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.     

具有无穷时滞的不确定大系统的鲁棒镇定

研究了具有无穷时滞的不确定微分－积分大系统的鲁棒稳定性问题，对每个子系统的应用稳定的局部状态反馈，利用Lyapunov函数法并结合不等式分析技巧，给出了具有无穷时滞的不确定微分－积分大系统的鲁棒全局一致近稳定的充分条件。     

Nonlinear time-varying controllers for robust stabilization

In this paper we consider the problem of robust stabilizallon for a certain class of plants with unstructured, infinite-dimensional uncertainty. We demonstrate that for the problem of robustly stabilizing this class of plants, linear time-invariant controllers perform as well as nonlinear time-varying (NLTV) ones. Ihis, in particular, implies that adaptive control laws offer no advantage as far as the problem of robust stabilization of this class of plants is concerned. As a corollary we demonstrate that the small-gain theorem is both necessary and sufficient for a certain class of NLTV operators.     

Topological feedback entropy and Nonlinear stabilization

It is well known in the field of dynamical systems that entropy can be defined rigorously for completely deterministic open-loop systems. However, such definitions have found limited application in engineering, unlike Shannon's statistical entropy. In this paper, it is shown that the problem of communication-limited stabilization is related to the concept of topological entropy, introduced by Adler et al. as a measure of the information rate of a continuous map on a compact topological space. Using similar open cover techniques, the notion of topological feedback entropy (TFE) is defined in this paper and proposed as a measure of the inherent rate at which a map on a noncompact topological space with inputs generates stability information. It is then proven that a topological dynamical plant can be stabilized into a compact set if and only if the data rate in the feedback loop exceeds the TFE of the plant on the set. By taking appropriate limits in a metric space, the concept of local TFE (LTFE) is defined at fixed points of the plant, and it is shown that the plant is locally uniformly asymptotically stabilizable to a fixed point if and only if the data rate exceeds the plant LTFE at the fixed point. For continuously differentiable plants in Euclidean space, real Jordan forms and volume partitioning arguments are then used to derive an expression for LTFE in terms of the unstable eigenvalues of the fixed point Jacobian.     

Nonlinear feedback control with global stabilization

Hamilton-Jacobi-Bellman theory is shown to provide a unified framework for nonlinear feedback control laws for special classes of nonlinear systems. These classes include Jurdjevic-Quinn type systems, as well as minimum phase systems with relative degree {1, 1, ..., 1}. Several examples are given to illustrate these results. For the controlled Lorenz equation, results obtained by Vincent and Yu are extended. Next, for spacecraft angular velocity stabilization with two torque inputs, a family of nonlinear feedback control laws that globally asymptotically stabilize angular velocity is established. Special cases of this family of control laws include generalizations of the locally stabilizing control laws of Brockett and Aeyels to global stabilization as well as the globally stabilizing control laws of Irving and Crouch and Byrnes and Isidori. Finally, the results are applied to spacecraft angular velocity stabilization with only one torque input. These last results extend control laws given by Outbib and Sallet.     

Robust stabilization of infinite dimensional systems by finite dimensional controllers

The problem of robustly stabilizing an infinite dimensional system with transfer function G, subject to an additive perturbation Δ is considered. It is assumed that: G ε ₀(σ) of systems introduced by Callier and Desoer [3]; the perturbation satisfies |W₁ΔW₂| < ε, where W₁ and W₂ are stable and minimum phase; and G and G + Δ have the same number of poles in ⁺. Now write W₁GW₂=G₁ + G₁, where G₁ is rational and totally unstable and G₂ is stable. Generalizing the finite dimensional results of Glover [12] this family of perturbed systems is shown to be stabilizable if and only if ε σ_min (G^*₁)( = the smallest Hankel singular value of G^*₁). A finite dimensional stabilizing controller is then given by where 2 is a rational approximation of G₂ such that

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) and K₁ robustly stabilizes G₁ to margin ε. The feedback system (G, K) will then be stable if |W₁ΔW₂| _∞< ε − Δ.     

Nonlinear system stabilization via hierarchical switching control

A nonlinear control system design framework predicated on a hierarchical switching controller architecture parameterized over a set of moving system equilibria is developed. Specifically, using equilibria-dependent Lyapunov functions, a hierarchical nonlinear control strategy is developed that stabilizes a given nonlinear system by stabilizing a collection of nonlinear controlled subsystems. The switching nonlinear controller architecture is designed based on a generalized lower semicontinuous Lyapunov function obtained by minimizing a potential function over a given switching set induced by the parameterized system equilibria. The proposed framework provides a rigorous alternative to designing gain-scheduled feedback controllers and guarantees local and global closed-loop system stability for general nonlinear systems     

Strong stabilization with infinite multivariable gain margin through linear periodic control

The problem of the strong stabilization with infinite gain margin, with the additional requirement of a prescribed rate of convergence of the free responses, is addressed for linear time-invariant discrete-time multivariable plants in the case when unknown different scalar gains act either on the inputs or on the outputs. Necessary and sufficient conditions for the solvability of the problem by means of a stable linear periodic discrete-time output feedback dynamic controller are derived. Algorithmic procedures are given for designing the proposed periodic controllers.     

Nonlinear robust stabilization of multi-parameter families of systems

We show that given any family of asymptotically stabilizable LTI systems depending continuously on a parameter that lies in some subset [a₁,b₁]××[a_p,b_p] of , there exists a C⁰ time-varying state feedback law v(t,x) (resp. a C⁰ time-invariant feedback law v(x)) which robustly globally exponentially stabilizes (resp. which robustly stabilizes, not asymptotically) the family. Further, if these systems are obtained by linearizing some nonlinear systems, then v(t,x) locally exponentially stabilizes these nonlinear systems. Finally, v(t,x) globally exponentially stabilizes any time-varying system which switches “slowly enough” between the given LTI systems.     

Nonlinear boundary stabilization of parallelly connected kirchoff plates

A system of nonlinearly coupled Kirchhoff plates is considered. It is shown that by applying nonlinear issipation on the boundary, the energy of the system decays to zero at a uniform rate.     

Nonlinear stochastic passivity,feedback equivalence and global stabilization

This paper addresses several important issues including stochastic passivity, feedback equivalence and global stabilization for a class of nonlinear stochastic systems. Based on a nonlinear stochastic Kalman–Yacubovitch–Popov (KYP) lemma, we investigate the relationship between a stochastic passive system and the corresponding zero‐output system. Different from the deterministic case, it is shown for the first time that feedback equivalence to a stochastic passive system requires a strong minimum‐phase condition, not the minimum‐phase one. Following the stochastic passivity theory, global stabilization results are established for a class of nonlinear stochastic systems with relative degree 1≤r<n. An example is presented to illustrate the effectiveness of our results. Copyright © 2011 John Wiley & Sons, Ltd.     

挠性卫星姿态非线性局部镇定控制

本文针对挠性卫星姿态机动和振动抑制问题,给出一种基于多项式平方和(sum of squares,SOS)的非线性局部镇定控制方法.根据姿态系统结构特征,在此基础上,采用SOS结合S-procedure理论,得出相应的非线性局部可镇定条件.该条件可借助有效凸优化工具进行检验,当优化问题可解时,可构造非线性姿态控制器的解析解.最后,将文中方法应用于某型挠性卫星姿态控制.仿真结果表明,在实现大角度姿态快速机动的同时,有效抑制了挠性附件振动.     

Nonlinear adaptive decentralized stabilization control for multimachine power systems

This paper presents a decentralized adaptive backstepping controller to dampen oscillations and improve the transient stability to parametric uncertainties in multimachine power systems. The proposed design on the i ^th synchronous generator uses only local information and operates without the need for remote signals from the other generators. The design of the nonlinear controller is based on a modified fourth-order nonlinear model of a synchronous generator, and the automatic voltage regulator model is considered so as to decrease the steady state voltage error. The construction of both the control law and the associated Lyapunov function is systematically designed within the design methodology. A 3-machine power system is used to demonstrate the effectiveness of the proposed controller over two other controllers, namely a conventional damping controller (power system stabilizer) and one designed using the feedback linearization techniques. Recommended by Editorial Board member Gang Tao under the direction of Editor Jae Weon Choi. This work was supported by the Korea Electrical Engineering and Science Research Institute, which is funded by Ministry of Commerce, Industry and Energy. Shan-Ying Li received the B.S. degrees in Computer Science and M.S. degree in Electrical Engineering from Northeast DianLi University, China, in 1997 and 2002, respectively. She obtained the Ph.D. degree in Electrical Engineering from Seoul National University, Korea, in 2008. She is a Post Doctor in North China Electric Power Research Institute, North China Grid Co., Ltd., China. Her research interests are in the areas of advanced control and stability applications on power systems. Sang-Seung Lee received the M.S.E.E. and Ph.D. degrees in Electrical Engineering at Seoul National University. Currently, he is with Power System Research Division of KESRI, Seoul National University, Korea. His interest areas are nonlinear/adaptive control theory, North-East Asia power system interconnection, distributed/small generation, distributed transmission/distribution load flow algorithm, regional/local energy system, PSS (power system stabilizer), and RCM (Reliability Centered Maintenance). Yong Tae Yoon was born in Korea on April 20, 1971. He received the B.S. degree, M.Eng. and Ph.D. degrees from M.I.T., USA in 1995, 1997, and 2001, respectively. Currently, he is an Assistant Professor in the School of Electrical Engineering and Computer Science at Seoul National University, Korea. His special field of interest includes electric power network economics, power system reliability, and the incentive regulation of independent transmission companies. Jong-Keun Park received the B.S. degree in Electrical Engineering from Seoul National University, Seoul, Korea in 1973 and the M.S. and Ph.D. degrees in Electrical Engineering from The University of Tokyo, Japan in 1979 and 1982, respectively. He is currently a Professor of School of Electrical Engineering, Seoul National University. In 1992, he attended as a Visiting Professor at Technology and Policy Program and Laboratory for Electromagnetic and Electronic Systems, Massachusetts Institute of Technology. He is a Senior Member of the IEEE, a Fellow of the IEE, and a Member of Japan Institute of Electrical Engineers (JIEE).     

An example for the switching delay feedback stabilization of an infinite dimensional system: The boundary stabilization of a string

For vibrating systems, a delay in the application of a feedback control may destroy the stabilizing effect of the control. In this paper we consider a vibrating string that is fixed at one end and stabilized with a boundary feedback with delay at the other end.We show that certain delays in the boundary feedback preserve the exponential stability of the system. In particular, we show that the system is exponentially stable with delays freely switching between the values 4L/c and 8L/c, where L is the length of the string and c is the wave speed.     

Stability and stabilization for discrete-time singular systems with infinite distributed delays and actuator failures

In this paper, a reliable H_?? controller is designed for discrete-time singular systems with infinite distributed delays. Firstly, a stability criterion is obtained based on linear matrix inequality (LMI). Then a controller is designed to ensure the singular time-delay system to be regular, causal and stable with disturbance attenuation level ?? for possible actuator failures. A numerical example is given to illustrate the validity of the proposed approach.     

Nonlinear stabilization of a thermal convection loop by state feedback

A nonlinear feedback control law that achieves global asymptotic stabilization of a 2D thermal convection loop (widely known for its “Lorenz system” approximation) is presented. The loop consists of viscous Newtonian fluid contained in between two concentric cylinders standing in a vertical plane. The lower half of the loop is heated while the upper half is cooled, which makes the no-motion steady state for the uncontrolled case unstable for values of the non-dimensional Rayleigh number R_a>1. The objective is to stabilize that steady state using boundary control of velocity and temperature on the outer cylinder. We discretize the original nonlinear PDE model in space using finite difference method and get a high order system of coupled nonlinear ODEs in 2D. Then, using backstepping design, we transform the original coupled system into two uncoupled systems that are asymptotically stable in l²-norm with homogeneous Dirichlet boundary conditions. The resulting boundary controls actuate velocity and temperature in the original coordinates. The control design is accompanied by an extensive simulation study which shows that the feedback control law designed on a very coarse grid (using just a few measurements of the flow and temperature fields) can successfully stabilize the actual system for a very wide range of the Rayleigh number.     

Nonlinear and hysteretic influence of piezoelectric actuators in AFMs on lateral dimension measurement

Piezoelectric actuators that are used in atomic force microscopes (AFM) have undesirable properties. The nonlinear and hysteretic characteristics of piezoelectric actuators introduce geometric deformations in the reconstructed AFM images. Due to these deformations, the quantitative interpretation of the absolute dimensions of surface features is difficult and often not accurate.A real-time measuring ‘Nano-metrological Atomic Force Microscope’ system equipped with an ultra-high resolution three-axis laser interferometer system is developed, in which the undesirable properties of piezoelectric actuators are compensated completely. Using this AFM and a one-dimensional (1D) grating reference standard with pitches of 240 nm, which is one of the widely used reference standards as nano-metrological lateral scales, the influences of nonlinear and hysteretic characteristics of piezoelectric actuators on image reconstruction and lateral dimension measurement are examined and compared quantitatively among three different measurement methods. The three measurement methods are: (1) the relative movement between probe tip and sample is controlled and measured directly by voltage signals applied on the XYZ scanner, the nonlinear and hysteretic characteristics of piezoelectric actuators are not compensated; (2) the relative movement between probe tip and sample is controlled by voltage signals applied on the XYZ scanner, but it is measured accurately by interferometers; (3) the relative movement between probe tip and sample in lateral directions are both controlled and measured accurately by interferometers. According to the comparison results, an accurate displacement control system is key to reduce the influences of undesirable properties of piezoelectric actuators and the developed AFM system with three-axis laser interferometer system is proved to eliminate the nonlinear and hysteretic characteristics of piezoelectric actuators completely.     

Nonlinear ultimate boundedness control and stabilization of a flexible robotic arm

We treat the question of control of an elastic robotic arm of two links. Of course, this approach can be extended to other elastic arms. A nonlinear ultimate boundedness controller (UBC) is synthesized such that in the closed-loop system the joint angle tracking error is uniformly bounded and tends to a certain small neighborhood of the origin. The controller includes a reference joint angle trajectory generator and integral error feedback. Although the joint angles are controlled using the UBC, elastic modes are excited. A feedback stabilizer is designed for the linearized model including the UBC about the terminal state that is switched only in the vicinity of the equilibrium state for stabilization. Simulation results are presented to show that in the closed-loop system including the UBC and the stabilizer accurate joint angle trajectory following and elastic mode stabilization are accomplished in the presence of uncertainty.     

Nonlinear versus linear control in the direct output feedback stabilization of linear systems

This paper investigates the use of nonlinear direct output feedback in the control of linear time-invariant systems. Inparticular, it is concerned with those nonlinear controllers which result in a closed loop system which is quadratically stable. That is, the closed loop nonlinear system is asymptotically stable and furthermore, a quadratic Lyapunov function can be used to establish this stability. The main result of this paper can be stated roughly as follows: Any linear system which can be made quadratically stable using nonlinear direct output feedback control can also be stabilized using linear direct output feedback control.     

Nonlinear control for global stabilization of multiple-integrator system by bounded controls

The global stabilization problem of the multiple-integrator system by bounded controls is considered. A nonlinear feedback law consisting of nested saturation functions is proposed. This type of nonlinear feedback law that is a modification and generalization of the result given in [1] needs only [（n ＋ 1）/2] （n is the dimensions of the system） saturation elements, which is fewer than that which the other nonlinear laws need. Furthermore, the poles of the closedloop system can be placed on any location on the left real axis when none of the saturation elements in the control laws is saturated. This type of nonlinear control law exhibits a simpler structure and can significantly improve the transient performances of the closed-loop system, and is very superior to the other existing methods. Simulation on a fourth-order system is used to validate the proposed method.