Active suspension H ∞ /generalized H 2 static output feedback control

Abstract

This paper proposes an approach of H_∞/generalized H₂ (GH₂) static output feedback control for vehicle active suspension. To address the conflicting performance requirements in active suspension, the H_∞ norm is minimized to optimize the ride comfort performance, while the GH₂ norm is designed to meet time-domain hard constraints, including suspension stroke, road-holding performance, and actuator saturation. As not all states of active suspension are measurable in practice, the static output feedback control is designed using suspension stroke and sprung mass velocity as feedback signals. An invertible matrix condition is introduced in the static output feedback control design, which transforms the control problem into a convex optimization problem that can be solved using linear matrix inequalities (LMIs). Simulation and hardware-in-the-loop (HiL) experiments are conducted on both bump and random road responses for active and passive suspension of a 2-degree-of-freedom quarter vehicle. The proposed active suspension H_∞/GH₂ static output feedback controller is compared with H_∞ state feedback controller and the existing controller solved by LMIs and genetic algorithms (GAs), demonstrating that the proposed strategy achieves better ride comfort performance under various road conditions while satisfying all time-domain hard constraints.

Keywords

active suspension static output feedback control HiL experiment

1. Introduction

Vehicle suspension control is a complex multi-objective control problem (Chen et al., 2003). The control objectives involve conflicting performance requirements, such as maximizing ride comfort, limiting suspension stroke, enhancing road-holding capacity, and satisfying actuator saturation constraints (Deshpande et al., 2017). Among these objectives, optimizing ride comfort is particularly crucial.

Compared to passive and semi-active suspension systems (Tseng and Hrovat, 2015), active suspension introduces an additional active force between the vehicle body and tires (Wang et al., 2019). The active force is generated by the actuator of active suspension system, which can effectively suppress vibrations caused by road disturbances and improve the overall performance of the vehicle (Park and Yim, 2021). In recent years, significant efforts have been devoted to the development of active suspension systems.

Optimal control is available for addressing the multi-objective control problem in active suspension, such as the techniques including Linear Quadratic Regulator (LQR) (Taghirad and Esmailzadeh, 1998), Linear Quadratic Gaussian (LQG) (Zhang et al., 2022), Model Predictive Control (MPC) (Song and Wang, 2020; Theunissen et al., 2020), etc. The controllers of LQR and LQG are often designed to optimize conflicting performance requirements in a single objective function, in which determining the weighting metrics is challenging (Chen and Guo, 2005).

Robust control is another crucial method for active suspension control, which includes state feedback control and output feedback control. State feedback control requires full measurability of system states, which is difficult to implement in practical systems (Akbari et al., 2010; Du and Zhang, 2007; Li et al., 2019; Wei et al., 2020). Although the approach based on state observer can reconstruct state quantities, it has some disadvantages such as high implementation costs, complex system design, and significant observation errors (Du et al., 2020). In contrast, output feedback control employs the output signal that can be directly measured as the feedback quantity.

Static output feedback control is widely used in practice due to its low cost and simple structure (Goyal et al., 2023; Kim et al., 2023). A static output feedback controller has been designed based on the H₂ norm of a vehicle quarter active suspension, which uses the displacement and velocity of the suspension stroke as output feedback (Camino et al., 1999). An H_∞ static output feedback controller of the half-vehicle active suspension has been designed (Wei et al., 2018). The driver seat acceleration, as well as the vehicle body acceleration and pitch acceleration, are simultaneously minimized to improve ride comfort. In the research above, either the H_∞ norm or the H₂ norm is utilized to minimize multiple performance requirements in a single objective function. The solution to the control problem often comes with a certain degree of conservatism.

To reduce the conservatism of minimizing multiple performance requirements in a single objective function, the combination of H_∞ and generalized H₂ (GH₂) has been utilized for multi-objective functional control (Du and Zhang, 2008; Liu and Zhao, 2009). The H_∞ norm is employed to describe the performance index of ride comfort, while the GH₂ norm is designed to represent time-domain constraints. A suboptimal H_∞/GH₂ static output feedback control approach has been proposed based on linear matrix inequalities (LMIs) and genetic algorithms (GAs) (Du and Zhang, 2008). By using GAs to search for possible control gain matrices and then resolving the LMIs together with the minimization optimization problem, H_∞/GH₂ static output feedback controllers are obtained. Numerical simulations demonstrate that the proposed approach can achieve similar active suspension performance compared with the state feedback control case. An H_∞/GH₂ static output feedback controller for active suspension has been developed utilizing the suspension stroke as the feedback signal (Liu and Zhao, 2009). By employing a differential evolutionary algorithm to determine the control gain, the H_∞ performance is achieved while considering GH₂ constraints. Although different searching algorithms are used to address the bilinear matrix inequality (BMI) in static output feedback control, obtaining a global optimal solution is not guaranteed due to the non-convex nature of the BMI problem.

This paper proposes an approach of H_∞/GH₂ static output feedback control that achieves the global optimal solution. The ride comfort is improved by optimizing an H_∞ norm, while a GH₂ norm is used to describe time-domain constraints such as suspension stroke, road-holding performance, and actuator saturation. To transform the control problem into a convex optimization problem, an invertible matrix condition is introduced for static output feedback control design. The global optimal solution of the H_∞/GH₂ static output feedback control law can then be solved using LMIs. The suspension stroke and vertical velocity of the vehicle body are chosen as output feedback signals. The proposed approach is compared with H_∞ state feedback control and an existing approach that obtains control gain matrices based on LMIs and GAs. Simulation and hardware-in-the-loop (HiL) experiments are conducted on both random and deterministic road surfaces, demonstrating that the proposed approach achieves better performance with simplified techniques.

This paper is organized as follows: Section 2 sets up the control problem based on the 2-degree-of-freedom (2-DOF) vehicle quarter suspension model, Section 3 designs the H_∞/GH₂ static output feedback controller for the active suspension, Section 4 presents comparative simulation and HiL experiments to demonstrate the superiority of the proposed approach, and Section 5 provides the conclusion.

Notation: $R^{n}$ represents the n-dimensional Euclidean space, $R^{n \times m}$ denotes the set of all n × m real matrices, ‖⋅‖ denotes the matrix 2-norm or the Euclidean vector norm, ‖G‖_∞ is H_∞ norm of transfer function G. For a real symmetric matrix Q, Q > 0 indicates that the matrix is positive definite, rank(Q) denotes the rank of the matrix Q, # denotes the transpose of a matrix block at a symmetric position, and I is a unit matrix of appropriate dimensions.

2. Problem setup

In this section, an active suspension model based on a 2-DOF quarter vehicle is established. Then the performance requirements and time-domain constraints are introduced. Finally, the control objective is proposed.

2.1. Suspension system modeling

In this subsection, both active suspension and passive suspension are established based on a 2-DOF quarter vehicle, in which the impact of vehicle load transfer on the suspension system is ignored (Gordon et al., 1991).

The diagrams of passive and active suspensions are shown in the left and right parts of Figure 1, respectively. The vehicle body mass is represented by the sprung mass m_s, and the tire is reduced to an elastic element with unsprung mass m_u and stiffness k_u. The passive suspension is simplified approximately to a linear spring and damping element between the sprung mass and unsprung mass, where the spring stiffness and the damping coefficient are k_s and c_s, respectively.

The passive suspension dynamics can be expressed as

{\begin{cases} m_{s} {\ddot{x}}_{s p} = - k_{s} (x_{s p} - x_{u p}) - c_{s} ({\dot{x}}_{s p} - {\dot{x}}_{u p}), \\ m_{u} {\ddot{x}}_{u p} = k_{s} (x_{s p} - x_{u p}) + c_{s} ({\dot{x}}_{s p} - {\dot{x}}_{u p}) - k_{u} (x_{u p} - x_{r p}), \end{cases}

(1)

where x_sp, x_up, and x_rp denote the vertical displacements of sprung mass, unsprung mass, and road surface, respectively. In (1), the expressions x_sp − x_up and x_up − x_rp represent the suspension stroke and tire deflection, the variables

{\dot{x}}_{s p}

and

{\dot{x}}_{u p}

denote the velocity of the sprung mass and unsprung mass, respectively.

The active suspension (right in Figure 1) introduces an extra actuator (red block) based on the passive suspension. The actuator generates an active force u_z, which is the control input and acts on both sprung and unsprung mass. Based on Newton’s second law of motion, the dynamics of active suspension can be represented as

{\begin{cases} m_{s} {\ddot{x}}_{s} = - k_{s} (x_{s} - x_{u}) - c_{s} ({\dot{x}}_{s} - {\dot{x}}_{u}) + u_{z}, \\ m_{u} {\ddot{x}}_{u} = k_{s} (x_{s} - x_{u}) + c_{s} ({\dot{x}}_{s} - {\dot{x}}_{u}) - k_{u} (x_{u} - x_{r}) - u_{z}, \end{cases}

(2)

where the variables x_s, x_u, and x_r denote the vertical displacements of sprung mass, unsprung mass, and road surface, respectively. The suspension stroke and tire deflection are represented by x_s − x_u and x_u − x_r, respectively. And the velocity of sprung mass and unsprung mass are

{\dot{x}}_{s}

and

{\dot{x}}_{u}

, respectively.

2.2. Performance requirements

The performance requirements for suspension design mainly include ride comfort, suspension stroke limit, road-holding, and actuator saturation. The evaluation index of ride comfort is the vertical acceleration of the vehicle body, that is, ${\ddot{x}}_{s}$ . And the root mean square (RMS) value of ${\ddot{x}}_{s}$ is introduced to evaluate ride comfort (Chen and Guo, 2005), that is

R M S ({\ddot{x}}_{s}) = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} {| {\ddot{x}}_{s} |}^{2}}

(3)

where N is the total number of sampling points.

Suspension stroke represents the relative displacement of the vehicle body and tires. When the suspension stroke exceeds a maximum value S_max, it will cause damage to the suspension and reduce the ride comfort. Therefore, the suspension stroke should be constrained as

\frac{| x_{s} - x_{u} |}{S_{\max}} \leq 1

(4)

Road-holding performance affects handling capacity of vehicle, the ground cannot provide sufficient tire force when the dynamic tire load k_u(x_u − x_r) is greater than the static tire load (m_s + m_u)g. So the ratio of dynamic tire load and static tire load (named as tire dynamic-to-static load ratio) should satisfy

\frac{k_{u} (x_{u} - x_{r})}{(m_{s} + m_{u}) g} \leq 1

(5)

In addition, the active force u_z should satisfy the actuator saturation constraint

\frac{| u_{z} |}{u_{z \max}} \leq 1

(6)

where u_z _max is the maximum value of

| u_{z} |

2.3. Control objective

Consider a system described by the state space equations

{\begin{cases} \dot{x} (t) = A x (t) + B_{1} ω (t) + B_{2} u (t), \\ z_{1} (t) = C_{1} x (t) + D_{11} ω (t) + D_{12} u (t), \\ z_{2} (t) = C_{2} x (t) + D_{21} ω (t) + D_{22} u (t), \\ y (t) = C_{3} x (t), \end{cases}

(7)

where

x \in R^{n_{x}}

is the state vector,

ω \in R^{n_{ω}}

is the disturbance input vector,

u \in R^{n_{u}}

denotes the control input,

z_{1} \in R^{n_{z_{1}}}

and

z_{2} \in R^{n_{z_{2}}}

are the vectors of performance output and constrained output, respectively. The measured output vector is

y \in R^{n_{y}}

and the measured output matrix

C_{3} \in R^{n_{y} \times n_{x}}

is with full rank of rows.

For the active suspension system, the state vector, road disturbance input, and control input in the system (7) are

x = {[x_{s} - x_{u} {\dot{x}}_{s} x_{u} - x_{r} {\dot{x}}_{u}]}^{T},

(8)

ω = {\dot{x}}_{r},

(9)

u = u_{z}

(10)

The performance output is defined by the vertical acceleration ${\ddot{x}}_{s}$ , by which the ride comfort performance is evaluated, that is

z_{1} = {\ddot{x}}_{s}

(11)

The performance requirements of (4)–(6) do not need to be minimized and should be designed as hard constraints. Thus, the constrained output of the active suspension system consists of the normalized suspension stroke, the road-holding, and the actuator saturation constraints, that is

z_{2} = {[\begin{array}{l} \frac{x_{s} - x_{u}}{S_{\max}} & \frac{k_{u} (x_{u} - x_{r})}{(m_{s} + m_{u}) g} & \frac{u_{z}}{u_{z \max}} \end{array}]}^{T}

(12)

Since the state variables of the active suspension system cannot be fully measured in practice, the suspension stroke and the sprung mass velocity are taken as the feedback quantities. The suspension stroke can be directly measured by displacement sensors, and the sprung mass velocity can be obtained by the body acceleration sensors (Du and Zhang, 2008). Then the measurement output is

y = {[x_{s} - x_{u} {\dot{x}}_{s}]}^{T}

(13)

The matrices in active suspension system (7) are

\begin{array}{l} A & = [\begin{array}{l} 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}}{m_{u}} \end{array}], \end{array}

\begin{array}{l} B_{1} & = {[\begin{array}{l} 0 & 0 & - 1 & 0 \end{array}]}^{T}, \end{array}

\begin{array}{l} B_{2} & = {[\begin{array}{l} 0 & \frac{1}{m_{s}} & 0 & - \frac{1}{m_{u}} \end{array}]}^{T}, \end{array}

\begin{array}{l} C_{1} & = [\begin{array}{l} - \frac{k_{s}}{m_{s}} & - \frac{c_{s}}{m_{s}} & 0 & \frac{c_{s}}{m_{s}} \end{array}], \end{array}

\begin{array}{l} C_{2} & = [\begin{array}{l} \frac{1}{S_{\max}} & 0 & 0 & 0 \\ 0 & 0 & \frac{k_{u}}{(m_{s} + m_{u}) g} & 0 \\ 0 & 0 & 0 & 0 \end{array}], \end{array}

\begin{array}{l} C_{3} & = [\begin{array}{l} \begin{array}{l} 1 & 0 & 0 & 0 \end{array} \\ \begin{array}{l} 0 & 1 & 0 & 0 \end{array} \end{array}], \end{array}

\begin{array}{l} D_{11} & = 0, D_{12} = \frac{1}{m_{s}}, \end{array}

\begin{array}{l} D_{21} & = {[\begin{array}{l} 0 & 0 & 0 \end{array}]}^{T}, D_{22} = {[\begin{array}{l} 0 & 0 & \frac{1}{u_{z \max}} \end{array}]}^{T}, \end{array}

where D₂₁ = 0 is supposed, that is, the disturbance input does not directly affect the constrained output of the system.

The active suspension control can be described as a multi-objective control problem with time-domain hard constraints. And the control objectives include improving ride comfort, that is, minimizing the response from the road disturbance to the vertical acceleration of the vehicle body. Meanwhile, the time-domain hard constraints should be satisfied.

3. H_∞/GH₂ static output feedback control

In this section, H_∞ norm and GH₂ norm are introduced to describe the performance output and the constrained output, respectively. And the H_∞/GH₂ static output feedback controller is designed.

3.1. H_∞ norm and GH₂ norm

Suppose the static output feedback control is

u (t) = F y (t)

(14)

where

F \in R^{n_{u} \times n_{y}}

is the matrix to be determined. Substitute u(t) = Fy(t) into (7), then the closed-loop system is obtained, that is

{\begin{cases} \dot{x} (t) = A_{c l} x (t) + B_{c l} ω (t), \\ z_{1} (t) = C_{1, c l} x (t) + D_{1, c l} ω (t), \\ z_{2} (t) = C_{2, c l} x (t) + D_{2, c l} ω (t), \end{cases}

(15)

where

A_{c l} = A + B_{2} F C_{3},

C_{1, c l} = C_{1} + D_{12} F C_{3},

C_{2, c l} = C_{2} + D_{22} F C_{3},

B_{c l} = B_{1},

D_{1, c l} = D_{11},

D_{2, c l} = D_{21}

Define the H_∞ norm of system (15) from disturbance input ω to performance output z₁ as

{‖ T_{ω z_{1}} ‖}_{\infty} : = \sup_{ω (t) \in L_{2}} \frac{{‖ z_{1} (t) ‖}_{2}}{{‖ ω (t) ‖}_{2}}

(16)

where ω(t) ∈ L₂ indicates that the disturbance input is an energy-bounded signal, that is

{‖ ω (t) ‖}_{2} : = \sqrt{\int_{0}^{\infty} {‖ ω (t) ‖}^{2} d t} < \infty

(17)

The H_∞ norm is the peak value of the maximum singular value of the system frequency response. When the input energy is bounded, the H_∞ norm describes the ratio of the system output signal energy to the input signal energy. The smaller the H_∞ norm is, the less the disturbance input of the system influences the performance output.

Lemma 1

(Boyd et al., 1994; Schereret and Weiland, 2011) For system (15), given a real number γ > 0, then the following conditions are equivalent

1. The system is asymptotically stable and ${‖ T_{ω z_{1}} ‖}_{\infty} < γ$ ;

2. There exists a matrix $P_{1} = {P_{1}}^{T} > 0$ such that

[\begin{array}{l} A_{c l}^{T} P_{1} + P_{1} A_{c l} & # & # \\ B_{c l}^{T} P_{1} & - I & # \\ C_{1, c l} & D_{1, c l} & - γ^{2} I \end{array}] < 0

(18)

Assuming that x(0) = 0, the GH₂ norm of the system from the disturbance input ω to the output z₂ is

{‖ T_{ω z_{2}} ‖}_{g} : = \sup_{ω (t) \in L_{2}} \frac{{‖ z_{2} (t) ‖}_{\infty}}{{‖ ω (t) ‖}_{2}}

(19)

When ω is a unit energy signal, the GH₂ norm represents the peak of the constrained output in the time domain.

Lemma 2

(Boyd et al., 1994; Schereret and Weiland, 2011) Suppose that D₂₁= 0 and x(0) = 0, for system (15),the following conditions are equivalent

1. The system is asymptotically stable and ${‖ T_{ω z_{2}} ‖}_{g} < 1$ ;

2. There exists a matrix $P_{2} = {P_{2}}^{T} > 0$ such that

[\begin{array}{l} A_{c l}^{T} P_{2} + P_{2} A_{c l} & # \\ B_{c l}^{T} P_{2} & - I \end{array}] < 0

(20a)

[\begin{array}{l} P_{2} & # \\ C_{2, c l} & I \end{array}] > 0

(20b)

3.2. Static output feedback controller design

H_∞/GH₂ static output feedback controller is to design the static output feedback gain F such that

1. The closed-loop system (15) is internally stable;

2. When the disturbance input is an energy-bounded signal, the H_∞ norm ${‖ T_{ω z_{1}} ‖}_{\infty}$ from the disturbance input ω(t) to the performance output z₁(t) is minimum and the GH₂ norm from ω(t) to the constrained output z₂(t) satisfies ${‖ T_{ω z_{2}} ‖}_{g} < 1$ .

In order to transform the control problem into a convex optimization problem that can be solved by LMIs, the following lemma is first introduced.

Lemma 3

Given a positive definite symmetric matrix $S \in R^{n \times n}$ and a full row rank matrix $N \in R^{m \times n}$ with m ≤ n, the matrix NSN^T is invertible.

Proof

Let the vector be $X \in R^{m}$ such that NSN^TX = 0, then X^TNSN^TX = 0, that is, ${(N^{T} X)}^{T} S (N^{T} X) = 0$ . Furthermore, N^TX = 0 since S is a positive definite symmetric matrix. Multiplying N left on both sides of the equation, NN^TX = 0 is obtained. As NN^T is invertible, then X = 0. In summary, NSN^TX = 0 has only zero solutions, that is, NSN^T is invertible.

The theorem for solving the H_∞/GH₂ static output feedback gain F is given below.

Theorem 1

Suppose there exist matrix $U \in R^{n_{u} \times n_{y}}$ , positive definite symmetries matrix $Q \in R^{n_{x} \times n_{x}}$ , and the scalar γ > 0 such that the following LMIs optimization problem

\underset{γ, Q, U}{minimize} γ^{2},

(21a)

subject to

[\begin{array}{l} A Q + Q A^{T} + B_{2} U C_{3} + {(B_{2} U C_{3})}^{T} & # & # \\ B_{1}^{T} & - I & # \\ C_{1} Q + D_{12} U C_{3} & D_{11} & - γ^{2} I \end{array}] < 0,

(21b)

[\begin{array}{l} Q & # \\ C_{2} Q + D_{22} U C_{3} & I \end{array}] > 0

(21c)

has the optimal solution (γ*, Q*, U*),

F^{*} = U^{*} {(V^{*})}^{- 1}

, and

V^{*} = (C_{3} Q^{*} C_{3}^{T}) {(C_{3} C_{3}^{T})}^{- 1}

, the closed-loop system (15) with the static output feedback control u = F*y satisfies

1. Internal stability;

2. The H_∞ performance from the road disturbance ω(t) to the performance output z₁(t) is less than γ*, and the GH₂ norm from ω(t) to the output z₂(t) is less than 1.

Proof

According to Lemma 1 and Lemma 2, in a multi-objective control framework, the conditions that A_cl is internally stable, ${‖ T_{ω z_{1}} ‖}_{\infty} < γ$ and ${‖ T_{ω z_{2}} ‖}_{g} < 1$ can be deduced to there exist matrices $P_{1} = {P_{1}}^{T} > 0$ and $P_{2} = {P_{2}}^{T} > 0$ such that

[\begin{array}{l} A_{c l}^{T} P_{1} + P_{1} A_{c l} & # & # \\ B_{c l}^{T} P_{1} & - I & # \\ C_{1, c l} & D_{1, c l} & - γ^{2} I \end{array}] < 0

(22a)

[\begin{array}{l} A_{c l}^{T} P_{2} + P_{2} A_{c l} & # \\ B_{c l}^{T} P_{2} & - I \end{array}] < 0

(22b)

[\begin{array}{l} P_{2} & # \\ C_{2, c l} & I \end{array}] > 0

(22c)

Let the matrix P₁= P₂= P, then substitute A_cl, B_cl, C_1,cl, C_2,cl, D_1,cl, and P into the matrix inequalities (22a) and (22c), there are

[\begin{array}{l} {(A + B_{2} F C_{3})}^{T} P + P (A + B_{2} F C_{3}) & # & # \\ B_{1}^{T} P & - I & # \\ C_{1} + D_{12} F C_{3} & D_{11} & - γ^{2} I \end{array}] < 0,

(23a)

[\begin{array}{l} P & # \\ C_{2} + D_{22} F C_{3} & I \end{array}] > 0

(23b)

Note that the linear matrix inequality (22b) is intrinsic to (22a).

Let Q = P⁻¹, multiply inequality (23a) left and right by diag ${Q, I, I}$ , and multiply inequality (23b) left and right by diag ${Q, I}$ , respectively, to obtain

[\begin{array}{l} A Q + Q A^{T} + B_{2} F C_{3} Q + {(B_{2} F C_{3} Q)}^{T} & # & # \\ B_{1}^{T} & - I & # \\ C_{1} Q + D_{12} F C_{3} Q & D_{11} & - γ^{2} I \end{array}] < 0,

(24a)

[\begin{array}{l} Q & # \\ C_{2} Q + D_{22} F C_{3} Q & I \end{array}] > 0

(24b)

Denote

V = (C_{3} Q C_{3}^{T}) {(C_{3} C_{3}^{T})}^{- 1}

, that is,

V C_{3} C_{3}^{T} = C_{3} Q C_{3}^{T}

, or

(V C_{3} - C_{3} Q) C_{3}^{T} = 0

, where

C_{3}^{T} \in R^{n_{x} \times n_{y}}

and

(V C_{3} - C_{3} Q) \in R^{n_{y} \times n_{x}}

. Since

C_{3} \in R^{n_{y} \times n_{x}}

and n_y ≤ n_x, the maximum value of rank(VC₃ − C₃Q) is n_y. Because (VC₃ − C₃Q)X= 0 has n_y non-zero solutions, there is VC₃ = C₃Q. According to Lemma 3,

C_{3} Q C_{3}^{T}

is invertible, so is V invertible.

Taking VC₃ = C₃Q and F = UV⁻¹ into matrix inequalities (24a) and (24b), the linear matrix inequalities (21b) and (21c) can be obtained. Suppose the optimization problem (21) is solved with an optimal solution (γ*, Q*, U*), then the conclusions (1)–(2) are satisfied in the closed-loop system (15).

Remark

For the external disturbance not exceeding unit energy, that is, ${‖ ω (t) ‖}_{2} \leq 1$ , the hard constraints are satisfied when condition (2) above is satisfied.

4. Simulation and HiL experiments

This section presents comparative experiments of active suspension between the proposed approach and the existing approaches, such as H_∞ state feedback control (Du and Zhang, 2007) and GAs active suspension which utilizes LMIs and GAs (Du and Zhang, 2008). Both simulation and HiL experiments are conducted on active and passive suspensions under different road disturbances and vehicle speeds, illustrating the superiority of the proposed H_∞/GH₂ static output feedback control to achieve better ride comfort with less feedback quantities and without need for a searching algorithm. The parameters are shown in Table 1 (Gordon et al., 1991).

Table 1.

Parameters of suspension and road.

Parameters	Symbol	Simulation	HiL	Unit
Sprung mass	m _s	320	2.45	kg
Unsprung mass	m _u	40	1	kg
Tire stiffness	k _u	200,000	2500	N/m
Spring stiffness	k _s	18,000	900	N/m
Damping coefficient	c _s	1000	7.5	N · s/m
Maximum active force	u _{z max}	1000	25	N
Maximum suspension stroke	S _max	0.08	0.03	m
Bump height	A _m	0.1	5	m
Bump length	L	0.032	1	m

4.1. Road disturbance

The deterministic road is represented by an isolated bump in an otherwise smooth road surface, that is, (Chen and Guo, 2005; Chen et al., 2007)

x_{r} (t) = {\begin{cases} \frac{A_{m}}{2} (1 - \cos \frac{2 π v}{L} t), & 0 \leq t \leq \frac{L}{v}, \\ 0, & t > \frac{L}{v}, \end{cases}

(25)

where v and t are the vehicle velocity and time, respectively, the height and length of the bump are A_m and L, respectively.

The random process with power spectral density (PSD) of Du et al. (2020) is taken for the random road (ISO 8608, 1995)

G_{x r} (n) = G_{x r} (n_{0}) {(\frac{n}{n_{0}})}^{- ω_{0}}

(26)

where G_xr(n) is PSD with the spatial frequency n and G_xr(n₀) is the spectral density with the reference spatial frequency n₀ = 0.1 m⁻¹. The variable ω₀ is frequency index which stands for the frequency structure of PSD. In this paper, the frequency index is chosen as ω₀ = 2, and the ground velocity can be generated by a white-noise signal

{\dot{x}}_{r} (t) = - 2 π n_{1} v x_{r} (t) + 2 π n_{0} \sqrt{G_{x r} (n_{0}) v} ω (t)

(27)

where n₁ = 0.01, and ω(t) is a Gaussian white-noise with a zero mean value and an intensity of 1. According to the grade of road roughness, the geometric mean values of G_xr(n₀) for commonly used B class and C class road are 64 × 10⁻⁶ and 256 × 10⁻⁶, respectively.

4.2. Simulation results

By solving the convex optimization problem (21), the results of proposed H_∞/GH₂ static output feedback controller gains F* and H_∞ performance γ* are obtained as

F^{*} = [- 18944 - 4535], γ^{*} = 0.7152

(28)

For comparison, a H_∞ state feedback control is designed to minimize conflicting performance requirements in a single objective function (Du and Zhang, 2007). Suppose that the system state variables are measurable in simulation, the controlled output z_z(t) is composed of ${\ddot{x}}_{s}$ , x_s − x_u, x_u − x_r and u_z. The control system is

{\begin{cases} \dot{x} (t) & = A x (t) + B_{1} ω (t) + B_{2} u (t), \\ z_{z} (t) & = C_{z 1} x (t) + D_{z 1} ω (t) + D_{z 2} u (t), \\ y_{z} (t) & = C_{z 2} x (t), \end{cases}

(29)

where x(t), ω(t), A, B₁, and B₂ are defined as system (7), u(t) is the control input, y_z(t) is the measured output, with

C_{z 1} = [\begin{array}{l} - \frac{k_{s}}{m_{s}} & - \frac{c_{s}}{m_{s}} & 0 & \frac{c_{s}}{m_{s}} \\ κ_{1} & 0 & 0 & 0 \\ 0 & 0 & κ_{2} & 0 \\ 0 & 0 & 0 & 0 \end{array}], C_{z 2} = I_{4 \times 4}

D_{z 1} = [\begin{array}{l} 0 \\ 0 \\ 0 \\ 0 \end{array}], D_{z 2} = [\begin{array}{l} \frac{1}{m_{s}} \\ 0 \\ 0 \\ κ_{3} \end{array}],

where κ₁ = 5/S_max, κ₂ = k_u/((m_s + m_u)g), and κ₃ = 8/u_max are weightings for the performance of suspension stroke, road-holding, and actuator saturation, respectively. The design of H_∞ state feedback control is to find a state feedback control u_k(t) = K_zx(t) such that the system is asymptotically stable and the H_∞ norm of system (29) from disturbance input ω to controlled output z_z is minimized.

The simulations are performed with both isolated bump excitation and random road disturbance input. Due to the page limit, partial results are presented in Figures 2 and 3, where the solid red lines indicate the response curves of the active suspension with proposed H_∞/GH₂ static output feedback controller (named as H_∞/GH₂ active suspension), the dotted green lines and dashed dot blue lines indicate H_∞ state feedback control (named as H_∞ active suspension) and GAs active suspension, respectively, the dashed black lines represent the response curves of passive suspension, and the dotted black lines indicate the maximum and minimum values of suspension stroke and active force.

The simulation results of isolated bump excitation with v = 50 km/h are shown in Figure 2, the sampling time is 0.02 s. As shown in Figure 2(a), the amplitude of ${\ddot{x}}_{s}$ for the H_∞/GH₂ active suspension is the smallest among suspensions, and the convergence time is shorter than that of the passive suspension, indicating that the proposed controller is effective in improving ride comfort. Moreover, Figure 2(b) shows that the passive suspension stroke exceed S_max, which implies mechanical damage in the actual system. Compared with the H_∞ state feedback control which requires the system state to be fully measurable and the active suspension controller which uses the searching algorithm such as GAs, the proposed H_∞/GH₂ active suspension can perform superior ride comfort.

The simulation results on B class random roads with v = 100 km/h are presented in Figure 3, all the active suspensions are optimized to minimize ${\ddot{x}}_{s}$ while satisfying time-domain hard constraints compared to the passive suspension.

To quantitatively evaluate the effectiveness of the proposed approach in improving ride comfort, RMS values of

{\ddot{x}}_{s}

are shown in Table 2, and the percentages of RMS values reduced by H_∞/GH₂ control compared to other control methods are shown in Table 3. In addition, to evaluate the amplitude of

{\ddot{x}}_{s}

on bump roads, peak-to-peak (P2P) values are calculated and shown in Table 4.

Table 2.

RMS values of vehicle body vertical acceleration (m/s²) in simulation.

Suspension	Bump	Bump	B class	C class
Suspension	50(km/h)	30(km/h)	100(km/h)	45(km/h)
H_∞/GH₂	1.6806	1.3132	1.3361	1.1184
H _∞	1.8278	1.5560	1.4716	1.2409
GAs	1.7610	1.5594	1.5530	1.2960
Passive	2.2517	2.5147	1.6398	1.3785

Table 3.

Reduction of RMS values by H_∞/GH₂ active suspension in simulation.

Suspension	Bump	Bump	B class	C class
Suspension	50(km/h)	30(km/h)	100(km/h)	45(km/h)
H _∞	8.1%	15.6%	9.2%	9.9%
GAs	4.6%	15.8%	14.0%	13.7%
Passive	25.4%	47.8%	18.5%	18.9%

Table 4.

P2P values of vehicle body vertical acceleration (m/s²) on bump roads.

Suspension	v = 50(km/h)	v = 30(km/h)
H_∞/GH₂	10.40	5.00
H _∞	10.92	7.05
GAs	10.81	7.18
Passive	11.09	10.18

As can be seen in Tables 2 –4, the proposed H_∞/GH₂ active suspension exhibits the most optimal ride comfort performance. Compared to H_∞ state feedback control, H_∞/GH₂ static output feedback control is able to reduce RMS values by at least 8.1% and satisfy all time-domain hard constraints. This is because the H_∞ state feedback control optimizes conflicting performance requirements in a single objective function, making the control problem conservative to a certain extent.

4.3. HiL experiments

In this subsection, comparative HiL experiments are conducted to further demonstrate the practical effectiveness of the proposed approach.

HiL experiments are conducted based on the active suspension test platform developed by Quanser^® company in Canada (Apkarian and Abdossalami, 2013), as shown in Figure 4.

The test platform mainly includes the active suspension system, displacement and acceleration sensors, a power amplifier, a data acquisition (DAQ) device, and a computer. The output signals suspension stroke and sprung mass velocity are obtained by a US Digital S1 single-ended optical shaft encoder and a dual-axis ADXL210 E accelerometer, respectively, with the sampling frequency 50 Hz. And the active force is generated by a high quality DC motor. The computer conducts controller design and data processing through Matlab/Simulink and the software QUARC^®. The control signals are sent to the active suspension system through the DAQ device and the power amplifier, while the feedback signals are received to complete the closed-loop control.

The controller gains and H_∞ performance are

F^{*} = [- 561.59 - 53.03], γ^{*} = 10.07

(30)

Due to the page limit, partial results with isolated bump road and random roads are presented in Figures 5 and 6, the legends are consistent with that in Figure 2.

The simulation results of bump road with v = 11 km/h are shown in Figure 5. As shown in Figure 5(a), the vertical acceleration amplitude and the convergence time of the proposed H_∞/GH₂ active suspension are smallest among three suspensions. Moreover, the dynamic-to-static load ratio of passive suspension exceeds the constraint (c.f. Figure 5(c)). On the contrary, the H_∞/GH₂ active suspension satisfies all time-domain constraints (c.f. Figures 5(b)–(d)).

Additionally, HiL experiments with B class and C class roads are conducted with v = 120 km/h and v = 90 km/h, respectively. The results of the C class road are presented in Figure 6, where the tire dynamic-to-static load ratio of passive suspension (dotted black lines) exceeds the constraint, while the H_∞/GH₂ active suspension (solid red lines) can reduce the vertical acceleration of the vehicle body and meet the constraints.

Remark

The control gains of GAs active suspension are largely influenced by the search space and iteration numbers of search algorithm GAs. Since the suspension parameters are different in simulation and HiL experiments, the search space and the solutions are different. Thus the deviations between the active forces in Figure 2(d) and Figure 5(d) are different.

The RMS values of

{\ddot{x}}_{s}

are shown in Table 5, and the RMS values reduction percentage of H_∞/GH₂ active suspension compared to those of GAs active suspension and passive suspension are shown in Table 6. Moreover, P2P values of bump roads are shown in Table 7. As shown in tables, the proposed approach can greatly reduce RMS and P2P values of

{\ddot{x}}_{s}

and improve ride comfort.

Table 5.

RMS values of vehicle body vertical acceleration (m/s²) in HiL experiments.

Suspension	Bump	Bump	B class	C class
Suspension	10(km/h)	11(km/h)	120(km/h)	90(km/h)
H_∞/GH₂	1.2618	1.5413	0.9912	1.5205
GAs	1.6831	1.7949	1.0961	1.8937
Passive	4.3020	5.8353	2.0772	3.0078

Table 6.

Reduction of RMS values by H_∞/GH₂ active suspension in HiL experiments.

Suspension	Bump	Bump	B class	C class
Suspension	10(km/h)	11(km/h)	120(km/h)	90(km/h)
GAs	25.0%	14.1%	9.6%	19.7%
Passive	70.7%	73.6%	52.3%	49.5%

Table 7.

P2P values of vehicle body vertical acceleration (m/s²) on bump roads.

Suspension	v = 10(km/h)	v = 11(km/h)
H_∞/GH₂	5.68	6.77
GAs	11.82	11.47
Passive	16.65	21.24

The controller gains and H_∞ performance of GAs active suspension are presented in Table 8. Note that the controller gains are obtained within the search space of [−1000, −200] and [−70, −30], which are set based on the existing optimal solution in (30). Careful selection of the search space is a necessary factor to obtain the suboptimal solution. However, the proposed approach does not require extensive searching while ensuring system performance.

Table 8.

Controller gains and H_∞ performance of GAs active suspension in HiL experiments.

Road	Speed(km/h)	F	γ
Bump	10	[−762.6 −68.8]	3.2671
Bump	11	[−261.4 −47.5]	3.2747
B class	120	[−543.8 −43.5]	3.4233
C class	90	[−649.2 −32.9]	3.4530

To further illustrate the impact of different search spaces and randomness of GAs, repeated experiments are conducted on a C class road surface with v = 90 km/h. The results are listed in Table 9, the search space of the first four experiments contains the optimal feedback control gain F*. As can be seen in Table 9, all the RMS values are larger than that obtained by the proposed approach. Moreover, the search space of the last two experiments does not contain F*, resulting in the controller gain is far from the optimal value, and RMS values are large. The average RMS value of the six experiments is 1.8390, which is larger than the RMS value of 1.5205 obtained by the proposed approach.

Table 9.

Results of repeated HiL experiments on C class road surface with v = 90 km/h.

Number	Search space of F₁ and F₂	F₁, F₂	γ	RMS (m/s²)
1	[−1000, −200], [−70, −30]	−976.2, −37.3	3.2839	1.9101
3	[−1000, −200], [−70, −30]	−438.2, −44.3	3.6729	1.6433
4	[−1000, −200], [−120, −20]	−995.5, −52.2	2.7537	1.8818
5	[−800, −200], [−40, −10]	−342.9, −31.2	6.6355	1.6741
6	[−1000, −600], [−40, −10]	−883.4, −29.2	10.8530	2.1626

5. Conclusion

In this paper, the multi-objective control problem with time-domain hard constraints for active suspension was transformed into a convex optimization problem that can be solved using LMIs. The H_∞ norm and the GH₂ norm were used to describe the optimization index and the hard constraints of the active suspension system, respectively. A static output feedback control was designed using measurable suspension stroke and sprung mass velocity as the feedback quantities. Comparative experiments were conducted with both determined and random road disturbances by simulation and HiL experiments. The experiment results showed the superiority of the proposed approach over the H_∞ state feedback control approach and the approach that relies on LMIs and searching algorithm in terms of enhancing ride comfort. Furthermore, the proposed approach can effectively meet all time-domain hard constraints.

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 Natural Science Foundation (NNSF) of China under Grant (No. U1964202), the Natural Science Foundation of Jilin Province (No. YDZJ202101ZYTS169), and the Foundation of Key Laboratory of Industrial Internet of Things and Networked Control (No. 2019FF01).

ORCID iDs

Zhuqing Shi

Shishuo Zhang

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Active suspension H ∞ /generalized H 2 static output feedback control

Abstract

Keywords

1. Introduction

2. Problem setup

2.1. Suspension system modeling

2.2. Performance requirements

2.3. Control objective

3. H ∞ /GH2 static output feedback control

3.1. H ∞ norm and GH2 norm

3.2. Static output feedback controller design

4. Simulation and HiL experiments

4.1. Road disturbance

4.2. Simulation results

4.3. HiL experiments

5. Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

References

3. H_∞/GH₂ static output feedback control

3.1. H_∞ norm and GH₂ norm