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Abstract

Due to the uncertainty in vehicle model parameters, the control performance of active suspensions often deteriorates, especially under extreme conditions such as high velocities and full loads. To address this challenge, this paper proposes a novel robust finite frequency H_∞ output feedback control strategy based on linear parameter varying (LPV) systems. First, we establish a new LPV model for the half-car active suspension control system, taking into account the uncertainties associated with vehicle velocity and sprung mass. Next, a robust finite frequency H_∞ output feedback controller is designed using a gain scheduling approach, formulated through a set of linear matrix inequalities (LMIs). Finally, simulation results demonstrate that the proposed velocity-dependent robust control strategy significantly reduces the root mean square (RMS) values of vertical acceleration by approximately 21% to 31% under various conditions of sprung mass and vehicle speed compared to traditional control methods.

Keywords

active suspension finite frequency linear parameter-varying system

Introduction

The suspension system is an essential component of a vehicle, serving not only to support the vehicle’s weight but also to dampen body vibrations. In automotive research, suspension control remains a common topic of investigation. Depending on the implementation of control, suspension systems are classified as passive, semi-active, or active.¹ Active suspensions are more frequently utilized than passive suspensions in vehicle comfort studies because of their ease of implementation, rapid change in control force, and controlled performance.² However, the active suspension control force has a distinct saturation limit, and the damping force can only be continuously varied within a limited range, making the control strategy extremely important to the performance of the active suspension system.³ Linear Quadratic Regulator (LQR) control, Model Predictive Control (MPC), and PID control are the three most commonly used control strategies.^4–7

Traditional active cab suspension systems are often designed under the assumption that system parameters remain constant, such as vehicle speed, mass, stiffness, and damping. These systems are typically designed using Linear Time-Invariant (LTI) models, which have demonstrated commendable control performance compared to passive suspension systems. For instance, Nguyen⁸ utilized a multi-loop algorithm to select optimal control parameters for sliding mode control under various vehicle vibration conditions, resulting in a significant reduction in body acceleration and displacement. Basaran⁹ modeled a commercial vehicle cab suspension system based on a half-car model and proposes a design method utilizing Lyapunov backstepping control. Hamza¹⁰ introduced a boundary model for Continuous Damping Control (CDC) dampers and develops an intelligent control method based on Artificial Neural Networks (ANN) for active truck suspension systems. However, these approaches encounter limitations in specific application scenarios where vehicle parameters fluctuate. To enhance the control performance of these systems, this study introduces Linear Parameter-Varying (LPV) control theory, which is better equipped to manage variability in vehicle parameters and improve the adaptability of the suspension system under diverse operating conditions. Since Shamma introduced LPV control theory,¹¹ it has become an important tool for addressing nonlinear system problems. The core of LPV control theory lay in approximating nonlinear systems as a set of parameter-dependent linear systems, allowing designers to utilize a variety of control methods from LTI systems to solve nonlinear control problems. Current research on LPV system control mainly focuses on three approaches: Linear Fractional Transformation (LFT), polytopic methods, and grid-based methods. The polytopic design approach has become a mainstream direction in the study of LPV systems due to its intuitive structure and the high degree of freedom it offers in controller design.¹² This method represents the system’s coefficient matrices in a polynomial or affine form. Tian¹³ and Peng¹⁴ studied path tracking and time-varying tire cornering stiffness uncertainties in autonomous vehicles based on LPV’s MPC and H_∞ gain-scheduled robust control strategies, significantly reducing computational complexity by simplifying the polytopic vertices.

In the design of vehicle suspension control systems, finite-frequency control strategies have gained attention due to their focus on vibration suppression within a specific frequency range. These strategies leverage the Kalman–Yakubovich–Popov (KYP) lemma to ingeniously transform performance and stability requirements in the frequency domain into time-domain conditions, which have been widely applied in filter design and control systems.^15–17 According to the ISO-2631 standard, humans are most sensitive to vertical vibrations in the 4–8 Hz range, prompting researchers to apply finite-frequency control strategies to vehicle suspension systems to significantly enhance ride comfort. Sun¹⁸ conducted in-depth research on finite-frequency control strategies. They applied the KYP lemma to optimize the H_∞ performance for the transfer function of road disturbances and control outputs in the 4–8 Hz frequency range, while also considering the time-domain constraints of actuator control force and suspension travel. Wang¹⁹ designed a robust fault-tolerant H_∞ output feedback controller for a full-vehicle active suspension system with finite-frequency constraints using LFT to process uncertain parameter matrices, offering a new solution for the reliability of the suspension system. Jing²⁰ further developed a robust H_∞ state feedback controller based on uncertainties in suspension stiffness, damping, and tire stiffness, enhancing the system’s adaptability to parameter changes. Jiang,²¹ in response to more complex uncertain conditions such as polyhedral uncertainty, external disturbances, state and input delays, and gain disturbances, proposed a non-fragile robust MPC strategy with delayed state compensation, providing ample design conditions for dealing with unobservable time-varying parameters in LPV systems.

During vehicle operation, the velocity is inherently variable, and the vehicle’s sprung mass is uncertain, varying with different loading conditions. These changes in vehicle model parameters can significantly affect ride comfort. Tsampardoukas²² analyzed the impact mechanisms of key parameters on system performance under different road input conditions using a half-car model. Abdelkareem²³ further quantified the combined effects of vehicle velocity, damping coefficient, and road input on ride comfort, road adhesion, and damper energy dissipation. The existing research on suspension control strategies indicates that in practical scenarios, vehicle mass and speed are variable parameters. These fluctuations can cause the suspension control system to deviate from its preset optimal state, reducing vibration isolation performance, especially under complex road conditions. While it is possible to design control gains for each set of fixed parameters, this approach is only practical for situations with limited parameter variations. Designing and storing control gains for all possible parameter variations, particularly when there are significant changes in vehicle mass and speed, is clearly impractical. Therefore, drawing on valuable insights from existing LPV control and finite frequency H_∞ control theories to devise an integrated control method that can handle dynamic parameter variations and enhance control accuracy is a key technical issue in the research on active suspension control strategies for vehicles

To enhance ride comfort in the presence of vehicle velocity and sprung mass uncertainties, this paper presents two primary research contributions:

1. The introduction of a novel half-car LPV system model, which is parameterized by real-time vehicle speed. This model dynamically adjusts control gains in response to variations in vehicle speed, effectively mitigating the adverse effects of speed fluctuations on vibration attenuation.

2. The development of a robust finite frequency output feedback H_∞ control strategy, aimed at suppressing vibrations within the frequency band critical to human comfort perception. This strategy is designed to be resilient against uncertainties in sprung mass, ensuring consistent performance across a range of vehicle load conditions.

Problem formulation

In previous research,^24–27 numerous complex suspension models have been proposed to simulate the dynamic properties of suspension systems. This paper employs a 4-degree-of-freedom (4-DOF) half-car model to analyze the dynamic response of the suspension, including the pitch and vertical motion of the sprung mass, as well as the vertical motion of the two unsprung masses. Figure 1 shows the structure of the suspension model. Based on the parameter values from research,²⁸ Table 1 lists the parameters used in the half-car model. The uncertainty range of the sprung mass is also considered to be M ∈ [400, 600] kg, while the range of variation of the forward velocity is v ∈ [10, 40] m/s.

Figure 1.

4-DOF half-car suspension system.

Table 1.

Parameters of the half-car suspension model.

Symbol and unit	Value	Symbol and unit	Value
M-(kg)	400 ∼ 600	I-(kg.m²)	910
m_f-(kg)	30	m_r-(kg)	40
L_a-(m)	1.25	L_b-(m)	1.45
k_f-(N/m)	10,000	k_r-(N/m)	10,000
k_tf-(Ns/m)	100,000	k_tr-(Ns/m)	100,000
c_f-(N.s/m)	1000	c_r-(N.s/m)	1000
z_max-(m)	0.08	u_max-(N)	1500

In the figure, M, m, and m_r represent the sprung mass of the vehicle and the unsprung masses of the front and rear axles, respectively. I denotes the pitch moment of inertia of the sprung mass. u_f and u_r represent the control force inputs of the active suspensions at the front and rear, respectively. Furthermore, z_f and z_r represent the vertical displacements of the unsprung masses. z_tf and z_tr represent the road surface displacement inputs corresponding to the front and rear wheels of the vehicle, respectively.

Let the vertical displacement of the sprung mass center of gravity be defined as z_c. Assuming that the pitch angle φ is sufficiently small, it holds that sin φ ≈ φ. Consequently, the suspension displacements at points z_cf and z_cr can be simplified as follows:

\begin{aligned} z_{c f} = z_{c} - l_{a} φ \\ z_{c r} = z_{c} + l_{b} φ \end{aligned}

(1)

where l_a and l_b denote the distances between the front and rear axles, respectively, which are referred to as the wheelbase lengths.

According to Newton’s second law, the dynamic equations of the suspension system are

\begin{aligned} M {\ddot{z}}_{c} & = F_{f} + F_{r} \\ I \ddot{φ} & = - l_{a} F_{f} + l_{b} F_{r} \\ m_{f} {\ddot{z}}_{f} & = - k_{t f} (z_{f} - z_{t f}) - F_{f} \\ m_{r} {\ddot{z}}_{r} & = - k_{t r} (z_{r} - z_{t r}) - F_{r} \end{aligned}

(2)

where F_f and F_r represent the suspension forces at the front and rear of the sprung mass, respectively, as illustrated below

\begin{aligned} F_{f} & = - k_{f} (z_{c f} - z_{f}) - c_{f} ({\dot{z}}_{c f} - {\dot{z}}_{f}) - u_{f} \\ F_{r} & = - k_{r} (z_{c r} - z_{r}) - c_{r} ({\dot{z}}_{c r} - {\dot{z}}_{r}) - u_{r} \end{aligned}

(3)

where the parameters k_f and k_r represent the stiffness coefficients of the front and rear suspensions for the sprung mass, respectively, while c_f and c_r denote the damping coefficients for the front and rear suspensions, respectively. Additionally, k_tf and k_tr signify the tire stiffness coefficients for the front and rear wheels, respectively.

The parameters for the state vector x of the half-car state space equation are as follows

\begin{aligned} x = {[z_{c f}, z_{c r}, z_{f}, z_{r}, {\dot{z}}_{c f}, {\dot{z}}_{c r}, {\dot{z}}_{f}, {\dot{z}}_{r}]}^{T} \end{aligned}

(4)

The road input w and the control input force u of the active suspension are described in the following way

\begin{aligned} u = [\begin{matrix} u_{f} \\ u_{r} \end{matrix}], w = [\begin{matrix} z_{t f} \\ z_{t r} \end{matrix}] \end{aligned}

(5)

In this article, the suspension model faces the following time domain constraints:

1. The suspension system is located between the body and the wheels and its mechanical components are limited by the vehicle structure. In order to prevent damage from overhandling, it is essential that the controller is designed with the suspension deflection limit in mind

\begin{aligned} | z_{c f} - z_{f} | \leq z_{\max} \\ | z_{c r} - z_{r} | \leq z_{\max} \end{aligned}

(6)

where z_max is the maximum permitted range of suspension movement.

2. There should be a limit to the amount of force that can be applied to the actuator, as the force provided by the active suspension actuator will eventually saturate

\begin{aligned} | u | \leq u_{\max} \end{aligned}

(7)

The objective of the design of the active suspension controller is to reduce the amount of body vibration, measured in terms of the acceleration of the sprung mass ${\ddot{z}}_{c}$ , provided that the time domain limitations mentioned above are respected. With this objective in mind, the following controllers can be characterized by their output variables

\begin{aligned} z_{1} & = {\ddot{z}}_{c} \\ z_{2} & = {[\begin{matrix} \frac{z_{c f} - z_{f}}{z_{\max}}, \frac{z_{c r} - z_{r}}{z_{\max}}, \frac{u_{f}}{u_{\max}}, \frac{u_{r}}{u_{\max}} \end{matrix}]}^{T} \end{aligned}

(8)

Then we can write the following as the equation of the half car model in its state space

\begin{aligned} \dot{x} & = A_{x} x + B_{1, x} w + B_{2, x} u \\ z_{1} & = C_{1, x} x + D_{12, x} u \\ z_{2} & = C_{2, x} x + D_{22, x} u \\ y & = C_{x} x \end{aligned}

(9)

If we assume that the road input z_tf, z_tr of the front and rear wheels represent the response of a first order linear filter to a white noise excitation input, as suggested by Prabakar et al.,²⁹ we obtain the following

\begin{aligned} {\dot{z}}_{t f} = a_{w} z_{t f} + b_{w} ϕ_{f} \\ {\dot{z}}_{t r} = a_{w} z_{t r} + b_{w} ϕ_{r} \end{aligned}

(10)

where a_w = −2πn₁v,

b_{w} = 2 π n_{0} \sqrt{G_{q} (n_{0}) v}

and n₀ is a constant (often 0.1) denoting the reference spatial frequency. The stimulation from the road has a lower spatial frequency limit, denoted by n₁, which is typically equal to 0.01. Where G_q (n₀) is the power spectral density of the road at the reference spatial frequency, ϕ_f, ϕ_r are Gaussian white noise with unit variance, and are related in time by a certain delay.

Defining the state vector as χ = [x,w]^T, (9) can be represented in state space as

\begin{aligned} \dot{χ} & = A χ + B_{1} ϕ + B_{2} u \\ z_{1} & = C_{1} χ + D_{12} u \\ z_{2} & = C_{2} χ + D_{22} u \\ y & = C χ \end{aligned}

(11)

where

\begin{aligned} A & = [\begin{matrix} A_{x} & B_{1, x} \\ 0 & A_{w} \end{matrix}], B_{1} = [\begin{matrix} 0 \\ B_{w} \end{matrix}], B_{2} = [\begin{matrix} B_{2, x} \\ 0 \end{matrix}], \\ C_{1} & = [\begin{matrix} C_{1, x} & 0 \end{matrix}], D_{12} = D_{12, x}, \\ C_{2} & = [\begin{matrix} C_{2, x} & 0 \end{matrix}], D_{22} = D_{22, x}, \\ A_{w} & = [\begin{matrix} a_{w} & 0 \\ 0 & a_{w} \end{matrix}], B_{w} = [\begin{matrix} b_{w} & 0 \\ 0 & b_{w} \end{matrix}] . \end{aligned}

There is a correlation between the matrices A, B₁ and the forward vehicle velocity v in (11). To ensure that there is only an affine relationship between the uncertain parameter θ and the system matrices A(θ), B₁(θ), we denote the time-varying uncertain parameter as $θ = {[v, \sqrt{v}]}^{T}$ .³⁰ The time-varying parameter θ can then be described as varying only within the polygon Θ defined by the vertices q₁, q₂, q₃ and q₄, that is,

\begin{aligned} θ \in Θ ≔ C o \{q_{1}, q_{2}, q_{3}, q_{4}\} = \{\sum_{i = 1}^{4} α_{i} q_{i}\} \end{aligned}

(12)

with α_i ≥ 0,

\sum_{i = 1}^{4} α_{i} = 1

and

\begin{aligned} q_{1} ≔ [\begin{matrix} v_{\max} \\ \sqrt{v_{\max}} \end{matrix}], q_{2} ≔ [\begin{matrix} v_{\max} \\ \sqrt{v_{\min}} \end{matrix}], q_{3} ≔ [\begin{matrix} v_{\min} \\ \sqrt{v_{\max}} \end{matrix}], q_{4} ≔ [\begin{matrix} v_{\min} \\ \sqrt{v_{\min}} \end{matrix}] . \end{aligned}

where Co means convex set. v_min and v_max denote the minimum and maximum values of vehicle velocity, respectively.

The system matrix associated with uncertain parameters can also be derived from a collection of polyhedral matrices, that is

\begin{aligned} (\begin{matrix} A (θ) \\ B_{1} (θ) \end{matrix}) ≔ C o \{(\begin{matrix} A_{i} \\ B_{1 i} \end{matrix})\} = \{\sum_{i = 1}^{4} α_{i} (\begin{matrix} A_{i} \\ B_{1 i} \end{matrix})\} \end{aligned}

(13)

where A_i = A(θ)|θ = q_i and B_1i = B₁(θ)|θ = q_i are the polyhedron matrix’s vertex coordinates.

It may also be shown that the positive scalar α_i is related to the online observable velocity v by

\begin{aligned} α_{1} ≔ a b, α_{2} ≔ a (1 - b), α_{3} ≔ (1 - a) b, α_{4} ≔ (1 - a) (1 - b) \end{aligned}

(14)

with

a = \frac{v - v_{\min}}{v_{\max} - v_{\min}}

b = \frac{\sqrt{v} - \sqrt{v_{\min}}}{\sqrt{v_{\max}} - \sqrt{v_{\min}}}

Changes in vehicle velocity over time are just one of several real world factors that can affect model parameters. The sprung mass of the semitrailer model is also a matter of some debate. For the same reason, we cannot learn anything about how a sprung mass actually works outside of its nominal operating range. By taking the minimum M_min and maximum M_max values of the sprung masses M, 1/M can be written as

\begin{aligned} \frac{1}{M} = \frac{1}{M_{0}} + λ \frac{1}{M_{1}} \end{aligned}

(15)

where λ is a scaler value with an unknown value in the range [−1, 1], and

\begin{aligned} \frac{1}{M_{0}} & = (\frac{1}{M_{\min}} + \frac{1}{M_{\min}}) / 2, \\ \frac{1}{M_{1}} & = (\frac{1}{M_{\max}} - \frac{1}{M_{\min}}) / 2 . \end{aligned}

In (11), the state-space matrix with uncertain sprung mass can be rewritten as follows

\begin{aligned} A (θ) = A^{0} (θ) + E_{1} λ H_{1}, \\ B_{2} = B_{2}^{0} + E_{1} λ H_{2}, \\ C_{1} = C_{1}^{0} + E_{2} λ H_{1}, \\ D_{12} = D_{12}^{0} + E_{2} λ H_{2} . \end{aligned}

(16)

where

A^{0} (θ) = A (θ) |_{M = M_{0}}

B_{2}^{0} = B_{2} |_{M = M_{0}}

C_{1}^{0} = C_{1} |_{M = M_{0}}

D_{12}^{0} = D_{12} |_{M = M_{0}}

with

$E_{1} = {[\begin{matrix} 0 & 0 & 0 & 0 & \frac{1}{M_{1}} & \frac{1}{M_{1}} & 0 & 0 & 0 & 0 \end{matrix}]}^{T},$

$H_{1} = [\begin{matrix} - k_{f} & - k_{r} & k_{f} & k_{r} & - c_{f} & - c_{r} & c_{f} & c_{r} & 0 & 0 \end{matrix}],$

$E_{2} = \frac{1}{M_{1}}, H_{2} = [\begin{matrix} - 1 & - 1 \end{matrix}] .$

Main methods

Direct sensing of suspension velocity is difficult in a real vehicle system. The aim of this research is to develop an output feedback H_∞ controller that dynamically modifies the control gain according to the real-time observed velocity v to improve the control performance.

\begin{aligned} \dot{ξ} & = A_{K} (θ) ξ + B_{K} (θ) y \\ u & = C_{K} (θ) ξ + D_{K} (θ) y \end{aligned}

(17)

where

ξ \in R^{n_{K}}

represents the controller’s state. The observable vehicle velocity is used to adaptively update the unknown controller matrices A_K(θ), B_K(θ), C_K(θ) and D_K(θ). Similarly, the control matrix can be characterized by a linear combination of finite vertex matrices as given by the polyhedron theory²⁸

\begin{aligned} Ω (θ) ≔ (\begin{matrix} A_{K} (θ) & B_{K} (θ) \\ C_{K} (θ) & D_{K} (θ) \end{matrix}) = \sum_{i = 1}^{4} α_{i} Ω_{i} = \sum_{i = 1}^{4} α_{i} (\begin{matrix} A_{K i} & B_{K i} \\ C_{K i} & D_{K i} \end{matrix}) \end{aligned}

(18)

By measuring the current vehicle velocity and calculating the controller’s parameter matrix Ω(θ) in real time, the control gain can be adapted to the changing vehicle velocity. Unlike conventional robust control strategies, which always consider worst-case solutions,^31–33 this approach considers all feasible outcomes.

By applying the controller (17) to the system (11), we obtain the closed-loop system defined by $\hat{x} = {[\begin{matrix} χ, ξ \end{matrix}]}^{T}$ , that is

\begin{aligned} \dot{\hat{x}} & = \hat{A} (θ) \hat{x} + \hat{B} (θ) ϕ \\ z_{1} & = {\hat{C}}_{1} (θ) \hat{x} \\ z_{2} & = {\hat{C}}_{2} (θ) \hat{x} \end{aligned}

(19)

where

\begin{aligned} \hat{A} (θ) = \bar{A} (θ) + E_{A} λ H_{A} (θ), \hat{B} (θ) = \bar{B} (θ), \\ {\hat{C}}_{1} (θ) = {\bar{C}}_{1} (θ) + E_{C} λ H_{A} (θ), {\hat{C}}_{2} (θ) = {\bar{C}}_{2} (θ), \end{aligned}

with

\begin{aligned} \bar{A} (θ) = [\begin{matrix} A^{0} (θ) + B_{2}^{0} D_{K} (θ) C & B_{2}^{0} C_{K} (θ) \\ B_{K} (θ) C & A_{K} (θ) \end{matrix}], \bar{B} (θ) = [\begin{matrix} B_{1} (θ) \\ 0 \end{matrix}], \\ E_{A} = [\begin{matrix} E_{1} \\ 0 \end{matrix}], H_{A} (θ) = [\begin{matrix} H_{1} + H_{2} D_{K} (θ) C & H_{2} C_{K} (θ) \end{matrix}], \end{aligned}

\begin{aligned} {\bar{C}}_{1} (θ) = [\begin{matrix} C_{1}^{0} + D_{12}^{0} D_{K} (θ) C & D_{12}^{0} C_{K} (θ) \end{matrix}], E_{C} = E_{2}, \end{aligned}

\begin{aligned} {\bar{C}}_{2} (θ) = [\begin{matrix} C_{2} + D_{22} D_{K} (θ) C & D_{22} C_{K} (θ) \end{matrix}] . \end{aligned}

Making the H-value of the closed-loop transfer function matrix G (jω) less than a certain constant γ while maintaining the asymptotic stability of the control system (19) is a common definition of the H_∞ control problem when the finite frequency constraint is taken into account.

\begin{aligned} \sup_{ω_{1} < ω < ω_{2}} ‖ G (j ω) ‖_{\infty} < γ \end{aligned}

(20)

where ω₁ and ω₂ are the lower and upper bounds of the concerned frequency constraints. G (jω) is the transfer function of the road input w to the controlled output z₁.

In addition, the controlled output z₂ must satisfy the following conditions in order to satisfy the time domain constraints of the active suspension (6) and (7)

\begin{aligned} | {z_{2}}_{i} | \leq 1, i = 1,2, \dots, 4 . \end{aligned}

(21)

Before introducing the idea of a finite-frequency robust H_∞ controller that takes into account the uncertainty of the sprung mass, we first offer the following lemmas and theorems.

Lemma 1

³⁴ Given positive scalars γ, η, and ρ, the closed-loop system represented by equation (19) is asymptotically stable, and the necessary and sufficient condition for $\sup_{ω_{1} < ω < ω_{2}} ‖ G (j ω) ‖_{\infty} < γ$ is that there exist real symmetric matrices P, P_s > 0, Q > 0 and general matrix F which satisfy the following inequalities:

\begin{align} [\begin{matrix} - {[F]}_{s} & F^{T} \hat{A} (θ) + P_{s} & F^{T} & F^{T} \hat{B} (θ) \\ * & - P_{s} & 0 & 0 \\ * & * & - P_{s} & 0 \\ * & * & * & - η I \end{matrix}] < 0 \end{align}

(22a)

\begin{align} [\begin{matrix} - Q & P + j ω_{c} Q - F & 0 & 0 \\ * & - ω_{1} ω_{2} Q + {[F^{T} \hat{A} (θ)]}_{s} & F^{T} \hat{B} (θ) & {\hat{C}}_{1}^{T} (θ) \\ * & * & - γ^{2} I & 0 \\ * & * & * & - I \end{matrix}] < 0 \end{align}

(22b)

\begin{align} [\begin{matrix} - I & \sqrt{ρ} {{\hat{C}}_{2} (θ)}_{j} \\ * & - P_{s} \end{matrix}] < 0, j = 1,2, \dots, 4, \end{align}

(22c)

where [F]_s stands for F + F^T, and ω_c = (ω₁ + ω₂)/2.

In Lemma 1, inequality (22a) represents the necessary and sufficient condition for the asymptotic stability of the system, (22b) denotes the frequency domain range established according to the KYP lemma, where ω₁ ≤ ω ≤ ω₂ satisfies the system’s H_∞ performance γ corresponding inequality, and (22c) indicates the necessary and sufficient condition for the system’s time-domain constraint inequality (21). The specific process of inequality proof can be referred to in the literature.¹⁸

Lemma 2

^19,34 Given matrices Y, E, and H of appropriate dimension, where Y is symmetric, then

Y + E N H + {(E N H)}^{T} < 0

(23)

is equivalent to

Y + ε E E^{T} + \frac{1}{ε} H^{T} H < 0,

(24)

for all matrices N satisfying N^TN < I, if ɛ > 0.

If we apply the Schur’s complement to the inequality (24), we get

[\begin{matrix} Y & ε E & H^{T} \\ * & - ε I & 0 \\ * & * & - ε I \end{matrix}] < 0

(25)

Theorem 1

The asymptotic stability of the system (19) and the necessary and sufficient conditions for $\sup_{ω_{1} < ω < ω_{2}} ‖ G (j ω) ‖_{\infty} < γ$ are the existence of symmetric matrices P, P_s > 0, Q > 0, a general matrix F and a set of constants ɛ₁ > 0, ɛ₂ > 0, and ɛ₃ > 0 such that the following inequality holds for given positive scalars γ, η, ρ:

\begin{align} [\begin{matrix} Φ_{1} & Φ_{2} \\ * & Φ_{3} \end{matrix}] < 0 \end{align}

(26a)

\begin{align} [\begin{matrix} Ψ_{1} & Ψ_{2} \\ * & Ψ_{3} \end{matrix}] < 0 \end{align}

(26b)

\begin{align} [\begin{matrix} - I & \sqrt{ρ} {{\bar{C}}_{2} (θ)}_{j} \\ * & - P_{s} \end{matrix}] < 0, j = 1,2, \dots, 4 . \end{align}

(26c)

where

$Φ_{1} = [\begin{matrix} - {[F]}_{s} & F^{T} \bar{A} (θ) + P_{s} & F^{T} & F^{T} \bar{B} (θ) \\ * & - P_{s} & 0 & 0 \\ * & * & - P_{s} & 0 \\ * & * & * & - η I \end{matrix}]$ ,

$Φ_{2} = [\begin{matrix} ε_{1} F^{T} E_{A} & 0 \\ 0 & H_{A}^{T} (θ) \\ 0 & 0 \\ 0 & 0 \end{matrix}], Φ_{3} = [\begin{matrix} - ε_{1} I & 0 \\ * & - ε_{1} I \end{matrix}]$ ,

$Ψ_{1} = [\begin{matrix} - Q & P + j ω_{c} Q - F & 0 & 0 \\ * & - ω_{1} ω_{2} Q + {[F^{T} \bar{A} (θ)]}_{s} & F^{T} \bar{B} (θ) & {\bar{C}}_{1}^{T} (θ) \\ * & * & - γ^{2} I & 0 \\ * & * & * & - I \end{matrix}]$ ,

$Ψ_{2} = [\begin{matrix} 0 & 0 & 0 & 0 \\ ε_{2} F^{T} E_{A} & ε_{3} H_{A}^{T} (θ) & H_{A}^{T} (θ) & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & E_{C} \end{matrix}]$ ,

$Ψ_{3} = [\begin{matrix} - ε_{2} I & 0 & 0 & 0 \\ * & - ε_{3} I & 0 & 0 \\ * & * & - ε_{2} I & 0 \\ * & * & * & - ε_{3} I \end{matrix}]$ .

Proof. Substituting $\hat{A} (θ) = \bar{A} (θ) + E_{A} λ H_{A} (θ)$ into the left-hand side of inequality (22a), we obtain the following result

$Φ_{1} + J_{1} λ L_{1} + L_{1}^{T} λ J_{1}^{T} < 0$ ,

with $J_{1} = [\begin{matrix} F^{T} E_{A} \\ 0 \\ 0 \\ 0 \end{matrix}]$ ,

and $L_{1} = [\begin{matrix} 0 & H_{A} (θ) & 0 & 0 \end{matrix}]$ .

According to the formulation of Lemma 2, the previous formula is clearly equivalent to the right side of the inequality (26a). Similarly, we can show that (26b) and (26c) correspond to (22b) and (22c), respectively. The proof is completed.

In (26a), (26b), and (26c) the values of $\bar{A}$ , ${\bar{C}}_{1}$ , and ${\bar{C}}_{2}$ depend on the unknown controller parameters. As a result, the matrix inequalities are all bilinear with respect to the general matrix F and the controller parameter matrices (A_K, B_K, C_K, D_K), making the conventional linear programming method inapplicable. Parameterizing the complex problem, replacing the original objective by simple linear parameters, is one approach to solving this problem. The exact procedure is as follows:

First, the matrix F and its inverse matrix are typically partitioned as follows

F = [\begin{matrix} Y & N_{1} \\ N_{2} & W \end{matrix}], F^{- 1} = [\begin{matrix} X & M_{1} \\ M_{2} & Z \end{matrix}]

(27)

where

Y, X \in R^{n}

and

W, Z \in R^{n_{K}}

are general nonsingular matrix.

If we define

Δ_{1} = [\begin{matrix} X & I \\ M_{2} & 0 \end{matrix}], Δ_{2} = [\begin{matrix} I & Y \\ 0 & N_{2} \end{matrix}]

(28)

according to F∗F⁻¹ = I, then we have FΔ₁ = Δ₂.

Replace the bilinear term in (26a), (26b) and (26c) with a single linear variable that can be rapidly solved in the following derivation. To facilitate comprehension, we now present the following portion of the variable substitution formula:

\begin{aligned} {\tilde{A}}_{i} & = Y^{T} (A_{i}^{0} + B_{2}^{0} D_{K i} C) X + N_{2}^{T} B_{K i} C X + Y^{T} B_{2}^{0} C_{K i} M_{2} + N_{2}^{T} A_{K i} M_{2} \\ {\tilde{B}}_{i} & = Y^{T} B_{2}^{0} D_{K} + N_{2}^{T} B_{K i} \\ {\tilde{C}}_{i} & = D_{K i} C X + C_{K i} M_{2} \\ {\tilde{D}}_{i} & = D_{K i} \end{aligned}

(29)

The following H_∞ control theorem for finite frequency robust output feedback can be derived by taking into account the uncertainty of the sprung mass and the influence of the time-varying vehicle velocity.

Theorem 2

For given positive scalars γ, η and ρ, the necessary and sufficient condition for the closed-loop system (19) to be asymptotically stable and $\sup_{ω_{1} < ω < ω_{2}} ‖ G (j ω) ‖_{\infty} < γ$ is the existence of symmetric matrices ${\tilde{P}}_{1}, {\tilde{P}}_{3}, {\tilde{P}}_{s 1}, {\tilde{P}}_{s 3}, {\tilde{Q}}_{1}, {\tilde{Q}}_{3}$ , a general matrix $X, Y, S, {\tilde{P}}_{2}, {\tilde{P}}_{s 2}, {\tilde{Q}}_{2}, {\tilde{A}}_{i}, {\tilde{B}}_{i}, {\tilde{C}}_{i}, {\tilde{D}}_{i}$ , i = 1, …, 4, and constants ɛ₁, ɛ₂, and ɛ₃ that make the following inequalities hold:

\begin{align} [\begin{matrix} Φ_{1, i} & Φ_{2, i} \\ * & Φ_{3, i} \end{matrix}] < 0 \end{align}

(30a)

\begin{align} [\begin{matrix} Ψ_{1, i} & Ψ_{2, i} \\ * & Ψ_{3, i} \end{matrix}] < 0 \end{align}

(30b)

\begin{align} [\begin{matrix} ϒ_{1, i} & ϒ_{2, i} \\ * & ϒ_{3, i} \end{matrix}] < 0 \end{align}

(30c)

where

$Φ_{1, i} = [\begin{matrix} - {[X]}_{s} & - (I + S) & A_{i}^{0} X + B_{2}^{0} {\tilde{C}}_{i} + {\tilde{P}}_{s 1} & A_{i}^{0} + B_{2}^{0} {\tilde{D}}_{i} C + {\tilde{P}}_{s 2} \\ * & - (Y^{T} + Y) & {\tilde{A}}_{i} + {\tilde{P}}_{s 2}^{T} & Y^{T} A_{i}^{0} + {\tilde{B}}_{i} C + {\tilde{P}}_{s 3} \\ * & * & - {\tilde{P}}_{s 1} & - {\tilde{P}}_{s 2} \\ * & * & * & - {\tilde{P}}_{s 3} \end{matrix}]$ ,

$Φ_{2, i} = [\begin{matrix} X & I & B_{1 i} & ε_{1} E_{1} & 0 \\ S^{T} & Y^{T} & Y^{T} B_{1 i} & ε_{1} Y^{T} E_{1} & 0 \\ 0 & 0 & 0 & 0 & X^{T} H_{1}^{T} + {\tilde{C}}_{i}^{T} H_{2}^{T} \\ 0 & 0 & 0 & 0 & H_{1}^{T} + C^{T} {\tilde{D}}_{i}^{T} H_{2}^{T} \end{matrix}],$

$Φ_{3, i} = [\begin{matrix} - {\tilde{P}}_{s 1} & - {\tilde{P}}_{s 2} & 0 & 0 & 0 \\ * & - {\tilde{P}}_{s 3} & 0 & 0 & 0 \\ * & * & - η I & 0 & 0 \\ * & * & * & - ε_{1} I & 0 \\ * & * & * & * & - ε_{1} I \end{matrix}],$

$Ψ_{1, i} = [\begin{matrix} - {\tilde{Q}}_{1} & - {\tilde{Q}}_{2} & {\tilde{P}}_{1} + j ω_{c} {\tilde{Q}}_{1} - X^{T} & {\tilde{P}}_{2} + j ω_{c} {\tilde{Q}}_{2} - S \\ * & - {\tilde{Q}}_{3} & {\tilde{P}}_{2}^{T} + j ω_{c} {\tilde{Q}}_{2}^{T} - I & {\tilde{P}}_{3} + j ω_{c} {\tilde{Q}}_{3} - Y \\ * & * & - ω_{1} ω_{2} {\tilde{Q}}_{1} + {[A_{i}^{0} X + B_{2}^{0} {\tilde{C}}_{i}]}_{s} & - ω_{1} ω_{2} {\tilde{Q}}_{2} + A_{i}^{0} + B_{2}^{0} {\tilde{D}}_{i} C + {\tilde{A}}_{i}^{T} \\ * & * & * & - ω_{1} ω_{2} {\tilde{Q}}_{3} + {[Y^{T} A_{i}^{0} + {\tilde{B}}_{i} C]}_{s} \end{matrix}],$

$Ψ_{2, i} = [\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ B_{1 i} & X^{T} {C_{1}^{0}}^{T} + {\tilde{C}}_{i}^{T} {D_{12}^{0}}^{T} & ε_{2} E_{1} & ε_{3} (X^{T} H_{1}^{T} + {\tilde{C}}_{i}^{T} H_{2}^{T}) & X^{T} H_{1}^{T} + {\tilde{C}}_{i}^{T} H_{2}^{T} & 0 \\ Y^{T} B_{1 i} & {C_{1}^{0}}^{T} + C^{T} {\tilde{D}}_{i}^{T} {D_{12}^{0}}^{T} & ε_{2} Y^{T} E_{1} & ε_{3} (H_{1}^{T} + C^{T} {\tilde{D}}_{i}^{T} H_{2}^{T}) & H_{1}^{T} + C^{T} {\tilde{D}}_{i}^{T} H_{2}^{T} & 0 \end{matrix}],$

$Ψ_{3, i} = [\begin{matrix} - γ^{2} I & 0 & 0 & 0 & 0 & 0 \\ * & - I & 0 & 0 & 0 & E_{2} \\ * & * & - ε_{2} I & 0 & 0 & 0 \\ * & * & * & - ε_{3} I & 0 & 0 \\ * & * & * & 0 & - ε_{2} I & 0 \\ * & * & * & * & * & - ε_{3} I \end{matrix}],$

Proof. If J₁ = diag{ $Δ_{1}^{T}$ , $Δ_{1}^{T}$ , $Δ_{1}^{T}$ , I, I, I}, J₂ = diag{ $Δ_{1}^{T}$ , $Δ_{1}^{T}$ , I, I, I, I, I, I}, J₃ = diag{ I, $Δ_{1}^{T}$ }, then the matrix parameters to be substituted for ${\tilde{A}}_{i}$ , ${\tilde{B}}_{i}$ , ${\tilde{C}}_{i}$ and ${\tilde{D}}_{i}$ in (26) are as follows:

$Δ_{1}^{T} F^{T} \bar{A} Δ_{1} = [\begin{matrix} A^{0} X + B_{2}^{0} \tilde{C} & A^{0} + B_{2}^{0} \tilde{D} C \\ \tilde{A} & Y^{T} A^{0} + \tilde{B} C \end{matrix}], Δ_{1}^{T} F Δ_{1} = [\begin{matrix} X^{T} & S \\ I & Y \end{matrix}],$

$S = X^{T} Y + M_{2}^{T} N_{2}, Δ_{1}^{T} F^{T} \bar{B} = [\begin{matrix} B_{1} \\ Y^{T} B_{1} \end{matrix}],$

${\bar{C}}_{1} Δ_{1} = [\begin{matrix} C_{1}^{0} X + D_{12}^{0} \tilde{C} & C_{1}^{0} + D_{12}^{0} \tilde{D} C \end{matrix}],$

$Δ_{1}^{T} P_{s} Δ_{1} = {\tilde{P}}_{s} = [\begin{matrix} {\tilde{P}}_{s 1} & {\tilde{P}}_{s 2} \\ * & {\tilde{P}}_{s 3} \end{matrix}], Δ_{1}^{T} P Δ_{1} = \tilde{P} = [\begin{matrix} {\tilde{P}}_{1} & {\tilde{P}}_{2} \\ * & {\tilde{P}}_{3} \end{matrix}],$

$Δ_{1}^{T} Q Δ_{1} = \tilde{Q} = [\begin{matrix} {\tilde{Q}}_{1} & {\tilde{Q}}_{2} \\ * & {\tilde{Q}}_{3} \end{matrix}],$

$Δ_{1}^{T} F^{T} E_{A} = [\begin{matrix} E_{1} \\ Y^{T} E_{1} \end{matrix}],$

$H_{A} Δ_{1} = [\begin{matrix} H_{1} X + H_{2} \tilde{C} & H_{1} + H_{2} \tilde{D} C \end{matrix}] .$

Finally, inequality (30a) is obtained by multiplying the left side of (26a) by J₁ and the right side by $J_{1}^{T}$ . Similarly, (30b) is obtained by multiplying the left and right sides of (26b) by J₂ and $J_{2}^{T}$ , and (30c) is obtained by multiplying the left and right sides of (26c) by J₃ and $J_{3}^{T}$ . The proof is completed.

Remark 1

Notice that the matrix to the left of the inequality (30b) has a term for the complex variable jω_c. According to the method in reference [³⁵], a matrix inequality with complex variables can be converted to a higher dimensional inequality as shown below, that is,

\begin{aligned} [\begin{matrix} S_{1} & S_{2} \\ - S_{2} & S_{1} \end{matrix}] \end{aligned}

(31)

where

$S_{1} = [\begin{matrix} Ψ_{1, i}^{'} & Ψ_{2, i} \\ * & Ψ_{3, i} \end{matrix}]$ ,

$Ψ_{1, i}^{'} = [\begin{matrix} - {\tilde{Q}}_{1} & - {\tilde{Q}}_{2} & {\tilde{P}}_{1} - X^{T} & {\tilde{P}}_{2} - S \\ * & - {\tilde{Q}}_{3} & {\tilde{P}}_{2}^{T} - I & {\tilde{P}}_{3} - Y \\ * & * & - ω_{1} ω_{2} {\tilde{Q}}_{1} + {[A_{i}^{0} X + B_{2}^{0} {\tilde{C}}_{i}]}_{s} & - ω_{1} ω_{2} {\tilde{Q}}_{2} + A_{i}^{0} + B_{2}^{0} {\tilde{D}}_{i} C + {\tilde{A}}_{i}^{T} \\ * & * & * & - ω_{1} ω_{2} {\tilde{Q}}_{3} + {[Y^{T} A_{i}^{0} + {\tilde{B}}_{i} C]}_{s} \end{matrix}]$ ,

$S_{2} = [\begin{matrix} 0 & 0 & ω_{c} {\tilde{Q}}_{1} & ω_{c} {\tilde{Q}}_{2} & 0 & \dots & 0 \\ 0 & 0 & ω_{c} {\tilde{Q}}_{2}^{T} & ω_{c} {\tilde{Q}}_{3} & 0 & \dots & 0 \\ - ω_{c} {\tilde{Q}}_{1} & - ω_{c} {\tilde{Q}}_{2} & 0 & 0 & 0 & \dots & 0 \\ - ω_{c} {\tilde{Q}}_{2}^{T} & - ω_{c} {\tilde{Q}}_{3} & 0 & 0 & 0 & \dots & 0 \\ 0 & 0 & 0 & 0 & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & 0 & 0 & \dots & 0 \end{matrix}] .$

Singular value decomposition always yields the invertible matrices M₂ and N₂ satisfying (30a), (30b), and (30c) since $S = M_{2}^{T} N_{2} + X^{T} Y$ . Therefore, the following formula can be used to derive the controller’s parameter matrix

\begin{aligned} D_{K i} & = {\tilde{D}}_{i} \\ C_{K i} & = ({\tilde{C}}_{i} - D_{K i} C X) {(M_{2})}^{- 1} \\ B_{K i} & = {(N_{2}^{T})}^{- 1} ({\tilde{B}}_{i} - Y^{T} B_{2}^{0} D_{K i}) \\ A_{K i} & = {(N_{2}^{T})}^{- 1} [{\tilde{A}}_{i} - Y^{T} (A_{i}^{0} + B_{2}^{0} D_{K i} C) X] {(M_{2})}^{- 1} - B_{K i} C X {(M_{2})}^{- 1} \\ - {(N_{2}^{T})}^{- 1} Y^{T} B_{2}^{0} C_{K i} \end{aligned}

(32)

A_Ki, B_Ki, C_Ki, and D_Ki can be pre-calculated off-line by combining the above expressions with the polyhedron vertex matrices A_i and B_1i in (13). The obtained current vehicle velocity v and (18) are used to calculate the velocity related controller matrices A_K, B_K, C_K, and D_K in the online stage.

Experiments and results

Using the half-car model with unknown sprung mass, the above method builds a velocity-dependent controller in which the controller matrix is changed adaptively based on the observed velocity. The simulated sprung mass and the associated uncertainty are given in Table 2.

Table 2.

Experiment settings.

Case number	Case 1	Case 2	Case 3	Case 4
λ	1	0.6	0.3	0

Frequency response

Here, we utilize MATLAB and the YALMIP toolbox to solve the positive definite programming problem (30a), (30b) and (30c). The control gain matrices A_Ki, B_Ki, C_Ki, D_Ki, i = 1, …, 4 are computed by setting the finite frequency constraint parameters to the values specified in Sun et al (2010).¹⁸ Specifically, we select w₁ = 4, w₂ = 8, η = 10,165 and ρ = 0.9. The control methods used for comparison in this section include:

1. Passive: where the control force is neglected, assuming a constant value of zero;

2. Entire: a dynamic output feedback H_∞ controller solved over the entire frequency range, as referenced in Sun et al.(2011)³⁶;

3. Proposed: the robust finite frequency H_∞ controller described in this study, which depends on vehicle velocity.

The comparison of different control methods aims to validate the effectiveness of the proposed approach in ensuring vibration control within the human sensitivity frequency range of 4–8 Hz, under time-varying parameters.

Figure 2 displays the frequency response of the sprung mass acceleration ${\ddot{z}}_{c}$ for the passive suspension, full-frequency controller, and the proposed finite frequency controller at forward velocities of 10 m/s, 20 m/s, 30 m/s, and 40 m/s. As illustrated, the proposed velocity-dependent finite frequency control strategy significantly reduces the peak acceleration of the sprung mass across all scenarios, particularly within the 4–8 Hz frequency range. A comparison of the frequency responses in the subplots for different unknown parameters λ reveals that λ = 0 corresponds to the largest peak in frequency response for the minimum sprung mass, while λ = 1 corresponds to the smallest peak for the maximum sprung mass. This observation aligns with the fact that increasing the sprung mass enhances the system’s inertial mass, thereby reducing the peak vertical acceleration of the vehicle. Additionally, the frequency response of the passive suspension system notably increases with velocity v. In contrast, both the entire frequency control and the proposed robust finite frequency control demonstrate greater stability than passive control, showing minimal variation with changes in vehicle velocity.

Figure 2.

Frequency responses of sprung mass acceleration. Passive (- -), Entire frequency (- · -), Finite frequency —. (a) v = 10 m/s, (b) v = 20 m/s, (c) v = 30 m/s, (d) v = 40 m/s.

Figure 3 presents a comparison of the H_∞ performance for passive control, entire frequency control, and the finite frequency controller calculated based on Theorem 2 under varying sprung mass and velocity conditions. In the figure, the sprung mass varies along the X-axis, velocity along the Y-axis, and the H_∞ performance value of the closed-loop system is displayed along the Z-axis to evaluate the impact of different parameters on the disturbance performance of the suspension system. The results indicate that under passive control conditions, the system’s H_∞ performance is relatively insensitive to changes in sprung mass, whereas velocity has a significantly greater effect on H_∞ performance than sprung mass. In contrast, both entire frequency control and finite frequency control strategies markedly improve the H_∞ performance metrics of the closed-loop system, demonstrating enhanced disturbance rejection capabilities. Notably, at higher velocity, the finite frequency control method achieves superior H_∞ performance compared to the entire frequency controller.

Figure 3.

Effect of sprung mass and velocity on H_∞ control performance. (a) Passive, (b) Entire frequency, and (c) Finite frequency.

Time response

The efficiency of the proposed controller in time-varying vehicle velocity scenarios is further evaluated by performing time-domain simulations for the two vehicle velocity variation circumstances shown in Figure 4, Scenario I and Scenario II, respectively.

Figure 4.

Two different scenarios of velocity changes. (a) Scenario I, (b) Scenario II.

First, we compare the proposed velocity-dependent finite frequency control method (LPV) with traditional linear time-invariant (LTI) finite frequency controllers by adding an additional constraint A_Ki = A_K, B_Ki = B_K, C_Ki = C_K, D_Ki = D_K, i = 1, …, 4 in inequality (30). The comparison includes LTI controllers operating at constant velocities of 10 m/s (LTI-10) and 40 m/s (LTI-40). Figures 5–8 present the root-mean-square (RMS) values of the sprung mass acceleration under ISO-B, ISO-C, ISO-D, and ISO-E road conditions. It should be noted that these simulations consider the sprung mass under the conditions specified in Case 1, without accounting for the uncertainty of sprung mass in other cases. The comparison is solely between the traditional constant-velocity finite-frequency H_∞ control method and the velocity-dependent robust finite-frequency H_∞ control method proposed in this study.

Figure 5.

Time-domain body acceleration responses on ISO-B road. (a) Scenario I, (b) Scenario II.

Figure 6.

Time-domain body acceleration responses on ISO-C road. (a) Scenario I, (b) Scenario II.

Figure 7.

Time-domain body acceleration responses on ISO-D road. (a) Scenario I, (b) Scenario II.

Figure 8.

Time-domain body acceleration responses on ISO-E road. (a) Scenario I, (b) Scenario II.

The results indicate that as the road quality increases, the RMS peak values of acceleration also exhibit an increasing trend. Notably, under ISO-E road conditions, the performance difference between the constant velocity and velocity-dependent control methods becomes negligible. This is attributed to the saturation of control inputs under high road inputs, where the effectiveness of either control method in reducing sprung mass acceleration is limited. In contrast, under lower road input conditions observed in ISO-B, ISO-C, and ISO-D, it is evident that the velocity-dependent finite frequency H_∞ control method effectively reduces the RMS values of sprung mass acceleration compared to the constant velocity control method.

Table 3 presents the root mean square (RMS) values of sprung mass acceleration under varying vehicle velocity conditions in Scenario I. The data indicate that the proposed velocity-dependent linear time-varying controller significantly outperforms the linear time-invariant controller designed for constant velocity. In the road scenarios of ISO-B, ISO-C, ISO-D, and ISO-E, the performance improvements of the finite-frequency controllers over the linear time-invariant system designed for a constant velocity of 10 m/s are approximately 31.28%, 31.28%, 27.77%, and 7.74%, respectively. This suggests that when dealing with high amplitudes of road unevenness, the output force of the controller tends to saturate, resulting in limited effectiveness regardless of the controller type used.

Table 3.

RMS value of sprung mass acceleration for Scenario I.

Road level	LTI − 10	LTI − 40	LPV
ISO-B	0.3223	0.3349	0.2215
ISO-C	0.6447	0.6698	0.4430
ISO-D	1.2978	1.3418	0.9374
ISO-E	2.9013	2.9208	2.6768

Table 4 presents the root mean square (RMS) values of sprung mass acceleration under varying vehicle velocity conditions in Scenario II. The results similarly indicate that the proposed velocity-dependent linear time-varying controller significantly outperforms the linear time-invariant controller designed for constant velocity. In the road scenarios of ISO-B, ISO-C, ISO-D, and ISO-E, the performance improvements of the finite-frequency controllers over the linear time-invariant system designed for a constant velocity of 10 m/s are approximately 27.49%, 27.51%, 21.73%, and 5.14%, respectively. Consistent with the results from Scenario I in Table 3, the effectiveness of the proposed robust finite-frequency control method is not significantly improved under the ISO-E road condition. Moreover, compared to the sinusoidal continuous velocity variations in Scenario I, the step changes in vehicle velocity in Scenario II show a slight decline in performance improvements across different road grades. This suggests that the proposed finite-frequency controller based on linear time-varying parameters is more suited for continuously varying conditions.

Table 4.

RMS value of sprung mass acceleration for Scenario II.

Road level	LTI − 10	LTI − 40	LPV
ISO-B	0.3419	0.3585	0.2479
ISO-C	0.6838	0.7170	0.4957
ISO-D	1.4016	1.4429	1.0970
ISO-E	3.2112	3.2201	3.0460

Furthermore, we evaluate the vibration control performance of the suspension system under ISO-A random road conditions. The reference spatial frequency for the ISO-A road is given by G_q (n₀) = 16 × 10⁻⁶ m³. Figures 9 and 10 illustrate the time-domain response of the vertical acceleration of the sprung mass ${\ddot{z}}_{c}$ for Scenario I and II under varying velocity. The results demonstrate that, compared to full-frequency control and passive control, the velocity-dependent finite frequency H_∞ controller calculated based on Theorem 2 yields the lowest peak vertical acceleration of the sprung mass. This finding is consistent with the results obtained from frequency domain analysis.

Figure 9.

Time-domain body acceleration responses for Scenario I. (a) Case 1, (b) Case 2, (c) Case 3, (d) Case 4.

Figure 10.

Time-domain body acceleration responses for Scenario II. (a) Case 1, (b) Case 2, (c) Case 3, (d) Case 4.

Figure 11 illustrates the variation in suspension dynamic stroke and actuator control force under changing vehicle velocities in Scenario I, with similar trends observed in Scenario II. The results indicate that, under ISO-A road conditions, both the suspension dynamic stroke and actuator control force remain below the maximum limit, specifically z₂ < 1. Therefore, inequality (30c) effectively ensures the desired time-domain constraints are met.

Figure 11.

Suspension deflection and control force under Scenario I. (a) front suspension deflection, (b) rear suspension deflection, (c) front control force, and (d) rear control force.

Tables 5 and 6 present the root mean square (RMS) values of the vertical acceleration of the sprung mass under different sprung mass conditions. The data indicate that the RMS value is lowest in Case 4 and highest in Case 1, which aligns with frequency domain results, confirming that an increase in sprung mass leads to a reduction in the average vertical acceleration. Compared to passive control, the proposed velocity-dependent finite frequency control strategy achieves a reduction of approximately 48% in the RMS value of vertical acceleration.

Table 5.

RMS value of sprung mass acceleration for Scenario I.

Case number	Passive	Entire frequency	Finite frequency
Case 1	0.2023	0.1241	0.1026
Case 2	0.1795	0.1492	0.0949
Case 3	0.1754	0.1570	0.0936
Case 4	0.1706	0.1682	0.0921

Table 6.

RMS value of sprung mass acceleration for Scenario II.

Case number	Passive	Entire frequency	Finite frequency
Case 1	0.2214	0.1352	0.1167
Case 2	0.1944	0.1585	0.1054
Case 3	0.1897	0.1663	0.1035
Case 4	0.1839	0.1777	0.1014

Conclusion

This study has developed a velocity-dependent robust finite frequency H_∞ control strategy for a half-car model, capable of meeting the vibration suppression objectives of active suspensions under various sprung mass conditions. Compared to traditional finite frequency H_∞ control methods with constant parameters, the proposed control strategy significantly reduces the root-mean-square (RMS) value of vertical acceleration by approximately 21% to 31% under varying velocity conditions. It also satisfies the predefined time-domain constraints on suspension displacement and control force. This addresses the decline in vibration suppression performance in the human sensitivity frequency range under uncertain sprung mass and vehicle velocity conditions. However, there are still some limitations in this study, as it focuses solely on the vertical acceleration of the sprung mass as the control objective and does not consider the optimization of pitch angular acceleration. In the future, we will consider multi-objective optimization methods to achieve a balance and compromise among multiple control objectives.

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: This work was supported by the National Key Technologies Research and Development Program of China (2018YFB0106200).

ORCID iDs

Qiangqiang Li

Zhiyong Chen

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Vehicle active suspension control considering parameter uncertainty and velocity variation

Abstract

Keywords

Introduction

Problem formulation

Main methods

Experiments and results

Frequency response

Time response

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

References