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Abstract

This paper presents a comfort-enhanced vibration control design approach for vehicle active suspensions with control delay. By introducing some novel relaxed inequalities, a less conservative bounded real lemma is derived such that the suspension system is asymptotically stable and has an enhanced vibration attenuation performance in the presence of road disturbance and control delay. In the control design, an augmented Lyapunov–Krasovskii functional is proposed to enlarge the degree of freedom of design. A suitable controller can be obtained via a sufficient condition with linear matrix inequality constraints. Compared with the previous methods based on frequently used inequalities and the traditional Lyapunov–Krasovskii functional, the proposed controller can provide better comfort performance for different control delays. Finally, the advantages of the proposed method are demonstrated using numerical examples.

Keywords

Vibration control active suspensions control delay augmented Lyapunov–Krasovskii functional Bessel–Legendre inequality

Introduction

The topic of vehicle suspension has been the subject of a large amount of research, due to its importance in automobile performance. A superior automotive suspension should consistently provide comfort, good handling, and be able to provide good road holding ability.^1,2 To improve the maneuverability of automotive suspensions, many researchers have explored various types of control strategies, such as active control, passive control, and semiactive control. Currently, robust control strategies exhibit good potential for solving vehicle suspension control problems. In recent years, several researchers have proposed advanced active suspension control strategies.^3–12 A nonlinear feedback state controller for the control of a quarter-car active suspension model with an extended state observer has been proposed.³ A robust output vibration controller was proposed for an active seat suspension system for heavy-duty vehicles.⁶ A robust fault-tolerant control scheme has been suggested for the active suspension of electric vehicles that used the installation of a dynamic vibration absorber.⁷ An energy-efficient adaptive control design was developed for a bioinspired active suspension.⁸ An adaptive control algorithm has been developed for a vehicle active suspension that had a dead zone and hysteresis nonlinearities.¹²

In addition, H_∞ control is also a valid method to enhance the performance of a suspension system. In recent years, it has attracted the attention of a large number of scholars.^13–22 A robust polytopic type L₂ controller has been designed for an uncertain quarter vehicle suspension.¹³ A noniteration feedback method has been developed for active suspension systems with finite-frequency constraint.¹⁴ It is a common phenomenon that input delay frequently happens in hydraulic systems and in-vehicle networks, and output performance may be severely affected due to input delay.^15,16 Therefore, the issue of vibration control problems in active suspension systems with input delay has attracted a lot of attention in the past few decades. For instance, Wang et al.¹⁷ studied the vehicle lateral output feedback control problem by considering delay and control saturation. A nonfragile control method was suggested for suspensions with input delay.²¹ These H_∞ design methods were all confirmed to be effective for systems with input delay.

Thanks to convenient tractability and easy extension, the Lyapunov–Krasovskii functional (LKF) and linear matrix inequality (LMI) methods have attracted great attention due to their ability to perform stability analyses of linear systems with control delays.^23–38 In the process of applying these methods, some integral terms need to be defined. In the previous suspension control studies, some methods were used for estimating the upper bound of these integral terms, including the Jensen, Park, and Moon inequalities.^19,21 However, applications of these frequently used methods are conservative, as these methods have difficulty in providing a tight upper bound of the integral term. Recently, some novel integral inequalities, such as the Wirtinger-based inequality, reciprocally convex inequality, and Bessel–Legendre inequality (BLI), have been suggested to reduce the estimated gap during criterion deriving.^29–38

Due to the advantages of these relaxed inequalities, they provide a promising way to design the robust H_∞ controller for delayed active suspension systems. However, related research is quite rare. In general, there are two difficulties in designing a controller using these relaxed inequalities. First, some traditional LKFs have been confirmed to be conservative, and thus choosing a proper LKF is important for control design.^31–35 Second, most of the design results are a nonlinear minimization problem and cannot be solved using the LMI technique. Currently, few researchers have published designs of a comfort-enhanced vibration controller for vehicle suspension by using these recently reported inequalities.

This paper focuses on the study of a comfort-enhanced vibration controller for delayed vehicle suspension. The main differences between this proposed method and the previous suspension control studies are summarized as follows: (1) a less conservative bounded real lemma is constructed by combining the second-order BLI with the delay-dependent ERCI approach; and (2) an augmented LKF is proposed to develop a LMI-based control criterion such that the degree of freedom of design can be effectively enlarged. Finally, the advantages of the comfort-enhanced design method are attested using simulation examples. The rest of this paper is arranged as follows. The problem description and some novel relaxed inequalities are given in the next section. A comfort-enhanced design method is presented in “Comfort-enhanced active vibration control” section. Simulation results are provided in “Simulation investigations” section. Conclusions are given in the final section. In Appendix 1, the proofs of the main design results are supplied.

Notation. In this paper, $col {\begin{matrix} • & \dots & • \end{matrix}} = {[\begin{matrix} •^{T} & \dots & •^{T} \end{matrix}]}^{T}$ , $2 • = • + •^{T}$ , ${•}_{i}$ refers to the $i$ th line of the matrix $•$ , $| | • | |$ refers to the Euclidean norm of $•$ , $θ_{\max} (•)$ refers to maximal eigenvalue of $•$ , and other notations are fairly standard.

Suspension model and problem description

A typical suspension model given in Figure 1 is considered in this study.¹⁸ The dynamic equations for the vehicle suspension model with control delay shown in Figure 1 can be given as

\begin{array}{l} m_{s} \frac{d^{2} z_{s} (t)}{d t^{2}} + c_{s} (\frac{d z_{s} (t)}{d t} - \frac{d z_{u} (t)}{d t}) + k_{s} (z_{s} (t) - z_{u} (t)) = u (t - d (t)) \\ m_{u} \frac{d^{2} z_{u} (t)}{d t^{2}} - c_{s} (\frac{d z_{s} (t)}{d t} - \frac{d z_{u} (t)}{d t}) - k_{s} (z_{s} (t) - z_{u} (t)) + c_{u} (\frac{d z_{u} (t)}{d t} - \frac{d z_{r} (t)}{d t}) \\ + k_{u} (z_{u} (t) - z_{r} (t)) = - u (t - d (t)) \end{array}

(1)

where

m_{s}

and

m_{u}

refer to the sprung mass and unsprung mass, respectively;

k_{s}

c_{s}

, and

u (t)

represent the suspension stiffness, damping, and active control force, respectively;

d (t)

denotes the control delay;

k_{u}

and

c_{u}

are the stiffness and damping of the tire;

z_{s} (t)

and

z_{u} (t)

represent the vertical displacement of the sprung mass and unsprung mass; and

z_{r} (t)

is the road vertical disturbance. For brevity,

z_{s} (t)

is replaced with

z_{s}

, and this is also done for the other symbols.

Figure 1.

Vehicle suspension model.

In general, the control delay is a time-varying continuous function satisfying

0 \leq d (t) \leq h, - μ \leq \frac{d d (t)}{d t} \leq μ

(2)

where

h

and

μ

are known positive constants,

h

denotes the upper bound of

d (t)

, and

μ

denotes the upper bound of

\frac{d d (t)}{d t}

In the design of the controller, comfort performance is a key objective that needs to be considered first. Thus, the vertical acceleration needs to be minimized as much as possible. In addition, small suspension deflection and good road holding stability are also important control objectives, and the requirements are the following

\begin{array}{l} | z_{s} - z_{u} | \leq z_{\max} \\ | k_{u} (z_{u} - z_{r}) | \leq (m_{s} + m_{u}) g \end{array}

(3)

where

z_{\max}

represents the maximum suspension travel limits and

g

represents the gravitational acceleration.

Therefore, the key performances can be given as

z_{1} (t) = \frac{d^{2} z_{s}}{d t^{2}}, z_{2} (t) = {[\begin{matrix} \frac{z_{s} - z_{u}}{z_{\max}} & \frac{k_{u} (z_{u} - z_{r})}{(m_{s} + m_{u}) g} \end{matrix}]}^{T}

(4)

where

z_{1} (t)

and

z_{2} (t)

denote the controlled outputs.

Let $x (t) = col {\begin{matrix} x_{1} (t) & x_{2} (t) & x_{3} (t) & x_{4} (t) \end{matrix}}$ and $ω (t) = \frac{d z_{r}}{d t}$ . Then, equation (1) is rewritten in the following form

\frac{d x (t)}{d t} = A x (t) + B u (t - d (t)) + B_{1} ω (t)

(5)

where

x_{1} (t) = z_{s} - z_{u}

x_{2} (t) = \frac{d z_{s}}{d t}

x_{3} (t) = z_{u} - z_{r}

x_{4} (t) = \frac{d z_{u}}{d t}

, and the matrices

A

B

B_{1}

are given by

A = [\begin{matrix} 0 & 1 & 0 & - 1 \\ - \frac{k_{s}}{m_{s}} & - \frac{c_{s}}{m_{s}} & 0 & \frac{c_{s}}{m_{s}} \\ 0 & 0 & 0 & 1 \\ \frac{k_{s}}{m_{u}} & \frac{c_{s}}{m_{u}} & - \frac{k_{u}}{m_{u}} & - \frac{c_{s} + c_{u}}{m_{u}} \end{matrix}], B = [\begin{matrix} 0 \\ \frac{1}{m_{s}} \\ 0 \\ \frac{- 1}{m_{u}} \end{matrix}], B_{1} = [\begin{matrix} 0 \\ 0 \\ - 1 \\ \frac{c_{u}}{m_{u}} \end{matrix}]

The controlled outputs in equation (4) can be expressed as

\begin{array}{l} z_{1} (t) = C_{1} x (t) + D_{1} u (t - d (t)) \\ z_{2} (t) = C_{2} x (t) \end{array}

(6)

where

C_{1} = [\begin{matrix} - \frac{k_{s}}{m_{s}} & - \frac{c_{s}}{m_{s}} & 0 & \frac{c_{s}}{m_{s}} \end{matrix}], D_{1} = \frac{1}{m_{s}}, C_{2} = [\begin{matrix} \frac{1}{z_{\max}} & 0 & 0 & 0 \\ 0 & 0 & \frac{k_{u}}{(m_{s} + m_{u}) g} & 0 \end{matrix}]

In the abovementioned model, $ω (t)$ is an energy-limited disturbance that belongs to $L_{2} [0, \infty]$ . In the definition of a state-feedback controller $u (t) = K x (t)$ , the aim of this study is to construct a feedback matrix, $K$ , such that the controlled suspension is asymptotically stable and has the following performances

{\begin{cases} \int_{0}^{\infty} {‖ z_{1} (t) ‖}^{2} d t \leq γ^{2} \int_{0}^{\infty} {‖ ω (t) ‖}^{2} d t \\ | {z_{2} (t)}_{i} | \leq 1, i = 1, 2 \end{cases}

(7)

where

\int_{0}^{\infty} {‖ ω (t) ‖}^{2} d t \leq ω_{\max} < + \infty

, and

γ

is the optimal vibration attenuation level.

Some novel relaxed inequalities are introduced in the following presentation.

Lemma 1 (BLI³³). Given a symmetric matrix $R > 0$ , scalars $a$ and $b$ $(b > a)$ , a differentiable vector $ϖ$ , and a well-defined integration $\int_{r_{1}}^{r_{2}} \frac{d ϖ^{T} (s)}{d t} R \frac{d ϖ (s)}{d t} d s$ , then

\int_{r_{1}}^{r_{2}} \frac{d ϖ^{T} (s)}{d t} R \frac{d ϖ (s)}{d t} d s \geq \frac{1}{r_{2} - r_{1}} ζ^{T} (r_{1}, r_{2}) \hat{\bar{R}} ζ (r_{1}, r_{2})

(8)

where

\hat{\bar{R}} = diag [R,3R,5R], ζ (r_{1}, r_{2}) = col {υ_{0}, υ_{1}, υ_{2}}

with

{\begin{cases} υ_{0} = ϖ (r_{2}) - ϖ (r_{1}) \\ υ_{1} = ϖ (r_{2}) + ϖ (r_{1}) + \frac{2}{r_{2} - r_{1}} \int_{r_{1}}^{r_{2}} ϖ (s) d s \\ υ_{2} = υ_{0} - \frac{6}{r_{2} - r_{1}} \int_{r_{1}}^{r_{2}} ϖ (s) d s + \frac{12}{{(r_{2} - r_{1})}^{2}} \int_{r_{1}}^{r_{2}} (r_{2} - s) ϖ (s) d s \end{cases}

Lemma 2 (delay-dependent ERCI³⁵). Let $R \in ℜ^{n \times n} (R > 0)$ and $ϖ_{i} \in ℜ^{n} (i = 1, 2)$ and $a \in (0, 1)$ . For any symmetric matrices, $X_{1}, X_{2} \in ℜ^{n \times n}$ , and general matrices, $Y_{1}, Y_{2} \in ℜ^{n \times n}$ , satisfying

[\begin{matrix} R & 0 \\ 0 & R \end{matrix}] - a [\begin{matrix} X_{1} & - Y_{1} \\ * & 0 \end{matrix}] - (1 - a) [\begin{matrix} 0 & - Y_{2} \\ * & X_{2} \end{matrix}] \geq 0

(9)

then

\frac{1}{a} ϖ_{1}^{T} R ϖ_{1} + \frac{1}{1 - a} ϖ_{2}^{T} R ϖ_{2} \geq ϑ (X, Y)

(10)

where

ϑ (X, Y) = ϖ_{1}^{T} [R + (1 - a) X_{1}] ϖ_{1} + ϖ_{2}^{T} (R + a X_{2}) ϖ_{2} + 2 ϖ_{1}^{T} [a Y_{1} + (1 - a) Y_{2}]

Lemma 3.³⁷ Consider a quadratic function $φ (d) = d^{2} g_{1} + d g_{2} + g_{3}$ , where $g_{_{i}} \in ℜ (i = 1, 2, 3)$ , and the function $φ (d) < 0$ for $\forall d \in [0, h]$ if and only if $φ (0) < 0$ , $φ (h) < 0$ and $- h^{2} g_{1} + φ (0) < 0$ .

Lemma 4.³⁹ Given positive scalars $η$ and $σ$ , and two compatible real matrices, $X^{- 1}$ and $\hat{\bar{K}}$ , there exists a real matrix, $K$ , such that

K^{T} K < η / σ^{2} I

(11)

if and only if

[\begin{matrix} η I & \hat{\bar{K}} \\ * & I \end{matrix}] > 0, X > σ I

(12)

holds, where

K = \hat{\bar{K}} X^{- 1}

A useful proposition is obtained using Lemmas 1 and 2 together, which is as follows.

Proposition 1. Given an $n \times n$ matrix $R > 0$ , if there exist symmetric matrices, $X_{1}, X_{2} \in ℜ^{3 n \times 3 n}$ , and general matrices, $Y_{1}, Y_{2} \in ℜ^{3 n \times 3 n}$ , such that

[\begin{matrix} \hat{\bar{R}} - X_{1} & Y_{1} \\ Y_{1}^{T} & R \end{matrix}] \geq 0, [\begin{matrix} \hat{\bar{R}} & Y_{2} \\ Y_{2}^{T} & R - X_{2} \end{matrix}] \geq 0

(13)

then

ℵ (t) \geq ξ^{T} (t) ϒ_{0} ξ (t)

(14)

where

ℵ (t) = h \int_{t - d (t)}^{t} \frac{d x^{T} (s)}{d t} R \frac{d x (s)}{d t} d s + h \int_{t - h}^{t - d (t)} \frac{d x^{T} (s)}{d t} R \frac{d x (s)}{d t} d s

ξ (t) = col {x (t), x (t - d (t)), x (t - h), ρ_{1} (t), ρ_{2} (t), ρ_{3} (t), ρ_{4} (t), ω (t), \frac{d x (t)}{d t}}

ϒ_{0} = Γ_{1}^{T} \hat{\bar{R}} Γ_{1} + (1 - a) Γ_{1}^{T} X_{1} Γ_{1} + Γ_{2}^{T} \hat{\bar{R}} Γ_{2} + a Γ_{2}^{T} X_{2} Γ_{2} + Γ_{1}^{T} [a Y_{1} + (1 - a) Y_{2}] Γ_{2}

\begin{array}{l} Γ_{1} = col {e_{2} - e_{3}, e_{2} + e_{3} - 2 e_{4}, e_{2} - e_{3} - 6 e_{4} + 12 e_{5}} \\ Γ_{2} = col {e_{1} - e_{2}, e_{1} + e_{2} - 2 e_{6}, e_{1} - e_{2} - 6 e_{6} + 12 e_{7}} \end{array}

\hat{\bar{R}} = diag {R,3 R,5 R}, a = (h - d (t)) / h

with

\begin{array}{l} ρ_{1} (t) = \int_{t - h}^{t - d (t)} \frac{x (s)}{h - d (t)} d s, ρ_{2} (t) = \int_{t - h}^{t - d (t)} \frac{(t - d (t) - s) x (s)}{{(h - d (t))}^{2}} d s, ρ_{3} (t) = \int_{t - d (t)}^{t} \frac{x (s)}{d (t)} d s \\ ρ_{4} (t) = \int_{t - d (t)}^{t} \frac{(t - s) x (s)}{d^{2} (t)} d s, e_{i} = [0_{n \times (i - 1) n}, I, 0_{n \times (9 - i) n}], i = 1, 2, \dots, 9 \end{array}

Proof. See Appendix 1.

Comfort-enhanced active vibration control

In this section, a new bounded real lemma is established based on an augmented LKF and the above lemmas. The main design results are as follows.

Theorem 1. Given the real scalars $h$ , $μ$ , $r$ , $γ$ , $ϑ_{2}$ , and $ϑ_{3}$ , if there exist symmetric matrices ${[P]}_{3 n \times 3 n} > 0$ , ${[Q]}_{3 n \times 3 n} > 0$ , $R > 0$ , $X_{1}$ , $X_{2}$ and general matrices $X$ , $\hat{\bar{K}}$ , $Y_{1}$ , $Y_{2}$ such that for $\frac{d d (t)}{d t} \in [- μ, μ]$

[\begin{matrix} {Ξ_{0} |}_{a = 0} & ε_{2}^{T} \\ * & - I \end{matrix}] < 0

(15)

[\begin{matrix} {Ξ_{0} |}_{a = 1} & ε_{2}^{T} \\ * & - I \end{matrix}] < 0

(16)

[\begin{matrix} - h^{2} g_{1} + {Ξ_{0} |}_{α = 1} & ε_{2}^{T} \\ * & - I \end{matrix}] < 0

(17)

[\begin{matrix} h Q_{22} & * \\ {C_{2}}_{i} X & 1 / r \end{matrix}] > 0, i = 1, 2

(18)

P_{22} > Q_{22}

(19)

[\begin{matrix} \hat{\bar{R}} - X_{1} & Y_{1} \\ * & \hat{\bar{R}} \end{matrix}] > 0, [\begin{matrix} \hat{\bar{R}} & Y_{2} \\ * & \hat{\bar{R}} - X_{2} \end{matrix}] > 0

(20)

where

\begin{array}{l} {Ξ_{0} |}_{α = 0} = {[L_{1} + h L_{2}]}^{T} P [L_{1} + h L_{2}] - L_{4}^{T} Q L_{4} - (1 - \frac{d d (t)}{d t}) L_{3}^{T} P L_{3} + (1 - \frac{d d (t)}{d t}) L_{5}^{T} Q L_{5} \\ + 2 L_{9}^{T} P [h L_{7} + h^{2} L_{8}] - e_{8}^{T} γ^{2} I e_{8} + 2 Θ + h^{2} e_{9}^{T} R e_{9} - {ϒ_{0} |}_{α = 0} \end{array}

\begin{matrix} {Ξ_{0} |}_{α = 1} = L_{1}^{T} P L_{1} - L_{4}^{T} Q L_{4} - (1 - \frac{d d (t)}{d t}) L_{3}^{T} P L_{3} + (1 - \frac{d d (t)}{d t}) {[L_{5} + h L_{6}]}^{T} Q [L_{5} + h L_{6}] \\ + 2 L_{12}^{T} Q [h L_{10} + h^{2} L_{11}] - e_{8}^{T} γ^{2} I e_{8} + 2 Θ + h^{2} e_{9}^{T} R e_{9} - {ϒ_{0} |}_{α = 1} \end{matrix}

\begin{array}{l} Θ = ϑ_{2} e_{1}^{T} A X e_{1} + ϑ_{2} e_{1}^{T} B \hat{\bar{K}} e_{2} + ϑ_{3} e_{1}^{T} X^{T} A^{T} e_{2} + ϑ_{2} e_{1}^{T} B_{1} e_{8} + e_{1}^{T} X^{T} A^{T} e_{9} - ϑ_{2} e_{1}^{T} X e_{9} \\ + ϑ_{3} e_{2}^{T} B \hat{\bar{K}} e_{2} + ϑ_{3} e_{2}^{T} B_{1} e_{8} + e_{2}^{T} {\hat{\bar{K}}}^{T} B^{T} e_{9} - ϑ_{3} e_{2}^{T} X e_{9} + e_{8}^{T} B_{1}^{T} e_{9} - e_{9}^{T} X e_{9} \end{array}

{ϒ_{0} |}_{a = 0} = Γ_{1}^{T} \hat{\bar{R}} Γ_{1} + Γ_{1}^{T} X_{1} Γ_{1} + Γ_{2}^{T} \hat{\bar{R}} Γ_{2} + 2 Γ_{1}^{T} Y_{2} Γ_{2}

{ϒ_{0} |}_{a = 1} = Γ_{1}^{T} \hat{\bar{R}} Γ_{1} + Γ_{2}^{T} \hat{\bar{R}} Γ_{2} + Γ_{2}^{T} X_{2} Γ_{2} + 2 Γ_{1}^{T} Y_{1} Γ_{2}

g_{1} = L_{2}^{T} P L_{2} + (1 - \frac{d d (t)}{d t}) L_{6}^{T} Q L_{6} + 2 L_{9}^{T} P L_{8} + 2 L_{12}^{T} Q L_{11}

P = [\begin{matrix} P_{11} & P_{12} & P_{13} \\ * & P_{22} & P_{23} \\ * & * & P_{33} \end{matrix}], Q = [\begin{matrix} Q_{11} & Q_{12} & Q_{13} \\ * & Q_{22} & Q_{23} \\ * & * & Q_{33} \end{matrix}], ε_{2} = (C_{1} X e_{1} + D_{1} \hat{\bar{K}} e_{2})

with

\begin{array}{l} L_{1} = col {e_{1}, e_{1}, 0}, L_{2} = col {0, 0, e_{6}}, L_{3} = col {e_{2}, e_{1}, 0}, L_{4} = col {e_{3}, e_{1}, 0} \\ L_{5} = col {e_{2}, e_{1}, 0}, L_{6} = col {0, 0, e_{4}}, L_{7} = col {e_{6}, e_{1}, 0}, L_{8} = col {0, 0, e_{7}} \\ L_{9} = col {0, e_{9}, (\frac{d d (t)}{d t} - 1) e_{2}}, L_{10} = col {e_{4}, e_{1}, 0}, L_{11} = col {0, 0, e_{5}}, L_{12} = col {0, e_{9}, - e_{3}} \end{array}

Then, under the state-feedback controller $u (t) = \hat{\bar{K}} X^{- 1} x (t)$ , the controlled system (5) is said to be asymptotically stable with an optimal vibration attenuation level $γ$ , and the controlled requirement (3) is satisfied when the upper bound of disturbance is $ω_{\max} = r / γ^{2}$ .

Proof. See Appendix 2.

It is clear that Theorem 1 can be directly solved by the standard LMI technique. If the above problem has a solution, then the gain can be given by $K = \hat{\bar{K}} X^{- 1}$ . Furthermore, the proposed LKF in Theorem 1 is feasible for fast-varying delay, i.e. $μ > 1$ . Therefore, it is more practical for vehicle control systems.

For comparison, a corollary based on traditional LKF is presented as follows.

Corollary 1. Given the real scalars $h$ , $μ$ , $r$ , $γ$ , if there exist symmetric matrices $\bar{P} > 0$ , $\bar{Q} > 0$ , $\bar{Z} > 0$ , $\bar{R} > 0$ , ${\bar{X}}_{1}$ , ${\bar{X}}_{2}$ and general matrices $\bar{K}$ , ${\bar{Y}}_{1}$ , ${\bar{Y}}_{2}$ such that for $\frac{d d (t)}{d t} \in [- μ, μ]$

[\begin{matrix} {\bar{Ξ}}_{0} - {{\bar{ϒ}}_{0} |}_{α = 1} & {\bar{ε}}_{2}^{T} & h {\bar{ε}}_{1}^{T} \\ * & - I & 0 \\ * & * & - \bar{P} {\bar{R}}^{- 1} \bar{P} \end{matrix}] < 0

(21)

[\begin{matrix} {\bar{Ξ}}_{0} - {{\bar{ϒ}}_{0} |}_{α = 0} & {\bar{ε}}_{2}^{T} & h {\bar{ε}}_{1}^{T} \\ * & - I & 0 \\ * & * & - \bar{P} {\bar{R}}^{- 1} \bar{P} \end{matrix}] < 0

(22)

[\begin{matrix} \bar{P} & * \\ {C_{2}}_{i} \bar{P} & 1 / r \end{matrix}] > 0, i = 1, 2

(23)

[\begin{matrix} \hat{R} - {\bar{X}}_{1} & {\bar{Y}}_{1} \\ * & \hat{R} \end{matrix}] > 0, [\begin{matrix} \hat{R} & {\bar{Y}}_{2} \\ * & \hat{R} - {\bar{X}}_{2} \end{matrix}] > 0

(24)

where

{\bar{Ξ}}_{0} = 2 e_{1}^{T} {\bar{ε}}_{1} + e_{1}^{T} (\bar{Q} + \bar{Z}) e_{1} - (1 - \frac{d d (t)}{d t}) e_{2}^{T} \bar{Q} e_{2} - e_{3}^{T} \bar{Z} e_{3} - e_{8}^{T} γ^{2} I e_{8}

{\bar{ε}}_{2} = C_{1} \bar{P} e_{1} + D_{1} \bar{K} e_{2}, {\bar{ε}}_{1} = A \bar{P} e_{1} + B \bar{K} e_{2} + B_{1} e_{8}, \hat{R} = diag {\bar{R},3 \bar{R}, 5 \bar{R}}

{{\bar{ϒ}}_{0} |}_{α = 0} = Γ_{1}^{T} \hat{R} Γ_{1} + Γ_{1}^{T} {\bar{X}}_{1} Γ_{1} + Γ_{2}^{T} \hat{R} Γ_{2} + 2 Γ_{1}^{T} {\bar{Y}}_{2} Γ_{2}

{{\bar{ϒ}}_{0} |}_{α = 1} = Γ_{1}^{T} \hat{R} Γ_{1} + Γ_{2}^{T} \hat{R} Γ_{2} + Γ_{2}^{T} {\bar{X}}_{2} Γ_{2} + 2 Γ_{1}^{T} {\bar{Y}}_{1} Γ_{2}

then, under the state-feedback controller

u (t) = \bar{K} {\bar{P}}^{- 1} x (t)

, the controlled system (5) is said to be asymptotically stable with an optimal vibration attenuation level,

γ

. The controlled requirement (3) is satisfied when the upper bound of disturbance is

ω_{\max} = r / γ^{2}

Proof. See Appendix 3.

If Corollary 1 is feasible, then the solution can be expressed as $K = \bar{K} {\bar{P}}^{- 1}$ .

Remark 1. In Corollary 1, in the view of $- \bar{P} {\bar{R}}^{- 1} \bar{P} \leq ρ^{2} \bar{R} - 2 ρ \bar{P}$ , the nonlinear term $- \bar{P} {\bar{R}}^{- 1} \bar{P}$ can be replaced by a linear term $ρ^{2} \bar{R} - 2 ρ \bar{P}$ , where $ρ$ is a positive scalar.^18,23 In addition, the cone complementary linearization algorithm can also be used for solving the above nonlinear minimization problem.

Remark 2. To maintain the control signal within the appropriate ranges, to avoid ill-conditioned systems, and to relax the solvability conditions, the entries of the controller matrix must be bounded.¹³ A way to do this is to restrict the singular value upper bound of the control gain $K$ , that is $K^{T} K < η / σ^{2} I$ , where $η$ and $σ$ are some positive scalars. According to Lemma 4, an LMI constraint is considered as follows

[\begin{matrix} η I & \hat{\bar{K}} \\ * & I \end{matrix}] > 0, X > σ I

(25)

Simulation investigations

In this section, the comfort-enhanced vibration control design approach is applied to the suspension model with the delay described in “Suspension model and problem description” section. The main parameters¹⁸ are given in Table 1. Furthermore, to illustrate the validity of the design approach, the traditional control methods derived from traditional LKF and the Jensen inequality are also compared in this section.

Table 1.

Parameters of the quarter-car suspension model.

$m_{s}$ (kg)	$m_{u}$ (kg)	$k_{s}$ (N/m)	$k_{u}$ N/m	$c_{s}$ (Ns/m)	$c_{u}$ (Ns/m)	$z_{\max}$ (m)
973	114	42,720	101,115	1095	14.6	0.1

Since the different design values, $ϑ_{2}$ and $ϑ_{3}$ , will lead to different results, for convenience, the design values are set as $ϑ_{2} = ϑ_{3} = 1$ . In fact, the design values can be further optimized using a genetic algorithm. Let $σ = 9$ , and the singular value upper bound of $K$ is set as $\sqrt{η / σ^{2}} = 5 \times 10^{4}$ .

For $r = 0.01$ , by applying Theorem 1 and Corollary 1, the vibration attenuation levels for different cases are recorded in Table 2. The results derived by the Jensen inequality are also tabulated. In Corollary 1, the design value, $ρ$ , has been optimized for an equitable comparison.

Table 2.

The value of $γ^{2}$ under different conditions.

Vibration attenuation level $γ^{2}$	Method
Vibration attenuation level $γ^{2}$	Theorem 1	Corollary 1	Jensen inequality
Case 1 $h = 1 ms, μ =0$	9.5701	11.743	11.687
Case 2 $h = 10 ms, μ =0$	14.159	15.325	16.236
Case 3 $h = 20 ms, μ =0$	24.042	37.516	39.213
Case 4 $h = 30 ms, μ =0$	29.660	96.388	100.59
Case 5 $h = 10 ms, μ =0 .5$	14.372	15.341	16.236
Case 6 $h = 20 ms, μ =2$	30.715	37.592	39.213
Case 7 $h = 30 ms, μ =5$	37.866	96.416	100.59
Case 8 $h = 40 ms, μ =5$	43.158	157.30	165.64
Case 9 $h = 50 ms, μ =5$	48.441	240.44	250.87
Case 10 $h = 100 ms, μ =5$	118.72	1218.1	1244.0

In Case 1, the control gains for the above three methods are given by

\begin{array}{l} K_{1} = 10^{4} \times [\begin{matrix} 4.2688 & 0.0871 & 0.0467 & - 0.1064 \end{matrix}] \\ K_{2} = 10^{4} \times [\begin{matrix} 0.1017 & - 1.6020 & - 3.6854 & - 0.0949 \end{matrix}] \\ K_{3} = 10^{4} \times [\begin{matrix} 0.0281 & - 1.6248 & - 3.6343 & - 0.0917 \end{matrix}] \end{array}

With the obtained controllers, the frequency-domain comparisons of $z_{1} (t)$ in Case 1 are illustrated in Figure 2. The other results for Cases 2–4 are illustrated in Figures 3 to 5. Based on these results, we can see that the comfort-enhanced controller provides a minimum singular value for the different cases, which means that enhanced performance can be obtained using the proposed controller in the whole frequency domain.

Figure 2.

Frequency responses of $z_{1} (t)$ in Case 1.

Figure 3.

Frequency responses of $z_{1} (t)$ in Case 2.

Figure 4.

Frequency responses of $z_{1} (t)$ in Case 3.

Figure 5.

Frequency responses of $z_{1} (t)$ in Case 4.

To further demonstrate the advantages of the vibration controller in time domain, a single shock road profile is employed and is given by^18,20

z_{r} (t) = {\begin{array}{l} \frac{a_{0}}{2} (1 - \cos (\frac{2 π υ_{0}}{l} t)), 0 \leq t \leq l / υ_{0} \\ 0, t > l / υ_{0} \end{array}

(26)

where

υ_{0}

a_{0}

, and

l

represent the vehicle speed of movement, the height of the bump, and the length of the bump, respectively. Here, the parameters of equation (26) are configured as

a_{0} = 60 mm, l = 5 m, and υ_{0} = 25 km / h

Under the condition of a single shock disturbance (26), the responses of the controlled outputs, including $z_{1}$ and $z_{2}$ in Cases 5 and 6, are shown in Figures 6 and 7, respectively. From these results, we can know that the controller designed using Theorem 1 achieves the best ride comfort, even when the input delay is large. Other performances, including the tire load ratio and the suspension deflection ratio, are much lower than one.

Figure 6.

Suspension performance responses in Case 5. (a) Road input, (b) car-body acceleration, (c) suspension deflection ratio, and (d) tire load ratio.

Figure 7.

Suspension performance responses in Case 6. (a) Road input, (b) car-body acceleration, (c) suspension deflection ratio, and (d) tire load ratio.

In the context of performance evaluation, ride comfort is closely related to the root mean square (RMS) value of $z_{1}$ , so the RMS results are plotted in Figures 8 and 9. The following random road profile is employed²¹

ω (t) = 2 π η_{0} \sqrt{G_{0} υ_{0}} ϕ (t)

(27)

Figure 8.

RMS results for a random road profile in Cases 5–7. FWM: free weighting matrix; RMS: root mean square.

Figure 9.

RMS results for a random road profile in Cases 8–10. FWM: free weighting matrix; RMS: root mean square.

Here, the parameters chosen are $η_{0} = 0.1 m^{- 1}, G_{0} = 1024 \times 10^{- 6} m^{3}, υ_{0} = 25 km / h$ . $ϕ (t)$ stands for the white noise process.

The RMS results for the different cases are shown in Figures 8 and 9. The free-weighting-matrix (FWM) approach based on traditional LKF is also compared in the figures. It is clear from the figures that the minimum RMS value can be obtained for the system controlled using the proposed control, which further exhibits the enhanced comfort performance of the designed control approach. Compared with passive control, the RMS value obtained using Theorem 1 decreases by more than 60% when the upper bound of delay reaches 100 ms, and other control methods seem to be weak in the case of a larger delay. On the contrary, the proposed controller is more tolerant in cases with a larger admissible delay bound. Moreover, from the results, it can be seen that the RMS values of Corollary 1 are almost equal to those of the Jensen inequality. In Case 10, the FWM is infeasible no matter what the design parameter.

From the above results, the following conclusions can be drawn:

The proposed controller can provide an enhanced vibration attenuation performance for different cases when compared with other frequently used design methods.

Although the relaxed integral inequalities are less conservative than the Jensen inequality, the controller designed using Corollary 1 seems to be as conservative as that of the controller based on the Jensen inequality, which means that the traditional LKF is unsuitable for performance-enhanced design.

Overall, the proposed vibration control design is less conservative in the design of the controller for delayed active suspension systems. Moreover, the vibration attenuation levels can be further decreased by tuning the design parameters $ϑ_{2}$ and $ϑ_{3}$ , and thus the comfort performance can be accordingly enhanced.

Conclusions

In this study, a comfort-enhanced vibration control method for delayed vehicle active suspensions was proposed. A less conservative bounded real lemma was established by using some novel relaxed inequalities. The control gain was directly derived using LMI optimization, and an augmented LKF was proposed to formulate the design criteria. Simulation results using two well-used road profiles indicated that the proposed controller achieved enhanced ride comfort, while the time-domain constraints were also satisfied when compared with controllers designed with frequently used inequalities and traditional LKF. The results further demonstrated that the proposed LKF significantly reduced design conservatism. In future work, we will consider control saturation and actuator dynamics for the suspension systems.

Footnotes

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: The work was supported by National Natural Science Foundation of China under Grant 51479073 and Qian-Ke-He platform talent [2017]5789-11.

Appendix 1. Proof of Proposition 1

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Comfort-enhanced vibration control for delayed active suspensions using relaxed inequalities

Abstract

Keywords

Introduction

Suspension model and problem description

Comfort-enhanced active vibration control

Simulation investigations

Conclusions

Footnotes

Declaration of conflicting interests

Funding

Appendix 1. Proof of Proposition 1

References