Modal exact linearization of a class of second-order switched nonlinear systems

This paper considers the problem of stabilizing single-input affine switched nonlinear systems. The main idea is to transform a switched nonlinear system to an equivalent controllable switched linear system. First, we define the notion of modal state feedback linearization. Then, we develop a set of conditions for modal state feedback linearizability of a certain class of second order switched nonlinear systems. Considering two special structures, easily verifiable conditions are proposed for the existence of suitable state transformations for modal feedback linearization. The results are constructive. Finally, the method is illustrated with two examples, including a Continuous Stirred Tank Reactor (CSTR) to demonstrate the applicability of the proposed approach.     

时变仿射非线性系统的动态补偿线性化

用微分几何方法讨论了时变仿射非线性系统的动态补偿线性化问题，给出了系统动态的反馈线性化的条件，它是状态反馈线性化的推广。     

New Approach to Linear Exact Model Matching for a Class of Nonlinear Systems

In this paper, a new approach to the linear exact model matching problem for a class of nonlinear systems, using static state feedback, is presented. This approach reduces the problem of determining the state feedback control law to that of solving a system of first-order partial differential equations. Based on these equations, two major issues are resolved: the necessary and sufficient conditions for the problem to have a solution and the general analytical expression for the feedback control law. Furthermore, the proposed approach is extended to solve the same problem via static output feedback.     

Input-output linearization with simultaneous decoupling by restricted state feedback

The problem of input-output linearization with simultaneousdecoupling via restricted state feedback for nonsquare nonlinearsystems is investigated in this paper. The problem is treatedon the basis of a new algebraic approach whose main featureis that it reduces the determination of the admissible state-feedbackcontrol laws to the solution of an algebraic and a first-orderpartial differential system of equations. Verifiable necessaryand sufficient conditions of an algebraic nature are establishedfor the solvability of the aforementioned problem, which relyon the solvability of these systems of equations. Moreover,an explicit expression for a special admissible restricted state-feedbackcontroller is analytically derived.     

非线性系统的动态反馈完全线性化

其中,为简单记,x(t)∈R~n,u(t)∈R~m,y(t)∈R~m,f(·),g(·)的诸列以及 h(·)的诸行是 x 的亚纯函数,即它们是 x 的解析函数环的分式域 (fraction field) 的元素.注意这个系统具有相同个数的输入和输出.本文考虑下述形式的动态反馈补偿器设计     

Some new results in differential algebraic control theory

In the differential algebra framework, static state and dynamicstate feedback linearization are considered. The relationshipbetween dynamic feedback and flatness defined by Fliess is discussed.The existence of an equivalent proper differential I–Osystem of a given differential I–O system is discussed,which is closely related to the choice of a proper fictitiousoutput in control design. The concept flatness and its relationwith dynamic feedback linearizability, controllability, observability,invertibility and minimal realization are discussed. Finally,it is demonstrated that many fundamental control concepts andtheir interrelationships can be incorporated into an extendedcontrol diagram.     

Nonlinear feedback control and systems of partial differential equations

Finding an equivalence between two feedback control systems is treated as a problem in the theory of partial differential equation systems. The mathematical aim is to embed the Jakubzyk-Respondek, Hunt-Meyer-Su work on feedback linearization in the general theory of differential systems due to Lie, Cartan, Vessiot, Spencer, and Goldschmidt. We do this by using the functor taking control systems into differential systems, and studying the equivalence invariants of such differential systems. After discussing the general case, attention is focussed on the special situation of most immediate practical importance, the theory of feedback linearization. In this case, the general system for feedback equivalence becomes a system of linear partial differential equations. Conditions are found that the general solution of this system may be described in terms of a Frobenius system and certain differential-algebraic operations.This work was supported by grant from the Ames Research Center of NASA and the Applied Mathematics Program of the National Science Foundation.     

Global Hopf bifurcation for differential equations with state-dependent delay

We develop a global Hopf bifurcation theory for a system of functional differential equations with state-dependent delay. The theory is based on an application of the homotopy invariance of S¹-equivariant degree using the formal linearization of the system at a stationary state. Our results show that under a set of mild conditions the information about the characteristic equation of the formal linearization with frozen delay can be utilized to detect the local Hopf bifurcation and to describe the global continuation of periodic solutions for such a system with state-dependent delay.     

Invariants for feedback equivalence and Cauchy characteristic multifoliations of nonlinear control systems

Linearization of a nonlinear feedback control system under nonlinear feedback is treated as a problem of equivalence-under the Lie pseudogroup of feedback transformations-of distributions on the product manifold of the state and control variables. The new feature of this paper is that it introduces the Cauchy characteristic sub-distributions of these distributions and their derived distributions. These Cauchy characteristic distributions are involutive and nested, hence define a Multifoliate Structure. A necessary condition for feedback equivalence of two nonlinear control systems is that these multifoliations be transformed under the feedback pseudogroup. For linear systems, this Cauchy characteristic multifoliate structuee is readily computed in terms of the (A, B)-matrix that defines the linear system. Assuming that the conditions for local feedback linearization are satisfied, the existence of a global feedback linearizing transformation is dependent on computing an element of the first cohomology group of the space with coefficients in the sheaf of groupoid of infinitesimal feedback automorphisms of the linear system. The theorem quoted above about the Cauchy characteristic multifoliations provides some information about this groupoid. It is computed explicitly and directly for control systems with one- or two-state dimensions. Finally, these Cauchy characteristic sub-distributions must inevitably play a role in the numerical or symbolic computational analysis of the Hunt-Su partial differential equations for the feedback-linearizing transformation.Senior Research Associate of the National Research Council at the Ames Research Center of NASA.     

Multivariable fuzzy control applied to the physical–chemical treatment facility of a Cellulose factory

Feedback linearization is a well-known technique in nonlinear control in which known system nonlinearities are canceled by the control input leaving a linear control problem. Feedback linearization requires an exact model for the system. Fundamental and advanced developments in neuro-fuzzy synergy for modeling and control are used to apply the feedback linearization control law on second-order plants. In the models that are used, the nonlinear plant is decomposed on six fuzzy systems necessary to apply the control signal to allow the following of a reference value. A practical application is also presented using a waste water plant. This method can be extended to multiple input–multiple output (MIMO) plants based on input–output data pairs collected directly from the plant.     

Formal analysis of the Cauchy problem for a system associated with the (2+1)-dimensional Krichever-Novikov equation

The singularity manifold equation of the Kadomtsev-Petviashvili equation, the so-called Krichever-Novikov equation, has an exact linearization to an overdetermined system of partial differential equations in three independent variables. We study in detail the Cauchy problem for this system as an example for the use of the formal theory of differential equations. A general existence and uniqueness theorem is established. Formal theory is then contrasted with Janet-Riquier theory in the formulation of Reid. Finally, the implications of the results for the Krichever-Novikov equation are outlined.     

Nonlinear feedback control of chaotic pendulum in presence of saturation effect

In present paper, a feedback linearization control is applied to control a chaotic pendulum system. Tracking the desired periodic orbits such as period-one, period-two, and period-four orbits is efficiently achieved. Due to the presence of saturation in real world control signals, the stability of controller is investigated in presence of saturation and sufficient stability conditions are obtained. At first feedback linearization control law is designed, then to avoid the singularity condition, a saturating constraint is applied to the control signal. The stability conditions are obtained analytically. These conditions must be investigated for each specific case numerically. Simulation results show the effectiveness and robustness of proposed controller. A major advantage of this method is its shorter chaotic transient time in compare to other methods such as OGY and Pyragas controllers.     

基于虚拟完整约束的欠驱动起重机控制方法

欠驱动系统的控制是非线性控制的一个重要领域,欠驱动系统指系统控制输入个数小于自由度个数的非线性系统.目前,欠驱动非线性系统动力学和控制研究的主要方法包括线性二次型最优控制方法和部分反馈线性化方法等,如何使系统持续的稳定在平衡位置一直是研究的难点.虚拟约束方法是指通过选择一个周期循环变化的变量作为自变量来设计系统的周期运动.该文以典型的欠驱动模型起重机为例,采用虚拟约束方法,使系统能够在平衡位置稳定或周期振荡运动.首先,通过建立虚拟约束,减少系统自由度变量;然后,通过部分反馈线性化理论推导出系统的状态方程;最后,通过线性二次调节器设计反馈控制器.仿真结果表明,重物在反馈控制下可以在竖直位置的附近达到稳定状态,反映了虚拟约束方法对欠驱动系统的有效性.     

永磁同步电机伺服系统的有限时间位置控制

研究了永磁同步电机伺服系统的位置跟踪问题.利用反馈线性化的思想,对永磁同步电机的数学模型进行分析,实现了电机模型的精确线性化和解耦.首先,将永磁同步电机位置跟踪系统采用反馈线性化技术变换为两个线性子系统,分别对其设计相应的基于连续状态反馈线性化的有限时间控制器,并设计了有限时间负载观测器来观测估计外部负载扰动.对永磁同步电机位置跟踪的闭环系统进行了稳定性的分析.与对应的渐近稳定控制的方案相比,基于有限时间的控制方案实现了永磁同步电机对期望信号的有限时间跟踪,获得了更好的动态响应和抗扰动性能.仿真结果表明了该控制方案的有效性.     

Some problems of linear differential equations on abstract spaces and unbounded perturbations of linear operator semigroup

This paper is a survey for development of linear distributed parameter system. At first we point out some questions existing in current study of control theory for the L^p linear system with an unbounded control operator and an unbounded observation operator, such as stabilization problem and observer theory that are closely relevant to state feedback operator. After then we survey briefly some results on relevant problems that are related to solvability of linear differential equations in general Banach space and semigroup perturbations. As a principle, we propose a concept of admissible state feedback operator for system (A, B). Finally we give an existence result of admissible state feedback operators, including semigroup generation and the equivalent conditions of admissibility of state feedback operators, for an L^p well-posed system.     

粘弹性板混沌振动的输出变量反馈线性化控制

研究了粘弹性板混沌振动的控制问题·　应用非线性系统精确线性化控制理论导出了一类非仿射控制系统的非线性反馈控制律·　建立了描述材料非线性的粘弹性板运动的数学模型并利用Ｃａｌｅｒｋｉｎ 方法进行简化·　采用相空间曲线和频率谱密度函数说明了在特定参数条件下系统将出现混沌振动,并以位移为输出变量将混沌振动控制为给定的周期运动·     

Exact model matching of generalized state space systems

The problem of exact model matching for generalized state space (GSS) systems via pure proportional state and output feedback is studied. The following two major issues are resolved here for the first time: The necessary and sufficient conditions for the problem to have a solution and the general analytical expressions for the exact modelmatching controller matrices. The important case of left invertible systems is treated separately wherein simple solutions are established for the above two major issues and results on structural properties of the closed-loop system are reported. Known results on model matching of regular systems are derived as a special case of the GSS systems results, thus unifying the solution of the exact model-matching problem of regular and singular systems.The work described in this paper has been partially funded by the General Secretariat for Research and Technology of the Greek Ministry of Industry, Research, and Technology and by the Heracles General Cement Company of Greece.     

Geometry of cascade feedback linearizable control systems

Cascade feedback linearization provides geometric insights on explicit integrability of nonlinear control systems with symmetry. A central piece of the theory requires that the partial contact curve reduction of the contact sub-connection be static feedback linearizable. This work establishes new necessary conditions on the equations of Lie type - in the abelian case - that arise in a contact sub-connection with the desired static feedback linearizability property via families of codimension one partial contact curves. Furthermore, an explicit class of contact sub-connections admitting static feedback linearizable contact curve reductions is presented, hinting at a possible classification of all such contact sub-connections. Key tools in proving, and stating, the main results of this paper are truncated versions of the total derivative and Euler operators. Additionally, the Battilotti-Califano system with three inputs is used as a clarifying example of both cascade feedback linearization and the new necessary conditions.     

Null controllability and global blowup controllability of ordinary differential equations with feedback controls

This paper concerns the problem of feedback null controllability and blowup controllability with feedback controls for ordinary differential equations. First, we study the feedback null controllability on a time-varying ordinary differential system by unbounded feedback operators. Then, the global exact blowup controllability with feedback controls is derived on a time-invariant ordinary differential system. Finally, we obtain the approximate null controllability by bounded feedback operators, and get the approximate blowup controllability with feedback controls for ordinary differential equations.     

Synchronization of non-autonomous chaotic systems with time-varying delay via delayed feedback control

In this paper, we investigate the synchronization of non-autonomous chaotic systems with time-varying delay via delayed feedback control. Using a combination of Riccati differential equation approach, Lyapunov-Krasovskii functional, inequality techniques, some sufficient conditions for exponentially stability of the error system are formulated in form of a solution to the standard Riccati differential equation. The designed controller ensures that the synchronization of non-autonomous chaotic systems are proposed via delayed feedback control and intermittent linear state delayed feedback control. Numerical simulations are presented to illustrate the effectiveness of these synchronization criteria.