Performance indices.

## 1. Introduction

Vehicle suspension systems are one of the most critical components of a vehicle and it have been a hot research topic due to their importance in vehicle performance. These systems are designed to provide comfort to the passengers to protect the chassis and the freigt [Zapateiro et al., 2012]. However, ride comfort, road holding and suspension deflection are often conflicting and a compromise of the requirements must be considered. Among the proposed solutions, active suspension is an approach to improve ride comfort while keeping suspension stroke and tire deflection within an acceptable level [Gao et al., 2010, Sun et al., 2011].

In semiactive suspension, the value of the damper coefficient can be controlled and can show reasonable performance as compared to that of an active suspension control. Besides, it does not require external energy. For instance, in the work by [Rashid et al., 2011] a semiactive suspension control of a quarter-car model using a hybrid-fuzzy-logic-based controlled is developed and implemented. [Wang et al., 2003] formulated a force-tracking PI controller for an MR-damper controlled quarter-car system. The preliminary results showed that the proposed semiactive force tracking PI control scheme could provide effective control of the sprung mass resonance as well as the wheel-hop control. Furthermore, the proposed control yields lower magnitudes of mass acceleration in the ride zone. [Yao et al., 2002] designed a semi active suspension system using a magnetorhelogical damper. The control law was formulated following the sky-hook technique in which the direction of the relative velocity between the sprung and unsprung masses is compared to that of the velocity of the unsprung mass. Depending on this result, an on-off action is performed. [Du et al., 2005] designed a semiactive static output

Backstepping is a recursive design for systems with nonlinearities not constrained by linear bounds. 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, 2010, Wang et al., 2009, Tong et al., 2009, Liu et al., 2010, Chen et al., 2010]. Backstepping control has also been explored in some works about suspension systems. For example, [Zapateiro et al., 2009a] designed a semiactive backstepping control combined with neural network (NN) techniques for a system with MR damper. 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., 2001] studied a hybrid control of active suspension systems for quarter-car models with two-degrees-of-freedom. This hybrid control was implemented by controlling the linear part with

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., 2004] analyzed

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., 2012]. VSC was firstly studied in the early 1950’s for systems represented by single-input high-order differential equations. A rise of interest became in the 1970’s because the robustness of VSC were step by step recognized. This control method can make the system completely insensitive to time-varying parameter uncertainties, multiple delayed state perturbations and external disturbances [Pai, 2010]. 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., 2012]), electric and electronic engineering (speed control of an induction motor drive [Barambones and Alkorta, 2011]). 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. The first strategy that we propose in this work,

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.

## 2. Suspension system model

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. The tyre subsystem is represented by the wheel mass

where:

Taking

Tyre subsystem:

Suspension subsystem:

where

where

where

## 3. Variable structure controller formulation

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 (

It is well known that the force generated by the MR damper cannot be commanded; only the voltage

where

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. More precisely, the shadowed area in Figure 2 is the area where

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 (

Variable structure controllers (VSC) are a very large class of robust controllers [Gao Hung, 1993]. 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:

where

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, 2002]. As a consequence, the stability of the closed-loop system is guaranteed.

## 4. Backstepping controller formulation

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. The adaptive backstepping controller will be designed in such a way that, for a given

where

In order to formulate the backstepping controller, the state space model (3) - (4) must be first written in strict feedback form [Krstic et al., 1995]. Therefore, the following coordinate transformation is performed [Karlsson et al., 2001]:

The system, represented in the new coordinates, is given by:

Tyre subsystem:

Suspension subsystem:

Substitution of the expression for

where

Assume that

Let

If the poles of the transfer functions (26) and (27) are in the left side of the

Finally, since

In order to begin with the adaptive backstepping design, we firstly define the following error variable and its derivative:

Now, the following Lyapunov function candidate is chosen:

whose first-order derivative is:

Equation (30) can be stabilized with the following virtual control input:

where

Therefore,

On the other hand, the derivatives of the errors of the uncertain parameter estimations are given by:

Now, an augmented Lyapunov function candidate is chosen:

Thus, by using (35) - (39) and the fact that

Now consider the following adaptation laws:

Substitution of (42) and (43) into (41) yields:

By choosing the following control law:

with

The objective of guaranteeing global boundedness of trajectories is equivalently expressed as rendering

or, equivanlently,

Then, it can be shown that

Thus, the adaptive backstepping controller satisfies both the H

The control force given by (45) can be used to drive an actively controlled damper. However, the fact that semiactive devices cannot inject energy into a system, makes necessary the modification of this control law in order to implement it with a semiactive damper; that is, semiactive dampers cannot apply force to the system, only absorb it. There are different ways to perform this [Zapateiro et al., 2009b, Arash et al., 2010]. In this work, we will calculate the MR damper voltage making use of its mathematical model. Thus, the following control law is proposed:

provided that

The same process followed to obtain the control law (45) can be used to demonstrate that the control law (4) does stabilize the system. Begin by replacing (6) into (44) in order to obtain:

Thus, by replacing the control law of (4) into (51) we also get

Finally, we can write the control law in terms of the state variables as follows:

## 5. Numerical simulations

In this section we will analyze the performance results obtained form simulations performed in Matlab/SImulink. The numerical values of the model that we used in this study. Thus:

We assume that the car has laser sensors that allow us to read the position of the sprung and unsprung masses. Since the velocities are needed for control implementation, these are obtained by first low-pass filtering the displacement readings and then applying a filter of the form

Index | Definition |

Norm. peak suspension deflection. | |

Norm. peak sprung mass velocity. | |

Norm. peak sprung mass acceleration. | |

Norm. suspension deflection ITSE. | |

Norm. sprung mass velocity ITSE. | |

Norm. sprung mass acceleration ITSE. | |

Maximum control effort. |

In the first scenario, the unevenness of the road was simulated by random vibration, as shown in Figure 4. This figure also compares the performance of the three

Index | Backstepping | |||

0.1288 | 0.1280 | 0.0993 | 0.5144 | |

0.1485 | 0.1488 | 0.2157 | 0.9800 | |

0.2205 | 0.2803 | 0.2803 | 1.2586 | |

0.0059 | 0.0058 | 0.0053 | 0.2317 | |

0.0090 | 0.0089 | 0.0181 | 0.5852 | |

0.0189 | 0.0187 | 0.0310 | 0.9615 | |

0.1268 | 0.1279 | 0.1471 | 0.4538 |

In the second scenario a bump on the road is simulated as seen in Figure 8. In this case, the VSC controllers have a similar performance and it happened in the previous scenario. The performance indices of Table 2 confirm this fact. In comparison, the

Index | Backstepping | |||

0.8317 | 0.8325 | 0.4584 | 0.4271 | |

1.1507 | 1.1505 | 1.3430 | 1.3892 | |

1.1157 | 1.2623 | 1.2623 | 1.3007 | |

0.1605 | 0.1625 | 0.2241 | 0.0703 | |

0.1827 | 0.1797 | 0.2681 | 0.4702 | |

0.4168 | 0.4113 | 0.5884 | 1.0308 | |

0.3613 | 0.3614 | 0.4100 | 0.4431 |

## 6. Conclusions

In this chapter we presented the problem of the vibration control in vehicles. One model-based and three variable structure controllers were analyzed and compared in order to study their performance during typical road disturbances. The performance of the controller were also analyzed for the particular situation in which the suspension system is made up of a magnetorheological damper, which is well-known to be a nonlinear device. All of the controllers performed satisfactorily at regulating the suspension deflection while keeping the acceleration, velocity and displacement variables within acceptable limits. One important result obtained in this work was that despite the simplicity of these controllers, they performed significantly better than the model-based controller. It is to be noted that further studies -theoretical and experimental- should be performed in order to get a better insight of the performance of such controllers and the possibilities of being used in real systems.

## Acknowledgement

M. Zapateiro has been supported by the ‘Juan de la Cierva’ fellowship from the Government of Spain. This study has also been partially supported by Secretaría de Estado de Investigación, Desarrollo e Innovación (formerly,Ministry of Science and Innovation) through DPI2011-27567-C02-02, DPI2011-28033-C03-01 and DPI 2011-27567-C02-01.