The ease with which backstepping incorporated uncertainties and unknown parameters contributed to its instant popularity and rapid acceptance. Applications of this technique have been recently reported ranging from robotics to industry or aerospace [ Chen Weng, , Wang et al.
Backstepping control has also been explored in some works about suspension systems.
Variable Structure Control of Complex Systems
For example, [ Zapateiro et al. In that work, the controller was formulated for an experimental platform, whose MR damper was modeled by means of an artificial neural network. The control input was updated with a backstepping controller. On the other hand, [ Nguyen et al. Some works on Quantitative Feedback Theory QFT applied to the control of suspension systems can be found in the literature. For instance, [ Amani et al. In this case, the nonlinearities were treated as uncertainties in the model so that the linear QFT could be applied to the control formulation.
As a result, similar performances between both classes of controllers were achieved.
In this chapter, we will analyze three model-free variable structure controllers for a class of semiactive vehicle suspension systems equipped with MR dampers. The variable structure control VSC is a control scheme which is well suited for nonlinear dynamic systems [ Glizer et al.
This control method can make the system completely insensitive to time-varying parameter uncertainties, multiple delayed state perturbations and external disturbances [ Pai, ]. Nowadays, research and development continue to apply VSC control to a wide variety of engineering areas, such as aeronautics guidance law of small bodies [ Zexu et al.
By using this kind of controllers, it is possible to take the best out of several different systems by switching from one to the other. In this case, the resulting algorithm can be viewed as the clipped control in [ Dyke et al. Finally, the last strategy presented is based on a time variable depending on the absolute value of the difference between the body angular velocity and the wheel angular velocity, and on the difference between the body angular position and the wheel angular position.
The study of the three variable structure controllers will be complemented with the comparison of a model-based controller which has been successfully applied by the authors in other works: backstepping. As it was mentioned earlier, backstepping is well suited to this kind of problems because it can account for robustness and nonlinearities.
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It has been used by the authors to analyze this particular problem [ Zapateiro et al. The chapter is organized as follows. Section 2 presents the mathematical details of the system to be controlled. In Section 3, the three variable structure controllers are developed. In Section 4, the backstepping control formulation details are outlined. Section 5 shows the numerical results, and in Section 6, the conclusions are drawn. The suspension system can be modeled as a quarter car model, as shown in Figure 1.
The system can be viewed as a composition of two subsystems: the tyre subsystem and the suspension subsystem.
Variable Structure Control of Complex Systems: Analysis and Design
The tyre subsystem is represented by the wheel mass m u while the suspension subsystem consists of a sprung mass, m s , that resembles the vehicle mass. This way of seeing the system will be useful later on when designing the model-based semi active controller. The compressibility of the wheel pneumatic is k t , while c s and k s are the damping and stiffness of the uncontrolled suspension system.
The quarter car model equations are given by:. Taking x 1 , x 2 , x 3 and x 4 as state variables allows us to formulate the following state-space representation:. The input u is given by:. In this study, we assume that the semiactive device is magnetorheological MR damper. It is modeled according to the following Bouc-Wen model [ Spencer et al. Feedback control radically alters the dynamics of a system: it affects its natural frequencies, its transient response as well as its stability. The MR damper of the quarter-car model considered in this study is voltage-controlled, so the voltage v is updated by a feedback control loop.
It is well known that the force generated by the MR damper cannot be commanded; only the voltage v applied to the current driver for the MR damper can be directly changed. One of the first control approaches involving an MR damper was proposed by [ Dyke et al. In this approach, the command voltage takes one of two possible values: zero or the maximum. This is chosen according to the following algorithm:. The sign part of equation 9 can be transformed in the following way:.
Finally, the full expression in equation 9 can be rewritten as a piecewise function in the following way:. This algorithm for selecting the command signal is graphically represented in Figure 2. Note that in that particular work, they used the voltage as the control signal because that is the way that current driver can be controlled.
In this paper we consider the same idea of changing the voltage. This control signal is computed according to the following control strategies, computed as a function of the sprung mass velocity x 4 , the unsprung mass velocity x 2 , and the suspension deflection x 3 :. The distinctive feature of VSC is that the structure of the system is intentionally changed according to an assigned law.
This can be obtained by switching on or cutting off feedback loops, scheduling gains and so forth. By using VSC, it is possible to take the best out of several different systems more precisely structures , by switching from one to the other. The control law defines various regions in the phase space and the controller switches between a structure and another at the boundary between two different regions according to the control law.
The three strategies presented in this section can be viewed as variable structure controllers, since the value of the control signal is set to be zero or one, as can be seen in the following transformations:. Semi-active control have two essential characteristics. The first is that the these devices offer the adaptability of active control devices without requiring the associated large power sources. The second is that the device cannot inject energy into the system; hence semi-active control devices do not have the potential to destabilize in the bounded input—bounded output sense the system [ Soong Spencer, ].
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As a consequence, the stability of the closed-loop system is guaranteed. In this section we present the formulation of a model-based controller. The objective, as explained in the Introduction, is to make a comparison between this model-based controller and the VSC controllers. We will appeal to the backstepping technique that has been developed in previous works for this kind of systems.
The objective is to design an adaptive backstepping controller to regulate the suspension deflection with the aid of an MR damper thus providing safety and comfort while on the road. In order to formulate the backstepping controller, the state space model 3 - 4 must be first written in strict feedback form [ Krstic et al. Therefore, the following coordinate transformation is performed [ Karlsson et al. Substitution of the expression for u 5 into 22 yields:.
Thus, the errors between the estimates and the actual values are given by:. From 21 - 22 , it can be shown that the transfer functions from d t and f mr t to z 1 t are:. If the poles of the transfer functions 26 and 27 are in the left side of the s plane, then we can guarantee the bounded input - bounded output BIBO stability of Z 1 s for any bounded input D s and F mr s.
Thus, the disturbance input d i t in 23 is also bounded. This boundedness condition will be necessary later in the controller stability condition.
In order to begin with the adaptive backstepping design, we firstly define the following error variable and its derivative:. Equation 30 can be stabilized with the following virtual control input:. Now define a second error variable and its derivative:.
On the other hand, the derivatives of the errors of the uncertain parameter estimations are given by:. Substitution of 42 and 43 into 41 yields:.
Analysis and Design
As stated earlier, the disturbance input d i is bounded as long as the poles of the transfer functions 26 and 27 are in the left side of the s plane. This implies that all the closed-loop trajectories have to remain bounded, as we wanted to show. Now, under zero initial conditions, from 46 we can write:. Seller Rating:. About this Item: Springer, Gebundene Ausgabe. Condition: Gebraucht.
Gebraucht - Sehr gut Leichte Lagerspuren - This book systematizes recent research work on variable-structure control. It is self-contained, presenting necessary mathematical preliminaries so that the theoretical developments can be easily understood by a broad readership. The text begins with an introduction to the fundamental ideas of variable-structure control pertinent to their application in complex nonlinear systems.
In the core of the book, the authors lay out an approach, suitable for a large class of systems, that deals with system uncertainties with nonlinear bounds. Its treatment of complex systems in which limited measurement information is available makes the results developed convenient to implement. Various case-study applications are described, from aerospace, through power systems to river pollution control with supporting simulations to aid the transition from mathematical theory to engineering practicalities.
The book addresses systems with nonlinearities, time delays and interconnections and considers issues such as stabilization, observer design, and fault detection and isolation. It makes extensive use of numerical and practical examples to render its ideas more readily absorbed. Variable-Structure Control of Complex Systems will be of interest to academic researchers studying control theory and its application in nonlinear, time-delayed an modular large-scale systems; the robustness of its approach will also be attractive to control engineers working in industries associate with aerospace, electrical and mechanical engineering.
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