In order to study the influence of input delay on the stability of Timoshenko beam system, a new controller is designed to compensate for the influence of input delay by using Backstepping method to stabilize the Timoshenko beam system with input delay and load at the boundary, thus a stable closed-loop system is obtained. Firstly, a time-delay-free system equivalent to the original time-delay system is given. Then, a Backstepping linear transformation is constructed, and it is proved that the linear transformation is bounded and reversible. Finally, the time-delay-free system is transformed into a stable target system by this transformation, and the corresponding controller is designed.
The results show that the time-delay-free system is equivalent to the target system, and its feedback control law can stabilize the original time-delay system. The research method solves the negative influence of input delay on elastic system, enriches the design method of controller and stability theory of distributed parameter control system, and has certain reference significance in engineering practice. First author’s brief introduction: Han Rumeng (1993 -), female, Cangzhou, Hebei Province, graduate student, mainly engaged in distributed parameter system research. Communication Author: Associate Professor Liu Dongyi.
In the field of aviation, ocean and civil engineering, elastic structures usually play a key role in connection and loading. Under the action of external disturbances and loads, these structures will vibrate and cause certain harm to engineering structures. For a long time, many scholars have devoted themselves to the design and stability analysis of controllers for elastic systems. The system is stabilized by resisting adverse factors such as time delay and external interference. Timoshenko beam system takes into account the effects of shear and rotation.
It can accurately describe the dynamic behavior of elastic rods and is a very accurate system model.
Journal of Hebei University of Science and Technology, No. 2, 2018, Han Rumeng, thermostatic element et al. Controller design and stability analysis of Timoshenko beam system with input delay have attracted considerable interest from many scholars.
In this paper, a Timoshenko beam with loads and input delays on the boundary is taken as the research object, and a new controller is designed to compensate for the effects of time delays by using Backstepping method, which makes the closed-loop system asymptotically stable.
系统模型如下:ρwtt(x,t)-κ(wxx-φx)(x,t)=0,Iρφtt(x,t)-EIφxx(x,t)-κ(wx-φ)(x,t)=0,mwtt(1,t) κ(wx-φ)(1,t)=u1(t-τ),Jφtt(1,t) EIφx(1,t)=u2(t-τ),w(0,t)=φ(0,t)=0,w(x,0)=w0(x),wt(x,0)=w1(x),φ(x,0)=φ0(x),φt(x,0)=φ1(x),u1(θ)=f1(θ),u2(θ)=f2(θ),θ∈(-τ,0),(1)其中:下標字母表 Partial differential for corresponding variables, x 0; function fi(theta) is bounded and measurable in appropriate space, i=1,2; w(x, t) represents the elastic deflection of the beam in its equilibrium state; _(x, t) represents the total rotation angle; u1(t) and u2(t) represent the boundary control forces and moments; p, k, Ip and EI represent the linear density, shear modulus of elasticity, and beam transverse, respectively. The moment of inertia and stiffness coefficient of the section. When the system has no time delay, i.
e. _ = 0, the output feedback control law: U1 (t) = – alpha1wt (1, t), U2 (t) = – alpha2phit (1, t), (2) can make the system (1) asymptotically stable [10]. When_0, that is, when the system has time delay, under which feedback control law, the system (1) can also be stabilized? This is the main consideration of this paper. For time-delay systems, the stability of one-dimensional wave equation for controllers such as alpha u(t) beta u(t-_) is studied in reference [11] and the so-called 1/2 rule is obtained. A new class of dynamic feedback controllers is designed in reference [12-14].
It is proved that the condition | a | | beta | can guarantee the stability of the closed-loop system. Based on Backstepping method [1518], a new kind of controller is designed in this paper.
Under the action of the feedback control law, the closed-loop system obtained is asymptotically stable. The author designs the controller of the original system by Backstepping method and gives the relevant stability conclusions. The main idea is to transform the stability of the original system into the stability of the target system by constructing a reversible bounded linear transformation.
So the transformation (4) is bounded and reversible.
Certificate is completed. Finally, Theorem 3 is proved.
It is proved that the delay-free system (3) is equivalent to the target system (11) and the system (12) by theorem 2. Lemma 2 shows that the target system is asymptotically stable, so the time-delay-free system (3) is asymptotically stable, that is, the feedback control law (13) can make the original system (1) asymptotically stable. Certificate is completed. Based on Backstepping method, a new controller is designed for Timoshenko beam system with load and input delay at the boundary. It is proved that the original system is asymptotically stable under the feedback control law. The research focuses on the design and stability analysis of the controller. The difficulty lies in the construction of the target system and the selection of the linear transformation. The control operators considered in this paper are bounded. When the control operators are unbounded, how should we consider them? Can such controllers be applied to high-dimensional system models? These are all problems to be studied in the future.