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Abstract

The active suspension systems have attracted increasing attentions because of its advantages in improving automotive dynamic performance. This paper focuses on the fuzzy robust non-fragile $H_{2} / H_{\infty}$ control problem for the nonlinear active suspension system with the time-varying actuator delay and control uncertainty. Firstly, a Takagi-Sugeno fuzzy suspension model is established to describe the highly nonlinear components in the suspension system. Then, a series of linear matrix inequalities are derived based on a novel parameter Lyapunov function to guarantee the asymptotic stability of the suspension systems. Also, a fuzzy $H_{2} / H_{\infty}$ controller is derived for the proposed fuzzy suspension model with the parallel distributed compensation method. Thirdly, a co-design criterion is proposed to obtain the fuzzy controller with the time varying delay and control uncertainty simultaneously. Finally, both simulations and experiments are carried out to demonstrate the effectiveness and robustness of the proposed controller.

Keywords

Active suspension system fuzzy system nonlinear control time varying delay robust control

Introduction

The suspension system is one of the significant parts of the cars.¹ It is composed of shock absorbers, connecting rods and springs. Its main function is to transmit the forces and moments between the wheels and the vehicle body, which has a great effect on improving the vehicle performance.² Due to the importance of the suspension system, many researches have been carried out. Nowadays, the semi-active suspension system³ and the active suspension system (ASS)^4,5 are widely used in modern vehicles. Further, the semi-active suspension and the ASS adopt different approaches to promote the ride comfort and the handing stability. However, the ride comfort, vehicle stability and road holding are in contradiction.⁶ Therefore, how to design a suspension system so that these performances can reach an optimal state is greatly significant. Considering the above problems, the ASS that combines the excellent control algorithms is a good choice. It has become the research focus of scholars.^7,8 As a result, numerous control methods have been applied to the system to improve the comprehensive suspension performance, such as model predictive control,^9,10 sliding model control,^11–13 adaptive control,^14–16 fault tolerant control,¹⁷ and robust control.^18–22 In particular, the robust control, such as $H_{2}$ and $H_{\infty}$ has been widely studied owing to its robustness.

Due to the transmission of information between control devices and the actuators, the situation of time delay will inevitably exist in the ASS.²³ In addition, the time delay is always changing owing to the changes in the environment. It is particularly important to note that the existence of time delay will deteriorate the control effect of the controller and even make the ASS unstable.^24,25 Therefore, this phenomenon must be taken into consideration. In Du et al.²⁶ and Karim Afshar,²⁷ the controllers were proposed to tackle the constant delay in the actuators. In Gao et al.²⁸ and Li et al.,²⁹ by transforming the sampling moment into the form of continuous time delay, a sampled-data H_∞ controller was proposed. In Li et al.,^30,31 the Lyapunov–Krasovskii functions were designed to cope with the actuators time delay questions. Therefore, the time delay problem deserves more researchers’ attention.

In reality, owing to the aging of actuators and other situations, the actuators usually appear control disturbance phenomenon. If uncertainties occur in the process of the system, these situations will seriously affect the control performance, making the suspension performance worse. Therefore, it is necessary to consider the uncertainties of the actuators. In recent years, linear fractional transformation (LFT) has been widely used to solve such problems.^32,33 Moreover, the ASS is a highly nonlinear system, such as the nonlinearity between damping and stiffness. In the recent studies, the Takagi-Sugeno (T-S) fuzzy approach has been extensively applied to settle the nonlinear characteristics of the ASS. Through a series of constructed linear subsystems, the original system is represented by weighting method, which can be easily solved by the linear system theory. In Li et al.^20,29,31 and Xu et al.,³³ taken the sprung mass and the un-sprung mass changing within a certain range into consideration, the T-S fuzzy model was established to deal with the parameter uncertainties of the suspension systems. In Zhang et al.,³⁴ based on the T-S fuzzy model, a novel fuzzy controller was proposed for the ASS subject to the sensor failure while the constrained stated were ensured simultaneously.

Through the above discussion, a fuzzy non-fragile $H_{2} / H_{\infty}$ controller for the nonlinear ASS is proposed with the time delay and the control uncertainty based on the T-S fuzzy model approach. The main contributions for this paper are summarized as follows:

A T-S fuzzy model is established to describe the high nonlinearity of the spring force and the damping force of ASS in the framework of $H_{2} / H_{\infty}$ control objectives.

A parameter Lyapunov function is designed to acquire less conservative sufficient conditions to satisfy both the stability and constraints of the closed-loop system.

A co-design criterion is proposed for the concerned nonlinear suspension system with the input time delay and control uncertainties.

The remaining of the article is organized as below. Section 2 shows the T-S fuzzy model and the constrained performance. Section 3 shows the design of the proposed controller. Section 4 presents the simulation and experiment results. Finally, conclusions of the paper are presented in Section 5.

Notation. Matrix $P^{T}$ represents the transpose of matrix P. Matrix $P^{- 1}$ denotes the inverse of matrix P. $R^{n}$ denotes the n-dimensional Euclidean space. The format $P > 0$ indicates that P is a positive matrix. The symmetric term in a matrix is denoted by * and the $diag (\cdot \cdot \cdot)$ represents a diagonal matrix. The identity matrix is denoted by I. The 2-norm of ω(t) is used by $‖ ω ‖_{2} = \sqrt{\int_{0}^{\infty} {| ω (t) |}^{2} dt}$ .

Problem formulation

Description of an ASS model

In this part, the suspension model is given in Figure 1. Assuming that the wheel has no lift and no slip, according to the second theorem of Newton’s classical mechanics, the ideal dynamics equations can be obtained as follows:

{\begin{matrix} m_{s} {\overset{\cdot\cdot}{z}}_{s} (t) + F_{s} (z_{s}, z_{u}, t) + F_{d} ({\overset{\cdot}{z}}_{s}, {\overset{\cdot}{z}}_{u}, t) - u (t - τ (t)) = 0 \\ m_{u} {\overset{\cdot\cdot}{z}}_{u} (t) - F_{s} (z_{s}, z_{u}, t) - F_{d} ({\overset{\cdot}{z}}_{s}, {\overset{\cdot}{z}}_{u}, t) + F_{ts} (z_{u}, z_{r}, t) \\ + F_{td} ({\overset{\cdot}{z}}_{u}, {\overset{\cdot}{z}}_{r}, t) + u (t - τ (t)) = 0 \end{matrix}

(1)

where m_s is the sprung mass; m_u is the un-sprung mass; F_d and F_s are the spring and damper forces, respectively; F_ts and F_td are tire force and damping force; z_u and z_s indicate the vertical displacement of the un-sprung mass and the sprung mass; z_r is the irregular road surface incentives; u represents the actuator force. Further, τ(t) represents the time varying delay of the actuator, which satisfies $0 \leq τ_{m} \leq τ (t) \leq τ_{M}, \overset{\cdot}{τ} (t) \leq η$

Figure 1.

Two-degree-of-freedom automobile active suspension model.

In the above formula, the mechanical properties of the suspension system and tires have the following forms:

\begin{matrix} F_{s} (z_{s}, z_{u}, t) = k_{s} (z_{s} (t) - z_{u} (t)) + k_{sn} (z_{s} (t) - z_{u} (t))^{3} \\ F_{d} ({\overset{\cdot}{z}}_{s}, {\overset{\cdot}{z}}_{u}, t) = {\begin{matrix} b_{e} ({\overset{\cdot}{z}}_{s} (t) - {\overset{\cdot}{z}}_{u} (t)), {\overset{\cdot}{z}}_{s} - {\overset{\cdot}{z}}_{u} \geq 0 \\ b_{c} ({\overset{\cdot}{z}}_{s} (t) - {\overset{\cdot}{z}}_{u} (t)), {\overset{\cdot}{z}}_{s} - {\overset{\cdot}{z}}_{u} < 0 \end{matrix} \\ F_{ts} (z_{u}, z_{r}, t) = k_{t} (z_{u} (t) - z_{r} (t)), F_{td} ({\overset{\cdot}{z}}_{u}, {\overset{\cdot}{z}}_{r}, t) \\ = b_{t} ({\overset{\cdot}{z}}_{u} (t) - {\overset{\cdot}{z}}_{r} (t)) \end{matrix}

(2)

where k_s is the spring stiffness coefficient of the linear part and k_sn is the stiffness coefficient of the cubic part; b_e and b_c are the tension and compression damping coefficient; k_t and b_t are the tire stiffness and damping coefficients.

Considering the convenience and availability of the state space equation, by setting the following state variable vectors and control outputs as:

\begin{matrix} x_{1} (t) = z_{s} (t) - z_{u} (t), x_{2} (t) = z_{u} (t) - z_{r} (t), \\ x_{3} (t) = {\overset{\cdot}{z}}_{s} (t), x_{4} (t) = {\overset{\cdot}{z}}_{u} (t), \\ z_{1} (t) = {\overset{\cdot\cdot}{z}}_{s} (t), \\ z_{2} (t) = {[\begin{matrix} \frac{z_{s} (t) - z_{u} (t)}{z_{max}} & \frac{k_{t} (z_{u} (t) - z_{r} (t))}{(m_{s} + m_{u}) g} \end{matrix}]}^{T} \end{matrix}

(3)

where $z_{s} (t) - z_{u} (t)$ denotes the ASS displacement; $z_{u} (t) - z_{r} (t)$ represents the wheel travel; ${\overset{\cdot}{z}}_{s} (t)$ represents the velocity of sprung mass; ${\overset{\cdot}{z}}_{u} (t)$ is the velocity of un-sprung mass; $z_{\max}$ indicates the maximum suspension displacement.

T-S fuzzy ASS modeling

According to the ASS model established in Section 2.1, the travel of the suspension varies within a certain range. When ${\overset{\cdot}{z}}_{s} - {\overset{\cdot}{z}}_{u} \geq 0$ , the extension damping of the system is used, and when ${\overset{\cdot}{z}}_{s} - {\overset{\cdot}{z}}_{u} < 0$ , the compression damping of the system is used.

Then, by defining two antecedent variables $α_{1} (t) = (z_{s} (t) - z_{u} (t))^{2} \in [z_{l}, z_{u}]$ and $α_{2} (t) = {\overset{\cdot}{z}}_{s} (t) - {\overset{\cdot}{z}}_{u} (t)$ . Based on the above two variables, the following expressions of the nonlinear characteristics can be obtained:

\begin{matrix} F_{s} (z_{s}, z_{u}, t) = (k_{s} + α_{1} (t) k_{sn}) (z_{s} (t) - z_{u} (t)) \\ F_{d} ({\overset{\cdot}{z}}_{s}, {\overset{\cdot}{z}}_{u}, t) = [b_{e} δ (α_{2} (t)) + b_{c} (1 - δ (α_{2} (t))] \end{matrix}

(4)

In the above formula, $δ (t)$ stands for the unit step function. The vehicle suspension dynamics equation in (1) can be obtained as:

\begin{matrix} \overset{\cdot}{x} (t) = A (t) x (t) + B_{1} (t) ω (t) + B_{2} (t) u (t - τ (t)) \\ z_{1} (t) = C_{1} (t) x (t) + D_{1} (t) u (t - τ (t)) \\ z_{2} (t) = C_{2} (t) x (t) \\ x (t) = φ (t), \forall t \in [- τ_{M}, 0] \end{matrix}

(5)

where

\begin{matrix} A (t) = [\begin{matrix} 0 & 0 & 1 & - 1 \\ 0 & 0 & 0 & 1 \\ - \frac{{\tilde{k}}_{s}}{m_{s}} & 0 & - \frac{{\tilde{b}}_{s}}{m_{s}} & \frac{{\tilde{b}}_{s}}{m_{s}} \\ \frac{{\tilde{k}}_{s}}{m_{u}} & - \frac{k_{t}}{m_{u}} & \frac{{\tilde{b}}_{s}}{m_{u}} & - \frac{{\tilde{b}}_{s} + b_{t}}{m_{u}} \end{matrix}], \\ B_{1} (t) = [\begin{matrix} 0 \\ - 1 \\ 0 \\ \frac{b_{t}}{m_{u}} \end{matrix}], B_{2} (t) = [\begin{matrix} 0 \\ 0 \\ \frac{1}{m_{s}} \\ - \frac{1}{m_{u}} \end{matrix}], \\ C_{1} (t) = [\begin{matrix} - \frac{{\tilde{k}}_{s}}{m_{s}} & 0 & - \frac{{\tilde{b}}_{s}}{m_{s}} & \frac{{\tilde{b}}_{s}}{m_{s}} \end{matrix}], \\ D_{1} (t) = [\frac{1}{m_{s}}], C_{2} (t) = [\begin{matrix} \frac{1}{z_{max}} & 0 & 0 & 0 \\ 0 & \frac{k_{t}}{(m_{s} + m_{u}) g} & 0 & 0 \end{matrix}] \end{matrix}

where ${\tilde{k}}_{s}$ represents $(k_{s} + α_{1} (t) k_{sn}) (z_{s} (t) - z_{u} (t))$ and ${\tilde{b}}_{s}$ represents $[b_{e} δ (α_{2} (t)) + b_{c} (1 - δ (α_{2} (t)))]$ . Considering that the values of $α_{1} (t)$ and $α_{2} (t)$ are bounded by their maximum values and minimum values, those fuzzy sets can be designed as follows:

\begin{matrix} {S_{11}, S_{12}, S_{13}, S_{14}} = {z_{l}, z_{l}, z_{u}, z_{u}} \\ {S_{21}, S_{22}, S_{23}, S_{24}} \\ = {α_{2} (t) \geq 0, α_{2} (t) < 0, α_{2} (t) \geq 0, α_{2} (t) < 0} \end{matrix}

Therefore, the nonlinear ASS can be represented using the T-S fuzzy method.

Model Rule i: IF $α_{1} (t)$ is S_1i, and $α_{2} (t)$ is S_2i,

THEN

\overset{\cdot}{x} (t) = A_{i} (t) x (t) + B_{2 i} (t) u (t - τ (t)) + B_{1 i} (t) ω (t)

z_{1} (t) = C_{1 i} (t) x (t) + D_{1 i} (t) u (t - τ (t))

z_{2} (t) = C_{2 i} (t) x (t)

According to the above fuzzy rules, $A_{i}, B_{1 i}, B_{2 i}, C_{1 i}, D_{1 i}$ and $C_{2 i}$ can be obtained by the corresponding $α_{1} (t)$ and $α_{2} (t)$ .

The whole T-S model can be presented as:

\begin{matrix} \overset{\cdot}{x} (t) = \sum_{i = 1}^{4} μ_{i} (α (t)) [A_{i} (t) x (t) + B_{2 i} (t) u \\ (t - τ (t)) + B_{1 i} (t) ω (t)] \\ z_{1} (t) = \sum_{i = 1}^{4} μ_{i} (α (t)) [C_{1 i} (t) x (t) + D_{1 i} (t) u (t - τ (t))] \\ z_{2} (t) = \sum_{i = 1}^{4} μ_{i} (α (t)) C_{2 i} (t) x (t) \\ x (t) = φ (t), \forall t \in [- τ_{M}, 0] \end{matrix}

(6)

where the membership function can be shown as follows:

\begin{matrix} μ_{1} (α (t)) = \frac{z_{u} - α_{1} (t)}{z_{u} - z_{l}} \times δ (α_{2} (t)), μ_{2} (α (t)) \\ = \frac{z_{u} - α_{1} (t)}{z_{u} - z_{l}} \times (1 - δ (α_{2} (t))), μ_{3} (α (t)) \\ = \frac{α_{1} (t) - z_{l}}{z_{u} - z_{l}} \times δ (α_{2} (t)), μ_{4} (α (t)) \\ = \frac{α_{1} (t) - z_{l}}{z_{u} - z_{l}} \times (1 - δ (α_{2} (t))) . \end{matrix}

Remark 1. The approach for the selection of the membership functions is shown in the previous work, that is, Li et al.³¹ In Li et al.,³¹ the author considered the uncertainty of the sprung mass and the un-sprung mass, and the membership functions are designed according to the uncertain parameters of $m_{s} (t)$ and $m_{u} (t)$ which bounded by their maximum values and minimum values. Similarly, given that the values $α_{1} (t)$ and $α_{2} (t)$ are bounded by their maximum values and minimum values, the membership functions are chosen as shown above.

The fuzzy weighting functions satisfies $\sum_{i = 1}^{4} μ_{i} (α (t)) = 1$ and $μ_{i} (α (t)) \geq 0$ . For simplicity, we can define $μ_{i} (α (t)) = μ_{i}$ .

Fuzzy robust H_∞ controller

Through the parallel distributed compensation (PDC) method, the following fuzzy robust H_∞ controller is given as below:

Model Rule i: IF $α_{1} (t)$ is S_1i, and $α_{2} (t)$ is S_2i,

THEN

u (t - τ (t)) = K_{j} x (t - τ (t))

The whole fuzzy controller can be presented as follows:

u (t - τ (t)) = \sum_{j = 1}^{4} μ_{j} K_{j} x (t - τ (t))

(7)

Then, the robust non-fragile controller can be described as

u (t - τ (t)) = \sum_{j = 1}^{4} μ_{j} (K_{j} + Δ K_{j}) x (t - τ (t))

(8)

Δ K_{j} = HF (t) E K_{j}

(9)

where K_j (j =1,2,3,4) denotes the controller gain matrices. $Δ K_{j}$ indicates the disturbance of the controller and $F (t)$ denotes an uncertain function which satisfies $F {(t)}^{T} F (t) \leq I$ . H and E stand for the real constant matrices.

Remark 2. The main idea of PDC method is to divide the mathematical model of nonlinear system into multiple linear subsystems, design local controller for each linear subsystem, and the design of controller has the same number of rules and conditional variables as the controlled fuzzy model. The global fuzzy controller is generated by normalized weight.

Finally, based on (6), (8), and (9), the nonlinear ASS can be obtained as follows:

\begin{matrix} \overset{\cdot}{x} (t) = \sum_{i = 1}^{4} \sum_{j = 1}^{4} μ_{i} μ_{j} [A_{i} (t) x (t) + B_{2 i} (t) (K_{j} + Δ K_{j}) x \\ (t - τ (t)) + B_{1 i} (t) ω (t)] \\ z_{1} (t) = \sum_{i = 1}^{4} \sum_{j = 1}^{4} μ_{i} μ_{j} [C_{1 i} (t) x (t) + D_{1 i} (t) \\ (K_{j} + Δ K_{j}) x (t - τ (t))] \\ z_{2} (t) = \sum_{i = 1}^{4} μ_{i} C_{2 i} (t) x (t) \\ x (t) = φ (t), \forall t \in [- τ_{M}, 0] \end{matrix}

(10)

Control objectives

Road holding, ride comfort and handling stability are important for the vehicle. The following targets are the control objectives:

Ride comfort: the control goal is to obtain the minimum value of ${\overset{\cdot\cdot}{z}}_{s} (t)$ as much as possible in the ASS. That is

\min {\overset{\cdot\cdot}{z}}_{s} (t)

(11)

Handling stability: the suspension disturbance cannot exceed its limit value. That is

| z_{s} (t) - z_{u} (t) | \leq z_{max}

(12)

Road holding: to make the vehicle wheel keep in contact with the road, the dynamic load is needed to be less than the static load. That is

k_{t} (z_{u} (t) - z_{r} (t)) < (m_{s} + m_{u}) g

(13)

Actuator limitation: due to the limitation of engine power, the output of the actuator is saturated. Therefore, the output force must be limited. That is

| u (t) | \leq u_{max}

(14)

Through the above analysis, a generalized $H_{2}$ performance satisfies the following constraints:

| {z_{2} {(t)}_{q}} | \leq 1, q = 1, 2

(15)

Main results

Stability analysis and $H_{\infty}$ performance constraints

This section presents a sufficient criterion for the proposed controller. The criterion can effectively keep the closed-loop system in (11) asymptotic stability. The constrained states are satisfied at the same time.

The lemmas applied in the following proof process are given in the Appendix A.

Theorem 1. For a prescribed $γ > 0$ , $ρ > 0$ , scalars $τ_{m}$ , $τ_{M}$ , $η$ , and K_j (j =1,2,3,4), considering the system in equation (6) with $Δ K_{j} = 0$ , the system (6) is asymptotically stable if the actuator time delay satisfies $0 \leq τ_{m} \leq τ (t) \leq τ_{M}$ , $τ^{.} (t) \leq η$ , the system satisfies $‖ z_{1} (t) ‖_{2} < γ ‖ ω (t) ‖_{2}$ and guarantee the condition $| {z_{2} {(t)}_{q}} | \leq 1$ and $| u (t) | \leq u_{\max}$ , if there are matrices $P = \sum_{r = 1}^{4} μ_{r} P_{r}$ >0, $Q_{j} = \sum_{r = 1}^{4} μ_{r} Q_{jr}$ >0 (j=1, 2, 3), $R_{l} = \sum_{r = 1}^{4} μ_{r} R_{lr}$ > 0, $N_{lij}$ , $M_{lij}$ (l=1,2,i,j $\in$ {1,2,3,4}) and any matrices $S_{n}$ (n =1,2,3) satisfying the LMIs:

Ψ^{ii} (l) = [\begin{matrix} Ψ_{11}^{ii} & * & * \\ Ψ_{21}^{ii} (l) & Ψ_{22} & * \\ Ψ_{31}^{ii} & 0 & Ψ_{33} \end{matrix}] < 0

(16)

Ψ^{ij} (l) = [\begin{matrix} Ψ_{11}^{ij} + Ψ_{11}^{ji} & * & * \\ Ψ_{21}^{ij} (l) + Ψ_{21}^{ji} (l) & 2 Ψ_{22} & * \\ Ψ_{31}^{ij} + Ψ_{31}^{ji} & 0 & 2 Ψ_{33} \end{matrix}] < 0

(17)

[\begin{matrix} - I & \sqrt{ρ} K_{j} \\ * & - u_{max}^{2} P \end{matrix}] < 0

(18)

[\begin{matrix} - I & \sqrt{ρ} {{C_{2}}_{i}}_{q} \\ * & - P \end{matrix}] < 0, q = 1, 2

(19)

l = 1, 2, i < j \in {1, 2, 3, 4},

\begin{matrix} Ψ_{11}^{ij} = [\begin{matrix} Λ_{1}^{ij} & * & * & * & * \\ R_{1} & Λ_{2}^{ij} & * & * & * \\ S_{1} B_{2 i} K_{j} + S_{2} A_{i} & - N_{1 ij} + N_{2 ij}^{T} & Λ_{3}^{ij} & * & * \\ 0 & 0 & - M_{1 ij} + M_{2 ij}^{T} & Λ_{4}^{ij} & * \\ P - S_{1}^{T} + S_{3} A_{i} & 0 & - S_{2}^{T} + S_{3} B_{2 i} K_{j} & 0 & Λ_{5} \end{matrix}] \\ Ψ_{21}^{ij} (1) = \sqrt{τ_{M} - τ_{m}} N_{ij}^{T}, Ψ_{21}^{ij} (2) = \sqrt{τ_{M} - τ_{m}} M_{ij}^{T}, Ψ_{31}^{ij} = [\begin{matrix} C_{1 i} & 0 & D_{1 i} K_{j} & 0 & 0 \\ B_{1 i}^{T} S_{1}^{T} & 0 & B_{1 i}^{T} S_{2}^{T} & 0 & B_{1 i}^{T} S_{3}^{T} \end{matrix}], \\ Ψ_{22} = diag (- R_{2}), Ψ_{33} = diag (- I, - γ^{2} I), \\ Λ_{1}^{ij} = Q_{1} + Q_{2} + Q_{3} - R_{1} + S_{1} A_{i} + A_{i}^{T} S_{1}^{T}, Λ_{2}^{ij} = - Q_{1} - R_{1} + N_{1 ij} + N_{1 ij}^{T}, \end{matrix}

\begin{matrix} Λ_{3}^{ij} = - (1 - η) Q_{2} - N_{2 ij} - N_{2 ij}^{T} + M_{1 ij} + M_{1 ij}^{T} \\ + S_{2} B_{i} K_{j} + K_{j}^{T} B_{i}^{T} S_{2}^{T}, Λ_{4}^{ij} = - Q_{3} - M_{2 ij} - M_{2 ij}^{T}, \end{matrix}

\begin{matrix} Λ_{5} = τ_{m}^{2} R_{1} + (τ_{M} - τ_{m}) R_{2} - S_{3} - S_{3}^{T}, N_{ij}^{T} \\ = [\begin{matrix} 0 & N_{1 ij}^{T} & N_{2 ij}^{T} & 0 & 0 \end{matrix}], M_{ij}^{T} \\ = [\begin{matrix} 0 & 0 & M_{1 ij}^{T} & M_{2 ij}^{T} & 0 \end{matrix}] \end{matrix}

Proof. The proof of Theorem 1 is given in the Appendix B.

Remark 3. Theorem 1 shows a sufficient condition to make the ASS (6) asymptotically stable and meet the control requirements. In the following part, the sufficient conditions for the calculation of the feedback gain of the fuzzy controller will be introduced.

Design of fuzzy controller

In this part, a sufficient condition is introduced to calculate the local feedback gain $K_{j}$ of the fuzzy controller.

Theorem 2. For a prescribed $γ > 0$ , $ρ > 0$ , scalars $τ_{m}$ , $τ_{M}$ , $η$ , ρ₂ and $ρ_{3} \neq 0$ , considering the system in equation (6) with $Δ K_{j} = 0$ , the system (6) is progressive stable if the actuator time delay satisfies $0 \leq τ_{m} \leq τ (t) \leq τ_{M}$ , $τ^{.} (t) \leq η$ , the system satisfies $‖ z_{1} (t) ‖_{2} < γ ‖ ω (t) ‖_{2}$ and guarantee the condition $| {z_{2} {(t)}_{q}} | \leq 1$ and $| u (t) | \leq u_{\max}$ , if there are matrices $\hat{P} = \sum_{r = 1}^{4} μ_{r} P_{r}$ >0, ${\hat{Q}}_{j} = \sum_{r = 1}^{4} μ_{r} Q_{jr}$ >0 (j =1,2,3), ${\hat{R}}_{l} = \sum_{r = 1}^{4} μ_{r} R_{lr}$ >0, Y_j, ${\hat{N}}_{lij}$ , ${\hat{M}}_{lij}$ (l=1,2,i,j $\in$ {1,2,3,4}), any matrix X and ${\hat{S}}_{n}$ (n =1,2,3) satisfying the LMIs:

{\hat{Ψ}}^{ii} (l) = [\begin{matrix} {\hat{Ψ}}_{11}^{ii} & * & * \\ {\hat{Ψ}}_{21}^{ii} (l) & {\hat{Ψ}}_{22} & * \\ {\hat{Ψ}}_{31}^{ii} & 0 & {\hat{Ψ}}_{33} \end{matrix}] < 0

(20)

{\hat{Ψ}}^{ij} (l) = [\begin{matrix} {\hat{Ψ}}_{11}^{ij} + {\hat{Ψ}}_{11}^{ji} & * & * \\ {\hat{Ψ}}_{21}^{ij} (l) + {\hat{Ψ}}_{21}^{ji} (l) & 2 {\hat{Ψ}}_{22} & * \\ {\hat{Ψ}}_{31}^{ij} + {\hat{Ψ}}_{31}^{ji} & 0 & 2 {\hat{Ψ}}_{33} \end{matrix}] < 0

(21)

[\begin{matrix} - I & \sqrt{ρ} Y_{j} \\ * & - u_{max}^{2} \hat{P} \end{matrix}] < 0

(22)

[\begin{matrix} - I & \sqrt{ρ} {{C_{2}}_{i}}_{q} X^{T} \\ * & - \hat{P} \end{matrix}] < 0, q = 1, 2

(23)

l = 1, 2, i < j \in {1, 2, 3, 4}

\begin{matrix} {\hat{Ψ}}_{11}^{ij} = [\begin{matrix} {\hat{Λ}}_{1}^{ij} & * & * & * & * \\ {\hat{R}}_{1} & {\hat{Λ}}_{2}^{ij} & * & * & * \\ B_{2 i} Y_{j} + ρ_{2} A_{i} X^{T} & - {\hat{N}}_{1 ij} + {\hat{N}}_{2 ij}^{T} & {\hat{Λ}}_{3}^{ij} & * & * \\ 0 & 0 & - {\hat{M}}_{1 ij} + {\hat{M}}_{2 ij}^{T} & {\hat{Λ}}_{4}^{ij} & * \\ \hat{P} - X + ρ_{3} A_{i} X^{T} & 0 & - ρ_{2} X + ρ_{3} B_{2 i} Y_{j} & 0 & {\hat{Λ}}_{5} \end{matrix}] \\ {\hat{Ψ}}_{21}^{ij} (1) = \sqrt{τ_{M} - τ_{m}} N_{ij}^{T}, {\hat{Ψ}}_{21}^{ij} (2) = \sqrt{τ_{M} - τ_{m}} M_{ij}^{T}, \\ {\hat{Ψ}}_{31}^{ij} = [\begin{matrix} C_{1 i} X^{T} & 0 & D_{1 i} Y_{j} & 0 & 0 \\ B_{2 i}^{T} & 0 & ρ_{2} B_{2 i}^{T} & 0 & ρ_{3} B_{2 i}^{T} \end{matrix}], {\hat{Ψ}}_{22} = diag (- {\hat{R}}_{2}), \\ {\hat{Ψ}}_{33} = diag (- I, - γ^{2} I), {\hat{Λ}}_{1}^{ij} = {\hat{Q}}_{1} + {\hat{Q}}_{2} + {\hat{Q}}_{3} - {\hat{R}}_{1} \\ + A_{i} X^{T} + {XA}_{i}^{T}, {\hat{Λ}}_{2}^{ij} = - {\hat{Q}}_{1} - {\hat{R}}_{1} + {\hat{N}}_{1 ij} + {\hat{N}}_{1 ij}^{T}, \end{matrix}

{\hat{Λ}}_{3}^{ij} = - (1 - η) {\hat{Q}}_{2} - {\hat{N}}_{2 ij} - {\hat{N}}_{2 ij}^{T} + {\hat{M}}_{1 ij} + {\hat{M}}_{1 ij}^{T} + ρ_{2} B_{i} Y_{j} + ρ_{2} Y_{j}^{T} B_{i}^{T}, {\hat{Λ}}_{4}^{ij} = - {\hat{Q}}_{3} - {\hat{M}}_{2 ij} - {\hat{M}}_{2 ij}^{T},

{\hat{Λ}}_{5} = τ_{m}^{2} {\hat{R}}_{1} + (τ_{M} - τ_{m}) {\hat{R}}_{2} - ρ_{3} X - ρ_{3} X^{T}, {\hat{N}}_{ij}^{T} = [\begin{matrix} 0 & {\hat{N}}_{1 ij}^{T} & {\hat{N}}_{2 ij}^{T} & 0 & 0 \end{matrix}], {\hat{M}}_{ij}^{T} = [\begin{matrix} 0 & 0 & {\hat{M}}_{1 ij}^{T} & {\hat{M}}_{2 ij}^{T} & 0 \end{matrix}]

Proof. Define ${\hat{S}}_{1} = X^{- 1}, {\hat{S}}_{2} = ρ_{2} X^{- 1}$ , and ${\hat{S}}_{3} = ρ_{3} X^{- 1}$ ,where $ρ_{2}$ is a real number and $ρ_{3} \neq 0$ , the matrix X is a non-singular matrix. Pre- and post-multiplying (20) and (21) by diag $(X X X X X X I I)$ and its transpose. Pre- and post-multiplying (22) and (23) by diag (I X) and its transpose. After that, defining $\hat{P} = XP X^{T} > 0$ , ${\hat{Q}}_{j} = X Q_{j} X^{T} > 0$ (i = 1,2,3), ${\hat{R}}_{l} = X R_{l} X^{T} > 0$ , ${\hat{N}}_{lij} = X N_{lij} X^{T}$ , ${\hat{M}}_{lij} = X M_{lij} X^{T} (l = 1, 2)$ , and $Y_{j} = K_{j} X^{T}$ , (20)−(23) can be obtained from (16) to (19). Therefore, the control gain $K_{j}$ can be obtained. The proof process is completed.

Remark 4. There are two various cases: slow time-varying and fast time-varying. Sakthivel et al.³⁵ proposed a controller to settle the slow time-varying delay problem which means $\overset{\cdot}{τ} (t) \leq 1$ . Obviously, the controller had considerable limitations. However, in this paper, Theorem 1 and Theorem 2 can be able to deal with both of them. Furthermore, even the parameter on the derivative η is unknown, we can set Q₂ = 0 to solve this problem.

Design of fuzzy non-fragile controller

In this part, taken the control uncertainty into account, the following theorem is introduced to further proposed the fuzzy non-fragile controller.

Theorem 3. For a prescribed $γ > 0$ , $ρ > 0$ , scalars $τ_{m}$ , $τ_{M}$ , $η$ , ρ₂ and $ρ_{3} \neq 0$ , considering the system in equation (10) with $Δ K_{j} = HF (t) E K_{j}$ , the system (10) is progressive stable when the actuator time delay satisfies $0 \leq τ_{m} \leq τ (t) \leq τ_{M}$ , $τ^{.} (t) \leq η$ , the system satisfies $‖ z_{1} (t) ‖_{2} < γ ‖ ω (t) ‖_{2}$ and guarantee the condition $| {z_{2} {(t)}_{q}} | \leq 1$ and $| u (t) | \leq u_{\max}$ , if there are matrices $\hat{P} = \sum_{r = 1}^{4} μ_{r} P_{r}$ >0, ${\hat{Q}}_{j} = \sum_{r = 1}^{4} μ_{r} Q_{jr}$ >0 (j =1,2,3), ${\hat{R}}_{l} = \sum_{r = 1}^{4} μ_{r} R_{lr}$ > 0, Y_j, ${\hat{N}}_{lij}$ , ${\hat{M}}_{lij}$ (l = 1, 2, i, j $\in$ {1,2,3,4}), matrix X and ${\hat{S}}_{n}$ (n = 1,2,3) satisfying the LMIs:

{\hat{Ψ}}^{ii} (l) = [\begin{matrix} {\hat{Ψ}}_{11}^{ii} & * & * & * & * \\ {\hat{Ψ}}_{21}^{ii} (l) & {\hat{Ψ}}_{22} & * & * & * \\ {\hat{Ψ}}_{31}^{ii} & 0 & {\hat{Ψ}}_{33} & * & * \\ {\hat{Ψ}}_{41}^{ii} & 0 & λ H^{T} D_{1 i}^{T} & - λ I & * \\ {\hat{Ψ}}_{51}^{ii} & 0 & 0 & 0 & - λ I \end{matrix}] < 0

(24)

{\hat{Ψ}}^{ij} (l) = [\begin{matrix} {\hat{Ψ}}_{11}^{ij} + {\hat{Ψ}}_{11}^{ji} & * & * & * & * \\ {\hat{Ψ}}_{21}^{ij} (l) + {\hat{Ψ}}_{21}^{ji} (l) & 2 {\hat{Ψ}}_{22} & * & * & * \\ {\hat{Ψ}}_{31}^{ij} + {\hat{Ψ}}_{31}^{ji} & 0 & 2 {\hat{Ψ}}_{33} & * & * \\ {\hat{Ψ}}_{41}^{ij} + {\hat{Ψ}}_{41}^{ji} & 0 & λ H^{T} (D_{1 i}^{T} + D_{1 j}^{T}) & - 2 λ I & * \\ {\hat{Ψ}}_{51}^{ij} + {\hat{Ψ}}_{51}^{ji} & 0 & 0 & 0 & - 2 λ I \end{matrix}] < 0

(25)

[\begin{matrix} - I & \sqrt{ρ} Y_{j} \\ * & - u_{max}^{2} \hat{P} \end{matrix}] < 0

(26)

[\begin{matrix} - I & \sqrt{ρ} {{C_{2}}_{i}}_{q} X^{T} \\ * & - \hat{P} \end{matrix}] < 0, q = 1, 2

(27)

l = 1, 2, i < j \in {1, 2, 3, 4}

\begin{matrix} {\hat{Ψ}}_{41}^{ij} = [\begin{matrix} λ H^{T} B_{2 i}^{T} & 0 & λ ρ_{2} H^{T} B_{2 i}^{T} & 0 & λ ρ_{3} H^{T} B_{2 i}^{T} \end{matrix}], \\ {\hat{Ψ}}_{51}^{ij} = [\begin{matrix} 0 & 0 & E Y_{j} & 0 & 0 \end{matrix}] \end{matrix}

Proof. Replace $K_{j} + Δ K_{j}$ for K_j with $Δ K_{j} = HF (t) E K_{j}$ in Theorem 2. By using lemma 1, the following equation can be obtained:

[\begin{matrix} {\hat{Ψ}}^{ij} (l) & * & * \\ λ {\tilde{H}}^{T} & - λ I & * \\ \tilde{E} & 0 & - λ I \end{matrix}] < 0

(28)

where

\begin{matrix} \tilde{H} = [\begin{matrix} {\hat{S}}_{1} B_{2 i} H & 0 & {\hat{S}}_{2} B_{2 i} H & 0 & {\hat{S}}_{3} B_{2 i} H & 0 & λ H^{T} D_{1 i}^{T} & 0 \end{matrix}], \\ \tilde{E} = [\begin{matrix} 0 & 0 & E Y_{j} & 0 & 0 & 0 & 0 & 0 \end{matrix}] \end{matrix}

Similarly, the proof process is similar to the Theorem 2, which is neglected here. Therefore, the proof process is finished.

Remark 5. In this paper, the matrices P, Q₁, Q₂, Q₃, R₁, R₂ are replaced by $\sum_{r = 1}^{4} μ_{r} P_{r}$ , $\sum_{r = 1}^{4} μ_{r} Q_{1 r}$ , $\sum_{r = 1}^{4} μ_{r} Q_{2 r}, \sum_{r = 1}^{4} μ_{r} Q_{3 r}, \sum_{r = 1}^{4} μ_{r} R_{1 r}, \sum_{r = 1}^{4} μ_{r} R_{2 r}$ in the process of designing the fuzzy controllers, respectively. As a result, the conservatism of the algorithm will be reduced compared with previous investigations.

Numerical simulations

In this section, numerical simulation and suspension plant experiments are carried out to verify the effectiveness of the robust nonfragile $H_{2} / H_{\infty}$ controller in the paper.

Simulation results

The suspension parameters are presented in Table 1. Three different cases are listed in Table 2 to verify the proposed controller performance. Then, the corresponding control laws can be calculated by Theorem 2 and 3. In Li et al.,²⁹ with the input delay approach, the active suspension system controller was constructed as a continuous control signal based on a discrete signal with an input time delay. To demonstrate the superiorities of the proposed control algorithm, the fuzzy sampled-data $H_{\infty}$ controller is used for comparison in Case 1.

Table 1.

Parameter description of quarter-car active suspensions model.

Symbol	Value	Unit	Symbol	Value	Unit
m_s	600	kg	m_u	60	kg
k_s	18,000	N/m	k_sn	1000	N/m ³
b_e	1500	Ns/m	b_c	1200	Ns/m
k_t	200,000	N/m	b_t	15	Ns/m
z_max	0.1	m	u_max	2500	N

Table 2.

Actuator time delays and uncertainties in Case 1, Case 2, and Case 3.

Cases	Delays	Uncertainties	$γ_{\min}$
Case 1	$τ$ = 20 ms	H = 0, F = 0,E = 0	8.5803
Case 2	$τ$ = 20 ms	H = 1, F = sint,E = 0.01	8.5895
Case 3	$τ$ = 20+10sin(100t) ms	H = 1, F = sint,E = 0.10	8.6282

A bump profile is used to demonstrate the effectiveness of the proposed controller, which can be described as follows:

z_{r} (t) = {\begin{matrix} \frac{A}{2} (1 - \cos (\frac{2 π V}{L} t)) \begin{matrix} , & \begin{matrix} if & 0 \leq t \leq \frac{L}{V} \end{matrix} \end{matrix} \\ 0 \begin{matrix} , & \begin{matrix} if & t > \frac{L}{V} \end{matrix} \end{matrix} \end{matrix}

(29)

where L = 5m represents the bump’s length, A = 60mm represent the bump’s height, and V = 35 km/h represents the forward speed of the car.

Figures 2 and 3 show the case of Case 1 and Case 2 during the bump response of the ASS, respectively. In Figure 2, the red line, the blue line and the black line denote the proposed controller, the compared controller and the passive suspension system. In Figure 2, to better highlight the characteristics of sampled-data control, the plots of Figure 2(a) and (d) are partially enlarged. It can be clearly observed the performance of sampled-data control and the sampling time is selected as 0.01 s. The proposed controller can yield a less sprung mass acceleration compared with the other two controllers and satisfy the constraint indexes of the suspension. It means that the proposed controller is superior than the compared controller and the passive system in improving ride comfort. In Figure 3, despite the uncertainty of the actuator, the ASS can still keep in a good performance compared with the actuator in normal condition in almost all respects. It denotes that even there is some uncertainty in the actuator, the proposed controller can make the suspension system stable and maintain an excellent performance of the system. Those simulation experiments demonstrate the effectiveness and robustness of the proposed controller.

Figure 2.

Bump response in Case 1: (a) sprung mass acceleration, (b)suspension travel, (c) tire deflection, and (d) actuator force.

Figure 3.

Bump response in Case 2: (a) sprung mass acceleration, (b) suspension travel, (c) tire deflection, and (d) actuator force.

When the vehicle is driving on the road at varying speeds, the road profile will become a non-stationary process. To further evaluate the performance of the ASS, a random road is utilized to verify the ASS performance of the proposed controller. According to the simulation method of filtering white noise, the non-stationary road profile can be expressed as:

{\overset{\cdot}{z}}_{r} (t) + (v_{0} + at) 2 π n_{c} z_{r} (t) = 2 π n_{0} \sqrt{G_{q} (n_{0}) (v_{0} + at)} ω (t)

(30)

where V₀ = 20 m/s represents the velocity, a = 4 m/s² represents the vehicle acceleration, n₀ = 0.1 m^--1 represents the normal spatial limiting frequency, n_c=0.01 m^--1 represents the road spatial limiting frequency, G_q(n₀) represents the road roughness coefficient, ω(t) means the white noise time series. The A-class road roughness coefficient is chosen for simulation experiments according to ISO-2361. Similarly, in Figure 4, compared with the controller in Li et al.,²⁹ the proposed controller can realize a smaller sprung mass acceleration. Figure 5 illustrates the actuator with uncertainty and time delay can make the same performance compared with the normal controller. It means that the proposed algorithm is more robust and practical. Both of them can keep the system stable and yield a less sprung mass acceleration. The non-fragile H₂/H_∞ controller can be able to improve the ride comfort. The mechanical limitations are also guaranteed at the same time. Therefore, the non-stationary response validates the effectiveness of the proposed ASS controller.

Figure 4.

Non-stationary response in Case 1: (a) sprung mass acceleration, (b) suspension travel, (c) tire deflection, and(d) actuator force.

Figure 5.

Non-stationary response in Case 3: (a) sprung mass acceleration, (b) suspension travel, (c) tire deflection, and(d) actuator force.

Moreover, the root mean square (RMS) can quantify the intensity of acceleration transmitted from a random road to the vehicle body, which is usually applied to reflect the ride comfort. Therefore, the RMS value is utilized to demonstrate the effect of the proposed controller. The RMS values can be described as follows:

RM S_{x} = \sqrt{(1 / T) \int_{0}^{T} x^{T} (t) x (t) dt}

(31)

The Table 3 shows the sprung mass acceleration RMS values and the reduced ratio of the proposed controller, the compared controller and the passive suspension in Case 1. It clearly shows the RMS values of the proposed controller can be decreased by 36.67% and the compared controller is decreased by 21.67%. It highlights the effectiveness of the proposed controller. The above simulation experiments illustrate that the proposed controller can realize a better comprehensive performance than the compared controller and the passive suspension.

Table 3.

RMS values of sprung mass acceleration in Case 1.

Suspension roughness	Passivesuspension	Active suspension
Suspension roughness	Passivesuspension	Comparedcontroller	Proposedcontroller
16 × 10^–6 m³	0.0060	0.0047	0.0038
64 × 10^–6 m³	0.0121	0.0093	0.0076
256 × 10^–6 m³	0.0241	0.0187	0.0152
1024 × 10^–6 m³	0.0481	0.0373	0.0305
Reduced rates		21.67%	36.67%

Suspension plant experiments

In this part, some suspension plant experiments are utilized to testify the effectiveness of the proposed controller. The experimental equipment is shown in Figure 6, and the diagram of the test plant is given in Figure 7. The plant parameters are presented in Table 4 according to Zhang et al.³⁶ For a better comparison, two different cases are listed in Table 5, and the sine wave disturbance and the pulse road input disturbance are designed to evaluate and verify the effectiveness and robustness of the proposed controller. By using a sine wave with a frequency of 3 Hz, the sine road input formula can be expressed as:

z_{r} (t) = h \sin (6 π t)

(32)

where h = 0.2cm represents the height of the vibration. The pulse road input formula with a period of 3 s can be expressed as:

z_{r} = {\begin{matrix} 0.02, if 0.1 \leq t \leq 1.7 \\ 0, else \end{matrix}

(33)

Table 4.

Parameter description of quarter-car active suspension test plant.

Symbol	Value	Unit	Symbol	Value	Unit
m_s	2.45	kg	m_u	1	kg
k_s	900	N/m	k_sn	10	N/m ³
b_e	8	Ns/m	b_c	7	Ns/m
k_t	2500	N/m	b_t	5	Ns/m
z_max	0.038	m	u_max	38.3	N

Table 5.

Actuator time delays and uncertainties in Case 4 and Case 5.

Cases	Delays	Uncertainties	$γ_{\min}$
Case 4	$τ$ = 10 ms	H = 0,F = 0, E = 0	30.5906
Case 5	$τ$ = 10+4sin(500t) ms	H = 1, F = sint,E = 0.01	31.4866

Figure 6.

The quarter suspension test plant.

Figure 7.

The diagram of test plant.

The pulse road input disturbance is given in Figure 8.

Figure 8.

Pulse road input simulation.

In Figure 9, the proposed controller can yield a much less sprung mass acceleration compared with the controller in Li et al.²⁹ and the passive suspension in Case 4, which denotes it can greatly contribute to realizing a better vehicle comprehensive performance and satisfy the physical constraints even if the actuator exists some time delays in the suspension system. It highlights the superiority of the designed algorithm. In addition, Figure 10 illustrates that the proposed controller is valid when the vehicle is driving on a road surface similar to the pulse form. It can provide better ride quality as compared with the passive suspension system. The RMS values and decreased ratios in Case 4 and Case 5 are presented in Table 6. It can be observed in Table 6 that the RMS values of the proposed controller can be largely decreased by 92.6% and the compared controller is decreased by 38.2% in Case 4. In Case 5, the RMS value is decreased by 76.3%.

Table 6.

RMS values of sprung mass acceleration on experiment plant in Case 4 and Case5.

Suspensioncase	Passivesuspension	Active suspension
Suspensioncase	Passivesuspension	Comparedcontroller	Proposedcontroller
Case 4	3.1138	1.9241(−38.2%)	0.2286(−92.6%)
Case 5	1.6246		0.3843(−76.3%)

Figure 9.

Sinusoidal response in Case 4: (a) sprung mass acceleration, (b) suspension travel, (c) tire deflection, and (d) actuator force.

Figure 10.

Pulse response in Case 5: (a) sprung mass acceleration, (b) suspension travel, (c) tire deflection, and (d) actuator force.

In conclusion, the whole experimental results manifest the effectiveness and robustness of the proposed controller.

Conclusions

In this work, the problem of fuzzy robust $H_{2} / H_{\infty}$ control for the nonlinear ASS with time varying delay and actuator uncertainties was investigated. A T-S fuzzy model was proposed to deal with the highly dynamic nonlinearity in the ASS, which is an effective approach to obtain the control gains. With the designed parameter Lyapunov function, the less conservatism of the proposed controller can be obtained. The stability and the constrained requirements of the ASS can be also assured. A co-design criterion was proposed to calculate the control law for the time delay and control uncertainty. Numerical simulations and the quarter suspension plant experiments both demonstrate the effectiveness and robustness of the proposed controller which can greatly improve the suspension performance. Considering the complexity of designing fuzzy controller, more attention will be paid to the fuzzy control to improve the flexibility in the future work.

Footnotes

Appendix A

Lemma 1. ³⁷ For a symmetric matrices L = L^T, matrices M and N of appropriate dimensions

(A.1)

L + MFN + N^{T} F^{T} M^{T} < 0

if F satisfies $F {(t)}^{T} F (t) \leq I$ , then there exists some $ε > 0$ , the inequality can be obtained as

(A.2)

L + ε M M^{T} + ε^{- 1} N^{T} N < 0

By Schur complement, the following formula can be written by

(A.3)

[\begin{matrix} L & ε M & N^{T} \\ * & - ε I & 0 \\ * & * & - ε I \end{matrix}] < 0

Lemma 2. ³⁸ Matrices Z₁, Z₂, and Γ denote constant matrices and $0 \leq τ_{m} \leq τ (t) \leq τ_{M}$ , then

(A.4)

(τ (t) - τ_{m}) Z_{1} + (τ_{M} - τ (t)) Z_{2} + Γ < 0

if and only if

(A.5)

(τ_{M} - τ_{m}) Z_{1} + Γ < 0

and

(A.6)

(τ_{M} - τ_{m}) Z_{2} + Γ < 0

Lemma 3. ³⁹ For matrix R₁ > 0, scalar $τ_{1} \leq τ (t) \leq τ_{2}$ , and vector $\overset{\cdot}{x} : [- τ_{2}, 0] \to R^{n}$ such that the inequality is well defined, then

(A.7)

\begin{matrix} (τ_{1} - τ_{2}) \int_{t - τ (t)}^{t - τ_{1}} {\overset{\cdot}{x}}^{T} (t) R_{1} \overset{\cdot}{x} (t) dt \leq {[\begin{matrix} x (t - τ_{1}) \\ x (t - τ (t)) \end{matrix}]}^{T} \\ [\begin{matrix} - R_{1} & R_{1} \\ R_{1} & - R_{1} \end{matrix}] [\begin{matrix} x (t - τ_{1}) \\ x (t - τ (t)) \end{matrix}] \end{matrix}

Appendix B

Proof. The Lyapunov-Krasovskii function is chosen as:

(B.1)

V (x_{t}) = V_{1} (x_{t}) + V_{2} (x_{t}) + V_{3} (x_{t})

where

V_{1} (x_{t}) = x^{T} (t) Px (t)

V_{2} (x_{t}) = \int_{t - τ_{m}}^{t} x^{T} (v) Q_{1} x (v) d v + \int_{t - τ (t)}^{t} x^{T} (v) Q_{2} x (v) d v + \int_{t - τ_{M}}^{t} x^{T} (v) Q_{3} x (v) d v

\begin{matrix} V_{3} (x_{t}) = τ_{m} \int_{t - τ_{m}}^{t} \int_{s}^{t} {\overset{\cdot}{x}}^{T} (v) R_{1} \overset{\cdot}{x} (v) dvds \\ + \int_{t - τ_{M}}^{t - τ_{m}} \int_{s}^{t} {\overset{\cdot}{x}}^{T} (v) R_{2} \overset{\cdot}{x} (v) dvds \end{matrix}

According to the Leibniz-Newton formula, the following formula always holds

(B.2)

x (t - τ_{m}) - x (t - τ (t)) - \int_{t - τ (t)}^{t - τ_{m}} \overset{\cdot}{x} (s) ds = 0

(B.3)

x (t - τ (t)) - x (t - τ_{M}) - \int_{t - τ_{M}}^{t - τ (t)} \overset{\cdot}{x} (s) ds = 0

Moreover, the following equation is also satisfied in (10)

(B.4)

\begin{matrix} \overset{\cdot}{x} (t) = \sum_{i = 1}^{4} \sum_{j = 1}^{4} μ_{i} μ_{j} \\ [A_{i} (t) x (t) + B_{2 i} (t) K_{j} x (t - τ (t)) + B_{1 i} ω (t)] \end{matrix}

Then, taking the derivative of the equations (B.1), we can get

(B.5)

{\overset{\cdot}{V}}_{1} (x_{t}) = 2 {\overset{\cdot}{x}}^{T} (t) Px (t)

(B.6)

\begin{matrix} {\overset{\cdot}{V}}_{2} (x_{t}) = x^{T} (t) (Q_{1} + Q_{2} + Q_{3}) x (t) - x^{T} (t - τ_{m}) Q_{1} \\ x (t - τ_{m}) - (1 - \overset{\cdot}{τ} (t)) x^{T} (t - τ (t)) Q_{2} x (t - τ (t)) \\ - x^{T} (t - τ_{M}) Q_{3} x (t - τ_{M}) \end{matrix}

(B.7)

\begin{matrix} {\overset{\cdot}{V}}_{3} (x_{t}) = {\overset{\cdot}{x}}^{T} (t) (τ_{m}^{2} R_{1} + (τ_{M} - τ_{m}) R_{2}) \overset{\cdot}{x} (t) \\ - τ_{m} \int_{t - τ_{m}}^{t} {\overset{\cdot}{x}}^{T} (s) R_{1} \overset{\cdot}{x} (s) ds - \int_{t - τ_{M}}^{t - τ_{m}} {\overset{\cdot}{x}}^{T} (s) R_{2} \overset{\cdot}{x} (s) ds \end{matrix}

By using (B.5)–(B.7) and applying the equations (B.2)–(B.4), we can get the following equation:

\begin{matrix} \overset{\cdot}{V} (t) \leq 2 {\overset{\cdot}{x}}^{T} (t) Px (t) + x^{T} (t) (Q_{1} + Q_{2} + Q_{3}) x (t) \\ - x^{T} (t - τ_{m}) Q_{1} x (t - τ_{m}) - (1 - η) x^{T} \\ (t - τ (t)) Q_{2} x (t - τ (t)) \\ - x^{T} (t - τ_{M}) Q_{3} x (t - τ_{M}) + {\overset{\cdot}{x}}^{T} (t) (τ_{m}^{2} R_{1} \\ + (τ_{M} - τ_{m}) R_{2}) \overset{\cdot}{x} (t) - τ_{m} \int_{t - τ_{m}}^{t} {\overset{\cdot}{x}}^{T} (s) R_{1} \overset{\cdot}{x} (s) ds \\ - \int_{t - τ_{M}}^{t - τ_{m}} {\overset{\cdot}{x}}^{T} (s) R_{1} \overset{\cdot}{x} (s) ds \\ + \sum_{i = 1}^{4} \sum_{j = 1}^{4} 2 μ_{i} μ_{j} ξ^{T} (t) N_{ij} [x (t - τ_{m}) - x (t - τ (t)) \end{matrix}

(B.8)

\begin{matrix} - \int_{t - τ (t)}^{t - τ_{m}} \overset{\cdot}{x} (s) ds] + \sum_{i = 1}^{4} \sum_{j = 1}^{4} 2 μ_{i} μ_{j} ξ^{T} (t) M_{ij} \\ [x (t - τ (t)) - x (t - τ_{M}) - \int_{t - τ_{M}}^{t - τ (t)} \overset{\cdot}{x} (s) ds] \\ + \sum_{i = 1}^{4} \sum_{j = 1}^{4} 2 μ_{i} μ_{j} ξ^{T} (t) S [\overset{\cdot}{x} (t) - (A_{i} (t) x (t) \\ + B_{2 i} (t) K_{j} x (t - τ (t)) + B_{1 i} ω (t))] \end{matrix}

where

(B.9)

\begin{matrix} ξ^{T} (t) = [\begin{matrix} x^{T} (t) x^{T} (t - τ_{m}) x^{T} (t - τ (t)) x^{T} (t - τ_{M}) {\overset{\cdot}{x}}^{T} (t) \end{matrix}], \\ S^{T} = [\begin{matrix} S_{1}^{T} 0 S_{2}^{T} 0 S_{3}^{T} \end{matrix}] \end{matrix}

By using lemma 3, we obtain

(B.10)

\begin{matrix} - τ_{m} \int_{t - τ_{m}}^{t} {\overset{\cdot}{x}}^{T} (t) R_{1} \overset{\cdot}{x} (t) dt \leq {[\begin{matrix} x (t) \\ x (t - τ_{m}) \end{matrix}]}^{T} \\ [\begin{matrix} - R_{1} & R_{1} \\ R_{1} & - R_{1} \end{matrix}] [\begin{matrix} x (t) \\ x (t - τ_{m}) \end{matrix}] \end{matrix}

The following inequalities can be obtained by using the method in Niamsup and Phat⁴⁰

(B.11)

\begin{matrix} - 2 \sum_{i = 1}^{4} \sum_{j = 1}^{4} μ_{i} μ_{j} ξ^{T} (t) N_{ij} \int_{t - τ (t)}^{t - τ_{m}} \overset{\cdot}{x} (s) ds \leq (τ (t) - τ_{m}) \\ \sum_{i = 1}^{4} \sum_{j = 1}^{4} μ_{i} μ_{j} ξ^{T} (t) N_{ij}^{T} R_{2}^{- 1} N_{ij} ξ (t) + \int_{t - τ (t)}^{t - τ_{m}} {\overset{\cdot}{x}}^{T} (s) R_{2} \overset{\cdot}{x} (s) ds \end{matrix}

(B.12)

\begin{matrix} - 2 \sum_{i = 1}^{4} \sum_{j = 1}^{4} μ_{i} μ_{j} ξ^{T} (t) M_{ij} \int_{t - τ_{M}}^{t - τ (t)} \overset{\cdot}{x} (s) ds \leq (τ_{M} - τ (t)) \\ \sum_{i = 1}^{4} \sum_{j = 1}^{4} μ_{i} μ_{j} ξ^{T} (t) M_{ij}^{T} R_{2}^{- 1} M_{ij} ξ (t) \\ + \int_{t - τ_{M}}^{t - τ (t)} {\overset{\cdot}{x}}^{T} (s) R_{2} \overset{\cdot}{x} (s) ds \end{matrix}

Then substituting (B.10–B.12) into (B.8), we can obtain

\begin{matrix} \overset{\cdot}{V} (t) \leq \sum_{i = 1}^{4} \sum_{j = 1}^{4} μ_{i} μ_{j} {ξ^{T} (t) \\ [Ψ_{11}^{ij} + (τ (t) - τ_{m}) N_{ij}^{T} R_{2}^{- 1} N_{ij} + (τ_{M} - τ (t)) \\ M_{ij}^{T} R_{2}^{- 1} M_{ij}] ξ (t) + 2 ξ^{T} (t) S B_{1 i} ω (t)} \end{matrix}

Furthermore, note that

(B.13)

\begin{matrix} z_{1}^{T} (t) z_{1}^{T} (t) = \sum_{i = 1}^{4} \sum_{j = 1}^{4} \sum_{k = 1}^{4} \sum_{l = 1}^{4} μ_{i} μ_{j} μ_{k} μ_{l} ξ^{T} (t) C_{ij}^{T} C_{ij} ξ (t) \\ \leq \sum_{i = 1}^{4} \sum_{j = 1}^{4} μ_{i} μ_{j} ξ^{T} (t) C_{ij}^{T} C_{ij} ξ (t) \end{matrix}

where

C_{ij} = [\begin{matrix} C_{1 i} & 0 & D_{1 i} K_{j} & 0 & 0 \end{matrix}]

considering the H_∞ performance indicators, we can obtain

(B.14)

\begin{matrix} \overset{\cdot}{V} (t) + z_{1}^{T} (t) z (t) - γ^{2} ω^{T} (t) ω (t) \leq \sum_{i = 1}^{4} \sum_{j = 1}^{4} μ_{i} μ_{j} \\ {ξ^{T} (t) [Ψ_{11}^{ij} + (τ (t) - τ_{m}) N_{ij}^{T} R_{2}^{- 1} N_{ij} \\ + (τ_{M} - τ (t)) M_{ij}^{T} R_{2}^{- 1} M_{ij}] ξ (t) \\ + 2 ξ^{T} (t) S B_{1 i} ω (t) + ξ^{T} (t) C_{ij}^{T} C_{ij} ξ (t) - γ^{2} ω^{T} (t) ω (t)} \\ \leq \sum_{i = 1}^{4} μ_{i}^{2} {[\begin{matrix} ξ^{T} (t) \\ ω (t) \end{matrix}]}^{T} [\begin{matrix} Γ_{1} & S B_{1 i} \\ B_{1 i}^{T} S^{T} & - γ^{2} I \end{matrix}] [\begin{matrix} ξ (t) \\ ω (t) \end{matrix}] \\ + \sum_{i, j = 1}^{4} \sum_{i < j} μ_{i} μ_{j} {[\begin{matrix} ξ (t) \\ ω (t) \end{matrix}]}^{T} \\ [\begin{matrix} Γ_{2} & S B_{1 i} + S B_{1 j} \\ B_{1 i}^{T} S^{T} + B_{1 j}^{T} S^{T} & - 2 γ^{2} I \end{matrix}] [\begin{matrix} ξ (t) \\ ω (t) \end{matrix}] \end{matrix}

where $Γ_{1} = Ψ_{11}^{ii} + C_{ii}^{T} C_{ii} + (τ (t) - τ_{m}) N_{ii}^{T} R_{2}^{- 1} N_{ii} +$ $(τ_{M} - τ (t)) M_{ii}^{T} R_{2}^{- 1} M_{ii}$ , $Γ_{2} = Ψ_{11}^{ij} + Ψ_{11}^{ji} + \frac{1}{2} (C_{ij}^{T} + C_{ji}^{T}) \times$ $(C_{ij} + C_{ji}) + \frac{1}{2} (τ (t) - τ_{m}) (N_{ij}^{T} + N_{ji}^{T}) R_{2}^{- 1} (N_{ij} + N_{ji}) + \frac{1}{2}$ $(τ_{M} - τ (t)) (M_{ij}^{T} + M_{ji}^{T}) R_{2}^{- 1} (M_{ij} + M_{ji}$ .

By using lemma 2, we can obtain

(B.15)

[\begin{matrix} Γ_{1} & S B_{1 i} \\ B_{1 i}^{T} S^{T} & - γ^{2} I \end{matrix}]

and

(B.16)

[\begin{matrix} Γ_{2} & S B_{1 i} + S B_{1 j} \\ B_{1 i}^{T} S^{T} + B_{1 j}^{T} S^{T} & - 2 γ^{2} I \end{matrix}]

Further, a generalized $H_{2}$ performance in (15) is established. Then, we obtain

(B.17)

\begin{matrix} \overset{\cdot}{V} (x_{t}) + z_{1}^{T} (t) z_{1} (t) - γ^{2} ω^{T} (t) ω (t) < 0 \\ J \leq \int_{0}^{\infty} [z_{1}^{T} (t) z_{1} (t) - γ^{2} ω^{2} (t) ω (t)] dt \\ + V (x (t)) |_{t = \infty} - V (x (t)) |_{t = 0} \end{matrix}

For all nonzero $ω \in L_{2} [0, \infty)$ , we have $V (0) = 0$ and $V (\infty) \geq 0$ , which means $‖ z_{1} ‖_{2} < γ ‖ ω ‖_{2}$ . In addition, if $\overset{\cdot}{V} (x_{t}) < 0$ , the closed-loop ASS (11) is asymptotically stable. From the Lyapunov function defined in (B.11), noting that the function is positive, then the following inequalities hold:

\begin{matrix} max_{t > 0} {| {z_{2} (t)}_{q} |}^{2} = max_{t > 0} {‖ x^{T} (t) {C_{2 i}}_{q}^{T} {C_{2 i}} x (t) ‖}_{2} \\ = max_{t > 0} {‖ x^{T} (t) P^{\frac{1}{2}} P^{- \frac{1}{2}} {C_{2 i}}_{q}^{T} {C_{2 i}}_{q} \times P^{- \frac{1}{2}} P^{\frac{1}{2}} ‖}_{2} \end{matrix}

< ρ \times θ_{max} (P^{- \frac{1}{2}} {C_{2 i}}_{q}^{T} {C_{2 i}}_{q} P^{- \frac{1}{2}}), q = 1, 2

\begin{matrix} max_{t > 0} | u (t) |^{2} = max_{t > 0} {‖ K_{j} x (t) ‖}_{2}^{2} = max_{t > 0} ‖ x^{T} (t) K_{j}^{T} K_{j} x (t) ‖^{2} \\ < ρ \times θ_{max} (P^{- \frac{1}{2}} K_{j}^{T} K_{j} P^{- \frac{1}{2}}) . \end{matrix}

where $θ_{\max}$ represents the maximum eigenvalue of the matrix. Therefore, from the above inequalities, the limitations can be guaranteed, if the inequality satisfies:

(B.18)

ρ \times (P^{- \frac{1}{2}} {C_{2 i}}_{q}^{T} {C_{2 i}}_{q} P^{- \frac{1}{2}}) < I, q = 1, 2

(B.19)

ρ \times (P^{- \frac{1}{2}} K_{j}^{T} K_{j} P^{- \frac{1}{2}}) < u_{max}^{2}

Then by Schur complement, (B.18) and (B.19) are equal to (25) and (26). The proof process is completed.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by National Natural Science Foundation of China (Grant No. 52175127), the Natural Science Foundation of Guangdong Province of China (Grant Nos. 2022A1515011495, 2021AA151500583), the Joint Project of Natural Science Foundation of Liaoning Province of China (Grant No. 2021-KF-11-02), the Fundamental Research Funds for the Central Universities (Grant No. N2203012), and the Research Grant of the University of Macau (Grant No. MYRG2020-00045-FST).

ORCID iDs

Pak Kin Wong

Jing Zhao

References

Lam

Cheung

. Control of vehicle suspension using an adaptive inverter. Proc IMechE, Part D: J Automobile Engineering 2015; 229: 1934–1943.

Zhou

Liu

, et al. Ride comfort optimization via speed planning and preview semi-active suspension control for autonomous vehicles on uneven roads. IEEE Trans Vehicular Technol 2020; 69: 8343–8355.

Wong

Zhao

. Practical multi-objective control for automotive semi-active suspension system with nonlinear hydraulic adjustable damper. Mech Syst Signal Process 2019; 117: 667–688.

Meng

Chen

Wang

, et al. Study on vehicle active suspension system control method based on homogeneous domination approach. Asian J Control 2021; 23: 561–571.

Lam

Cheung

. Multi-objective control for active vehicle suspension with wheelbase preview. J Sound Vib 2014; 333: 5269–5282.

Zhao

Wong

Xie

, et al. Design and control of an automotive variable hydraulic damper using cuckoo search optimized PID method. Int J Autom Technol 2019; 20: 51–63.

Lam

Cheung

. Motion-based active disturbance rejection control for a non-linear full-car suspension system. Proc IMechE, Part D: J Automobile Engineering 2018; 232: 616–631.

Zhang

Jing

. A bioinspired dynamics-based adaptive fuzzy SMC method for half-car active suspension systems with input dead zones and saturations. IEEE Trans Cybern 2021; 51: 1743–1755.

Sun

Yuan

Cai

, et al. Model predictive control of an air suspension system with damping multi-mode switching damper based on hybrid model. Mech Syst Signal Process 2017; 94: 94–110.

10.

Sun

Yin

, et al. Model predictive thrust force control for linear motor actuator used in active suspension. IEEE Trans Energy Convers 2021; 36: 3063–3072.

11.

Wang

Chang

Tian

. Extended state observer–based backstepping fast terminal sliding mode control for active suspension vibration. J Vib Control 2021; 27: 2303–2318.

12.

Abdul Zahra

Abdalla

. Design of fuzzy super twisting sliding mode control scheme for unknown full vehicle active suspension systems using an artificial bee colony optimization algorithm. Asian J Control 2021; 23: 1966–1981.

13.

Lin

. Fuzzy sliding mode control for active suspension system with proportional differential sliding mode observer. Asian J Control 2019; 21: 264–276.

14.

Zhang

Yan

, et al. Adaptive event-triggered fuzzy control for uncertain active suspension systems. IEEE Trans Cybern 2019; 49: 4388–4397.

15.

Liu

Chen

. Adaptive sliding mode control for uncertain active suspension systems with prescribed performance. IEEE Trans Syst Man Cybern Syst 2021; 51: 6414–6422.

16.

Zhao

Wang

Wong

, et al. Multi-objective frequency domain-constrained static output feedback control for delayed active suspension systems with wheelbase preview information. Nonlinear Dyn 2021; 103: 1757–1774.

17.

Xue

Jin

. Parameter-dependent actuator fault estimation for vehicle active suspension systems based on RBFNN. Proc IMechE, Part D: J Automobile Engineering 2021; 235: 2540–2550.

18.

Park

Cheng

. Robust fuzzy delayed sampled-data control for nonlinear active suspension systems with varying vehicle load and frequency-domain constraint. Nonlinear Dyn 2021; 105: 2265–2281.

19.

Jeyasenthil

Yoon

Choi

, et al. Robust semiactive control of a half-car vehicle suspension system with magnetorheological dampers: quantitative feedback theory approach with dynamic decoupler. Int J Robust Nonlinear Control 2021; 31: 1418–1435.

20.

Xie

Zhao

, et al. Static-output-feedback based robust fuzzy wheelbase preview control for uncertain active suspensions with time delay and finite frequency constraint. IEEE/CAA J Autom Sin 2021; 8: 664–678.

21.

Qin

Liu

, et al. Adaptive robust control for active suspension systems: targeting nonholonomic reference trajectory and large mismatched uncertainty. Nonlinear Dyn 2021; 104: 3861–3880.

22.

Shao

Naghdy

. Reliable fuzzy H_∞ control for active suspension of in-wheel motor driven electric vehicles with dynamic damping. Mech Syst Signal Process 2017; 87: 365–383.

23.

Wang

Jia

Zhang

. Stability and stabilization of T–S fuzzy time-delay system via relaxed integral inequality and dynamic delay partition. IEEE Trans Fuzzy Syst 2021; 29: 2829–2843.

24.

Zhao

Wong

, et al. Chassis integrated control for active suspension, active front steering and direct yaw moment systems using hierarchical strategy. Veh Syst Dyn 2017; 55: 72–103.

25.

Chen

Cai

Pan

. Experimental study of delayed feedback control for a flexible plate. J Sound Vib 2009; 322: 629–651.

26.

Zhang

Lam

. Parameter-dependent input-delayed control of uncertain vehicle suspensions. J Sound Vib 2008; 317: 537–556.

27.

Karim Afshar

Javadi

Jahed-Motlagh

. Robust control of an active suspension system with actuator time delay by predictor feedback. IET Control Theory Appl 2018; 12: 1012–1023.

28.

Gao

Sun

Shi

. Robust sampled-data H_∞ control for vehicle active suspension systems. IEEE Trans Control Syst Technol 2010; 18: 238–245.

29.

Jing

Lam

, et al. Fuzzy sampled-data control for uncertain vehicle suspension systems. IEEE Trans Cybern 2014; 44: 1111–1126.

30.

Xie

Wong

, et al. Robust nonfragile H_∞ optimum control for active suspension systems with time-varying actuator delay. J Vib Control 2019; 25: 2435–2452.

31.

Xie

Zhao

, et al. Fuzzy finite-frequency output feedback control for nonlinear active suspension systems with time delay and output constraints. Mech Syst Signal Process 2019; 132: 315–334.

32.

Xie

Wong

, et al. Nonfragile H_∞ control of delayed active suspension systems in finite frequency under nonstationary running. J Dyn Syst Meas Control 2019; 141: 1001–1016.

33.

Xie

Zhao

, et al. Robust non-fragile finite frequency H_∞ control for uncertain active suspension systems with time-delay using T-S fuzzy approach. J Franklin Inst 2021; 358: 4209–4238.

34.

Zhang

, et al. Finite frequency fuzzy H control for uncertain active suspension systems with sensor failure. IEEE/CAA J Autom Sin 2018; 5: 777–786.

35.

Sakthivel

Arunkumar

Mathiyalagan

, et al. Robust reliable control for uncertain vehicle suspension systems with input delays. J Dyn Syst Meas Control 2015; 137: 1013–1025.

36.

Zhang

Jing

Wang

. Bioinspired nonlinear dynamics-based adaptive neural network control for vehicle suspension systems with uncertain/unknown dynamics and input delay. IEEE Trans Ind Electron 2021; 68: 12646–12656.

37.

She

, et al. Delay-dependent criteria for robust stability of time-varying delay systems. Automatica 2004; 40: 1435–1439.

38.

Tian

Yue

Zhang

. Delay-dependent robust H_∞ control for T–S fuzzy system with interval time-varying delay. Fuzzy Sets Syst 2009; 160: 1708–1719.

39.

Peng

Tian

. Delay-dependent robust stability criteria for uncertain systems with interval time-varying delay. J Comput Appl Math 2008; 214: 480–494.

40.

Niamsup

Phat

. State feedback stabilization of linear descriptor time-varying delay systems. Trans Inst Meas Contr 2020; 42: 2191–2197.

Fuzzy robust non-fragile control for nonlinear active suspension systems with time varying actuator delay

Abstract

Keywords

Introduction

Problem formulation

Description of an ASS model

T-S fuzzy ASS modeling

Fuzzy robust H∞ controller

Control objectives

Main results

Stability analysis and H ∞ performance constraints

Design of fuzzy controller

Design of fuzzy non-fragile controller

Numerical simulations

Simulation results

Suspension plant experiments

Conclusions

Footnotes

Appendix A

Appendix B

Declaration of conflicting interests

Funding

ORCID iDs

References

Fuzzy robust H_∞ controller

Stability analysis and $H_{\infty}$ performance constraints