Robust generation of almost‐periodic oscillations in a class of nonlinear systems

This paper deals with the problem of generating, by state feedback, stable oscillations in high order nonlinear systems. The desired oscillations are robust in the presence of disturbances, such as unmodelled dynamics and bounded noise signals, which result in bounded deviations from the nominal target orbit. The method consists of two steps. First, a globally attractive oscillation is generated in a nominal second‐order subsystem. Based on a partition of the state space and solving the Lyapunov equation on each part, a strict Lyapunov function is obtained that ensures exponential convergence, even in the presence of disturbances, to a ring‐shaped region containing the target limit cycle. Then, the oscillation stabilizing controller and the strict Lyapunov function are extended to arbitrary order systems, via backstepping. Notwithstanding backstepping is intended for cascade systems, the acquired ability to deal with unmodelled dynamics permits the analytical treatment of non‐triangular structures, as is illustrated with the Ball and Beam example. Copyright © 2006 John Wiley & Sons, Ltd.     

Global complete observability and output-to-state stability imply the existence of a globally convergent observer

We consider systems which are globally completely observable and output-to-state stable. The former property guarantees the existence of coordinates such that the dynamics can be expressed in observability form. The latter property guarantees the existence of a state norm observer and therefore the possibility of bounding any continuous state functions. Both properties allow to conceptually build an observer from an approximation of an exponentially attractive invariant manifold in the space of the system state and an output driven dynamic extension. The proposed observer provides convergence to zero of the estimation error within the domain of definition of the solutions. The work of A. Astolfi is partly supported by the Leverhulme Trust.     

有未知控制系数和零动态的高阶非线性系统的自适应控制设计

主要研究一类控制系数未知和有不可测零动态的高阶非线性系统的全局自适应镇定问题. 进一步放宽了对零动态的约束条件, 通过定义一个恰当的未知参数, 把连续自适应状态反馈控制器的动态阶数降到最低(仅一维). 通过结合增加幂次积分方法, 相关的自适应技术以及交换能量函数思想, 给出控制器的设计步骤. 所设计的控制器确保闭环系统的所有状态是全局一致有界的, 且原系统的状态收敛到零.     

Control of the Underactuated Inertia Wheel Inverted Pendulum for Stable Limit Cycle Generation

This paper deals with a control approach dedicated to stable limit cycle generation for underactuated mechanical systems. The proposed approach is based on partial nonlinear feedback linearization and dynamic control for optimal periodic reference trajectories tracking. The computation of the reference trajectories is performed in order to optimize the behavior of the whole dynamics of the system and especially its zero dynamics at the end of each cycle. Simulation results as well as experiments show the performance and the efficiency of the proposed control scheme.     

Limit cycle-based exact estimation of FOPDT process parameters under input/output disturbances: a state-space approach

A set of novel explicit expressions for the identification of stable, unstable and integrating first-order plus dead time process dynamics is presented. In the absence of sensor noise/static load disturbances, an autonomous relay control system with an asymmetrical relay induces a smooth limit cycle at the process output. Dynamic model parameters are estimated using the set of proposed expressions which depend on the parameters of limit cycle output and its derivatives. However, in practice, the process output is generally corrupted by the measurement noise, thereby rendering an erroneous identification of process dynamics. Furthermore, static load disturbance during an identification test also induces an asymmetrical limit cycle output resulting in inaccurate measurements. Hence, a fast Fourier transform technique and biased relay feedback methods are implemented to obviate the problem of asymmetries and chattering in the limit cycle output yielding the original limit cycle and its subsequent derivatives. The proposed method has been validated, considering five typical examples from literature. An extensive comparison study with existing approaches based on Nyquist plots demonstrate the efficacy of the method presented.     

Global stability of relay feedback systems

For a large class of relay feedback systems (RFS) there will be limit cycle oscillations. Conditions to check existence and local stability of limit cycles for these systems are well known. Global stability conditions, however, are practically nonexistent. The paper presents conditions in the form of linear matrix inequalities (LMIs) that, when satisfied, guarantee global asymptotic stability of limit cycles induced by relays with hysteresis in feedback with linear time-invariant (LTI) stable systems. The analysis consists in finding quadratic surface Lyapunov functions for Poincare maps associated with RFS. These results are based on the discovery that a typical Poincare map induced by an LTI flow between two hyperplanes can be represented as a linear transformation analytically parametrized by a scalar function of the state. Moreover, level sets of this function are convex subsets of linear manifolds. The search for quadratic Lyapunov functions on switching surfaces is done by solving a set of LMIs. Although this analysis methodology yields only a sufficient criterion of stability, it has proved very successful in globally analyzing a large number of examples with a unique locally stable symmetric unimodal limit cycle. In fact, it is still an open problem whether there exists an example with a globally stable symmetric unimodal limit cycle that could not be successfully analyzed with this new methodology. Examples analyzed include minimum-phase systems, systems of relative degree larger than one, and of high dimension. Such results lead us to believe that globally stable limit cycles of RFS frequently have quadratic surface Lyapunov functions     

Stabilization of a relay system containing three-position element and dead time†

An exact method of analysis of non-linear systems containing a three-position relay element and a dead-time element is given. Such a system shows two limit cycles, an unstable limit cycle and a stable limit cycle. Expressions are derived to evaluate the period and amplitude of the oscillations.

Using the phase-plane concept it is shown that the given system can be stabilized by adding a zero to the second-order transfer function of the rational part.     

Adaptive inverse dynamics control of robots with uncertain kinematics and dynamics

It has been about two decades since the first globally convergent adaptive tracking controller was derived for robots with dynamic uncertainties. However, not until recently has the problem of concurrent adaptation to both the kinematic and dynamic uncertainties found its solution. This adaptive controller belongs to passivity-based control. Though passivity-based controllers have many attractive properties, in general, they are not able to guarantee the uniform performance of the robot over the entire workspace. Even in the ideal case of perfect knowledge of the manipulator parameters, the closed-loop system remains nonlinear and coupled. Thus the closed-loop tracking performance is difficult to quantify, while the inverse dynamics controllers can overcome these deficiencies. Therefore, in this work, we will develop a new adaptive Jacobian tracking controller based on the inverse manipulator dynamics. Using the Lyapunov approach, we have proved that the end-effector motion tracking errors converge asymptotically to zero. Simulation results are presented to show the performance of the proposed controller.     

New results and examples in nonlinear feedback stabilization

Using the general methodology of nonlinear zero dynamics we derive globally stabilizing state feedback laws for broad classes of nonlinear systems. While it is tempting to reinterpret this design philosophy in terms of high gain feedback, we present an example of a globally stabilizable nonlinear feedback system which cannot be stabilized, even locally, by any output feedback law - in particular, by a high gain law.     

一类带不确定输入动态非线性系统的输出反馈鲁棒镇定

研究一类带不确定输入动态非线性系统的输出反馈鲁棒镇定问题.通过在高增益观测器引入新的设计参数,改进了通常的高增益反馈控制的设计方法.在输入动态满足零相对阶最小相位的假设下,基于非分离设计原则给出了动态输出反馈控制器的设计方法,所设计的控制器实现了对任意可允许不确定输入动态的全局鲁棒镇定.     

带有不确定性参数的多变量非线性系统的半全局实用镇定

研究一类含有零动态和不确定性参数的多输入多输出非线性系统的镇定问题,通过构造一个组合Lyapunov函数,对非线性动态部分设计了一个控制器可半全局实用镇定整个闭环系统的平衡点.仿真实例说明了所采用方法的有效性.     

A coupled gradient network approach for static and temporalmixed-integer optimization

Utilizes the ideas of artificial neural networks to propose new solution methods for a class of constrained mixed-integer optimization problems. These new solution methods are more suitable to parallel implementation than the usual sequential methods of mathematical programming. Another attractive feature of the proposed approach is that some global search mechanisms may be easily incorporated into the computation, producing results which are more globally optimal. To formulate the solution method proposed in this paper, a penalty function approach is used to define a coupled gradient-type network with an appropriate architecture, energy function and dynamics such that high-quality solutions may be obtained upon convergence of the dynamics. Finally, it is shown how the coupled gradient net may be extended to handle temporal mixed-integer optimization problems, and simulations are presented which demonstrate the effectiveness of the approach.     

Global output-feedback stabilization for a class of stochastic non-minimum-phase nonlinear systems

In this paper, the problem of output-feedback stabilization is investigated for the first time for a class of stochastic nonlinear systems whose zero dynamics may be unstable. Under the assumption that the inverse dynamics of the system is stochastic input-to-state stabilizable, a stabilizing output-feedback controller is constructively designed by the integrator backstepping method together with a new reduced-order observer design and the technique of changing supply functions. It is shown that, under small-gain type conditions for small signals, the resulting closed-loop system is globally asymptotically stable in probability. The obtained results extend the existing methodology from deterministic systems to stochastic systems. An example is given to demonstrate the main features and effectiveness of the proposed output-feedback control scheme.     

Input output linearization approach to state observer design fornonlinear system

Presents a state observer for a class of nonlinear systems based on the input output linearization. While the previous result presented state observers for nonlinear systems of full relative degree, we proposed a procedure fur the design of nonlinear state observers which do not require the hypothesis of full relative degree. Assuming that there exists a global state observer for internal dynamics and that some functions are globally Lipschitz, we can design a globally convergent state observer. It is also shown that if the zero dynamics are locally exponentially stable, then there exists a local state observer. An example is given to illustrate the proposed design of nonlinear state observers     

基于矩阵分解的多变量鲁棒自适应反推控制

针对含有输入未建模动态的一类MIMO系统,在高频增益矩阵的顺序主子式的符号已知的前提下,给出了多变量自适应反推控制器的设计.严格地证明了对一类未建模动态,闭环适应系统的所有信号都是全局一致有界的,且输出渐近收敛于零.     

Fully analytical solution to discrete behavior of hybrid zero dynamics in limit cycle walking with constraint on impact posture

This paper proposes a fully analytical solution to the discrete behavior of hybrid zero dynamics (HZD) in limit cycle walking with constraint on impact posture. First, we introduce a passive rimless wheel and explain the stability principle through derivations of the analytical transition functions of the state error for the stance and collision phases. Second, we consider an active rimless wheel driven by a steady control input for investigating the stability of semi-passive dynamic walking, and propose a method for analytically deriving the transition function for the stance phase without including unknown parameters. We then numerically investigate the solution accuracy and discuss how the discrete behavior of the HZD changes according to the control parameters. Furthermore, we extend the analysis to level walking of an underactuated rimless wheel with a torso and show that the discrete behavior of the HZD can be determined in the same manner.     

Limit cycle stability analysis and adaptive control of a multi-compartment model for a pressure-limited respirator and lung mechanics system

Acute respiratory failure due to infection, trauma or major surgery is one of the most common problems encountered in intensive care units, and mechanical ventilation is the mainstay of supportive therapy for such patients. In this article, we develop a general mathematical model for the dynamic behaviour of a multi-compartment respiratory system in response to an arbitrary applied inspiratory pressure. Specifically, we use compartmental dynamical system theory and Poincaré maps to model and analyse the dynamics of a pressure-limited respirator and lung mechanics system, and show that the periodic orbit generated by this system is globally asymptotically stable. Furthermore, we show that the individual compartmental volumes, and hence the total lung volume, converge to steady-state end-inspiratory and end-expiratory values. Finally, we develop a model reference direct adaptive controller framework for the multi-compartmental model of a pressure-limited respirator and lung mechanics system where the plant and reference model involve switching and time-varying dynamics. We then apply the proposed adaptive feedback controller framework to stabilise a given limit cycle corresponding to a clinically plausible respiratory pattern.     

Existence of limit cycle and stabilization of induction motor via new nonlinear state observer

In this paper the boundedness of the solutious of the bilinear and nonlinear differential equations, which describe the dynamic behavior of an ideal three-phase squirrel cage induction motor, is shown using a Lyapunov function. It is then proved by sampling combined with a digital simulation that an unstable machine has limit cycle. Utilizing these results a new bilinear and nonlinear reduced-order state observer, which is globally asymptotically stable, is constructed to estimate the unmeasurable state variables. By using this observer a new two-step procedure for stabilizing an unstable machine, which has a limit cycle, is proposed. This scheme can be easily implemented resulting in an asymptotically stable overall system. These results are numerically verified by simulation.