Non-smooth Feedback Control for a Class of Constraint Systems

Abstract

Beginning from the second-order input constraint systems, the paper pays attention to study the control prob- lem for a class of the constraint systems with the nalyzes the stability of the second-order input constraint systems under the condition of linear state feedback. On the basis of that, it gives the non-smooth feedback control law of the third-order state constraint systems. At the same time, it sets up the relationship between system's steady state and its initial condi- tions. Finally, it has verified the validity of the conclusion by simulation. In the daily life, systems usually work under the condi- tion of some constraints or limits which is often showed by one link from systems that has been affected by a certain degree of the characteristics of saturation nonlinearity. Influ- enced by saturation nonlinearity, this kind of systems will make the analysis and design of the systems difficult because it is not only deferent from normal one in system perfor- mance, but also difficult to adopt common means to judge its stability, especially for the situation that the change of some system state affected by the saturation nonlinearity, under which it is very difficult to use existing methods to analyze the stability of the systems. Therefore, judging from the re- searches on saturation nonlinearity control problem, there is not an effective theoretical framework to solve the stability of the systems and the design. However, although saturation nonlinearity on independent variable is continuous, it is not continuous for the derivative of variable. So that is a kind of non- smooth nonlinearity which has the system with satura- tion nonlinear. What's more, in essence, it is a class of non- smooth system, whose control problem can be considered from non-smooth control theories and design methods. Just from the perspective of non-smooth feedback design view, this paper studies that the change of system state has the con- trol problem of saturation non-linear constraint systems. In fact, as for the control problem of the system with sat- uration nonlinearity, there has been a lot of research achievements published. Reference (1) gives the sufficient and necessary condition of system stability for the reliability problem of a class of linear saturation nonlinear system. That condition shows that, in the case that the state of linearity system is completely affected by saturation nonlinearity, the system would make the global asymptotic stability come true if the system matrix met certain conditions. Especially for the second-order system, it is required that System Matrix A belongs to Hurwitz's with a line which has diagonally domi- nant characteristic. Reference (2), based on Reference (1), further discusses the convergence problem for the second- order system. In regard to the study of the second-order sys- tem, it can be found that Reference (3) studies detailed the track of a class of the second-order system under the condi- tion of saturation constraints (4). And input limited system is discussed in Reference (4) where there are the design meth- ods that linear system gets the maximum convergence rate of elliptic invariant sets in the sate of saturation constraints. These research results generally focus on a class of systems whose saturation nonlinearity has an effect on all the system states, and the restricted states of many systems are usually limited in practice (5-7). T hus, the conclusion above can not be used directly. This paper studies a class of systems which is in the lim- ited state. The systems have three states, among which there is a kind of state-saturation nonlinearity-which is often seen under the condition of actuator torque limited. Firstly, the paper starts with the feedback designs with the state of the second-order input constraint systems and draws the conclu- sion of the stability of the systems that are in the status of linearity. And then, the linear state feedback controller gotten by the second-order system directly extends to the target sys- tem and gets non-smooth feedback controller. At the same time, it further discusses the relationship between system sta- bility and initial conditions and establishes the relationship between the two. Finally, it has verified the conclusion under different initial conditions by numerical simulation.