Output feedback stabilization of the angular velocity of a rigid body

The problem of stabilization of the angular velocity of a rigid body using only two control signals and partial state information is addressed. It is shown that if any two (out of three) states are measured the system is not asymptotically stabilizable with (continuous) dynamic output feedback. Nevertheless, we prove that practical stability is achievable if the measurable states fulfill a certain structural property, and that, under the same structural condition, a hybrid control law yielding exponential convergence can be constructed. Finally, we also study some geometric features of the Euler’s equations and the connection between local strong accessibility and local observability.     

Robust output feedback stabilization of the angular velocity of a rigid body

The problem of semi-global stabilization of the angular velocity of an underactuated rigid body in the presence of model errors is addressed and solved using a smooth, time-varying, dynamic, output feedback control law. The proposed scheme improves some of the existing results and provides guidelines for a general stabilization strategy applicable to systems which are not exponentially stabilizable. Simulations results are included.     

Global feedback stabilization of the angular velocity of a symmetric rigid body

In this paper we prove that the angular velocity equations of a symmetric rigid body can be globally stabilized by means of linear feedback when two control torques act on the body.     

Stabilizability of the angular velocity of a rigid body revisited

We prove that the jurdjevic-Quinn techniques for smooth stabilization can be applied to the equation of the angular velocity of a rigid body. An explicit polynomial control is given for the symmetric case.     

Asymptotic stabilization of the position of a rigid body with fixed point without velocity measurements

The problem of stabilization of a rigid body with a fixed point in reference to its velocity and position is considered. It is assumed that the rigid body is controlled by three external torques. The proposed controlling torques need neither the measurements of the velocity nor the information about the inertial parameters. Such a control may be realized by the on-line solution of the auxiliary differential equations.The stability of the closed loop nonlinear system is established by means of the Lyapunov function and Barbashin's theorem. The proposed control may be applied to stabilize the satellites controlled by the reactive jets.     

角速度约束下的刚体飞行器鲁棒有限时间姿态镇定

针对角速度状态受限条件下的刚体飞行器姿态镇定控制问题,提出一种基于扰动观测器的时变状态增益有限时间姿态控制方法.针对基于修正型罗德里格斯参数(MRPs)描述的刚体飞行器姿态控制模型,首先,利用齐次性理论并充分考虑到系统的模型结构特点,设计一种带有角速度约束项的有限时间姿态控制器,使得系统有限时间镇定;其次,在初始状态满足受限条件的情况下,角速度在任意时刻都可以被约束在期望的范围内;然后,针对存在外部干扰的姿态环动力学系统,提出一种带扰动估计补偿的复合有限时间姿态镇定控制器;最后,通过与其他两种控制方法的仿真比较,验证了所提出控制方案的有效性和优越性.     

Robust stabilization of nonlinear cascaded systems

This paper considers the problem of robust stabilization for an uncertain nonlinear system which is a cascaded interconnection of two subsystems. Both of the subsystems are allowed to be nonlinear, multi-variable, and containing uncertain parameters. We present a new approach to designing stabilizing controllers which assure both robust global asymptotic stability and local quadratic stability. Compared with existing results, the assumptions required for such robust stabilizing controllers to exist are significantly simplified.     

Robust stabilization of singularly perturbed nonlinear systems

Consider a singularly perturbed nonlinear system endowed with a control input and suppose a static nonlinear feedback law is designed. We show that the operations ‘compute the reduced order system’ (i.e., let the singular perturbation parameter μ = 0) and ‘close the feedback loop’ commute, i.e. the closed loop reduced-order system is unambiguously determined. We then show that, if the reduced order system associated with the original system is stabilizable or has uncertainties matched with the input (a condition frequently used in the design of robust controllers), then the closed loop reduced-order system enjoys the same property. As shown by an example, this result can be used to simplify the structure of a composite controller.     

Stabilization by smooth feedback of the angular velocity of a rigid body

Stabilization by smooth feedback of the null solution of Euler's angular velocity equations is investigated. Assume there is one external torque, aligned with a principal axis. The main result is that there exists a stabilizing smooth feedback law if the moment of inertia of the rigid body along that principal axis is either larger or smaller than the remaining two. The feedback law is robust with respect to changes in the parameters defining the control law.     

Vision-based control for rigid body stabilization

This paper addresses the problem of stabilizing to a desired equilibrium point an eye-in-hand system, which consists of a single camera mounted on a rigid body free to move on . It is assumed that there is a collection of landmarks fixed in the environment and that the image coordinates of those landmarks are provided to the system by an on-board CCD camera. The proposed method addresses not only the problem of stabilization but also that of maintaining feature visibility along the system’s trajectory. The resulting solution consists of a feedback control law based on the current and desired image coordinates and reconstructed attitude and depth ratio information, which guarantees that (i) the desired equilibrium point is an almost global attractor; (ii) a set of necessary conditions for feature visibility holds throughout the system’s trajectories; and (iii) the image of a predefined feature point is kept inside the camera’s field of view.     

Comments on the stabilizability of the angular velocity of a rigid body

This paper discusses the stabilizability by smooth feedback of the angular velocity of a rigid body. First we show how to asymptotically stabilize the system with a single control. Then it is shown that a single control aligned with a principal axis cannot asymptotically stabilize the system. Finally we present some new results on stabilization by means of scalar feedback control.     

Global asymptotic stabilization of the attitude and the angular rates of an underactuated non-symmetric rigid body

The paper deals with the global stabilization of both the attitude and the angular velocities of an underactuated rigid body. First a stability theorem is proven for a class of systems; subsequently, the equations describing the physics of the rigid body are presented, showing that the rigid body belongs to the considered class of systems, and a sufficient condition for the application of the theorem to the stability of the rigid body equilibrium is pointed out. Finally, some simulation results are reported showing the effectiveness of the proposed methodology.     

Robust output feedback stabilization via risk-sensitive control

We consider a problem of robust linear quadratic Gaussian (LQG) control for discrete-time stochastic uncertain systems with partial state measurements. For a finite-horizon case, the problem was recently introduced by Petersen et al. (IEEE Trans. Automat. Control 45 (2000) 398). In this paper, an infinite horizon extension of the results of Petersen et al. (IEEE Trans. Automat. Control 45 (2000) 398) is discussed. We show that for a broad class of uncertain systems under consideration, a controller constructed in terms of the solution to a specially parameterized risk-sensitive stochastic control problem absolutely stabilizes the stochastic uncertain system.     

Robust stabilization of the current profile in tokamak plasmas using sliding mode approach in infinite dimension

This paper deals with the robust stabilization of the spatial distribution of tokamak plasmas current profile using a sliding mode feedback control approach. The control design is based on the 1D resistive diffusion equation of the magnetic flux that governs the plasma current profile evolution. The feedback control law is derived in the infinite dimensional setting without spatial discretization. Numerical simulations are provided and the tuning of the controller parameters that would reject uncertain perturbations is discussed. Closed loop simulations performed on realistic test cases using a physics based tokamak integrated simulator confirm the relevance of the proposed control algorithm in view of practical implementation.     

Robust output regulation with nonlinear exosystems

For over a decade, the solvability of the nonlinear robust output regulation problem relies on the assumption that the exosystem is linear and neutrally stable. Thus, the only exogenous signal that can be accommodated by the existing theory is a combination of finitely many step functions and sinusoidal functions. In this paper, we will show that it is possible to find controllers that can admit exogenous signals produced by nonlinear exosystems. An example with the well known van der Pol oscillator as the exosystem is given to illustrate our approach.     

Robust quantized feedback stabilization of linear systems

This paper investigates the feedback stabilization problem for SISO linear uncertain control systems with saturating quantized measurements. In the fixed quantization sensitivity framework, we propose a time varying control law able to effectively account for the presence of saturation, which is often the main source of instability, designed using sliding mode techniques. Such controller is proved able to stabilize the plant both in the presence and in the absence of quantization.     

On global stabilization of Burgers’ equation by boundary control

Although often referred to as a one-dimensional “cartoon” of Navier–Stokes equation because it does not exhibit turbulence, the Burgers equation is a natural first step towards developing methods for control of flows. Recent references include Burns and Kang [Nonlinear Dynamics 2 (1991) 235–262], Choi et al. [J. Fluid Mech. 253 (1993) 509–543], Ito and Kang [SIAM J. Control Optim. 32 (1994) 831–854], Ito and Yan [J. Math. Anal. Appl. 227 (1998) 271–299], Byrnes et al. [J. Dynam. Control Systems 4 (1998) 457–519] and Van Ly et al. [Numer. Funct. Anal. Optim. 18 (1997) 143–188]. While these papers have achieved tremendous progress in local stabilization and global analysis of attractors, the problem of global asymptotic stabilization has remained open. This problem is non-trivial because for large initial conditions the quadratic (convective) term – which is negligible in a linear/local analysis – dominates the dynamics. We derive nonlinear boundary control laws that achieve global asymptotic stability. We consider both the viscous and the inviscid Burgers’ equation, using both Neumann and Dirichlet boundary control. We also study the case where the viscosity parameter is uncertain, as well as the case of stochastic Burgers’ equation. For some of the control laws that would require the measurement in the interior of the domain, we develop the observer-based versions.     

A control energy approach for the stabilization of the rigid beam balanced by the cart

In this paper we present a control strategy for the stabilization of a rigid beam balanced by a cart. The control problem consists of horizontally balancing the beam by moving the cart backwards and forwards along the beam. The cart movement is restricted to always remain over the beam. The control strategy is based on the interconnection and damping assignment passivity based controller. The stability analysis is carried out using LaSalles's theorem. Copyright © 2008 John Wiley & Sons, Ltd.     

Robust PI stabilization of a class of chemical reactors

The aim of this letter is to study the stabilization of a class of continuous-stirred tank reactors in the face of control input saturations and uncertain chemical kinetics. A first-order compensator with modeling error compensation is designed, and its ability to stabilize the reactor temperature at an arbitrary setpoint is proven. It is shown that the resulting controller is equivalent to a standard PI compensator with antireset windup scheme.