Linear State Feedback Research Articles

Robust Optimal Output-Tracking Control of Constrained Mechanical Systems With Application to Autonomous Rovers

This article presents a robust optimal output-tracking control strategy for underactuated mechanical systems whose motion is restricted by mixed holonomic and nonholonomic constraints. Autonomous rovers/cars and unmanned underwater/aerial vehicles are a few examples of such systems that must often operate in harsh environments and under uncertain conditions. We present a comprehensive control analysis of this large class of nonlinear systems, including the existing studies on local reachability and static state feedback linearization. We also propose a local observability analysis of the feedback transformed input–output linearized systems. Based on the input–output linearization of the holonomically restricted nominal system, we develop a sliding mode control strategy that is robust against the projected effects of uncertainties and disturbances on the system’s output. Asymptotic stability of the output toward a bounded desired trajectory is proven using Lyapunov’s direct method, while the system’s internal stability (in the sense of boundedness) is investigated based on the notion of tracking-error zero dynamics. Time-dependent bounded matched uncertainties in the inertia parameters and disturbance forces arising in the unrestricted system are considered in this study. We propose an optimal sliding manifold according to the finite-horizon linear-quadratic regulator design problem with split boundary value conditions. The developed control strategy is implemented on a six-wheel autonomous Lunar rover in a simulation environment and its performance is compared with that of an optimal proportional–integral–derivative feedback, feedforward controller. The optimal sliding mode controller shows superior performance in trajectory tracking with acceptable control actions.

Full state feedback linearization for a haulage trolley system: a novel differentiable manifold approach

The manuscript presents a manifold-based full-state linearization for a haulage trolley. The key result from the manifold approach is representing the linearized system as a function of the tangent vectors of the nonlinear system. The haulage trolley is linearized, and the system's performance is studied. Three case studies are interrogated; the nonlinear system, a regulator control problem, and a tracking control problem. A settling time of 2 s is set for the control problem, and a reference is set to 1 rad/s for the tracking problem. The results confirm that linear control methods work on the controllable linearized haulage trolley system.

Simulation analysis of VSC reactive power compensation device

Reactive power compensators based on three-phase voltage source converters continuously absorb and output adjustable reactive power. The mathematical model of the three-phase voltage source converter in a direct quadrature (DQ)-frame is analyzed. A decoupling method of the DC-side voltage and reactive power is designed through the state feedback linearization controlling strategy in non-linear differential geometry theory, and the PI controller parameters are set. Finally, the control method is verified effectiveness by Electromagnetic transient simulation software PSCAD/EMTDC. Simulation results show that the developed control method possesses better steady-state and dynamic behavior. The power factor of the compensation point is improved significantly.

Domain stabilization in probability in a fixed time for nonlinear stochastic systems via feedback control

AbstractThis paper is concerned with the domain stabilization in probability in a fixed time for nonlinear stochastic systems. More specifically, given a target domain , a probability and a fixed time , we aim to find suitable state feedback control laws such that the trajectory of the resulting closed‐loop system, starting from outside , will reach the target domain in the fixed time with the minimum probability . To this end, we prove a local existence and uniqueness result of solutions to nonlinear stochastic systems under mild conditions, and derive upper bounds of the mean of the first hitting time with respect to under various rigorous conditions. By these results, a Lyapunov‐based controller design is given to guarantee the domain stability for nonlinear stochastic systems with affine control inputs. Besides, inequalities of quadratic forms focusing on semi‐linear stochastic systems are used to design a linear state feedback controller for achieving the domain stability in probability. Several examples are given for demonstration.

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

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

Exact Feedback Linearization of a Multi-Variable Controller for a Bi-Directional DC-DC Converter as Interface of an Energy Storage System

DC microgrids have shown to be a good approach for better accommodating stochastic renewable energy sources (RES) and for the charging of electric vehicles (EVs) at the distribution level. For this, fast-acting energy storage units (ESSs) are essential. This requires that both the bi-directional power converter topology and the control scheme present the right set of features. The ESS discussed in this paper consists of a new DC-DC converter based on a tapped inductor (TI) for a higher voltage gain at moderate duty cycles. The direction of the current in its intermediate inductor does not need to be reversed for power flow reversal, leading to a faster action. Moreover, it can employ a multi-state and multi-variable modulation scheme that eliminates the right half-plane (RHP) zero, common in boost-type converters. In order to achieve good dynamic performance across a wide range of operating points, a control scheme based on feedback linearization is developed. This paper presents the modeling of the five-switch DC-DC converter operating in the tri-state buck–boost mode. A systematic approach for deriving control laws for the TI current and output voltage based on exact state feedback linearization is discussed. The performance of the proposed control scheme is verified by simulation for a supercapacitor (SC)-based ESS. It is compared to that of a conventional control scheme for a dual-state buck–boost mode with cascaded PI controllers designed based on small-signal models. The results show that both control schemes work similarly well at the operating point that the conventional control scheme was designed for. However, only the proposed scheme allows the SC-based ESS to control the current injected into the DC microgrid with the voltage of the SC varying between the expected range of rated to half-rated.

Modeling and Nonlinear Control of a Quadcopter for Stabilization and Trajectory Tracking

This paper presents an adequate mathematical representation of a quadcopter’s system dynamics and effective control techniques. A quadcopter is an unmanned aerial vehicle (UAV) that is able to do vertical take-off and landing. This study presents a nonlinear quadcopter system’s mathematical modeling and control for stabilization and trajectory tracking. The mathematical model of the system dynamics of the quadcopter is derived using Newton and Euler equations with proper references to the appropriate frame or coordinate system. A PD control algorithm is developed for the nonlinear system for stabilization. Another nonlinear control technique called full state feedback linearization (FBL) using nonlinear dynamic inversion (NDI) is developed and implemented on the quadcopter system. However, there is a problem with the normal approach of the complete derivation of the full state FBL system using NDI as gathered from the literature review. In such an approach, the PD controller that was used for attitude stabilization was able to stabilize the angles to zero states, but the position variables cannot be stabilized because the state variables are not observable. Thus, a new approach where the position variables are mapped to the angle variables which are controllable so as to drive all states to zero stability was proposed in this study. The aim of the study was achieved but the downside is that it takes a longer time to achieve this stability so it is not efficient and should only be considered when absolute zero stability is the aim without considering time efficiency. The study further investigates the problem of nonlinear quadcopter system’s mathematical modelling and control for stabilization and trajectory tracking using the feedback linearization (FBL) technique combined with the PD controller. The proposed control algorithms are implemented on the quadcopter model using MATLAB and analyzed in terms of system stabilization and trajectory tracking. The PD controller produces satisfactory results for system stabilization, but the FBL system combined with the PD controller performs better for trajectory tracking of the quadcopter system.

Sliding-Mode Control of Full-State Constraint Nonlinear Systems: A Barrier Lyapunov Function Approach

This study presents the design of a robust control based on the sliding-mode theory to solve both; the stabilization and the trajectory tracking problems of nonlinear systems subjected to a class of full-state restrictions. The selected nonlinear system satisfies a standard Lagrangian structure affected by nonparametric uncertainties. A barrier Lyapunov function is used to ensure the state constraints by designing a time-varying gain, which guarantees the fulfillment of the predefined state constraints even under external perturbations. The proposed design methodology for the barrier sliding-mode control (BSMC) ensures the convergence of the sliding surface in finite time to the origin. Consequently, the asymptotic convergence of the states to the corresponding equilibrium point is achieved. The finite-time stability of the origin in the closed-loop system with the proposed controller has been demonstrated using the second Lyapunov stability method. The suggested controller was evaluated on a two-link robotic manipulator. Then, the obtained results showed better stabilization and tracking performances (while the restrictions are satisfied) than the traditional first-order sliding-mode or linear state feedback controllers.

Unified Control Parameterization Approach for Finite-Horizon Feedback Control With Trajectory Shaping

This article presents control parameterization as a unifying framework for designing a linear feedback control law that achieves the finite-time transfer of output as well as trajectory shaping. Representing control input as a linear combination of independent basis functions allows wide variability in the resultant feedback control laws through the selection of the number and types of basis functions. Given an array of basis functions that meets the trajectory shaping necessities, the unified design approach proceeds with the determination of the coefficients so that the predicted trajectory attains the desired output at the final time. The input evaluated with the coefficients found at each instance essentially turns out to be a linear state feedback policy with an additional feedforward term and time-dependent gains, which is appropriate for practical use. The unified control parameterization approach lends itself well to missile guidance applications with the expandability and direct trajectory shaping capability that it provides. To emphasize the expandability of the framework, this study revisits the trajectory shaping guidance laws from the control parameterization viewpoint and shows how the notion of specifying input basis functions not only generalizes various existing methods but also enables further extensions. Furthermore, an application to integrated guidance and control illustrates the strength of the design process in handling the shaping requirements more directly through the construction of appropriate basis.

Fastest containment control of discrete-time multi-agent systems using static linear feedback protocol

This paper focuses on the fastest containment control of discrete-time single-input linear multi-agent systems (MASs) over directed graphs. By proving that the largest convergence region for containment control is identical to the largest gain margin of the discrete-time linear quadratic regulator (LQR) in the sense of a radial projection, the necessary and sufficient condition on the containment control problem is derived using a standard Riccati design method. Inspired by this, we introduce a speed factor to characterize the worst-case convergence speed of the containment control and show that the convergence speed becomes faster as the speed factor decreases. The analytical solutions of the fastest convergence speed are obtained, and static linear distributed protocols are designed using the standard algebra Riccati equation to achieve the fastest convergence speed of the containment control. Finally, a numerical example is given to verify the validity of the developed theoretical results.

Non-overshoot tracking control for positive continuous-time linear systems

This article focuses on state and output tracking control for positive continuous-time linear systems without overshoot. Linear state feedback controllers are designed respectively to ensure both the system state, and output can unilateral track the reference state and output. In addition, using Lyapunov direct method, the exponential convergence rate of tracking system is calculated. A numerical example is given to demonstrate the effectiveness of the proposed method.

Design, modeling, and control of a variable stiffness elbow joint

New technological advances are changing the way robotics are designed for safe and dependable physical human–robot interaction and human-like prosthesis. Outstanding examples are the adoption of soft covers, compliant transmission elements, and motion control laws that allow compliant behavior in the event of collisions while preserving accuracy and performance during motion in free space. In this scenario, there is growing interest in variable stiffness actuators (VSAs). Herein, we present a new design of an anthropomorphic elbow VSA based on an architecture we developed previously. A robust dynamic feedback linearization algorithm is used to achieve simultaneous control of the output link position and stiffness. This actuation system makes use of two compliant transmission elements, characterized by a nonlinear relation between deflection and applied torque. Static feedback control algorithms have been proposed in literature considering purely elastic transmission; however, viscoelasticity is often observed in practice. This phenomenon may harm the performance of static feedback linearization algorithms, particularly in the case of trajectory tracking. To overcome this limitation, we propose a dynamic feedback linearization algorithm that explicitly considers the viscoelasticity of the transmission elements, and validate it through simulations and experimental studies. The results are compared with the static feedback case to showcase the improvement in trajectory tracking, even in the case of parameter uncertainty.

Design and application of models reference adaptive control (MRAC) on ball and beam

This paper presents the implementation of an adaptive control approach to the ball and beam system (BBS). The dynamics of a BBS are non-linear, and in the implementation, the uncertainty of the system's parameters may occur. In this research, the linear state-feedback model reference adaptive control (MRAC) is used to synchronize the states of the BBS with the states of the given reference model. This research investigates the performance of the MRAC method for a linear system that is applied to a non-linear system or BBS. In order to get a faster states convergence response, we define the initial condition of the feedback gains. In addition, the feedback gains are limited to get less oscillation response. The results show the error convergence is improved for the different sets of the sinusoidal reference signal for the MRAC with modified feedback gains. The ball position convergence improvement of MRAC with modified feedback gains for sinusoidal reference with an amplitude of 0.25, 0.5, and 0.75 are 35.1 %, 36 %, and 52.4 %, respectively.

Assignment of the upper Bohl exponent for linear periodic control systems in Hilbert spaces

A linear continuous-time control system with time-varying linear bounded operator coefficients in a Hilbert space is considered. The controller in the system has the form of linear state feedback with a time-varying linear bounded gain operator function. We study the problem of arbitrary assignment of the upper Bohl exponent by state feedback control. The property of exact controllability for the open-loop system is studied, some necessary and sufficient conditions are obtained for exact controllability. For periodic systems, it is proved that if the open-loop system is exactly controllable then one can shift the upper Bohl exponent of the closed-loop system by any pregiven number with respect to the upper Bohl exponent of the free system by means of linear state feedback with a periodic gain operator function. This implies arbitrary assignability of the upper Bohl exponent by linear state feedback. Finally, an illustrative example is presented.

On interval observer design for active Fault Tolerant Control of Linear Parameter-Varying systems

This paper proposes an active Fault Tolerant Control (FTC) scheme for polytopic uncertain Linear Parameter-Varying (LPV) systems subject to uncertainties and actuator faults. First, a fault estimation interval observer is designed to estimate the system state and the actuator fault. A novel approach is developed using the L ∞ norm to attenuate the effects of the uncertainties and to improve the accuracy of the proposed observer. Then, based on the fault estimation information, the FTC strategy is designed using a linear state feedback control law and H ∞ technique to compensate actuator faults and maintain system performance and stability, even under faulty conditions. Finally, the effectiveness of the proposed method is demonstrated by its application to a vehicle lateral dynamic nonlinear model. • Fault Tolerant Control using interval state estimation and robust stabilization. • Set-membership framework for the design of Linear Parameter Varying systems. • Fault estimation observer design for descriptor system. • L ∞ norm-based design method to attenuate the effect of fault and uncertainties and to improve the estimation performance.

Hidden and Coexisting Attractors in a Novel 4D Hyperchaotic System with No Equilibrium Point

The investigation of chaotic systems containing hidden and coexisting attractors has attracted extensive attention. This paper presents a four-dimensional (4D) novel hyperchaotic system, evolved by adding a linear state feedback controller to a 3D chaotic system with two stable node-focus points. The proposed system has no equilibrium point or two lines of equilibria, depending on the value of the constant term. Complex dynamical behaviors such as hidden chaotic and hyperchaotic attractors and five types of coexisting attractors of the simple 4D autonomous system are investigated and discussed, and are numerically verified by analyzing phase diagrams, Poincaré maps, the Lyapunov exponent spectrum, and its bifurcation diagram. The short unstable cycles in the hyperchaotic system are systematically explored via the variational method, and symbol codings of the cycles with four letters are realized based on the topological properties of the trajectory projection on the 2D phase space. The bifurcations of the cycles are explored through a homotopy evolution approach. Finally, the novel 4D system is implemented by an analog electronic circuit and is found to be consistent with the numerical simulation results.

Linear active disturbance rejection control for a dual unified power quality conditioner

In this paper, a linear active disturbance rejection control (LADRC) method is presented for a dual unified power quality conditioner (iUPQC). Compared with the traditional PI controller that requires double closed-loop control and decoupling design, the presented LADRC control method with Quasi-resonant (QR) feedforward has the benefits to simplify controller structure, and improve system steady-state, dynamic and anti-interference performance under the conditions with system uncertainty and external disturbances. By considering the unfavorable factors as control disturbances, the LADRC control method is able to obtain ideal performance through utilizing linear error state feedback (LESF) and dynamic disturbances compensation. In addition, in order to reduce DC link voltage fluctuation and eliminate double frequency ripple, a super capacitor is connected to DC link through a bidirectional DC/DC converter. By controlling the PWM signal of converter, the DC link capacitor can be charged or discharged to keep its voltage stable. Finally, the feasibility and effectiveness of this method are verified by MATLAB/SIMULINK simulation on a three-phase power distribution system.

Stabilization of the cart-inverted-pendulum system using trivial state-feedback to output-feedback control conversion

A new linear observer-free output-feedback controller with five adjustable parameters is proposed to stabilize the cart-inverted-pendulum system (CIP) at the unstable equilibrium point. The controller architecture is deduced from a trivial conversion of the linear state-feedback controller that is obtained using a two-step method. First, based on a set of cart change variables, a slightly modified state-feedback controller is developed. Then, the output-feedback controller is obtained through the judicious combination of the cart step reference input internal model and a convenient open-loop state estimator with the above modified state-feedback controller. The local stability of the output-based control system is conducted using the signature formulas method to get simplified conditions. A partial single parameter tuning method and optimal global single parameter tuning method are proposed for adjusting the controller gains to maximize a new efficiency-based objective function. Numerical simulations are first conducted to reveal the simplicity of output-feedback controller design using the partial tuning method, where the state-feedback gains are assumed to be known. Then, an optimal output-feedback controller is designed using the global tuning method. The proposed output-feedback controller is equivalent in terms of performance efficiency to the best five-parameter output-feedback two PID controller.

First Order Approximation of Model Predictive Control Solutions for High Frequency Feedback

The lack of computational power on mobile robots is a well-known challenge when it comes to implementing a real-time MPC scheme to perform complex motions. Currently the best solvers are barely able to reach 100 Hz for computing the control of a whole-body legged model, while modern robots are expecting new torque references in less than 1 ms. This problem is usually tackled by using a handcrafted low-level tracking control whose inputs are the low-frequency trajectory computed by the MPC. We show that a linear state feedback controller naturally arises from the optimal control formulation and can be used directly in the low-level control loop along with other sensitivities of relevant time-varying parameters of the problem. When the optimal control problem is solved by DDP, this linear controller can be computed for cheap as a by-product of the backward pass, and corresponds in part to the classical Riccati gains. A side effect of our proposition is to show that Riccati gains are valuable assets that must be used to achieve an efficient control and that they are not stiffer than the optimal control scheme itself. We propose a complete implementation of this idea on a full-scale humanoid robot and demonstrate its importance with real experiments on the robot Talos.

Spectral stabilization of linear transport equations with boundary and in-domain couplings

In this work, the problem of stabilization of general systems of linear transport equations with indomain and boundary couplings is investigated. It is proved that the unstable part of the spectrum is of finite cardinal. Then, using the pole placement theorem, a linear full state feedback controller is synthesized to stabilize the unstable finite dimensional part of the system. Finally, by a careful study of semigroups, we prove the exponential stability of the closed-loop system. As a by product, the linear control constructed before is saturated and a fine estimate of the basin of attraction is given.