Smooth State Feedback Control Research Articles

Smooth Time-Varying Feedback Control of Nonholonomic Systems with Applications to the Attitude Control of Underactuated Spacecraft

This paper studies the control of chained nonholonomic systems by using smooth time-varying feedback. First, by using a time-varying state transformation, the chained nonholonomic system is transformed into a linear time-varying system. Second, with the aid of some properties of a class of parametric Lyapunov equations and time-varying systems theory, smooth time-varying state feedback controller is developed to ensure the exponential convergence of both the system states and controls. Different from existing results, the design parameters in the proposed control strategy are independent of the initial conditions of the system. Third, as an application of the proposed smooth time-varying feedback, the attitude control problem for underactuated axisymmetric spacecraft is addressed. Finally, a numerical simulation is presented to validate the effectiveness of the proposed method.

H∞$$ {H}_{\infty } $$ bumpless transfer control of switched systems with time‐varying delays

AbstractThis paper investigates the bumpless transfer control (HBTC) problem of switched systems with time‐varying delays. A novel smooth switching state feedback controller is designed to eliminate the bumps existing in the classical switched state feedback control strategy. By the use of average dwell time (ADT) and multiple Lyapunov–Krasovskii functionals, some solvability conditions for the HBTC problem are derived. The control gain matrices can be calculated by solving linear matrix inequalities(LMIs). A numerical example is employed to show the validity of the proposed results.

Stabilization control of quadrotor helicopter through matching solution by controlled Lagrangian method

AbstractThis paper implements the controlled Lagrangian method on nonlinear quadrotor system with two degrees of underactuation. A Lagrangian system based on virtual angles is developed for the quadrotor. Based on the model, the matching conditions presented by nonlinear partial differential equations are explicitly solved for the nonlinear quadrotor system without decoupling or division. Besides, it is demonstrated that a constant controlled kinetic matrix could be utilized for energy shaping and simplify the matching conditions since gyroscopic force terms are omitted. Smooth state feedback control laws guaranteeing asymptotic stability of quadrotor system are obtained. Stability analysis and range of the control parameters are deduced based on the reshaped closed‐loop energy. Simulation for a quadrotor helicopter shows the feasibility and efficiency of the theoretical results.

Energy shaping control for systems with underactuation degrees two by controlled Lagrangian method

AbstractThis article investigates underactuated systems' energy shaping control by controlled Lagrangian (CL) method. The existence of CL based control depends on a set of matching conditions, which are usually in terms of partial differential equations (PDEs). In this work, the matching conditions are derived and simplified for underactuation‐degrees‐two systems. Moreover, the solvability conditions are provided for a class of systems satisfying Assumptions 1 and 2, based on which the controlled energy could be explicitly solved. By the proposed matching solution method, the desired equilibrium can be effectively assigned. The proposed matching controller is implemented for a strongly coupled single rigid body system with six degrees of freedom for the first time. Smooth state feedback control laws guaranteeing asymptotic stability of the system are presented. The range of the control parameters could be obtained based on the positive definiteness of reshaped closed‐loop energy. Simulations for comparison with relate work show the effectiveness of the theoretical results.

Output Feedback Stabilization for a Class of Uncertain High-Order Nonlinear Systems

We investigate output feedback stabilization for a class of high-order nonlinear systems whose output function and nonlinear terms are unknown. First, a smooth state feedback control law is designed by adding a power integrator technique. Next, we design a high-order observer to estimate the unmeasurable state, and allocate gains of the observer one by one in an iterative way. Finally, a dynamic output compensator is achieved such that the closed-loop system converges to the equilibrium point quick. Two examples are provided to demonstrate the effectiveness of the proposed method.

Controlling the variable length pendulum: Analysis and Lyapunov based design methods

The analysis and design of methods to damp the swing of the variable length pendulum by adjusting its length are presented here. To analyze the theoretical limits of such Coriolis force based damping, a comprehensive open-loop numerical analysis is performed for a two-dimensional model having the string length as the controlled input. Further, for a four dimensional model, having the force applied to the string as the controlled input, a smooth static state feedback controller is designed using backstepping. Results are verified both in simulations and through extensive laboratory experiments, and compared with previously published results achievable using an identical experimental setting.

Adaptive neural tracking control for high angle of attack maneuver with average dwell time

SummaryThis article attempts to study the high angle of attack maneuver from the perspective of switched system control. In view of the complex aerodynamic characteristics, an improved longitudinal attitude motion model is presented, which is a switched stochastic nonstrict feedback nonlinear system with distributed delays. The significant design difficulty is the completely unknown diffusion and drift terms and distributed delays with all state variables. Based on a technical lemma and neural networks, an improved smooth state feedback control law for nonstrict feedback systems is proposed without any growth assumptions. To eliminate the influence of distributed delays, an improved Lyapunov–Krasovskii function is constructed, which skillfully removes the constraint of the upper bound of the delay change rate. Then, by combining the average dwell‐time scheme and stochastic backstepping technique, an adaptive neural network tracking control law is designed, which extends a newly proposed switched system stability condition to the stochastic switched system. Theoretical analysis and flight control simulation experiments are provided to illustrate the effectiveness of the proposed control method.

Smooth State Feedback Control of a New Nonholonomic Manipulator Coping With Singularities

This paper presents a smooth state feedback control algorithm for set point stabilization of a three-link planar nonholonomic manipulator (NHM). Compared to known control algorithms for such a manipulator, the one presented here is able to control the NHM in its singular positions. In the control algorithm synthesis, we consider the kinematic model of the NHM. Our control algorithm solution is based on an extension of models state variables. Smooth point stabilization is achieved by implementing the transverse function approach. The control algorithm synthesis is performed with utilization of a Lie group representation. Because the Lie algebra associated with the NHM is not nilpotent, we apply a homogeneous nilpotent approximation. Analytical results have been reinforced by simulations and experiments, which were executed using a newly built three-link planar NHM. To the best of our knowledge, this manipulator is the third of its kind in the world, but due to its distinct kinematic structure, it is a unique mechanism.

Adaptive state feedback control for nonlinear systems with multiple interval time-varying delays and non-symmetric dead-zone input: A Lyapunov-Krasovskii approach

This paper addresses the stability analysis and adaptive state feedback control for a class of nonlinear discrete-time systems with multiple interval time-varying delays and non-symmetric dead-zone input. The multiple interval time-varying delays uncertainties are bounded by the nonlinear function with unknown coefficients. The non-symmetric dead-zone input without the knowledge of the dead-zone parameters is considered for the nonlinear system. The smooth adaptive state feedback controller is designed. By introducing the new Lyapunov-Krasovskii functional, it can be seen that the solutions of the closed-loop error system converge to an adjustable bounded region. In general, the proposed adaptive state feedback control strategy does not require the knowledge of maximum and minimum values for the characteristic slopes, and the knowledge of all system states. The control design conditions are relaxed because the system output is available for the controller design. Finally, three simulation examples are performed to show the effectiveness of the proposed methods.

A combined NN and dynamic gain-based approach to further stabilize nonlinear time-delay systems

This paper focuses on the state-feedback control problem for a class of high-order nonlinear systems with unknown time delay and control coefficients. Based on a novel dynamic gain-based backstepping technique and radial basis function neural network (RBF NN) approximation approach, the restrictions on high-order and nonlinearities are removed or further relaxed. Under these weaker conditions, a smooth state-feedback controller is skillfully constructed with only one adaptive parameter. In addition, the knowledge of time delay, NN nodes and weights is not necessary to be known a priori. It is proven that the designed controller can render the closed-loop system be semi-globally uniformly ultimately bounded. Finally, both practical and numerical examples are shown to demonstrate the effectiveness of the proposed scheme.

Smooth output feedback stabilization for a class of nonlinear systems with time‐varying powers

SummaryThis paper investigates the global asymptotic stabilization problem for a class of nonlinear systems with time‐varying powers. First, adding a power integrator technique is revamped to design a smooth state feedback controller, which is implementable with only upper and lower bounds of the time‐varying powers. When the system state is not fully available and the time‐varying power is exactly known, a smooth output feedback controller constituted by a state feedback and a nonlinear state observer is constructed to globally stabilize the system. Copyright © 2017 John Wiley & Sons, Ltd.

Smooth output feedback stabilization for nonlinear systems with time-varying powers

Abstract: This paper considers the global asymptotic stabilization problem for a class of nonlinear systems with time-varying powers. With only upper and lower bounds of the time-varying power, a smooth state feedback control law solving the problem is frst developed via the so-called adding a power integrator technique. When the system state is not fully available and the time-varying power is exactly known, a smooth output feedback control law constituted by a nonlinear state observer and a state feedback is constructed for achieving the stabilization task. In both cases, systematic approaches are explicitly presented for designing gains of control law and observer.

Smooth Switching Output Tracking Control for LPV Systems

This paper is concerned with the problem of smooth switching state feedback controller design for aircraft dynamic systems with multiple operating points. Based on the theory of robust control, a single state feedback controller which considers smooth switching is constructed. With the constant controller, the output response can be considerably improved in the switching process when the flight condition changes. An example for autopilot design of an F-18 aircraft is given to illustrate the effectiveness of the proposed approach.

Backstepping Robust H∞ Control for a Class of Uncertain Switched Nonlinear Systems Under Arbitrary Switchings

AbstractThis paper addresses global robust H ∞ control for a class of switched nonlinear systems with uncertainty under arbitrary switchings. Each subsystem is in lower triangular form. The uncertainties are assumed to be in a known compact set. The backstepping design technique is used to design a smooth state feedback controller that renders the associated closed‐loop switched system globally robustly asymptotically stable and imposes a pre‐specified upper bound to the L 2‐gain under arbitrary switchings. An example is provided to demonstrate the efficacy of the design approach.

Discrete-time control of chained non-holonomic systems

A discrete-time time-varying smooth state feedback controller is presented for the set point control of non-holonomic chained systems. The control law drives the system state to zero with an exponential convergent rate without any assumption on the initial state and the sampling rate. Specifically, the discretised model is broken up into two subsystems, and the first control input is explicitly designed such that the second subsystem is transformed into a perturbed linear time-varying system by introducing a novel state transformation, for which classical linear control techniques can be easily adopted to drive the state to the origin. Simulation results are provided to validate the presented approach.

On the Stabilization of Nonholonomic Mechanical Systems via Immersion and Invariance

AbstractThis work presents some results on the application of the Immersion & Invariance (I&I) approach proposed for stabilization of nonlinear systems to the case of mechanical systems with kinematic constraints. We consider only smooth state–feedback control laws and thus, by Brockett's necessary condition, restrict our attention to deriving an I&I controller that achieves smooth stabilization of an equilibrium manifold. As is well–known, the design of I&I controllers requires the solution of some partial differential equations. A second contribution of our work is to obviate this obstacle, giving explicit control laws for a class of nonholonomic mechanical systems, which include the knife-edge and the rolling wheel. The approach is illustrated with the well-known knife-edge example.

Output Tracking of High-Order Stochastic Nonlinear Systems with Application to Benchmark Mechanical System

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This note considers output tracking of high-order stochastic nonlinear systems without imposing any restriction on the high-order and the drift and diffusion terms. By using the backstepping design technique, a smooth state-feedback controller is given to guarantee that the solution process is bounded in probability and the error signal between the output and the reference signal can be regulated into a small neighborhood of the origin in probability. A practical example of stochastic benchmark mechanical system and simulation are provided to demonstrate the effectiveness of the control scheme. </para>

State-feedback stabilization for stochastic high-order nonlinear systems with a ratio of odd integers power

This paper investigates the problem of globally asymptotically stable in probability by state-feedback for a class of stochastic high-order nonlinear systems with a ratio of odd integers power. By extending the adding a power integrator technique and choosing an appropriate Lyapunov function, a linear smooth state-feedback controller is explicitly constructed to render the system globally asymptotically stable in probability. Furthermore, we address the problem of state-feedback inverse optimal stabilization in probability. A simulation example is provided to show the effectiveness of the proposed approach.

Controller design for 2-DOF underactuated mechanical systems based on controlled Lagrangians and application to Acrobot control

On the basis of controlled Lagrangians, a controller design is proposed for underactuated mechanical systems with two degrees of freedom. A new kinetic energy equation (K-equation) independent of the gyroscopic forces is found due to the use of their property. As a result, the necessary and sufficient matching condition comprises the new K-equation and the potential energy equation (P-equation) cascaded, the regular condition, and the explicit gyroscopic forces. Further, for two classes of input decoupled systems that cover the main benchmark examples, the new K-equation, respectively, degenerates from a quasilinear partial differential equation (PDE) into an ordinary differential equation (ODE) under some choice and into a homogeneous linear PDE with two kinds of explicit general solutions. Benefiting from one of the general solutions, the obtained smooth state feedback controller for the Acrobots is of a more general form. Specifically, a constant fixed in a related paper by the system parameters is converted into a controller parameter ranging over an open interval along with some new nonlinear terms involved. Unlike what is mentioned in the related paper, some categories of the Acrobots cannot be stabilized with the existing interconnection and damping assignment passivity based control (IDA-PBC) method. As a contribution, the system can be locally asymptotically stabilized by the selection of the new controller parameter except for only one special case.

Controller design for mechanical systems with underactuation degree one based on controlled Lagrangians method

This article extends controller design based on the controlled Lagrangians method from two to n degrees of freedom (DOF) mechanical systems with underactuation degree one. Importantly, a new kinetic energy equation (K-equation) is found from which the gyroscopic forces are separated due to the use of their property. Further, along with them chosen as an explicit solution, the other K-equations are satisfied for any regular controlled kinetic energy. As our main contribution, a sufficient matching condition is obtained which comprises the new K-equation and one P-equation (potential energy equation) cascaded, the regular condition and the explicit gyroscopic forces. Accordingly, the matching condition has an advantage for the fewest partial differential equations (PDEs) to be solved and the explicit gyroscopic forces. With all unknown functions of the controlled kinetic energy, the new K-equation can be simplified further under some choices on these functions and/or assumptions on specific original systems. Finally, taking the advantage, we obtain a nonlinear smooth state feedback control law that achieves local asymptotic stabilisation for a class of 3-DOF Pendubots in a vertical plane.