Finite Frequency Vibration Control for Polytopic Active Suspensions via Dynamic Output Feedback

This paper presents a disturbance attenuation strategy for active suspension systems with frequency band constraints, where dynamic output feedback control is employed in consideration that not all the state variables can be measured on-line. In view of the fact that human are sensitive to the virbation between 4–8Hz in vertical direction, the H ∞ control based on generalized Kalman-Yakubovich-Popov (KYP) lemma is developed in this specific frequency, in order to achieve the targeted disturbance attenuation. Moreover, practical constraints required in active suspension design are guaranteed in the whole time domain. At the end of the paper, the outstanding performance of the system using finite frequency approach is confirmed by simulation.


Introduction
By reason of rough road conditions, passengers in the car are often in a vibration environment which negatively impacts the comfort, mental, and physical health of them, and suspensions are crucial to attenuate the disturbance transferred to passengers [1][2][3].Hence, various approaches various approaches have been developed that aim to enhance suspensions' performance such as adaptive control [4], robust control [5,6], and fuzzy control [7].Generally speaking, there are three types of suspensions: passive, semiactive, and active suspensions.Compared with the other two kinds of suspensions, active suspensions have a greater potential to improve ride comfort and to guarantee the ride safety due to the existence of active actuator.
There are three performance requirements for active suspension systems.One is the ride comfort, which requires isolation of vibration from road; another one is handling performance mainly described by road holding, which restricts the hop of wheel in order to ensure continuous contact of wheels to road; the last one is the sprung displacement which limits the suspension stroke within an allowable band.
However, these requirements are usually conflicting.For instance, a large suspension displacement may exist if a better rider comfort performance is required .A variety of control strategies have been applied to cope with this conflict [8][9][10][11][12].In particular, because the  ∞ norm index can measure the vibration attenuation performance of system appropriately [13], many suspension problems are considered by  ∞ control theory [14][15][16][17][18][19][20][21][22].In this paper, the handling performance and the suspension stroke are regarded as constraints, and the ride comfort is deemed as the main index to optimize.
Although various control strategies have been applied to promote ride comfort performance of suspension systems, few of them notice the fact that due to the human body structure and other factors people are more sensitive to disturbances in 4-8 Hz than other frequency in vertical direction (ISO2361).Therefore, it is considerable to develop a finite frequency strategy to reduce the negative effect caused by disturbances in 4-8 Hz.The generalized Kalman-Yakubovich-Popov (KYP) lemma, which has been used to solve practical problems [23,24], is applied to achieve the finite frequency control to active suspensions.
It should be mentioned that the parameters of passengers including model for suspension system could vary due to the mass change of passengers, so how to guarantee the performance of suspension with varying parameters is worth discussing.Meanwhile, considering that the mass of passengers

Quarter Car Suspension Model
The model of a quarter car suspension is shown in Figure 1.  and   stand for sprung and unsprung mass, respectively.  ,   , and   denote the sprung, unsprung displacement, and disturbance displacement from the road, respectively.  ,   ,   , and   are the stiffnesses and dampings of the suspension system, respectively.The input of the controller is denoted by .
Based on the law of Newton, the motion equation of suspension can be denoted as Define the following state variables and the disturbance input: Then ( 1) is equivalent to where Define which reflects the acceleration of   that contains the body mass of passengers and seat.In the design of control law for suspension system, body acceleration is the main index that needs to be optimized.The handling performance requires continuous contact of wheel to road, which means that the suspension system needs to guarantee that dynamic tire load is less than static load, namely, The stroke of the suspension could not be so large that it may exceed the maximum, which can be formulated as The state space expression is described integrally as Mathematical Problems in Engineering 3 where , , and  1 are same with the definition in (4), and 1 () reflects the acceleration output of   ;  2 () represents for the relative (normalized) constraints output; () stands for the output of measurable states.
The parameter-varying model is depicted as the following polytopic form: where and   max ,   min stand for the maximum and minimum of   , respectively.The form of dynamic output feedback controller is described as Substituting ( 12) into (8), we get where Denote as the transfer function from () to  1 ().
The  ∞ norm of transfer function matrix  is applied to depict the ride comfort performance of suspension system, which is defined as where The control target is summarized as follows.
For a certain , design a dynamic output feedback in the form of (12) which satisfies (I) The closed loop system is asymptotically stable.(II) The finite frequency (from  1 to  2 )  ∞ norm from road disturbance to vehicle acceleration is less than .namely, (III) The relative constraint responses shown in the following can be satisfied as long as the disturbance energy is less than the maximum of the 2-norm of disturbance input denoted as  max .that is,

Dynamic Output Feedback Controller Design for Polytopic Suspension System
In this section, we will derive three theorems for the design of output feedback controller that satisfies (19) and one theorem of full frequency controller used for comparison.

Finite Frequency Design
Theorem 1.For the given parameters , ,  > 0, if there exist symmetric matrices (  ), positive definite symmetric matrices   (  ), (  ), and general matrix (  ) satisfying where then a dynamic output feedback controller exists, which satisfies the requirements of (I), (II) and (III) with  max = /.

Applying reciprocal projection theorem (see Appendix A) and choosing 𝑆
Multiplying the above inequality from both the left and the right sides by   (  ), we get which guarantees obviously.From the standard Lyapunov theory for continuous-time linear system, the closed-loop system ( 13) is asymptotically stable with () = 0.By substitution, inequality ( 21) is equivalent to where Based on projection lemma (see Appendix A), inequality (30) is equivalent to Noting that inequality (33) is eternal establishment, we just need to consider inequality (32), which is equivalent to where Rewrite inequality (34) as and then we obtain Applying generalized KYP lemma (see Appendix A), we get where sup Select as the Lyapunov function, we obtain Applying the following inequality and ( 28) to (42), we get Integrate (44) from 0 to : Substituting ( 41) into (45) with (0) = 0, we get which is equivalent to where where  max stands for the maximum eigenvalue, we can therefore guarantee constraints mentioned in (19) as long as which is equivalent to (22) by applying Schur complement.
Before giving a convex expression which can be solved by LMI Toolbox, we firstly perform transformation to inequalities (20), (21), and (22).
Decompose matrix (  ) for convenience in the following form: According to the literature [25], we assume that both (  ) and (  ) are invertible without loss of generality. Denote and perform the congruence transformation to inequalities (20), (21), and ( 22) by respectively.Denote Mathematical Problems in Engineering 7 then we get the following theorem.
Theorem 2. For the given parameters , ,  > 0, if there exist symmetric matrix P(  ), positive definite symmetric matrices P (  ), Q(  ), and general matrices Â (  ), B (  ), Ĉ (  ), D (  ), Ŵ(  ), (  ), (  ), (  ) satisfying then a dynamic output feedback controller exists, which satisfies the requirements of (I), (II) and (III) with  max = /.The corresponding controller in the form of (12) can be given by (57) Though a parameter-dependent controller can be designed via Theorem 2, it is difficult to obtain targeted matrices in real time as   varies.Therefore, we give a tractable LMI-based theorem as follows.
Theorem 3.For the given parameters , , and  > 0, if there exist symmetric matrix P , positive definite symmetric matrices P , Q , and general matrices Â , B , Ĉ , D , Ŵ ,   ,   ,   ( = 1, 2), satisfying where then a dynamic output feedback controller exists, which satisfies the requirements of (I), (II) and (III) with  max = /.The corresponding controller in the form of ( 12) can be given by where and   ( = 1, 2) can be calculated by (11).
Proof.we just prove that inequality (58) is sufficient to inequality (54) for simplicity.Denote which stands for the left of inequality (54).Inequality (58) is equivalent to where Assume that and then we get which is negative definite by inequality (65), that is, Remark 4. (  ), (  ) should be chosen to meet the definition, that is, However, the value of (  ) and (  ) can be chosen variously for the given (  ), (  ), and (  ).In this paper, we use the singular value decomposition approach.
Remark 5. Based on [26], for real matrices  1 and  2 , Thusly, (59) can be converted into real matrix inequality by defining Remark 6.The matrices of dynamic output feedback controller   (  ),   (  ),   (  ),   (  ), and Lyapunov matrix   (  ) are dependent nonlinearly on parameter   .For instance, as can be seen in the above deduction, matrix   (  ) in Theorem 3 can be formulated as where D1 and D2 are the corresponding matrix solutions of LMIs in Theorem 3, and the value of matrix   (  ) may vary with the change of   .
Remark 7. Inequalities from (58) to (60) can be simplified from 12 LMIs to 4 LMIs if the following matrix variables are chosen as and  1 =  2 .However, this simplification will keep the matrices for the designed controller constant for all   , and an invariant Lyapunov function, rather than a parameterdependent one, has to be used for the whole domain, which will bring a larger conservativeness than Theorem 3.
The same approach mentioned in Remark 4 is used to obtain the value of   (  ) and   (  ), which should meet the following equation:

A Design Example
In this section, we will show how to apply the above theorems to design finite frequency controller and full frequency controller for a specific suspension system.The parameter for the suspension is shown in Table 1.
We first choose the following parameters:  = 1,  = 10000,  max = 0.1 m,  1 = 8rad/s, and  2 = 16rad/s.Then, the matrices in ( 12) can be calculated by applying Theorem 3  Then, the curves of maximum singular values of systems using open-loop, finite frequency (parameter-dependent), and full frequency controllers are shown in Figure 2. Compared with the other two curves (the open-loop system and the system with full frequency controller), the system with finite frequency controller has the least  ∞ norm of the three systems in 4-8 Hz, which indicates that the finite frequency controller has a better effect on the attenuation of targeted frequency disturbance.
In order to examine the performance of finite frequency controller, we assume that the disturbance is in the form of where ,  stand for the amplitude and the frequency of disturbance, respectively, and  = 1/.Suppose that  is 0.4 m and  is 5 Hz, and then the body acceleration and the relative constraints responses to this disturbance are shown in Figures 3, 4, and 5, respectively.
It is obvious that the body acceleration response for finite frequency controller decreases faster with respect to time than the other two controllers, and at the same time, both the relative dynamic tire load response and relative suspension stroke response are within the allowable range, which satisfy requirement (III), namely, |{ 2 ()}  | < 1,  = 1, 2.

Conclusion
In this paper, we manage to design a dynamic output feedback controller for active suspensions with practical   constraints included.This controller particularly diminishes disturbance at 4-8 Hz, which is the frequency sensitive band for human.Besides, for the reason that the controller is a parameter-dependent one, it has a smaller conservativeness than controller designed on the basis of quadratic stability and constant parameter feedback.The excellent performance of the closed-loop system with finite frequency controller has been demonstrated by simulation.The parameter matrices of the dynamic output feedback controller for Theorem 8:

Figure 1 :
Figure 1: The quarter car model.

Figure 2 :
Figure 2: The curves of maximum singular values.

2 )Figure 3 :
Figure 3: The time response of body acceleration.

Figure 4 :
Figure 4: The time response of relative dynamic tire load.

Figure 5 :
Figure 5: The time response of relative suspension stroke.

Table 1 :
The model parameters of active suspensions.