Feedback control of actuation‐constrained moving structure carrying Timoshenko beam

Abstract

This article deals with tracking control and active vibration suppression of a hybrid rigid‐elastic vibrational mobile system with a nonlinear elastic foundation in the presence of actuator bandwidth limitation and control input saturation. The system consists of a rigid rectilinearly mobile frame on an elastic foundation, and a flexible Timoshenko beam clamped to it with a heavy attached concentrated mass. The hybrid partial differential equation (PDE)‐ordinary differential equation (ODE) system of the plant is obtained using Hamilton's principle and then is converted to a system of ODEs using the assumed mode method (AMM). Next, the model will be discretized in the time domain using the two‐step Adams‐Bashforth technique. Considering the first‐order dynamics to cover the bandwidth limitation of the actuator, a discrete‐time sliding mode controller (DSMC) is designed based on a positive definite Lyapunov functional. Following that, an innovative optimal predictive algorithm is proposed to compensate for the effects of input saturation. The proposed algorithm is based on a positive definite cost function and using an intelligent regime of optimizations and decisions, minimizes the cost function in a predefined prediction horizon. The vibratory behavior of the system also is included in the mentioned cost function. Consequently, the overall control scheme called optimal rigid‐elastic interaction control (OREIC) will be able to follow the desired tracking trajectory, suppress the beam vibrations, and compensate for the effect of bandwidth limitation as well as the effect of input saturation. The performance of the OREIC algorithm is compared with a saturated DSMC controller and the simulation results demonstrate the efficiency of the method in vibration control of mobile rigid‐elastic systems subjected to actuator bandwidth frequency and saturation limitations.

Nonlinear Tracking Control Algorithm for Dynamical Output Systems Manipulated by the Hardly Constrained Oscillatory Actuator

“…According to the theorems of fexible beam vibrations, in the simplest case, instead of Structural Control and Health Monitoring a beam, a spring with a stifness coefcient based on the beam defection equation can be considered. According to the well-known theories of Euler-Bernoulli or Timoshenko [11], this equivalent stifness coefcient corresponds to the frst vibrational mode of the fexible beam system. On the other hand, since the beam support base is connected to a nonlinear impedance confguration, it is no longer possible to analytically determine and solve the governing equations, which are a set of coupled PDE and ODE equations, by the method of separation of variables or the popular technique of assume modes [5].…”

“…In this situation, an important concern for the plausible performance of the specifed objective is to minimize the vibrations transmitted to the equipment mounted on the fexible arm. In such cases, because the control forces are applied to the moving support and are not applied directly to the equipment coordinate, the defned output of the control system has linear or nonlinear dynamics based on the measured state variables [5,11]. However, in many feedback control systems, the relationship between output and system state variables is often algebraic and linear.…”

“…Terefore, with the problem of the presence of actuator bandwidth limitation, the maximum vibrational frequency that can be eliminated according to the mentioned actuation restrictions should be examined [12,15]. Finally, it can be concluded that the recent research studies in the feld of designing nonlinear feedback control systems mainly focus on solving the problem of designing Lyapunov stable control systems in the presence of hard constraints such as system switching, the output constraint, the small gain theorem, unmodeled dynamics, the dead zone of the actuator, the restrictions imposed due to the communication between the sensors, and the presence of time delay in measuring the quantities or in commanding the actuators [11,13,14,16]. Tis article is also among the research studies carried out with this point of view in order to give more possibility of practical implementation.…”

“…(i) In the design of controllers based on optimization methods and model prediction algorithms, the computational load increases so much that the realtime implementation of such algorithms becomes practically impossible and these methods remain at the theoretical level [11].…”

“…(ii) In the design of control systems based on artifcial intelligence, if the adaptive mechanism adjusts the parameters of the artifcial intelligent system in realtime, it may train the network in such a way that the intelligence of the system forgets the previous training due to the occurrence of nonlinear behaviors of the system [11].…”