (LQR) decoupling control method based on a linear extended state observer (LESO) was proposed. The system was decoupled into pitch and yaw subsystems, treated the internal forces and external disturbances of each subsystem as total disturbances, and estimated the total disturbances with LESO. The control law was divided into two parts. The first part was the total disturbance compensator, while the second part was the linear state feedback controller. The simulation results show that the overshoot of the controlled system in the dynamic process is nearly 0 rad, reaching the design value very smoothly. Moreover, when the controlled system is in a stable state, the control precision is within 0.005%.

Optimization of the Model Predictive Control Update Interval Using Reinforcement Learning

Abstract In control applications there is often a compromise that needs to be made with respect to the complexity and performance of the controller, and the computational resources that are available. For instance, the typical hardware platform in embedded control applications is a microcontroller with limited memory and processing power, and for battery powered applications the control system can account for a significant portion of the energy consumption. We propose a controller architecture in which the computational cost is explicitly optimized along with the control objective. This is achieved by a three-part architecture where a high-level, computationally expensive controller generates plans, which a computationally simpler controller executes by compensating for prediction errors, while a recomputation policy decides when the plan should be recomputed. In this paper, we employ model predictive control (MPC) as the high-level plan-generating controller, a linear state feedback controller as the simpler compensating controller, and reinforcement learning (RL) to learn the recomputation policy. Simulation results for the classic control task of balancing an inverted pendulum show that not only is the total processor time reduced by 60% — the RL policy is even able to uncover a non-trivial synergistic relationship between the MPC and the state feedback controller - improving the control performance by 20% over the MPC alone.

Simple adaptive control strategy for a class of systems subjected to real parametric uncertainty with application to unified chaotic system

In this paper, a relatively simplified adaptive control strategy for set of chaotic systems is presented. The results utilise the basic linear state feedback control strategy blended with algebraic...

Simple adaptive control strategy for a class of systems subjected to real parametric uncertainty with application to unified chaotic system

In this paper, a relatively simplified adaptive control strategy for set of chaotic systems is presented. The results utilise the basic linear state feedback control strategy blended with algebraic matrix Riccati equation (AMRE) to derive the control mechanism for chaotic systems operating in uncertain environment. The proposed approach utilises lesser number of control inputs in comparison to methods available in literature. Moreover, the controller structure turns out to be simple linear combination of states of system for the chaotic systems belonging to the proposed class, which are otherwise handled with nonlinear controller design strategies. The Lyapunov stability approach is exploited to analytically derive the control function and the parametric updation laws for uncertain parameters. Extensive simulation results are elaborated for set of chaotic systems to demonstrate effectiveness and validity of the presented approach.

Dynamics and control of the modified generalized Korteweg–de Vries–Burgers equation with periodic boundary conditions

The paper deals with the dynamics and control of the modified generalized Korteweg–de Vries–Burgers equation (MGKdVB) with periodic boundary conditions. First, the dynamics of the MGKdVB equation is studied using the Fourier Galerkin and the Karhunen–Loeve (K–L) Galerkin methods. The Fourier Galerkin approach is used to generate a system of nine ordinary differential equations (ODEs) from a partial differential equation (PDE), and the K–L Galerkin method is used as a model reduction technique for nonlinear systems to derive a reduced-order system of two ODEs that imitates the dynamics of the MGKdVB equation. It is shown that the two-dimensional reduced-order ODE system based on the K–L Galerkin method is superior to the Fourier Galerkin n-dimensional ODE system for any dimension $$n<2N+1$$ , where $$N=2$$ is the number of unstable eigenvalues, and it is comparable to the traditional higher nine-dimensional Fourier Galerkin system. Then, in order to enhance the stability of the MGKdVB equation, we propose to use state feedback linearization control schemes for both systems and show that the reduced-order system based on K–L Galerkin method is less expensive to control as compared to the Fourier Galerkin system. Finally, numerical simulations of the controlled systems are shown to illustrate the developed theory.

Optimization approach to the constrained regulation problem for linear continuous-time fractional-order systems

Abstract This paper deals with the Constrained Regulation Problem (CRP) for linear continuous-times fractional-order systems. The aim is to find the existence conditions of linear feedback control law for CRP of fractional-order systems and to provide numerical solving method by means of positively invariant sets. Under two different types of the initial state constraints, the algebraic condition guaranteeing the existence of linear feedback control law for CRP is obtained. Necessary and sufficient conditions for the polyhedral set to be a positive invariant set of linear fractional-order systems are presented, an optimization model and corresponding algorithm for solving linear state feedback control law are proposed based on the positive invariance of polyhedral sets. The proposed model and algorithm transform the fractional-order CRP problem into a linear programming problem which can readily solved from the computational point of view. Numerical examples illustrate the proposed results and show the effectiveness of our approach.

Analytical guidance for circular orbit transfers with staging of space propulsion systems

This paper considers the combination of an analytical reference trajectory with linear state feedback control to allow for autonomous guidance and control of a spacecraft for coplanar circle-to-circle transfers with a stage-based propulsion system. Staging of electric propulsion components, such as tanks and thrusters, could allow small spacecraft to achieve high-ΔV capabilities with current propulsion technology. In order to utilize these propulsion systems, further developments in guidance and control of such spacecraft are required due to limitations in computational power and communications. Analytical approximations for low-thrust trajectories could allow for computationally simple guidance and control of autonomous spacecraft for circle-to-circle transfers around large central bodies. Many trajectories have been developed for conventional propulsion systems, based either on their shape or input thrust, and applied for preliminary mission design. A previously developed analytical trajectory is extended to account for the effects of staging propulsion system components. In order to stabilize the trajectory in the presence of disturbances, a linear state feedback control law is designed with linear quadratic regulator methods. Finally, a methodology for determining the correct phasing between the spacecraft and a target object is developed and is practical to implement on power-limited computers. The use of the analytical reference trajectory is simulated on an orbit transfer from low-Earth orbit to geostationary orbit.

Global Stabilization of the Spacecraft Rendezvous System by Delayed and Bounded Linear Feedback

This article investigates the global stabilization problem of the circular orbit rendezvous system with actuator saturation and time-delay. By decomposing the linearized relative motion equations into a cascade of neutral stable linear systems, linear state feedback controllers are proposed in the presence of both actuator saturation and/or time-delay. The global stability of the closed-loop system is proved. Optimal feedback gain is also obtained in the delay-free case. Simulation results are given to show the effectiveness of the presented methods.

Global Mittag–Leffler Stabilization of Fractional-Order BAM Neural Networks with Linear State Feedback Controllers

In this paper, the global Mittag–Leffler stabilization of fractional-order BAM neural networks is investigated. First, a new lemma is proposed by using basic inequality to broaden the selection of Lyapunov function. Second, linear state feedback control strategies are designed to induce the stability of fractional-order BAM neural networks. Third, based on constructed Lyapunov function, generalized Gronwall-like inequality, and control strategies, several sufficient conditions for the global Mittag–Leffler stabilization of fractional-order BAM neural networks are established. Finally, a numerical simulation is given to demonstrate the effectiveness of our theoretical results.

A New 4D Chaotic System with Coexisting Hidden Chaotic Attractors

A new 4D chaotic system with infinitely many equilibria is proposed using a linear state feedback controller in the Sprott C system. Although the new 4D chaotic system has only two nonlinear terms, it has rich dynamic characteristics, such as hidden attractors and coexisting attractors. Besides, the freedom of offset boosting of a variable is achieved by adjusting a controlled constant. The dynamic characteristics of the new chaotic system are fully analyzed from the aspects of phase portraits, bifurcation diagrams, Lyapunov exponents and Poincaré maps. The corresponding analogue electronic circuit is designed and implemented to verify the new 4D chaotic system. By taking advantage of the complex dynamic properties of the new chaotic system, a random number generator algorithm is proposed.

Robust backstepping output voltage controller for standalone voltage‐sourced converters

This work presents a linear state-feedback controller using a backstepping design approach for output voltage regulation of 3 ϕ voltage-sourced converters feeding to customers' loads in a stand-alone AC microgrid system. Irrespective of load type and its variations, parameter uncertainties, and other disturbances, the controller is robust enough to achieve a regulated voltage magnitude within the prescribed bounds and exact frequency tracking. The criterion for gain selection using Lyapunov analysis is given. With global exponential convergence feature and better coordination among inner current and outer voltage control loops, improved transient response is achieved. Controller features are examined by eigenvalue analysis and extensive simulation studies. Experimental results demonstrate the feasibility of the proposed controller implementation.

Nonlinear vibration analysis in precision motion stage with PID and time-delayed feedback controls

We investigate the control of friction-induced vibration in a precision motion stage under the effect of the LuGre friction dynamics. We consider a lumped parameter model of the precision motion stage with PID and linear time-delayed state feedback control acting in the direction of the motion of the stage. Linear stability analysis reveals the criticality of integral gain in the stability and, accordingly, the existence of multiple stability lobes and codimension-2 Hopf points for a given choice of system parameters. The nature of the bifurcation is determined by an analytical study using the method of multiple scales and harmonic balance. We observe the existence of both subcritical and supercritical Hopf bifurcations in the system, depending on the choice of control parameters. Hence, the nonlinearity due to dynamic friction model could both be stabilizing or destabilizing in nature, and therefore, stick-slip nonlinearity is essential to capture the global behavior of the system dynamics. Furthermore, numerical bifurcation analysis of the system reveals the existence of period-doubling bifurcation near the Hopf points. We observe complicated solutions such as period-4, quasi-periodic, large-amplitude stick-slip limit cycles along with chaotic attractor in the system.

Nonlinear guaranteed cost attitude control via dynamic gain design

This paper investigates the guaranteed cost attitude control problem of a rigid-body subject to parameter uncertainty problem. The traditional LQR and guaranteed cost control (GCC) methods rest on linear systems and then entail model linearization in the application of attitude control by assuming small-angle rotations, since rigid-body attitude systems are highly nonlinear. In contrast, the proposed method effectively accounts for the nonlinearities inherent in the attitude dynamics of the rigid-body and ensures the attitude system with asymptotic stability and a guaranteed cost index. A linear state-feedback controller of the classical proportional/derivative (PD) structure is provided, which is off-line designed by some linear matrix inequality (LMI) conditions. Moreover, by virtue of the dynamic gain technology, the conditions are of simple structure, which thereby are easy to solve and eligible to turn into some optimization problems for optimizing the cost index and/or controller gains. The optimization problems are tractable to handle computationally and thus the optimal solution can be ensured. The developed theoretical results are illustrated by simulations.

A new adaptive synchronization and hyperchaos control of a biological snap oscillator

• Chaos control and synchronization of a biological snap oscillator is investigated. • Optimal and adaptive controllers are designed to overcome hyperchaotic behaviors. • An adaptive control scheme is introduced to synchronize two identical snap oscillators. • A new fractional model is developed for the biological system under study. • Efficient control strategies are used in fractional sense for the purpose of control and synchronization. The purpose of this paper is to analyze and control the hyperchaotic behaviours of a biological snap oscillator. We mainly study the chaos control and synchronization of a hyperchaotic model in both the frameworks of classical and fractional calculus, respectively. First, the phase portraits of the considered model and its hyperchaotic attractors are analyzed. Then two efficacious optimal and adaptive controllers are designed to compensate the undesirable hyperchaotic behaviours. Moreover, applying an efficient adaptive control procedure, we generally synchronize two identical biological snap oscillator models. Finally, a new fractional model is proposed for the considered oscillator in order to acquire the hyperchaotic attractors. Indeed, the fractional calculus leads to more realistic and flexible models with memory effects, which could help us to design more efficient controllers. Considering this feature, we apply a linear state-feedback controller as well as an active control scheme to control hyperchaos and achieve synchronization, respectively. The related theoretical consequences are numerically justified via the obtained simulations and experiments.

Implementation of Linear Controllers via Active Disturbance Rejection Control Structure

Linear active disturbance rejection control (LADRC) takes the controlled plant as a cascaded integral model and treats all other plant information and external disturbances as a generalized disturbance, and uses an linear extended state observer (ESO) to estimate the generalized disturbance and incorporates it in a linear state feedback control law to reject it quickly. This ESO-based LADRC controller is a fixed-structure controller as long as the (relative) order of the controlled system is specified. This article shows that any linear finite-dimensional controller can be implemented via the LADRC structure; thus, it is a general-purposed control structure in that it can be applied to any control system where linear control is sufficient. Case study shows that the implementation not only retains the disturbance rejection performance of the original linear controller but also improves its tracking performance due to the two-freedom-of-degree property of LADRC.

Flight envelope estimation for helicopters under icing conditions via the zonotopic reachability analysis

This paper presents a new approach to estimating a flight envelope for a helicopter flying under clean and icing conditions based on a zonotopic reachability analysis. The helicopter is stabilized with a state-feedback linear controller to maintain its attitude around a trim point. Given the sets of possible initial conditions and disturbance inputs, and control effectiveness indices, this analysis results in reachable sets of state variables of the helicopter for a finite time interval. The reachability analysis is thus useful for predicting whether or not the helicopter evolves into an unsafe flight condition when it undergoes degraded control effectiveness ensued from the icing condition. The merit of this approach is shown through a case study about the Raptor-90 unmanned helicopter stabilized with either the linear quadratic regulator (LQR) or the H∞ controller in a hover-flight mode. This helicopter is considered to be affected by both symmetric and asymmetric icing conditions. The case study shows that the H∞ controller outperforms the LQR controller as the former is able to cope with greater loss of control effectiveness caused by the icing conditions. Moreover, as compared to the longitudinal control input, the lateral and directional control inputs are more susceptible to adverse effects of the icing conditions. These results verify not only the effects of ice contamination upon the helicopter dynamics and control, but also the capability of the proposed approach for flight envelope estimation.

Memoryless linear feedback control for a class of upper-triangular systems with large delays in the state and input

This paper studies the problem of global asymptotic stabilization for a class of upper-triangular systems with large delays in the state and input. Using the idea of feedback domination, we develop a Lyapunov-Krasovskii method for the design of time-invariant, memoryless linear state and output feedback controllers, achieving global asymptotic stabilization of the closed-loop systems under a linear growth condition. The resulting controllers are different from predictor-based or delay-dependent controllers in the sense that they are delay-free and easier to be implemented due to the nature of “memoryless” feedback. Two examples are presented to demonstrate the effectiveness of the proposed delay-free state and output feedback control strategies.