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

Abstract

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 4–8 Hz 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.

1. 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 [1–3]. 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 [8–12]. In particular, because the 𝐻_∞ norm index can measure the vibration attenuation performance of system appropriately [13], many suspension problems are considered by 𝐻_∞ control theory [14–22]. 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 can be accessed on line, in this paper, the suspension model is described as a polytope, and then a parameter-dependent control law is proposed to assure the above performance requirements, which can also reach a lower conservativeness than control law based on quadratic stability and constant parameter feedback at the same time. In addition, though state feedback may attain a superior performance compared with static output feedback, measuring some states may bring burden to the systems. Thusly, constructing dynamic output feedback is desirable, which can achieve a relative enhanced performance and meanwhile reduce state-measuring sensors.

The paper is organized as follows. In Section 2 the state-space model for quarter car suspension is presented. In Section 3 the theorems which can be used to design the dynamic feedback controller are illustrated. The simulation is presented in Section 4, and the conclusion is in Section 5.

Notation. For a matrix 𝑃, 𝑃^𝑇, 𝑃^∗, 𝑃⁻¹, 𝑃^−𝑇, and  𝑃^⟂ stand for its transpose, conjugate transpose, inverse, transposed inverse, and orthogonal complement, respectively; sym(𝑃) denotes 𝑃+𝑃^𝑇. 𝑃>0(𝑃<0) means that matrix 𝑃 is positive (negative) definite. For a matrix, {⋅}_𝑖 stands for the 𝑖th line of the matrix. ‖𝐺(𝑗𝜔)‖_∞ stands for the 𝐻_∞ norm of transfer function matrix 𝐺. For matrices 𝑃 and 𝑄, 𝑃⊗𝑄 means the Kronecker product. In symmetric block matrices or complex matrix expressions, we use an asterisk (∗) to represent a term that is induced by symmetry.

2. Quarter Car Suspension Model

The model of a quarter car suspension is shown in Figure 1. 𝑚_𝑠 and 𝑚_𝑢 stand for sprung and unsprung mass, respectively. 𝑧_𝑠, 𝑧_𝑢, and 𝑧_𝑟 denote the sprung, unsprung displacement, and disturbance displacement from the road, respectively. 𝑘_𝑠, 𝑘_𝑡, 𝑐_𝑠, and 𝑐_𝑡 are the stiffnesses and dampings of the suspension system, respectively. The input of the controller is denoted by 𝑢.

Figure 1 

The quarter car model.

Based on the law of Newton, the motion equation of suspension can be denoted as 𝑚_𝑠𝑧̈_𝑠(𝑡)+𝑐_𝑠𝑧̇_𝑠(𝑡)−𝑧̇_𝑢(𝑡)+𝑘_𝑠𝑧_𝑠(𝑡)−𝑧_𝑢(𝑡(𝑡),𝑚)=𝑢_𝑢𝑧̈_𝑢(𝑡)−𝑐_𝑠𝑧̇_𝑠(𝑡)−𝑧̇_𝑢(𝑡)−𝑘_𝑠𝑧_𝑠(𝑡)−𝑧_𝑢(𝑡)+𝑘_𝑡𝑧_𝑢(𝑡)−𝑧_𝑟(𝑡)−𝑐_𝑡𝑧̇_𝑟(𝑡)−𝑧̇_𝑢(𝑡)=−𝑢(𝑡). (1)

Define the following state variables and the disturbance input: 𝜉₁(𝑡)=𝑧_𝑠(𝑡)−𝑧_𝑢(𝑡),𝜉₂(𝑡)=𝑧_𝑢(𝑡)−𝑧_𝑟(𝑡),𝜉₃(𝑡)=𝑧̇_𝑠(𝑡),𝜉₄(𝑡)=𝑧̇_𝑢(𝑡),𝑤(𝑡)=𝑧̇_𝑟(𝑡). (2) Then (1) is equivalent to 𝜉̇(𝑡)=𝐴𝑚_𝑠𝜉(𝑡)+𝐵𝑚_𝑠𝑢(𝑡)+𝐵₁𝑚_𝑠𝑤(𝑡), (3) where 𝜉(𝑡)=𝜉₁(𝑡)𝜉₂(𝑡)𝜉₃(𝑡)𝜉₄(𝑡)^𝑇,𝐴𝑚_𝑠=−𝑘001−10001_𝑠𝑚_𝑠𝑐0−^𝑠𝑚_𝑠𝑐_𝑠𝑚_𝑠𝑘_𝑠𝑚_𝑢−𝑘_𝑡𝑚_𝑢𝑐_𝑠𝑚_𝑢−𝑐_𝑠+𝑐_𝑡𝑚_𝑢,𝐵𝑚_𝑠=100𝑚_𝑠−1𝑚_𝑢^𝑇,𝐵₁𝑚_𝑠=𝑐0−10^𝑡𝑚_𝑢^𝑇. (4) Define 𝑧_𝑜1(𝑡)=𝑧̈_𝑠(𝑡), (5) which reflects the acceleration of 𝑚_𝑠 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, 𝑘_𝑡𝑧_𝑢(𝑡)−𝑧_𝑟(𝑡)<𝑚_𝑠+𝑚_𝑢𝑔. (6)

The stroke of the suspension could not be so large that it may exceed the maximum, which can be formulated as 𝑧_𝑠(𝑡)−𝑧_𝑢(𝑡)<𝑧_max. (7)

The state space expression is described integrally as 𝜉̇(𝑡)=𝐴𝑚_𝑠𝜉(𝑡)+𝐵𝑚_𝑠𝑢(𝑡)+𝐵₁𝑤(𝑡),𝑧_𝑜1(𝑡)=𝐶₁𝑚_𝑠𝜉(𝑡)+𝐷₁𝑚_𝑠𝑢(𝑡),𝑧_𝑜2(𝑡)=𝐶₂𝑚_𝑠𝜉(𝑡),𝑦(𝑡)=𝐶𝜉(𝑡), (8) where 𝐴, 𝐵, and 𝐵₁ are same with the definition in (4), and 𝐶₁𝑚_𝑠=−𝑘_𝑠𝑚_𝑠𝑐0−^𝑠𝑚_𝑠𝑐_𝑠𝑚_𝑠,𝐶₂𝑚_𝑠=1𝑧_max0𝑘000_𝑡𝑚_𝑠𝑔+𝑚_𝑢𝑔,𝐷00₁𝑚_𝑠=1𝑚_𝑠,𝐶=.100001000010 (9)

𝑧_𝑜1(𝑡) reflects the acceleration output of 𝑚_𝑠; 𝑧_𝑜2(𝑡) represents for the relative (normalized) constraints output; 𝑦(𝑡) stands for the output of measurable states.

The parameter-varying model is depicted as the following polytopic form: 𝐴𝑚_𝑠,𝐵𝑚_𝑠,𝐵₁𝑚_𝑠,𝐶₁𝑚_𝑠,𝐶₂𝑚_𝑠,𝐷₁𝑚_𝑠=²_𝑖=1𝜆_𝑖𝐴_𝑖,𝐵_𝑖,𝐵_1𝑖,𝐶_1𝑖,𝐶_2𝑖,𝐷_1𝑖, (10) where 𝐴₁,𝐵₁,𝐵₁₁,𝐶₁₁,𝐶₂₁,𝐷₁₁=𝐴𝑚_𝑠max,𝐵𝑚_𝑠max,𝐵₁𝑚_𝑠max,𝐶₁𝑚_𝑠max,𝐶₂𝑚_𝑠max,𝐷₁𝑚_𝑠max,𝐴₂,𝐵₂,𝐵₁₂,𝐶₁₂,𝐶₂₂,𝐷₁₂=𝐴𝑚_𝑠min,𝐵𝑚_𝑠min,𝐵₁𝑚_𝑠min,𝐶₁𝑚_𝑠min,𝐶₂𝑚_𝑠min,𝐷₁𝑚_𝑠min,𝜆₁=1/𝑚_𝑠−1/𝑚_𝑠min1/𝑚_𝑠max−1/𝑚_𝑠min,𝜆₂=1/𝑚_𝑠max−1/𝑚_𝑠1/𝑚_𝑠max−1/𝑚_𝑠min, (11) and 𝑚_𝑠max, 𝑚_𝑠min stand for the maximum and minimum of 𝑚_𝑠, respectively.

The form of dynamic output feedback controller is described as 𝜂̇(𝑡)=𝐴_𝐾𝑚_𝑠𝜂(𝑡)+𝐵_𝐾𝑚_𝑠𝑦(𝑡),𝑢(𝑡)=𝐶_𝐾𝑚_𝑠𝜂(𝑡)+𝐷_𝐾𝑚_𝑠𝑦(𝑡). (12) Substituting (12) into (8), we get 𝑥̇(𝑡)=𝐴𝑚_𝑠𝑥(𝑡)+𝐵𝑚_𝑠𝑤(𝑡),𝑧_𝑜1(𝑡)=𝐶₁𝑚_𝑠𝑥(𝑡),𝑧_𝑜2(𝑡)=𝐶₂𝑚_𝑠𝑥(𝑡), (13) where 𝑥(𝑡)=𝜉(𝑡)𝜂(𝑡),𝐴𝑚_𝑠=𝐴𝑚_𝑠+𝐵𝑚_𝑠𝐷_𝐾𝑚_𝑠𝐶𝐵𝑚_𝑠𝐶_𝐾𝑚_𝑠𝐵_𝐾𝑚_𝑠𝐶𝐴_𝐾𝑚_𝑠,𝐵𝑚_𝑠=𝐵₁𝑚_𝑠0,𝐶₁𝑚_𝑠=𝐶₁𝑚_𝑠+𝐷₁𝑚_𝑠𝐷_𝐾𝑚_𝑠𝐶𝐷₁𝑚_𝑠𝐶_𝐾𝑚_𝑠,𝐶₂𝑚_𝑠=𝐶₂𝑚_𝑠0. (14) Denote 𝐺𝑗𝜔=𝐶₁𝑚_𝑠𝑗𝜔𝐼−𝐴𝑚_𝑠⁻¹𝐵𝑚_𝑠, (15) as the transfer function from 𝑤(𝑡) to 𝑧_𝑜1(𝑡).

The 𝐻_∞ norm of transfer function matrix 𝐺 is applied to depict the ride comfort performance of suspension system, which is defined as 𝐺𝑗𝜔_∞=sup_{0∉𝑤∈𝐿2[0,+∞)}𝑧_𝑜2𝐿2‖𝑤‖_𝐿2, (16) where 𝑧_𝑜2𝐿2=₀^+∞𝑧_𝑜2(𝑡)²𝑑𝑡^1/2,‖𝑤‖_𝐿2=₀^+∞‖𝑤(𝑡)‖²𝑑𝑡^1/2. (17)

The control target is summarized as follows.

For a certain 𝛾, 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 𝜔₁ to 𝜔₂) 𝐻_∞ norm from road disturbance to vehicle acceleration is less than 𝛾. namely, 𝐺𝑗𝜔_{∞𝜔1<𝜔<𝜔2}<𝛾. (18)(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 𝑤_max. that is, 𝑧_𝑜2(𝑡)_𝑘<1,𝑘=1,2. (19)

3. 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.

3.1. Finite Frequency Design

Theorem 1. For the given parameters 𝛾,𝜂,𝜌>0, if there exist symmetric matrices 𝑃(𝑚_𝑠), positive definite symmetric matrices 𝑃_𝑆(𝑚_𝑠), 𝑄(𝑚_𝑠), and general matrix 𝑊(𝑚_𝑠) satisfying−sym𝑊𝑚_𝑠−𝑊^𝑇𝑚_𝑠𝐴𝑚_𝑠+𝑃_𝑆𝑚_𝑠−𝑊^𝑇𝑚_𝑠−𝑊^𝑇𝑚_𝑠𝐵𝑚_𝑠∗−𝑃_𝑆𝑚_𝑠00∗∗−𝑃_𝑆𝑚_𝑠0Ω∗∗∗−𝜂𝐼<0,₁Ω₂∗Ω₃√<0,−𝐼𝜌𝐶₂𝑚_𝑠𝑘∗−𝑃_𝑆𝑚_𝑠<0,𝑘=1,2, (20)(21)(22)whereΩ₁=−𝜔₁𝜔₂𝑄𝑚_𝑠−sym𝐴^𝑇𝑚_𝑠𝑊𝑚_𝑠𝑃𝑚_𝑠−𝑗𝜔_𝑐𝑄𝑚_𝑠+𝑊^𝑇𝑚_𝑠𝑃𝑚_𝑠+𝑗𝜔_𝑐𝑄𝑚_𝑠+𝑊𝑚_𝑠−𝑄𝑚_𝑠,Ω₂=−𝑊^𝑇𝑚_𝑠𝐵𝑚_𝑠𝐶^𝑇₁𝑚_𝑠,Ω00₃=−𝛾²,𝜔𝐼00−𝐼_𝑐=𝜔₁+𝜔₂2, (23)then a dynamic output feedback controller exists, which satisfies the requirements of (I), (II) and (III) with 𝑤_max=𝜌/𝜂.

Proof. By using Schur complement, inequality (20) is equivalent to Γ−𝑊^𝑇𝑚_𝑠𝐴𝑚_𝑠+𝑃_𝑆𝑚_𝑠∗−𝑃_𝑆𝑚_𝑠<0, (24) where 1Γ=𝜂𝑊^𝑇𝑚_𝑠𝐵𝑚_𝑠𝐵^𝑇𝑚_𝑠𝑊𝑚_𝑠+𝑊^𝑇𝑚_𝑠𝑃_𝑆⁻¹𝑚_𝑠𝑊𝑚_𝑠+sym𝑊𝑚_𝑠. (25)
Multiplying (24) by diag{−𝑊^−𝑇(𝑚_𝑠),𝑃_𝑆^−𝑇(𝑚_𝑠)} from the left side and by diag{−𝑊⁻¹(𝑚_𝑠),𝑃_𝑆⁻¹(𝑚_𝑠)} from the right side, respectively, we obtain1𝜂𝐵𝑚_𝑠𝐵^𝑇𝑚_𝑠+𝑃_𝑆⁻¹𝑚_𝑠−sym𝐿𝑚_𝑠𝐴𝑚_𝑠𝑃_𝑆⁻¹𝑚_𝑠+𝐿^𝑇𝑚_𝑠∗−𝑃_𝑆⁻¹𝑚_𝑠<0, (26)where 𝐿=−𝑊⁻¹.
Applying reciprocal projection theorem (see Appendix A) and choosing 𝑆^𝑇(𝑚_𝑠)=𝐴(𝑚_𝑠)𝑃_𝑆⁻¹(𝑚_𝑠), Ψ(𝑚_𝑠)=(1/𝜂)𝐵(𝑚_𝑠)𝐵^𝑇(𝑚_𝑠), inequality (26) is equivalent to 𝐴𝑚_𝑠𝑃_𝑆⁻¹𝑚_𝑠+𝑃_𝑆⁻¹𝑚_𝑠𝐴^𝑇𝑚_𝑠+1𝜂𝐵𝑚_𝑠𝐵^𝑇𝑚_𝑠<0. (27)
Multiplying the above inequality from both the left and the right sides by 𝑃_𝑆(𝑚_𝑠), we get 𝑃_𝑆𝑚_𝑠𝐴𝑚_𝑠+𝐴^𝑇𝑚_𝑠𝑃_𝑆𝑚_𝑠+1𝜂𝑃_𝑆𝑚_𝑠𝐵𝑚_𝑠𝐵^𝑇𝑚_𝑠𝑃_𝑆𝑚_𝑠<0, (28) which guarantees 𝑃_𝑆𝑚_𝑠𝐴𝑚_𝑠+𝐴^𝑇𝑚_𝑠𝑃_𝑆𝑚_𝑠<0, (29) obviously. From the standard Lyapunov theory for continuous-time linear system, the closed-loop system (13) is asymptotically stable with 𝑤(𝑡)=0.
By substitution, inequality (21) is equivalent to 𝐼𝐹_𝐵𝑚_𝑠Ω𝑚_𝑠𝐼𝐹_𝐵𝑚_𝑠^𝑇+sym𝐹_𝐴𝑚_𝑠𝑊𝑚_𝑠𝑅<0, (30) where 𝐹_𝐴𝑚_𝑠=−𝐴^𝑇𝑚_𝑠𝐼−𝐵^𝑇𝑚_𝑠,𝐹_𝐵𝑚_𝑠=𝐶^𝑇₁𝑚_𝑠00,,Ω𝑚𝑅=𝐼00_𝑠=𝑇Φ⊗𝑃_𝑠𝑚_𝑠+Ψ⊗𝑄𝑚_𝑠0𝑇0Π^𝑇,Π=𝐼00−𝛾²𝐼,Φ=,0110Ψ=−1𝑗𝜔_𝑐−𝑗𝜔_𝑐−𝜔₁𝜔₂,𝑇=.0𝐼00𝐼000000𝐼00𝐼0 (31)
Based on projection lemma (see Appendix A), inequality (30) is equivalent to 𝐹^⟂_𝐴𝑚_𝑠𝐼𝐹_𝐵𝑚_𝑠Ω𝑚_𝑠𝐼𝐹_𝐵𝑚_𝑠^𝑇𝐹^⟂_𝐴𝑚_𝑠^𝑇𝑅<0,_𝑇⟂𝐼𝐹_𝐵𝑚_𝑠Ω𝑚_𝑠𝐼𝐹_𝐵𝑚_𝑠^𝑇𝑅^𝑇⟂𝑇<0. (32)(33)
Noting that inequality (33) is eternal establishment, we just need to consider inequality (32), which is equivalent to 𝐹^𝑇𝑚_𝑠Ω𝑚_𝑠𝐹𝑚_𝑠<0, (34) where 𝐹𝑚_𝑠=𝐼𝐴^𝑇𝑚_𝑠0𝐶^𝑇₁𝑚_𝑠0𝐵^𝑇𝑚_𝑠𝐼0^𝑇. (35)
Rewrite inequality (34) as 𝐴𝑚_𝑠𝐵𝑚_𝑠𝐼0𝐶₁𝑚_𝑠00𝐼^𝑇Φ⊗𝑃𝑚_𝑠+Ψ⊗𝑄𝑚_𝑠0×0Π𝐴𝑚_𝑠𝐵𝑚_𝑠𝐼0𝐶₁𝑚_𝑠00𝐼<0, (36) and then we obtain 𝐴𝑚_𝑠𝐵𝑚_𝑠𝐼0^𝑇Φ⊗𝑃𝑚_𝑠+Ψ⊗𝑄𝑚_𝑠×𝐴𝑚_𝑠𝐵𝑚_𝑠+𝐼0𝐶₁𝑚_𝑠00𝐼^𝑇×Π𝐶₁𝑚_𝑠00𝐼<0. (37)
Applying generalized KYP lemma (see Appendix A), we get Ξ^∗ΠΞ<0,𝜔₁<𝜔<𝜔₂, (38) where Ξ=𝐶₁𝑚_𝑠00𝐼𝑗𝜔𝐼−(𝐴𝑚_𝑠⁻¹𝐵𝑚_𝑠𝐼, (39) namely, sup_{𝜔1<𝜔<𝜔2}𝐺𝑗𝜔_∞<𝛾. (40)
Select 𝑉(𝑡)=𝑥^𝑇(𝑡)𝑃_𝑆𝑚_𝑠𝑥(𝑡) (41) as the Lyapunov function, we obtain 𝑉̇(𝑡)=2𝑥^𝑇(𝑡)𝑃_𝑆𝑚_𝑠𝐴𝑚_𝑠𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑃_𝑆𝑚_𝑠𝐵𝑚_𝑠𝑤(𝑡). (42)
Applying the following inequality 2𝑥^𝑇(𝑡)𝑃_𝑆𝑚_𝑠𝐵𝑚_𝑠𝑤(𝑡)≤1𝜂𝑥^𝑇(𝑡)𝑃_𝑠𝑚_𝑠𝐵𝑚_𝑠𝐵^𝑇𝑚_𝑠𝑃_𝑆𝑚_𝑠(𝑥)𝑡+𝜂𝑤^𝑇(𝑡)𝑤(𝑡),∀𝜂>0, (43) and (28) to (42), we get 𝑉̇(𝑡)≤𝜂𝑤^𝑇(𝑡)𝑤(𝑡). (44)
Integrate (44) from 0 to 𝑡: 𝑉(𝑡)−𝑉(0)≤𝜂^𝑡₀𝑤^𝑇(𝑡)𝑤(𝑡)𝑑𝑡=𝜂‖𝑤(𝑡)‖₂≤𝜂𝑤_max. (45)
Substituting (41) into (45) with 𝑉(0)=0, we get 𝑥^𝑇(𝑡)𝑃_𝑆𝑚_𝑠𝑥(𝑡)≤𝑉(0)+𝜂𝑤_max=𝜌, (46) which is equivalent to 𝑥^𝑇(𝑡)𝑥(𝑡)≤𝜌, (47) where 𝑥(𝑡)=𝑃_𝑆^1/2(𝑚_𝑠)𝑥(𝑡).
Noting that max𝑧_𝑜2(𝑡)_𝑘²=max𝑥^𝑇(𝑡)𝑃_𝑆^−1/2𝑚_𝑠𝐶₂𝑚_𝑠^𝑇_𝑘×𝐶₂𝑚_𝑠𝑘𝑃_𝑆^−1/2𝑚_𝑠𝑥(𝑡)₂≤𝜌𝜃_max𝑃_𝑆^−1/2𝑚_𝑠𝐶₂𝑚_𝑠^𝑇_𝑘𝐶₂𝑚_𝑠𝑘𝑃_𝑆^−1/2𝑚_𝑠, (48) where 𝜃_max stands for the maximum eigenvalue, we can therefore guarantee constraints mentioned in (19) as long as 𝜌𝑃_𝑆^−1/2𝑚_𝑠𝐶₂𝑚_𝑠^𝑇_𝑘𝐶₂𝑚_𝑠𝑘𝑃_𝑆^−1/2𝑚_𝑠<𝐼, (49) 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 𝑊(𝑚_𝑠) for convenience in the following form: 𝑊𝑚_𝑠=𝑋𝑚_𝑠𝑌𝑚_𝑠𝑈𝑚_𝑠𝑉𝑚_𝑠,𝑊⁻¹𝑚_𝑠=𝑀𝑚_𝑠𝐺𝑚_𝑠𝐻𝑚_𝑠𝐿𝑚_𝑠. (50)

According to the literature [25], we assume that both 𝑈(𝑚_𝑠) and 𝐻(𝑚_𝑠) are invertible without loss of generality.

Denote Δ𝑚_𝑠=𝐼𝑀𝑚_𝑠0𝐻𝑚_𝑠, (51) and perform the congruence transformation to inequalities (20), (21), and (22) by 𝐽₁𝑚_𝑠=diagΔ^𝑇𝑚_𝑠,Δ^𝑇𝑚_𝑠,Δ^𝑇𝑚_𝑠,𝐽,𝐼₂𝑚_𝑠=diagΔ^𝑇𝑚_𝑠,Δ^𝑇𝑚_𝑠,𝐽,𝐼,𝐼₃𝑚_𝑠=diag𝐼,Δ^𝑇𝑚_𝑠, (52) respectively.

Denote𝑄𝑚_𝑠=Δ^𝑇𝑚_𝑠𝑄𝑚_𝑠Δ𝑚_𝑠,𝑃𝑚_𝑠=Δ^𝑇𝑚_𝑠𝑃𝑚_𝑠Δ𝑚_𝑠,𝑃_𝑆𝑚_𝑠=Δ^𝑇𝑚_𝑠𝑃_𝑆𝑚_𝑠Δ𝑚_𝑠,𝐴𝑚_𝑠=Δ^𝑇𝑚_𝑠𝑊^𝑇𝑚_𝑠𝐴𝑚_𝑠Δ𝑚_𝑠=𝑋^𝑇𝑚_𝑠𝐴𝑚_𝑠+𝐵_𝐾𝑚_𝑠𝐶𝐴_𝐾𝑚_𝑠𝐴𝑚_𝑠+𝐵𝑚_𝑠𝐷_𝐾𝑚_𝑠𝐶𝐴𝑚_𝑠𝑀𝑚_𝑠+𝐵𝑚_𝑠𝐶_𝐾𝑚_𝑠,𝐵𝑚_𝑠=Δ^𝑇𝑚_𝑠𝑊^𝑇𝑚_𝑠𝐵𝑚_𝑠=𝑋^𝑇𝑚_𝑠𝐵₁𝑚_𝑠𝐵₁𝑚_𝑠,𝐶₁𝑚_𝑠=𝐶₁𝑚_𝑠Δ𝑚_𝑠=𝐶₁𝑚_𝑠+𝐷₁𝑚_𝑠𝐷_𝐾𝑚_𝑠𝐶𝐶₁𝑚_𝑠𝑀𝑚_𝑠+𝐷₁𝑚_𝑠𝐶_𝐾𝑚_𝑠,𝐶₂𝑚_𝑠=𝐶₂𝑚_𝑠Δ𝑚_𝑠=𝐶₂𝑚_𝑠𝐶₂𝑚_𝑠𝑀𝑚_𝑠,𝑊𝑚_𝑠=Δ^𝑇𝑚_𝑠𝑊𝑚_𝑠Δ𝑚_𝑠=𝑋^𝑇𝑚_𝑠𝑍𝑚_𝑠𝐼𝑀𝑚_𝑠, (53)then we get the following theorem.

Theorem 2. For the given parameters 𝛾,𝜂,𝜌>0, if there exist symmetric matrix 𝑃(𝑚_𝑠), positive definite symmetric matrices 𝑃_𝑆(𝑚_𝑠), 𝑄(𝑚_𝑠), and general matrices 𝐴_𝐾(𝑚_𝑠), 𝐵_𝐾(𝑚_𝑠), 𝐶_𝐾(𝑚_𝑠), 𝐷_𝐾(𝑚_𝑠), 𝑊(𝑚_𝑠), 𝑀(𝑚_𝑠), 𝑋(𝑚_𝑠), 𝑍(𝑚_𝑠) satisfyingsym𝑊𝑚_𝑠−𝐴𝑚_𝑠+𝑃_𝑆𝑚_𝑠−𝑊^𝑇𝑚_𝑠−𝐵𝑚_𝑠∗−𝑃_𝑆𝑚_𝑠𝑃00∗∗−_𝑆𝑚_𝑠0∗∗∗−𝜂𝐼<0,−𝜔₁𝜔₂𝑄𝑚_𝑠−sym𝐴𝑚_𝑠𝑃𝑚_𝑠−𝑗𝜔_𝑐𝑄𝑚_𝑠+𝑊^𝑇𝑚_𝑠−𝐵𝑚_𝑠𝐶^𝑇₁𝑚_𝑠𝑃𝑚_𝑠+𝑗𝜔_𝑐𝑄𝑚_𝑠+𝑊𝑚_𝑠−𝑄𝑚_𝑠−𝐵00_𝑇𝑚_𝑠0−𝛾²𝐶𝐼0₁𝑚_𝑠√00−𝐼<0,−𝐼𝜌𝐶₂𝑚_𝑠𝑘∗−𝑃_𝑆𝑚_𝑠<0,𝑘=1,2, (54)(55)(56)then a dynamic output feedback controller exists, which satisfies the requirements of (I), (II) and (III) with 𝑤_max=𝜌/𝜂.
The corresponding controller in the form of (12) can be given by 𝐷_𝐾𝑚_𝑠=𝐷_𝐾𝑚_𝑠,𝐶_𝐾𝑚_𝑠=𝐶_𝐾𝑚_𝑠−𝐷_𝐾𝑚_𝑠𝐶𝑀𝑚_𝑠𝐻⁻¹𝑚_𝑠,𝐵_𝐾𝑚_𝑠=𝑈^−𝑇𝑚_𝑠𝐵_𝐾𝑚_𝑠−𝑋^𝑇𝑚_𝑠𝐵𝑚_𝑠𝐷_𝐾𝑚_𝑠,𝐴_𝐾𝑚_𝑠=𝑈^−𝑇𝑚_𝑠𝐴_𝐾𝑚_𝑠−𝑋^𝑇𝑚_𝑠𝐴𝑚_𝑠𝑀𝑚_𝑠−𝑋^𝑇𝑚_𝑠𝐵𝑚_𝑠𝐷_𝐾𝑚_𝑠𝐶𝑀𝑚_𝑠−𝑈^𝑇𝑚_𝑠𝐵_𝐾𝑚_𝑠𝐶𝑀𝑚_𝑠−𝑋^𝑇𝑚_𝑠𝐵𝑚_𝑠𝐶_𝐾𝑚_𝑠𝐻𝑚_𝑠×𝐻⁻¹𝑚_𝑠. (57)

Though a parameter-dependent controller can be designed via Theorem 2, it is difficult to obtain targeted matrices in real time as 𝑚_𝑠 varies. Therefore, we give a tractable LMI-based theorem as follows.

Theorem 3. For the given parameters 𝛾,𝜂, and 𝜌>0, if there exist symmetric matrix 𝑃_𝑖, positive definite symmetric matrices 𝑃_𝑆𝑖,𝑄_𝑖, and general matrices 𝐴_𝐾𝑖,𝐵_𝐾𝑖,𝐶_𝐾𝑖,𝐷_𝐾𝑖,𝑊_𝑖,𝑀_𝑖,𝑋_𝑖,𝑍_𝑖 (𝑖=1,2), satisfying 𝐽₁𝐽₂∗𝐽₃𝐾<0,1≤𝑖≤𝑗≤2,₁+𝑗𝐾₂𝐾₃∗𝐾₄√<0,1≤𝑖≤𝑗≤2,−𝐼𝜌𝐶_2𝑖𝑗+𝐶_{2𝑗𝑖𝑘}∗−𝑃_𝑆𝑖−𝑃_𝑆𝑗<0,𝑘=1,2;1≤𝑖≤𝑗≤2, (58)(59)(60) where 𝐽₁=sym𝑊_𝑖+𝑊_𝑗−𝐴_𝑖𝑗−𝐴_𝑗𝑖+𝑃_𝑆𝑖+𝑃_𝑆𝑗∗−𝑃_𝑆𝑖−𝑃_𝑆𝑗,𝐽₂=−𝑊^𝑇_𝑖−𝑊^𝑇_𝑗−𝐵_𝑖𝑗−𝐵_𝑗𝑖,𝐽00₃=−𝑃_𝑆𝑖−𝑃_𝑆𝑗0,𝐾∗−2𝜂𝐼₁=−𝜔₁𝜔₂𝑄_𝑖+𝑄_𝑗−sym𝐴_𝑖𝑗+𝐴_𝑗𝑖𝑃_𝑖+𝑃_𝑗+𝑊^𝑇_𝑖+𝑊^𝑇_𝑗𝑃_𝑖+𝑃_𝑗+𝑊_𝑖+𝑊_𝑗−𝑄_𝑖−𝑄_𝑗,𝐾₂=0−𝜔_𝑐𝑄_𝑖+𝑄_𝑗𝜔_𝑐𝑄_𝑖+𝑄_𝑗0,𝐾₃=−𝐵_𝑖𝑗−𝐵_𝑗𝑖𝐶^𝑇_1𝑖𝑗+𝐶^𝑇_1𝑗𝑖,𝐾00₄=−2𝛾²,𝐴𝐼00−2𝐼_𝑖𝑗=𝑋^𝑇_𝑖𝐴_𝑗+𝐵_𝐾𝑖𝐶𝐴_𝐾𝑖𝐴_𝑖+𝐵_𝑖𝐷_𝐾𝑗𝐶𝐴_𝑖𝑀_𝑗+𝐵_𝑖𝐶_𝐾𝑗,𝐵_𝑖𝑗=𝑋^𝑇_𝑖𝐵_1𝑗𝐵_1𝑖,𝐶_1𝑖𝑗=𝐶_1𝑖+𝐷_1𝑖𝐷_𝐾𝑗𝐶𝐶_1𝑖𝑀_𝑗+𝐷_1𝑖𝐶_𝐾𝑗,𝐶_2𝑖𝑗=𝐶_2𝑖𝐶_2𝑖𝑀_𝑗,𝑊_𝑖=𝑋^𝑇_𝑖𝑍_𝑖𝐼𝑀_𝑖, (61) then a dynamic output feedback controller exists, which satisfies the requirements of (I), (II) and (III) with 𝑤_max=𝜌/𝜂.
The corresponding controller in the form of (12) can be given by 𝐷_𝐾𝑚_𝑠=𝐷_𝐾𝑚_𝑠,𝐶_𝐾𝑚_𝑠=𝐶_𝐾𝑚_𝑠−𝐷_𝐾𝑚_𝑠𝐶𝑀𝑚_𝑠𝐻⁻¹𝑚_𝑠,𝐵_𝐾𝑚_𝑠=𝑈^−𝑇𝑚_𝑠𝐵_𝐾𝑚_𝑠−𝑋^𝑇𝑚_𝑠𝐵𝑚_𝑠𝐷_𝐾𝑚_𝑠,𝐴_𝐾𝑚_𝑠=𝑈^−𝑇𝑚_𝑠𝐴_𝐾𝑚_𝑠−𝑋^𝑇𝑚_𝑠𝐴𝑚_𝑠𝑀𝑚_𝑠−𝑋^𝑇𝑚_𝑠𝐵𝑚_𝑠𝐷_𝐾𝑚_𝑠𝐶𝑀𝑚_𝑠−𝑈^𝑇𝑚_𝑠𝐵_𝐾𝑚_𝑠𝐶𝑀𝑚_𝑠−𝑋^𝑇𝑚_𝑠𝐵𝑚_𝑠𝐶_𝐾𝑚_𝑠𝐻𝑚_𝑠×𝐻⁻¹𝑚_𝑠, (62) where 𝐴_𝐾𝑚_𝑠,𝐵_𝐾𝑚_𝑠,𝐶_𝐾𝑚_𝑠,𝐷_𝐾𝑚_𝑠,𝑀𝑚_𝑠,𝑋𝑚_𝑠,𝑍𝑚_𝑠=²_𝑖=1𝜆_𝑖𝐴_𝐾𝑖,𝐵_𝐾𝑖,𝐶_𝐾𝑖,𝐷_𝐾𝑖,𝑀_𝑖,𝑋_𝑖,𝑍_𝑖, (63) and 𝜆_𝑖(𝑖=1,2) can be calculated by (11).

Proof. we just prove that inequality (58) is sufficient to inequality (54) for simplicity.
DenoteΨ𝑚_𝑠=sym𝑊𝑚_𝑠−𝐴𝑚_𝑠+𝑃_𝑆𝑚_𝑠−𝑊^𝑇𝑚_𝑠−𝐵𝑚_𝑠∗−𝑃_𝑆𝑚_𝑠𝑃𝑚00∗∗−_𝑠0∗∗∗−𝜂𝐼, (64)which stands for the left of inequality (54).
Inequality (58) is equivalent to Ψ_𝑖𝑗+Ψ_𝑗𝑖Ψ<0,1≤𝑖<𝑗≤2,_𝑖𝑖<0,𝑖=1,2, (65) where Ψ_𝑖𝑗=sym𝑊_𝑖−𝐴_𝑖𝑗+𝑃_𝑆𝑖−𝑊^𝑇_𝑖−𝐵_𝑖𝑗∗−𝑃_𝑆𝑖𝑃00∗∗−_𝑆𝑖0,∗∗∗−𝜂𝐼1≤𝑖≤𝑗≤2. (66) Assume that 𝐴_𝐾𝑚_𝑠,𝐵_𝐾𝑚_𝑠,𝐶_𝐾𝑚_𝑠,𝐷_𝐾𝑚_𝑠,𝑀𝑚_𝑠,𝑋𝑚_𝑠,𝑍𝑚_𝑠=²_𝑖=1𝜆_𝑖𝐴_𝐾𝑖,𝐵_𝐾𝑖,𝐶_𝐾𝑖,𝐷_𝐾𝑖,𝑀_𝑖,𝑋_𝑖,𝑍_𝑖, (67) and then we get Ψ𝑚_𝑠=²²_𝑖=1𝑗=1𝜆_𝑖𝜆_𝑗Ψ_𝑖𝑗=²_𝑖=1𝜆²_𝑖Ψ_𝑖𝑖+𝜆₁𝜆₂Ψ₁₂+Ψ₂₁, (68) which is negative definite by inequality (65), that is, Ψ𝑚_𝑠<0. (69)

Remark 4. 𝑈(𝑚_𝑠), 𝐻(𝑚_𝑠) should be chosen to meet the definition, that is, 𝑈^𝑇𝑚_𝑠𝐻𝑚_𝑠=𝑍𝑚_𝑠−𝑋^𝑇𝑚_𝑠𝑀𝑚_𝑠. (70)
However, the value of 𝑈(𝑚_𝑠) and 𝐻(𝑚_𝑠) can be chosen variously for the given 𝑍(𝑚_𝑠), 𝐻(𝑚_𝑠), and 𝑀(𝑚_𝑠). In this paper, we use the singular value decomposition approach.

Remark 5. Based on [26], for real matrices 𝑆₁ and 𝑆₂, 𝑆₁+𝑗𝑆₂<0 is equivalent to ^𝑆1𝑆2_{−𝑆2𝑆1}<0. Thusly, (59) can be converted into real matrix inequality by defining 𝑆₁=𝐾₁𝐾₃∗𝐾₄,𝑆₂=𝐾₂000. (71)

Remark 6. The matrices of dynamic output feedback controller 𝐴_𝐾(𝑚_𝑠), 𝐵_𝐾(𝑚_𝑠), 𝐶_𝐾(𝑚_𝑠), 𝐷_𝐾(𝑚_𝑠), and Lyapunov matrix 𝑃_𝑆(𝑚_𝑠) are dependent nonlinearly on parameter 𝑚_𝑠. For instance, as can be seen in the above deduction, matrix 𝐷_𝐾(𝑚_𝑠) in Theorem 3 can be formulated as 𝐷_𝐾𝑚_𝑠=𝐷_𝐾𝑚_𝑠=²_𝑖=1𝜆_𝑖𝐷_𝐾𝑖=𝜆₁𝐷_𝐾1+𝜆₂𝐷_𝐾2=1/𝑚_𝑠−1/𝑚_𝑠min1/𝑚_𝑠max−1/𝑚_𝑠min𝐷_𝐾1+1/𝑚_𝑠max−1/𝑚_𝑠1/𝑚_𝑠max−1/𝑚_𝑠min𝐷_𝐾2, (72) where 𝐷_𝐾1 and 𝐷_𝐾2 are the corresponding matrix solutions of LMIs in Theorem 3, and the value of matrix 𝐷_𝐾(𝑚_𝑠) may vary with the change of 𝑚_𝑠.

Remark 7. Inequalities from (58) to (60) can be simplified from 12 LMIs to 4 LMIs if the following matrix variables are chosen as 𝑃₁=𝑃₂, 𝑃_𝑆1=𝑃_𝑆2, 𝑄₁=𝑄₂, 𝐴_𝐾1=𝐴_𝐾2, 𝐵_𝐾1=𝐵_𝐾2, 𝐶_𝐾1=𝐶_𝐾2, 𝐷_𝐾1=𝐷_𝐾2, 𝑊₁=𝑊₂, 𝑀₁=𝑀₂, 𝑋₁=𝑋₂, and 𝑍₁=𝑍₂. However, this simplification will keep the matrices for the designed controller constant for all 𝑚_𝑠, and an invariant Lyapunov function, rather than a parameter-dependent one, has to be used for the whole domain, which will bring a larger conservativeness than Theorem 3.

3.2. Full Frequency Design

Based on the theorem of [27], we formulate Theorem 8 without proof for simplicity.

Theorem 8. For the given parameter 𝛾>0, if there exist positive definite symmetric matrices 𝑌_𝐶𝑖,𝑋_𝐶𝑖 and general matrices 𝐴_𝐶𝑖,𝐵_𝐶𝑖,𝐶_𝐶𝑖,𝐷_𝐶𝑖, (𝑖=1,2), satisfying sym𝐴_𝑒𝑖𝑗+𝐴_𝑒𝑗𝑖𝐵_𝑒𝑖𝑗𝐶^𝑇_{𝑒1𝑖𝑗}+𝐶^𝑇_{𝑒1𝑗𝑖}∗−2𝛾²√𝐼0∗∗−2𝐼<0,1≤𝑖≤𝑗≤2,−2𝐼𝜌𝐶_{𝑒2𝑖𝑗}+𝐶_{𝑒2𝑗𝑖𝑘}∗−𝑃_𝐶𝑖−𝑃_𝐶𝑗<0,𝑘=1,2;1≤𝑖≤𝑗≤2, (73)(74) where 𝐴_𝑒𝑖𝑗=𝐴_𝑖𝑋_𝐶𝑗+𝐵_𝑖𝐶_𝐶𝑗𝐴_𝑖+𝐵_𝑖𝐷_𝐶𝑗𝐶𝐴_𝐶𝑖𝑌_𝐶𝑖𝐴_𝑗+𝐵_𝐶𝑖𝐶,𝐵_𝑒𝑖𝑗=𝐵_1𝑖𝑌_𝐶𝑖𝐵_1𝑗,𝐶_{𝑒1𝑖𝑗}=𝐶_1𝑖𝑋_𝐶𝑗+𝐷_1𝑖𝐶_𝐶𝑗𝐶_1𝑖+𝐷_1𝑖𝐷_𝐶𝑗𝐶,𝐶_{𝑒2𝑖𝑗}=𝐶_2𝑖𝑋_𝐶𝑗𝐶_2𝑖,𝑃_𝐶𝑖=𝑋_𝐶𝑖𝐼𝐼𝑌_𝐶𝑖, (75) then a dynamic output feedback controller exists, which satisfies (I) and (III), and ‖𝐺(𝑗𝜔)‖_∞<𝛾 with 𝑤_max=𝜌/𝜂.
The corresponding controller in the form of (12) can be given by 𝐷_𝐾𝑚_𝑠=𝐷_𝐶𝑚_𝑠,𝐶_𝐾𝑚_𝑠=𝐶_𝐶𝑚_𝑠−𝐷_𝐶𝑚_𝑠𝐶𝑋_𝐶𝑚_𝑠𝑀_𝐶^−𝑇𝑚_𝑠,𝐵_𝐾𝑚_𝑠=𝑁_𝐶⁻¹𝑚_𝑠𝐵_𝐶𝑚_𝑠−𝑌_𝐶𝑚_𝑠𝐵𝑚_𝑠𝐷_𝐶𝑚_𝑠𝐴_𝐾,𝑚_𝑠=𝑁_𝐶⁻¹𝑚_𝑠𝐴_𝐶𝑚_𝑠−𝑌_𝐶𝑚_𝑠𝐴𝑚_𝑠𝑋_𝐶𝑚_𝑠−𝑌_𝐶𝑚_𝑠𝐵𝑚_𝑠𝐷_𝐶𝑚_𝑠𝐶𝑋_𝐶𝑚_𝑠−𝑁_𝐶𝑚_𝑠𝐵_𝐾𝑚_𝑠𝐶𝑋_𝐶𝑚_𝑠−𝑌_𝐶𝑚_𝑠𝐵𝑚_𝑠𝐶_𝐾𝑚_𝑠𝑀^𝑇_𝐶×𝑀_𝐶^−𝑇𝑚_𝑠, (76)(77)(78)(79) where 𝐴_𝐶𝑚_𝑠,𝐵_𝐶𝑚_𝑠,𝐶_𝐶𝑚_𝑠,𝐷_𝐶𝑚_𝑠,𝑋_𝐶𝑚_𝑠,𝑌_𝐶𝑚_𝑠=²_𝑖=1𝜆_𝑖𝐴_𝐶𝑖,𝐵_𝐶𝑖,𝐶_𝐶𝑖,𝐷_𝐶𝑖,𝑋_𝐶𝑖,𝑌_𝐶𝑖, (80) and 𝜆_𝑖(𝑖=1,2) can be calculated by (11).

The same approach mentioned in Remark 4 is used to obtain the value of 𝑁_𝐶(𝑚_𝑠) and 𝑀_𝐶(𝑚_𝑠), which should meet the following equation: 𝑁_𝐶𝑚_𝑠𝑀^𝑇_𝐶𝑚_𝑠=𝐼−𝑌_𝐶𝑚_𝑠𝑋_𝐶𝑚_𝑠. (81)

4. 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.

Table 1 

The model parameters of active suspensions.

We first choose the following parameters: 𝜌=1, 𝜂=10000, 𝑧_max=0.1m, 𝜔₁=8𝜋rad/s, and 𝜔₂=16𝜋rad/s. Then, the matrices in (12) can be calculated by applying Theorem 3 and Remark 7 for finite frequency controller and Theorem 8 for full frequency controller (shown in Appendix B) with the least guaranteed 𝐻_∞ performances 𝛾=2.54 for Theorem 3, 𝛾=2.73 for Remark 6, and 𝛾=10.98 for Theorem 8.

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 𝐻_∞ 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.

Figure 2 

The curves of maximum singular values.

In order to examine the performance of finite frequency controller, we assume that the disturbance is in the form of 𝑤(𝑡)=[][],𝐴sin2𝜋𝑓𝑡𝑡∈0,𝑇0𝑡∉0,𝑇 (82) where 𝐴,𝑓 stand for the amplitude and the frequency of disturbance, respectively, and 𝑇=1/𝑓.

Suppose that 𝐴 is 0.4 m and 𝑓 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.

Figure 3 

The time response of body acceleration.

Figure 4 

The time response of relative dynamic tire load.

Figure 5 

The time response of relative suspension stroke.

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, |{𝑧_𝑜2(𝑡)}_𝑘|<1,  𝑘=1,2.

5. 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.

Appendices

A. Related Lemmas

Lemma A.1 (generalized KYP lemma [28]). Let matrices Θ, 𝐹, Φ, and Ψ be given, Denote by 𝑁_𝜔 the null space of 𝑇_𝜔𝐹, where 𝑇_𝜔=𝐼−𝑗𝜔𝐼. The inequality 𝑁^∗_𝜔Θ𝑁_𝜔<0,𝜔∈𝜔₁𝜔₂ (A.1) holds, if and only if, there exist 𝑃, 𝑄>0 such that 𝐹^∗(Φ⊗𝑃+Ψ⊗𝑄)𝐹<0, (A.2) where Φ=0110,Ψ=−1𝑗𝜔_𝑐−𝑗𝜔_𝑐−𝜔₁𝜔₂, (A.3) with 𝜔_𝑐=(𝜔₁+𝜔₂)/2.

Lemma A.2 (projection lemma [25] ). Let Γ, Λ, and Θ be given, there exists a matrix satisfying sym(ΓΛΘ)+Θ<0, (A.4) if and only if, the following two conditions hold: Γ^⟂ΘΓ^⟂𝑇<0,Λ^𝑇⟂ΘΛ^𝑇⟂𝑇<0. (A.5)

Lemma A.3 (reciprocal projection lemma [25]). Let 𝑃 be any given positive definite matrix. The following statements are equivalent.(1)Ψ+𝑆+𝑆^𝑇<0.(2)The LMI problem Ψ+𝑃−𝑊_𝑠𝑆^𝑇+𝑊^𝑇∗−𝑃<0 (A.6) is feasible with respect to 𝑋.

B. Controller Matrices

The parameter matrices of the dynamic output feedback controller for Theorem 3:𝐴_𝐾=−14.821−0.12731−1.14240.00255−228.44−136.9213.7010.020302038.1151.59−21.603−0.475982.4991×10⁹−7.4319×10⁷4.2012×10⁶−1.0171×10⁶,𝐵_𝐾=99.943109.0844.969−3170.6−2956.8−891.61−632.001156.4−1125.8−1.0944×10⁸−1.8461×10⁹−7.9445×10⁷,𝐶_𝐾=,𝐷3814.4−686.191758.93.5961_𝐾=−1.8676×10⁵−2.0429×10⁵−86675. (B.1)

The parameter matrices of the dynamic output feedback controller for Theorem 8: 𝐴_𝐾=,𝐵−12.73426.22521.367−0.31502−36.28614.95436.9840.69500−111.01−84.507−65.487−0.0771−45995−42118−29117−1623.6_𝐾=−793.37−56.495−136.301770.9122.15113.03−90.8553448.7−802.396.4417×10⁵7.1058×10⁶−2.7515×10⁵,𝐶_𝐾=,𝐷518.19−604.03−590.4110.015_𝐾=254221720.82609.8. (B.2)