Stabilization of minimum phase nonlinear systems by dynamic outputfeedback

In this paper, the stabilization problem of minimum phase nonlinear systems by dynamic output feedback is investigated. It is shown that if the zero dynamics of an affine nonlinear system is asymptotically stable according to the kth approximation, then the system is stabilizable by dynamic output feedback. The proof is constructive, which provides a method to design the stabilizer. The result is applied to investigating the stabilization problem of the rotating stall in axial flow compressors     

On semi-global stabilization of minimum phase nonlinear systems without vector relative degrees

Recently, we developed a structural decomposition for multiple input multiple output nonlinear systems that are affine in control but otherwise general. This structural decomposition simplifies the conventional backstepping design and allows a new backstepping design procedure that is able to stabilize some systems on which the conventional backstepping is not applicable. In this paper we further exploit the properties of such a decomposition for the purpose of solving the semi-global stabilization problem for minimum phase nonlinear systems without vector relative degrees. By taking advantage of special structure of the decomposed system, we first apply the low gain design to the part of system that possesses a linear dynamics. The low gain design results in an augmented zero dynamics that is locally stable at the origin with a domain of attraction that can be made arbitrarily large by lowering the gain. With this augmented zero dynamics, backstepping design is then applied to achieve semi-global stabilization of the overall system.     

Adaptive stabilization of single-input single-output delay systems

For the problem of adaptive stabilization of linear systems with unknown, noncommensurate, real delays, the existence of a smooth controller that regulates the output to zero is proven. The assumptions imposed on the system are natural generalizations of the familiar minimum phase and relative degree one conditions in nondelay systems.     

离散时间非线性最小相位系统的动态输出反馈镇定

研究了离散时间非线性最小相位系统的动态输出反馈镇定.首先对离散时间非线性系 统引入了逼近渐近稳定性的概念.基于此概念,提出了一种动态补偿器设计的新方法.主要结果 是,如果一非线性系统的零动态是逼近渐近稳定的,则能用动态输出反馈镇定.动态补偿器的设 计是构造性的.     

非线性最小相位系统输出反馈镇定的一个注记

讨论了单输入单输出非线性最小相位系统的动态输出反馈镇定.通过加积分器和非 线性变换将系统化为一种标准形式,并基于标准形式的线性部分提出了动态补偿器的设计方 法.然后根据得到的中心流形的表达式和稳定性定理,在零动态流形为一维时,证明了闭环系 统的渐近稳定性,最后给出了一个零动态不具有齐次渐近稳定性但仍能动态输出反馈镇定的 非线性最小相位系统的例子.     

具有扰动输入的不确定性非线性系统的输出调节极限性能

本文研究了一类具有扰动输入的不确定性非线性系统的输出调节问题, 给出了该类系统在最差的不确定性参数和扰动输入情况下系统输出调节的极限性能. 所讨论的非线性系统是可镇定非最小相位系统, 并且该系统的零动态由“鲁棒输入对状态稳定(robust input-to-state stable)部分”和“不稳定但可镇定部分”组成. 假设系统的不确定性参数和扰动输入分别以非线性函数和仿射形式同时出现在系统零动态的鲁棒输入对状态稳定部分和系统的可线性化部分, 而且其可线性化部分的不确定性具有下三角形结构形式. 该系统输出调节问题的性能以其输出信号能量作为度量. 对于上述非线性系统, 在最差的不确定性参数和扰动输入情况下, 输出调节问题的极限性能只取决于镇定其零动态“不稳定部分”所需的最小能量.     

STABILIZATION OF A CLASS OF NONLINEAR NON‐MINIMUM PHASE SYSTEMS

This paper tackles the problem of stabilization of a class of non‐minimum phase nonlinear systems which have zero dynamics with an eigenvalue zero of multiplicity 2. By adding some new terms, called cross terms, we are able to generalize the concept of the Lyapunov function with a homogeneous derivative along the trajectory, which was introduced in [4], to produce a suitable Lyapunov function. The Lyapunov function assures that the stability of an approximate system, which consists of some lower order terms of a nonlinear system with an eigenvalue zero of multiplicity 2, implies the stability of the whole system. Applying these to non‐minimum phase zero dynamics of nonlinear systems with such a center, a sufficient condition and a design method of state feedback control are obtained for stabilizing the systems.     

Output feedback control of a class of nonminimum‐phase nonlinear systems

The problem of global stabilization by output feedback is investigated in this paper for a class of nonminimum‐phase nonlinear systems. The system under consideration has a cascade configuration that consists of a driven system known as the inverse dynamics and a driving system. It is proved that although the zero dynamics may be unstable, there is an output feedback controller, globally stabilizing the nonminimum‐phase system if both driven and driving systems have a lower‐triangular form and satisfy a Lipschitz‐like condition, and the inverse dynamics satisfy a stronger version of input‐to‐state stabilizability condition. A design procedure is provided for the construction of an n‐dimensional dynamic output feedback compensator. Examples and simulations are also given to validate the effectiveness of the proposed output feedback controller. Copyright © 2016 John Wiley & Sons, Ltd.     

Robust output feedback stabilization via a small gain theorem

In this paper, we give sufficient conditions for designing robust globally stabilizing controllers for a class of uncertain systems, consisting of ‘nominal’ nonlinear minimum phase systems perturbed by uncertainties which may affect the equilibrium point of the nominal system (‘biased’ systems). The constructive proof combines a systematic step-by-step procedure, based on H_∞ arguments, with a small gain theorem, recently proved for nonliner systems. At each step, one finds two Lyapunov functions, one for a state-feedback problem and the other one for an output injection problem. Combining these two functions, one derives at each step a Lyapunov function candidate for solving an ouptut feedback stabilization problem. This approach allows one to put into a unified framework many existing results on robust output feedback stabilization. © 1998 John Wiley & Sons, Ltd.     

一类线性不可观非线性系统的动态输出反馈镇定

本文研究一类不可观非线性系统的动态输出反馈镇定,基于逼近渐近稳定性的概念,给出了动态输出反馈可镇定的充分条件,本文主要结果的直接推论是零动太逼近渐近稳定的最小相位系统能用动态输出反馈镇定,本文的方法也能处理非最小相位系统。     

A stabilization method for steady motions with zero roots in the closed system

Based on previous results, we consider stabilization problems for both non-asymptotic stability and asymptotic stability with respect to all variables for equilibrium positions and stationary motions of mechanical systems with redundant coordinates. The linear stabilizing control is defined by the solution of a linear–quadratic stabilization problem for an allocated linear controllable subsystem of as small dimension as possible. We find sufficient conditions under which a complete nonlinear system closed by this control is ensured asymptotic stability despite the presence of at least as many zero roots of the characteristic equation as the number of geometric relations. We prove a theorem on the stabilization of the control equilibrium applied only with respect to redundant coordinates and constructed from the estimate of the phase state vector obtained by a measurement of as small dimension as possible.     

仿射非线性奇异系统的反馈控制与稳定化

研究非线性奇异系统的反馈稳定化问题，首先给出仿射非线性奇异系统反馈稳定化的概念；然后利用零动态算法构造的局部坐标变换给出仿射非线性奇异系统的一种标准型，并将其用于研究仿射非线性奇异系统的反馈控制和系统稳定化问题；最后证明了对于正则仿射非线性奇异系统，当其零动态渐近稳定时，该系统可通过反馈控制实现系统的稳定化。     

Stabilizability conditions of multivariable uncertain systems viaoutput feedback control

A stabilizability condition is established for stabilizing multivariable uncertain dynamical systems with matched uncertainties by linear static output feedback control. It is shown that under the assumptions of minimum phase and nonsingular high-frequency gain for the nominal plant, stabilization with linear static output feedback control is achievable. A necessary and sufficient condition is also obtained for the existence of constant output feedback which makes the closed-loop system strictly positive real     

Stabilization of one class of nonlinear systems

Consideration was given to the problem of stabilization for one class of systems that are nonlinear and uncontrollable in the first approximation. The stabilization problem was solved by considering the nonlinear approximation. The stabilizing control was obtained by the method of Lyapunov function which was constructed as a quadratic form. To determine matrix of this quadratic form, a singular matrix Lyapunov equation was solved. For the system of nonlinear approximation, the stabilizing control was determined explicitly. It was proved that the resulting control solves the problem of stabilizing the original nonlinear system. An ellipsoidal estimate of the attraction domain of the zero stationary point was given.     

Nonlinear output regulation for invertible nonlinear MIMO systems

In this paper, we address the problem of output regulation for a broad class of multi‐input multi‐output (MIMO) nonlinear systems. Specifically, we consider input–affine systems, which are invertible and input–output linearizable. This class includes, as a trivial special case, the class of MIMO systems which possess a well‐defined vector relative degree. It is shown that if a system in this class is strongly minimum phase, in a sense specified in the paper, the problem of output regulation can be solved via partial‐state feedback or via (dynamic) output feedback. The result substantially broadens the class of nonlinear MIMO systems for which the problem in question is known to be possible. Copyright © 2015 John Wiley & Sons, Ltd.     

Construction of control Lyapunov functions for damping stabilization of control affine systems

This paper considers the stabilization of nonlinear control affine systems that satisfy Jurdjevic–Quinn conditions. We first obtain a differential one-form associated to the system by taking the interior product of a non vanishing two-form with respect to the drift vector field. We then construct a homotopy operator on a star-shaped region centered at a desired equilibrium point that decomposes the system into an exact part and an anti-exact one. Integrating the exact one-form, we obtain a locally-defined dissipative potential that is used to generate the damping feedback controller. Applying the same decomposition approach on the entire control affine system under damping feedback, we compute a control Lyapunov function for the closed-loop system. Under Jurdjevic–Quinn conditions, it is shown that the obtained damping feedback is locally stabilizing the system to the desired equilibrium point provided that it is the maximal invariant set for the controlled dynamics. The technique is also applied to construct damping feedback controllers for the stabilization of periodic orbits. Examples are presented to illustrate the proposed method.     

Stabilization of positive linear systems

We consider stabilization of equilibrium points of positive linear systems which are in the interior of the first orthant. The existence of an interior equilibrium point implies that the system matrix does not possess eigenvalues in the open right half plane. This allows to transform the problem to the stabilization problem of compartmental systems, which is known and for which a solution has been proposed already. We provide necessary and sufficient conditions to solve the stabilization problem by means of affine state feedback. A class of stabilizing feedbacks is given explicitly.     

Superstable Linear Control Systems. II. Design

The notion of superstability introduced in [1] is applied to the design of stabilizing and optimal controllers. It is shown that a static output feedback controller which ensures superstability of the closed-loop system can be found (provided it exists) by means of linear programming (LP) techniques; finding a superstable matrix in the given affine family is a generalization of this problem. The ideology of superstability is also shown to be fruitful in optimal and robust control. This is exemplified by the problems of rejection of bounded disturbances, optimization of the integral performance index which involves absolute values (rather than squares) of the control and state, and by stabilization of an interval matrix family and simultaneous stabilization.     

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.