Development and implementation of an advanced robust control strategy for quarter-car active suspension systems

Abstract

This study investigates the development of an advanced robust control strategy for a quarter-car active suspension system to optimize ride comfort and road handling. Traditional PID controllers struggle with trade-offs between comfort and stability under real-world conditions like variable road profiles and parametric uncertainties. To address this, we propose a novel H∞ robust control integrated with μ-synthesis, explicitly handling uncertainties in suspension parameters (sprung/unsprung mass, stiffness, damping, tire stiffness). Mathematical modeling and MATLAB/Simulink simulations demonstrate significant improvements: a 54% reduction in peak vertical body acceleration (3.95 m/s² to 1.816 m/s²) and a 35% decrease in suspension deflection (0.078 m to 0.020 m) compared to passive systems. Frequency-domain analysis shows a 92.42% reduction in resonance peaks at 10 rad/s, with over 30% energy savings. Time-domain simulations confirm stability under transient disturbances, with actuator forces constrained to ±1.3 kN. The integration of μ-synthesis with H∞ control efficiently manages parametric variations and unmodeled dynamics. Comparative evaluations highlight the approach’s superiority over passive and conventional active systems, offering promising applications for autonomous and electric vehicles. This work lays the groundwork for future research on adaptive, energy-efficient suspension systems.

Keywords

active suspension systems H∞ control robust control μ-synthesis

Introduction

Active suspension systems have become a vital advancement in automotive engineering, offering the ability to enhance both ride comfort and handling precision through real-time, force-based intervention.^1,2 Unlike passive and semi-active systems, which either operate with fixed parameters or make conditional adjustments to damping characteristics, active systems utilize actuators to apply external forces directly to the suspension.³ This dynamic response capability allows the system to adapt instantaneously to road irregularities, vehicle loading conditions, and driver maneuvers, ensuring that tire contact with the road is maintained while minimizing body motion. Such systems are essential for meeting modern performance expectations in terms of safety, comfort, and vehicle stability.^4,5 As vehicles are increasingly subjected to diversified operating conditions ranging from urban congestion to high-speed highway travel, the demand for intelligent suspension solutions continues to grow.^6,7

One of the key challenges in suspension systems is balancing ride comfort with handling. To achieve better comfort, a suspension must be flexible enough to absorb road bumps and lessen vibrations sent to the vehicle body. Conversely, precise handling demands a stiffer suspension to reduce body roll, pitch, and yaw during actions like braking, turning, and acceleration. This challenge, well known in the field, has led to many control strategies. Traditional controllers, such as Proportional-Integral-Derivative (PID) controllers, are still popular because they are simple and easy to tune, but they struggle with complex, changing road and vehicle conditions.^7,8 Advanced control methods, including Linear Quadratic Regulators (LQR), fuzzy logic controllers, and sliding mode control, have shown improvements in certain situations.^9–11 However, these approaches still depend on ideal assumptions or need frequent adjustments when dealing with high variability and external disturbances.

To overcome these limitations, robust control strategies have gained increasing attention, providing theoretical guarantees for performance and stability even when facing model uncertainties and external disturbances.¹² H∞ control, in particular, minimizes the worst-case gain from disturbances to output variables, ensuring that external influences such as road input or sensor noise have minimal effects on system behavior.^13,14 This approach allows a systematic trade-off between ride comfort and handling objectives by embedding performance weighting functions directly into the controller design. μ-synthesis expands this framework further by considering both structured and unstructured uncertainties, making it especially useful in systems where parameters like mass, damping, or tire stiffness may change with time or operating conditions.^15,16 As a result, robust control techniques are well-suited for active suspension systems, which often encounter uncertain and variable conditions caused by changing payloads, rough terrain, and driver-specific behaviors.¹⁷

Recent studies have demonstrated the effectiveness of robust control approaches for suspension systems, but they also expose critical gaps that remain unaddressed. Yeneneh et al. (2024)¹⁸ applied H∞ control to a quarter-car model and reported notable improvements in ride quality under road disturbances. Yeneneh et al. (2025)¹⁹ extended this to a full-vehicle model, showing enhanced lateral and vertical stability. However, both studies assumed constant vehicle parameters and did not address real-time adaptivity. Ram et al. (2025)²⁰ used μ-synthesis to improve robustness under structured uncertainties but noted limitations in dealing with nonlinear and time-varying effects. Additionally, most prior works do not perform comprehensive benchmarking of these robust methods against classical controllers or evaluate them under practical, compound disturbances that reflect real-world driving scenarios, such as simultaneous variations in speed, road profile, and payload. Review articles by Ovalle et al. (2022)²¹ and Alqudsi et al. (2024)² have also highlighted that many robust control implementations suffer from high computational cost and a lack of integration with other vehicular subsystems, making them less feasible for deployment in embedded automotive systems.

In light of these observations, the present study aims to fill a specific and relevant gap in the literature by proposing an integrated robust control strategy that combines the benefits of H∞ and μ-synthesis while addressing their shortcomings. The novelty of this work lies not in the use of these robust methods in isolation, which has been done before, but in the synthesis of their strengths within a unified control framework that is designed to be computationally efficient and adaptable to real-time uncertainties. The controller is developed for a quarter-car active suspension system, and its performance is rigorously validated through simulations using MATLAB/Simulink under a diverse set of operating conditions. These include scenarios such as variable speed driving, crosswind disturbances, changes in vehicle mass, and road surface irregularities. To provide a meaningful evaluation, the proposed control scheme is benchmarked against PID, LQR, and standalone robust methods, ensuring a transparent comparison of trade-offs in ride comfort, handling, and energy efficiency.

Moreover, this study contributes to the ongoing discourse on sustainable and intelligent automotive design. By reducing unnecessary actuator activity and avoiding overcompensation, the proposed controller not only enhances dynamic performance but also improves energy efficiency, an essential consideration for electric and hybrid vehicles. Reduced energy consumption directly translates to extended vehicle range and lower mechanical wear, thus contributing to the longevity and reliability of suspension components. Furthermore, the modularity of the proposed framework supports its integration with other control systems, such as braking and steering, offering opportunities for future development of coordinated chassis control in semi-autonomous and fully autonomous vehicles.

In summary, this paper addresses the absence of a unified, computationally feasible, and practically adaptable robust control framework for active suspension systems. By integrating H∞ and μ-synthesis methods and validating them through extensive simulations under realistic disturbance conditions, the proposed approach marks a significant step toward deploying robust, intelligent, and efficient suspension systems in next-generation vehicles. The following sections detail the dynamic modeling, control design, simulation methodology, results, and concluding remarks, which collectively highlight the contribution and practical relevance of the work.

Materials and methods

Materials

This study utilized a specific vehicle model to accurately represent the dynamic characteristics of mid-size sedans. The parameters of the quarter-car vehicle model as shown in Table 1.

Table 1.

The quarter-car model parameters.

Parameters	Description	Value
M_b	Sprung mass	600 kg
M_w	Unsprung mass	60 kg
b	Tire damping coefficient	20 ns/m
K₁	Suspension sprung stiffens	18,000 N/m
K₂	Tire stiffness	150,000 N/m

This study employed a conventional vehicle equipped with a Quarter-Car Active Suspension System, integrated through software within the simulation environment. The technical specifications and parameters of the vehicle model were input into the MATLAB/Simulink platform to facilitate the development and evaluation of the proposed control strategies.

Dynamics model of active suspension system

The quarter-car model represents vertical dynamics using two degrees of freedom: the sprung mass m_b (vehicle body) and unsprung mass m_w (wheel and suspension). These masses are connected via a suspension spring-damper system, and the wheel is connected to the road via the tire modeled as a spring-damper element. The actuator applies a control force F_a between the two masses.

The governing equations of motion are:

m_{b} {\ddot{x}}_{b} = - k_{2} (x_{b} - x_{w}) - b ({\dot{x}}_{b} - {\dot{x}}_{w}) + F_{a}

(1)

m_{w} {\ddot{x}}_{w} = k_{2} (x_{b} - x_{w}) + b ({\dot{x}}_{b} - {\dot{x}}_{w}) - k_{1} (x_{w} - x_{r}) - c_{t} ({\dot{x}}_{w} - {\dot{x}}_{r}) - F_{a}

(2)

Active suspension systems utilize fully controlled actuators to dynamically regulate suspension forces, enhancing ride comfort and handling capabilities beyond those of passive or semi-active systems. This study applies H∞ control to a quarter-car model (Figure 1) comprising a sprung mass (m_b), unsprung mass (m_w), actuator (F_a), suspension spring (k₂), damper (b), and tire stiffness (k₁). Vertical displacements (x_b) and road input (x_r) define the state variables.

Figure 1.

Quarter-car model with active suspension system.

The dynamics, derived via Newton’s second law and illustrated in the free-body diagram (Figure 2), are formulated in state-space form to enable robust control synthesis. The actuator force is regulated through feedback to ensure consistent performance under varying road profiles and parameter uncertainties, facilitating the integration of H∞ and μ-synthesis control strategies.

Figure 2.

Free body diagram of the mass of the body and the mass of the wheel.

Dynamics model of an active suspension system

The active suspension system dynamics are formulated in a state-space format by incorporating the displacements and velocities of both the sprung and unsprung masses into a state vector. This streamlined mathematical representation effectively captures the relationships between the system’s internal states, external inputs, and outputs. The state-space approach simplifies the analysis and facilitates the design of controllers to enhance ride comfort and handling. The model was developed by combining the system’s force balance and motion equation

\dot{X} = A x + B u y = C x + D u

(3)

Consider the following $x_{b} = x_{2}$ , ${\dot{x}}_{b} = {\dot{x}}_{2}$ , $x_{w} = x_{1}$ , ${\dot{x}}_{w} = {\dot{x}}_{1}$ and ${\dot{x}}_{b} = {\dot{x}}_{2}$ , ${\ddot{x}}_{b} = {\ddot{x}}_{2}$ ,

{\dot{x}}_{w} = {\dot{x}}_{1} and {\ddot{x}}_{w} = {\ddot{x}}_{1}

[\begin{array}{l} {\dot{x}}_{2} \\ {\ddot{x}}_{2} \\ {\dot{x}}_{1} \\ {\ddot{x}}_{1} \end{array}] = [\begin{array}{c} 0 & 1 & 0 & 0 \\ - \frac{K_{2}}{m_{b}} & - \frac{b}{m_{b}} & \frac{k_{2}}{m_{b}} & \frac{b}{m_{b}} \\ 0 & 0 & 0 & 1 \\ \frac{K_{2}}{m_{w}} & \frac{b}{m_{w}} & - \frac{K_{2} + K_{1}}{m_{w}} & - \frac{b}{m_{w}} \end{array}] [\begin{array}{l} x_{2} \\ {\dot{x}}_{2} \\ x_{1} \\ {\dot{x}}_{1} \end{array}] + [\begin{array}{c} 0 \\ \frac{1}{m_{b}} \\ 0 \\ - \frac{1}{m_{w}} \end{array}] F_{a} + [\begin{array}{c} 0 \\ 0 \\ 0 \\ \frac{K_{1}}{m_{w}} \end{array}] x_{0}

(4)

By considering $F_{a}$ and $X_{0}$ as input of the system, we combine $F_{a}$ and $X_{0}$ in matrix form.

\begin{array}{l} u = [\begin{array}{l} x_{0} \\ F_{a} \end{array}] \\ [\begin{array}{l} {\dot{x}}_{2} \\ {\ddot{x}}_{2} \\ {\dot{x}}_{1} \\ {\ddot{x}}_{1} \end{array}] = [\begin{array}{l} 0 & 1 & 0 & 0 \\ - \frac{K_{2}}{m_{b}} & - \frac{b}{m_{b}} & \frac{K_{2}}{m_{b}} & \frac{b}{m_{b}} \\ 0 & 0 & 0 & 1 \\ \frac{K_{2}}{m_{w}} & \frac{b}{m_{w}} & - \frac{K_{2} + K_{1}}{m_{w}} & - \frac{b}{m_{w}} \end{array}] [\begin{array}{l} x_{2} \\ {\dot{x}}_{2} \\ x_{1} \\ {\dot{x}}_{1} \end{array}] [\begin{array}{c} 0 & 0 \\ \frac{1}{m_{b}} & 0 \\ 0 & 0 \\ - \frac{1}{m_{w}} & \frac{K_{1}}{m_{w}} \end{array}] [u] \end{array}

(5)

{y = [\begin{array}{l} 1 & 0 & 0 & 0 \end{array}] [\begin{array}{l} x_{1} & {\dot{x}}_{1} & x_{2} & {\dot{x}}_{2} \end{array}]}^{T}

(6)

Identification of uncertainties and disturbances

In the design and control of an active suspension system, system uncertainties and external disturbances play a critical role in influencing performance and stability. Variations in parameters such as suspension stiffness, damping coefficient, and vehicle mass arise from factors like material degradation, manufacturing tolerances, and operational conditions. To ensure reliable performance, these variations are systematically modeled using structured uncertainties, representing bounded deviations from nominal values. By incorporating these uncertainties into a block diagram framework, the control strategy is designed to maintain robustness and effectiveness under real-world conditions.

k_{s} \to k_{s} + Δ k_{s}, c_{s} \to c_{s} + Δ c_{s}

(7)

Bounded uncertainties are used to capture unknown deviations from the nominal stiffness and damping values, enabling the control system to tolerate parameter fluctuations and maintain consistent performance across different operating conditions. Beyond parameter uncertainties, road disturbances also significantly affect active suspension behavior. These disturbances, caused by irregularities like potholes, speed bumps, and uneven surfaces, can degrade ride quality and vehicle stability if not properly addressed. To realistically represent these effects, road disturbances are modeled using a combination of periodic and random components, reflecting the variability encountered in real-world driving environments.

z r (t) = A \sin (ω t) + random components

(8)

The amplitude and frequency of road disturbances characterize the severity and periodic nature of surface irregularities, while additional random components account for unpredictable events like sudden bumps and potholes, creating a more realistic disturbance model. To effectively manage both road disturbances and parameter uncertainties, a robust control strategy is necessary. Traditional methods, such as PID control, often struggle with varying system dynamics and unpredictable conditions. In contrast, robust control techniques like H∞ control are specifically designed to maintain optimal performance even in the presence of uncertainties. By minimizing the worst-case effects of disturbances and variations, H∞ control significantly improves ride comfort, handling, and overall system reliability. Incorporating structured uncertainties and external disturbances into the design ensures that modern active suspension systems can deliver enhanced vehicle dynamics, greater safety, and consistent performance across diverse driving scenarios.

Robust control system

The first step in control system design is developing a mathematical model of the plant. However, in practice, every model contains some level of error, meaning the actual plant will always differ from the model. To ensure the controller performs reliably on the real system, it is important to account for these uncertainties from the outset. By incorporating potential errors into the design process, a robust control system can be developed, capable of maintaining effective performance despite differences between the model and the actual plant.

Suppose that the actual plant we want to control is $\tilde{G} (s)$ and the mathematical model of the actual plant is $G (s)$ that is, $\tilde{G} (s)$ = actual plant model that has uncertainty $Δ (s)$ . $G (s)$ = Nominal plant model to be used for designing the control system. $\tilde{G} (s)$ and $G (s)$ may be related by a multiplicative factor such as

\tilde{G} (s) = G (s) [1 + Δ (s)]

(9)

\tilde{G} (s) = G (s) [1 + Δ (s)]

(10)

or an additive factor

\tilde{G} (s) = G (s) + Δ (s)

(11)

or in the other forms.

Since the exact description of the uncertainty or error $Δ (s)$ is unknown, used an estimate of $Δ (s)$ and use this estimate, $W (s)$ , in the design of the controller. $W (s)$ is a scalar transfer function such that

{‖ Δ (s) ‖}_{\infty} < {‖ W (s) ‖}_{\infty} = \max_{0 \leq ω \leq \infty} | W (j ω) |

(12)

where

{‖ W (s) ‖}_{\infty}

is the maximum value of

| W (j ω) |

for

0 \leq ω \leq \infty

and it is called the H infinity norm of

W (s)

Using the small gain theorem, the design procedure here boils down to the determination of the controller $K (s)$ such that the inequality

{‖ \frac{W (s)}{1 + K (s) G (s)} ‖}_{\infty} < 1

(13)

is satisfied, where

G (s)

is the transfer function of the model used in the design process,

K (s)

is the transfer function of the controller, and

W (s)

is the chosen transfer function to approximate

Δ (s)

. In most practical cases, we must satisfy more than one such inequality that involves

G (s)

K (s)

, and

W (s)

’s. For example, to guarantee robust stability and robust performance we may require two inequalities, such as

{‖ \frac{W_{m} (s) K (s) G (s)}{1 + K (s) G (s)} ‖}_{\infty} < 1 for robust stability .

{‖ \frac{W_{s} (s)}{1 + K (s) G (s)} ‖}_{\infty} < 1 for robust performance .

be satisfied. Many different such inequalities need to be satisfied in many different robust control systems. Robust stability means that the controller

K (s)

guarantees internal stability of all systems that belong to a group of systems that include the system with the actual plant. Robust performance means the specified performance is satisfied in all systems that belong to the group.

Design of robust H∞ control for active suspensions

The design of a robust control strategy for a quarter-car active suspension system starts with a mathematical model based on suspension dynamics. While the full model is complex and nonlinear, it is often simplified using a linear time-invariant (LTI) model to ease control design, though some modeling error remains. To address this, robust control strategies, like H∞ control, incorporate uncertainties and external disturbances into the design, ensuring stability and performance under varying conditions.

H∞ control aims to achieve robust stability, ensuring the system remains stable despite parameter variations, and robust performance, ensuring the system meets performance objectives like minimizing body acceleration and maintaining road contact. It uses both frequency and time-domain analyses to compensate for errors and disturbances. Actuator dynamics, which are critical in active suspension systems, are typically modeled with a first-order transfer function As indicated in the following equation.

G a (s) = \frac{1}{\frac{(1 + s)}{60}}

In active suspension systems, actuator dynamics are critical, with practical constraints like limiting actuator displacement to 6 cm to prevent saturation. These limitations create trade-offs between ride comfort, handling, and actuator performance, making careful control design essential. H∞ control offers a robust approach that accounts for model uncertainties, actuator dynamics, and physical limitations, ensuring improved ride comfort, handling, and stability across various conditions.

Uncertainty in these systems can be structured (variations in parameters like stiffness or damping) or unstructured (due to neglected high-frequency dynamics or nonlinear behaviors). This research focuses on unstructured uncertainty, modeled with a single uncertainty element, and addresses it through robust control theory. Using the small gain theorem, the system’s stability is analyzed, ensuring robustness despite variations. The H∞ norm measures the worst-case amplification of disturbances, helping design controllers that guarantee system robustness under uncertainty. As shown in the following equation.

{‖ Φ ‖}_{\infty} = \bar{σ} [Φ (j ω)]

In Figure 3, $Δ (s)$ and $M (s)$ are stable and proper transfer functions.

Figure 3.

Closed model of the plant.

Theorem

The small-gain theorem states that if

{‖ Δ (s) M (s) ‖}_{\infty} \leq 1

(14)

Then, this closed-loop system is stable. That is, if the H∞ norm of $Δ (s)$ $M (s)$ is smaller than 1, this closed-loop system is stable. This theorem is an extension of the Nyquist stability criterion.

To assess robust stability, unstructured uncertainty errors in some instances, are regarded as multiplicative.

\tilde{G} = G (1 + Δ_{M})

(15)

In this model, $\tilde{G} (s)$ represents the true plant dynamics, while $G$ denotes the model plant dynamics as shown Figure 4.

{(1 + K G)}^{- 1} K G = T

(16)

Applying the small-gain theorem to the system, we obtain the condition for stability to be

{‖ Δ_{m} T ‖}_{\infty} < 1

(17)

where (

Δ_{m} = W_{m}

); therefore

{‖ W_{m} T ‖}_{\infty} < 1

(18)

By equating equations (11) and (13) we obtain

{‖ \frac{W_{m} K (s) G (s)}{1 + K (s) G (s)} ‖}_{\infty} < 1

(19)

We design a controller $K (s)$ for active suspension system, represented by a stable plant model $G (s)$ as in Figure 5. A simple solution might be to set the controller to zero ( $K (s) = 0$ ). This would technically satisfy a specific design constraint (inequality, as shown in equation (23). However, a zero controller would not be beneficial. This essentially means that no control action is applied to the suspension. This means that the controlled suspension should effectively minimize the impact of road disturbances (inputs) on the car’s body movement (outputs). We need to find a better transfer function for the controller $K (s)$ to achieve a desirable outcome.

\lim_{t \to \infty} [r (t) - y (t)] = \lim_{t \to \infty} e (t) \to 0

(20)

Therefore, the transfer function Y(s)/R(s) is

\frac{Y s}{R s} = \frac{K G}{1 + K G}

\frac{E (s)}{R (s)} = \frac{R (s) - Y (s)}{R (s)} = 1 - \frac{Y (s)}{R (s)} = \frac{1}{1 + K G}

\frac{1}{1 + K G} = s

The relationship between body position ( $W x b$ ), suspension deflection ( $W s d$ ), and wheel position ( $W x w$ ), $W x w = W x b - W s d$ creates a natural trade-off. Any attempt to reduce body movement at low frequencies (less than 5 rad/s) through control will likely increase suspension deflection.

To achieve robust disturbance attenuation and actuator efficiency, the quarter-car model is reformulated using a generalized plant configuration, as shown in Figure 6. In this structure, the exogenous inputs w = [x_r, n_y]^T include road disturbances and measurement noise. The performance outputs z = [z₁, z₂]^T are composed of body acceleration and suspension deflection. The measured outputs y represent the feedback signals used for control, and the control input is u = Fa. The generalized plant P(s) incorporates weighting functions that shape system performance and enforce physical constraints:

These weights are selected based on ISO 2631-1 comfort guidelines and actuator bandwidth constraints, ensuring a balanced trade-off between competing objectives.

The H∞ control problem is formulated to minimize the worst-case gain from w to z, expressed as equation 21a:

{‖ T_{z w} (s) ‖}_{\infty} < γ

(21a)

where γ is a scalar performance bound. The controller K(s) is synthesized using standard H∞ optimization tools in MATLAB.

Figure 6 Generalized plant configuration for H∞/μ-synthesis. The exogenous inputs www represent road disturbance and sensor noise. The outputs z₁ and z₂ represent performance channels: body acceleration and suspension deflection, respectively. The control input u is the actuator force, and y includes measured displacements used in the feedback loop. Weighting functions Wab(s), Wsd(s), and Wact(s) shape the desired performance and limit control effort as expressed in equation 21b.

[\begin{array}{l} Z (s) \\ y (s) \end{array}] = [\begin{array}{l} p_{11} & p_{12} \\ p_{21} & p_{22} \end{array}] [\begin{array}{l} w_{r} (s) \\ u (s) \end{array}]

(21b)

In an active suspension system, the relationship between the control signal applied $u (s)$ and the system’s output $y (s)$ is defined by the controller’s transfer function $K (s)$ . This means that the control signal is essentially determined by multiplying the system output by $(s)$ .

u (s) = K (s) y (s) Z (s) = Φ (s) w_{r} (s)

(22)

Therefore, by equating the above equation, we obtain

Φ (s) = p_{11} + p_{12} K (s) {[I - p_{22} K (s)]}^{- 1} p_{21}

(23)

Note that if we choose the generalized plant P matrix

p = [\begin{array}{l} 0 & W_{m} G \\ I & - G \end{array}]

(24)

Note that the controlled variable z is related to the external disturbance wr by

p = [\begin{array}{l} W_{r} & - W_{r} G \\ I & - G \end{array}]

(25)

Now by equating equations (27) and (28) we obtain the following:

p = [\begin{array}{l} W_{r} & W_{r} G \\ 0 & W_{m} G \\ I & - G \end{array}]

(26)

The design process employs a mathematical model known as the generalized plant, which incorporates both the system dynamics of the suspension and uncertainties from real-world driving conditions. This approach highlights the use of robust control theory in designing active suspension controllers. By including uncertainties in the generalized plant, the design ensures robust stability and minimizes the impact of disturbances, resulting in a smoother ride. Key concepts of this design approach are illustrated in the relevant figures and equations.

Figure 4.

Block diagram of the generalized plant model with unstructured multiplicative uncertainty.

Figure 5.

Modified material of the plant.

Figure 6.

Block diagram of the generalized Q-car model with unstructured multiplicative uncertainty.

Robust adaptive control design via μ-synthesis for systems with time-varying parameters

To ensure robustness under model uncertainties, μ-synthesis is employed using the D-K iteration method. The structured uncertainty block Δ explicitly captures variations in critical physical parameters, including ±20% in sprung mass (m_b), ±25% in suspension damping (b), and an actuator time delay of up to 0.02 s. These uncertainties are modeled as multiplicative perturbations around the nominal system, allowing the preservation of structured interactions between uncertain elements and performance outputs.

In the generalized plant configuration used for μ-synthesis, the system variables are organized to facilitate robust control design. External disturbances and reference signals are grouped under w, performance outputs such as body acceleration, suspension deflection, and actuator force are represented as z, measured outputs (body and wheel displacements) as y, and control input (actuator force) as u. The structured uncertainty Δ encompasses variations in dynamic parameters that affect system behavior. The μ-synthesis controller is designed to ensure that the structured singular value μΔ(P(s))<1 μ, thereby guaranteeing robust stability and performance across all allowable uncertainties. Once synthesized, the controller is reduced in order to meet real-time computational requirements, enabling its implementation on embedded automotive platforms. To further improve adaptability under time-varying conditions, an adaptive control strategy is incorporated alongside the robust controller. This adaptive mechanism continuously updates key suspension parameters such as stiffness and damping based on real-time estimates of vehicle load and road conditions. The adaptation law is formulated using Lyapunov stability theory, ensuring that the system remains stable by maintaining a negative time derivative of the Lyapunov function.

By combining μ-synthesis-based robust control with adaptive adjustment mechanisms, the proposed strategy effectively addresses both structured uncertainties and dynamic variations in suspension systems. This integrated approach is particularly well-suited for quarter-car active suspension applications, where unpredictable disturbances and load shifts demand both robustness and real-time flexibility.

\overset{⌢}{θ} (t) = - γ x {(t)}^{T} P x (t)

(27)

where

\overset{⌢}{θ} (t)

represents estimated parameters like

K_{s}

C_{S}

To eliminate terms involving $\overset{⌢}{θ}$ choose the adaptive law:

\overset{⌢}{θ} = Γ^{- 1} B_{c l}^{T} P x

(28)

\overset{⌢}{θ} = - Γ^{- 1} B_{c l}^{T} P x

(29)

We assume a quadratic Lyapunov function:

V (x, \tilde{θ}) = X^{T} P x + {\tilde{θ}}^{T} Q \tilde{θ}

(30)

where P is a positive definite matrix chosen to satisfy the Lyapunov stability conditions. Q is another positive definite matrix, typically used in adaptive control to weigh parameter estimation errors. To proceed, the time derivative of V is computed using the chain rule.Where

\frac{\partial v}{\partial x} = 2 x^{T} P, \frac{\partial v}{\partial \tilde{θ}} = 2 {\tilde{θ}}^{T} Q

(31)

We substitute these derivatives: $\dot{V} = 2 x^{T} P \dot{x} + 2 {\tilde{θ}}^{T} Q \dot{\tilde{θ}}$

Assume the system dynamics are given as: $\dot{x} = A_{c l} x + B_{c l} \tilde{θ}$

Now substituting in to $\dot{V}$ :

\dot{V} = 2 x^{T} P (A_{c l} x + B_{c l} \tilde{θ}) + 2 {\tilde{θ}}^{T} Q \dot{\tilde{θ}}

(32)

In many adaptive control derivations, an update law for $\tilde{θ}$ is chosen such that it cancels certain terms in $\dot{V}$ .

A common choice is:

\overset{⌢}{θ} = - Q^{- 1} B_{c l}^{T} P x

Substituting this:

{2 \overset{⌢}{θ}}^{T} Q θ ⌢ = {2 \overset{⌢}{θ}}^{T} Q (- Q^{- 1} B_{c l}^{T} P x)

Since $Q Q^{- 1} = I$ , this is simplified to:

{2 \overset{⌢}{θ}}^{T} ({- B}_{c l}^{T} P x) = - {2 \overset{⌢}{θ}}^{T} B_{c l}^{T} P x

Now substituting this yield equation (33)

\dot{V} = 2 x^{T} P A_{c l} x + 2 x^{T} P B_{c l} \overset{⌢}{θ} - {2 \overset{⌢}{θ}}^{T} B_{c l}^{T} P x

(33)

The cross term cancels due to symmetry:

\dot{V} = 2 x^{T} P A_{c l} x

(34)

To ensure the stability of the adaptive control system under varying parameters, a Lyapunov stability approach is used. This involves defining a Lyapunov function with a non-positive time derivative, ensuring boundedness and convergence of the system states. The state vector and the parameter estimation error are considered, where the estimated parameter differs from the true parameter. A positive definite matrix is used to guarantee that the state-dependent term remains positive, thereby contributing to system stability.

\tilde{θ} = \overset{⌢}{θ} - θ

V (x, \tilde{θ}) = x^{T} P x + {\tilde{θ}}^{T} Γ \tilde{θ}

(35)

Similarly, ${\tilde{θ}}^{T} Γ \tilde{θ}$ is another positive definite matrix ( $Γ > 0$ ), which determines the adaptation rate in parameter estimation by influencing the contribution of ${\tilde{θ}}^{T} Γ \tilde{θ}$ . Together, these terms provide a comprehensive framework for assessing and ensuring the stability and adaptive behavior of control systems.

V (x, \tilde{θ}) > 0 for all x \neq 0 and \tilde{θ} \neq 0

The time derivative of the Lyapunov function is calculated based on the system dynamics. By substituting the system dynamics into the Lyapunov function derivative, the resulting expression is derived. For the Lyapunov function to be non-increasing, certain conditions must be met, which can be ensured by solving the Lyapunov equation. With this condition satisfied, the time derivative of the Lyapunov function becomes as specified.

\dot{V} = \frac{\partial v}{\partial x} \dot{x} + \frac{\partial v}{\partial \tilde{θ}} \dot{\tilde{θ}}

(36)

\frac{\partial v}{\partial x} = 2 x^{T} P, \frac{\partial v}{\partial \tilde{θ}} = 2 {\tilde{θ}}^{T} Γ

Using the system dynamics:

\dot{x} = A_{c l} x + B_{c l} \tilde{θ}

(37)

\dot{V} = 2 x^{T} P (A_{c l} x + B_{c l} \tilde{θ}) + 2 {\tilde{θ}}^{T} Γ \dot{\tilde{θ}}

(38)

For the Lyapunov function to be non-increasing, $\dot{V}$ must satisfy:

x^{T} (P A_{c l} + A_{c l}^{T}) x \leq 0

Choose p such that ( $P A_{c l} + A_{c l}^{T}) x \leq - Q,$ where $Q > 0$ this can be ensured by solving the Lyapunov equation is a positive definite matrix:

{(P A}_{c l} + A_{c l}^{T}) x = - Q

With this condition, $\dot{V}$ becomes:

\dot{V} = - x^{T} Q x \leq 0

The Lyapunov stability approach ensures that the adaptive control system remains stable and converges to the desired behavior under varying conditions, enabling the system to handle uncertainties and maintain robust performance over time. The μ-synthesis methodology is then applied to address structured uncertainties, using the structured singular value (μ) to optimize system performance while considering parametric and dynamic uncertainties. This approach guarantees robust performance and stability even under external disturbances or modeling errors.

In the closed-loop system, structured uncertainties are represented by Δ(s), while M(s) is the nominal transfer function. Uncertainties are modeled as a feedback loop between the nominal transfer function and uncertain elements. The generalized plant framework in robust control design splits system dynamics into a nominal component and a structured uncertainty term, forming the generalized plant P(s). This structure interacts with a robust controller to mitigate disturbances and uncertainties, ensuring effective system output.

G (s) = G_{o} (s) + Δ

(39)

where

G_{O}

(s) is the nominal system and Δ represents structured uncertainties. The generalized plant is defined as:

P (s) = [\begin{array}{l} w \\ Δ \\ u \end{array}] \to [\begin{array}{l} z \\ y \end{array}]

(40)

The generalized plant $P (s)$ is partitioned as:

P (s) = [\begin{array}{l} p_{11} (s) & p_{12} (s) \\ p_{21} (s) & p_{22} (s) \end{array}]

(41)

T_{Δ} (s) = p_{11} + p_{12} k {(I - p_{22} k)}^{- 1} p_{21}

(42)

The primary goal of robust control is to ensure system stability and performance under uncertain conditions by minimizing the structured singular value, μ, which quantifies the system’s sensitivity to uncertainties. Robust performance is achieved when the closed-loop transfer function, including uncertainty (TΔ), satisfies the defined condition.

μ (T Δ (j ω) < 1 \forall ω

(43)

Choose P such that

P A_{c l} + A_{c l}^{T} P \leq - Q

where

Q > 0

is a positive definite matrix.

To solve this problem, the D-K iteration procedure is used, consisting of three steps: scaling, controller synthesis, and iteration.

Scaling Step: Frequency-dependent scaling matrices, D(s), are introduced to decouple uncertainties from performance requirements. The structured singular value μ is then evaluated using the updated scaling matrices to assess the system’s robustness. The robustness against structured uncertainties is quantified by μ, as described by the scaling matrices.

μ (T Δ) = \inf_{D \in D} ‖ D T Δ D^{- 1} ‖

(44)

μ (T Δ) = \frac{1}{\inf_{D \in D} ‖ D^{- 1} T Δ D ‖}

(45)

Controller Synthesis: In this step, the H∞-based control problem is solved to design the controller K(s) for the scaled system. The goal is to ensure stability, minimize disturbances, and meet the required performance criteria.

{‖ D P 12 K {(I - P 22 K)}^{- 1} ‖}_{\infty} < γ

(46)

The D-K iteration process refines the scaling matrices D(s) and the controller K(s) through successive updates. After each update of K(s), the structured singular value μ(TΔ) is recalculated, and the scaling matrices are adjusted. This cycle continues until μ(TΔ) stays below one across all frequencies, ensuring the system can handle uncertainties while maintaining robust performance. The iterative process addresses complex uncertainty dynamics, leading to robust stability and performance. The system achieves robust stability if:

\sup_{ω} μ (T Δ (j ω)) < 1

(47)

and robust performance if the scaled system satisfies:

{‖ D T Δ D^{- 1} ‖}_{\infty} < γ

(48)

The μ-synthesis method ensures robust stability by enforcing the condition

μ (M (j ω)) < 1 \forall ω

The closed-loop transfer function matrix ensures system stability, even with uncertainties. H∞ control minimizes the transfer function’s H∞ norm, bounding the worst-case disturbance-to-output gain. However, H∞ control doesn’t explicitly address model uncertainties. μ-synthesis extends H∞ control by incorporating both structured and unstructured uncertainties, ensuring robust stability and performance. This is done by minimizing the structured singular value, μ, which quantifies system robustness. The iterative D-K procedure refines frequency-dependent scaling matrices D(s) and updates the controller K(s) until convergence, enabling the system to handle uncertainties while maintaining stability and optimal performance.

Results and discussions

This section presents the validation of the proposed H∞ and μ-synthesis-based control strategy for a quarter-car active suspension system using comprehensive time- and frequency-domain simulations. The quarter-car model consists of two masses representing the chassis and wheel assembly, connected via spring-damper elements, and is actuated by a hydraulic actuator modeled as a first-order system with a maximum displacement of 0.07 m. Key state variables include body displacement, wheel displacement, road disturbance input, and actuator force. The control system is designed to address challenges such as frequency-dependent uncertainties, sensor noise, and the inherent trade-off between ride comfort and suspension travel. Notably, the system exhibits complex dynamics, including imaginary-axis zeros, which complicate feedback control.

The robust control framework minimizes the worst-case effect of disturbances using H∞ performance metrics, while μ-synthesis further accounts for structured uncertainties. Performance weights and a high-pass filter are employed to shape system sensitivity, with a scalar tuning parameter (β) allowing adjustment between comfort and handling objectives. This enables the controller to dynamically prioritize passenger comfort, suspension deflection, or road-holding, depending on the driving conditions. Comparative analysis against fuzzy PID, sliding mode, and bioinspired controllers demonstrates superior performance in terms of adaptability, energy efficiency, and compliance with ISO standards, confirming the proposed method’s practical relevance in intelligent suspension systems.

Frequency domain analysis of suspension response to road disturbance and actuator input

Figure 7 presents the frequency response characteristics of the quarter-car active suspension system under two excitation conditions: road disturbance input (solid green line) and actuator force response (magenta line). A distinct resonance peak is evident around 10 rad/s in the road disturbance response, with a maximum magnitude of approximately 74.75 dB, corresponding to the system’s natural frequency. This significant amplification of oscillations underscores the susceptibility of the suspension system to resonance-induced vibrations in the absence of active control. In comparison, the actuator force response reaches a lower peak magnitude of approximately 49.25 dB at the same frequency, indicating the controller’s effectiveness in mitigating resonant amplification. Additionally, the actuator input demonstrates notable attenuation at lower frequencies, with negative dB values reflecting its capacity to suppress low-frequency vibrations and enhance ride comfort. Beyond the resonance region, the response stabilizes, further confirming the controller’s robustness in managing higher-frequency disturbances.

Figure 7.

Frequency response of active suspension to road disturbance and actuator force.

Overall, the analysis in Figure 7 highlights the essential role of active control in suppressing vibration peaks, minimizing road-induced perturbations, and maintaining dynamic stability. These findings underscore the importance of robust control strategies in achieving a balanced trade-off between ride comfort and road-holding performance in modern suspension systems.

Figure 8 demonstrates the frequency response of suspension deflection (Wsd) under road disturbance and actuator force inputs, highlighting the controller’s effectiveness in regulating relative motion between the sprung and unsprung masses. A sharp resonance peak is observed around 10 rad/s in response to road excitation, reaching approximately 46.35 dB, which reflects significant amplification of suspension deflection at the system’s natural frequency. In stark contrast, the actuator force response at this frequency is attenuated to below −300 dB, confirming the controller’s ability to suppress resonance-induced oscillations with high precision.

Figure 8.

Suspension deflection frequency response.

Across a wide frequency range, the controlled system consistently exhibits reduced deflection amplitude compared to the passive response. At low frequencies (below 5 rad/s), minor attenuation in actuator response indicates effective damping of long-wavelength road inputs, contributing to improved ride comfort. Beyond the resonance region (above 20 rad/s), the road-induced response stabilizes, while the actuator force remains minimal, further supporting the system’s stability and control efficiency under high-frequency excitations.

These results substantiate the claim that the proposed active suspension strategy significantly reduces suspension deflection across all critical frequency ranges. The robust attenuation at resonance and effective control across both low and high frequencies confirm the system’s capability to enhance ride comfort, reduce suspension travel, and maintain vehicle stability under varying dynamic conditions.

Frequency response characteristics of actuators subject to dynamic uncertainty

Figure 9 presents the Bode magnitude plot of the control force (F_c) under nominal conditions (red band) and ±20% structured parameter uncertainty (black band). Across the frequency range, minimal deviation between the nominal and uncertain responses confirms the robustness of the μ-synthesis controller. At low frequencies (<1 rad/s), the magnitude remains near 0 dB, indicating minimal amplification. In the mid-frequency range (1–100 rad/s), moderate variations appear, reflecting controlled sensitivity to dynamic changes. At higher frequencies (>100 rad/s), the magnitude steadily decreases, reaching below −60 dB, demonstrating effective attenuation of high-frequency disturbances.

Figure 9.

Frequency response analysis of active suspension control.

These results validate the controller’s ability to maintain stability and performance despite mass and damping uncertainties. The robust control design ensures consistent force regulation, improved ride comfort, and enhanced road-holding capability under varying system conditions.

Suspension performance trade-offs: Passive and active control strategies

Figure 10 illustrates the frequency response characteristics of a quarter-car active suspension system under road disturbances, comparing the dynamic behavior of passive and active suspension strategies. The passive system displays distinct resonance peaks near 10 rad/s and 100 rad/s, indicating substantial amplification of road inputs that compromise ride comfort and handling. In contrast, the active suspension significantly attenuates these resonance peaks, particularly at low frequencies (below 100 rad/s), demonstrating effective vibration suppression and improved dynamic performance. At higher frequencies (above 100 rad/s), the active system exhibits a gradual increase in magnitude, indicating a trade-off between low-frequency vibration mitigation and high-frequency disturbance rejection. This trend suggests that while the active suspension enhances low-frequency behavior, it may require increased control effort and face potential stability challenges at higher frequencies.

Figure 10.

Frequency response of passive versus active suspension.

Overall, Figure 10 highlights the effectiveness of active suspension control in reducing road-induced disturbances, enhancing ride comfort and road-holding, while also emphasizing the inherent trade-offs associated with frequency-dependent control performance.

Ride comfort and handling trade-offs in active suspension systems

To ensure real-time implementability, the high-order μ-synthesis controller (initially 12th order) was reduced to a 6th-order approximation using balanced truncation without significantly compromising performance. This reduced-order controller preserves the essential dynamics and satisfies robustness margins under uncertainty, making it suitable for embedded implementation in automotive electronic control units (ECUs).

Moreover, to validate robustness against parameter variability, the model was tested over 100 Monte Carlo simulations, where the sprung mass, suspension damping, and actuator delay were randomly varied within ±20% of their nominal values. Performance metrics (RMS body acceleration, suspension deflection, and actuator energy) remained within acceptable bounds across all trials, confirming the consistency and robustness of the proposed control approach.

A frequency domain h∞ control approach

Figure 11 illustrates the influence of different control strategies passive, comfort-oriented, balanced, and handling-focused, on the frequency response of a quarter-car active suspension system, evaluated through sprung mass displacement (Wxb), suspension working space (Wsd), and body acceleration (Wab).

Figure 11.

Frequency response comparison of passive and active suspension strategies.

In the sprung mass displacement (Wxb) response, the passive suspension exhibits a pronounced resonance around 10 rad/s, indicating substantial body motion. All active strategies effectively suppress this peak, with the comfort-oriented strategy achieving the most significant reduction, thereby enhancing ride comfort at lower frequencies. For the suspension working space (Wsd), the passive system shows excessive suspension travel at resonance. Active control strategies mitigate this effect, with the comfort strategy maintaining smoother suspension motion across the frequency range. However, the handling strategy introduces a slight increase in suspension deflection at higher frequencies (above 100 rad/s), reflecting a prioritization of road-holding performance.

The body acceleration response (Wab) similarly shows a significant resonance for the passive system at around 10 rad/s. Active strategies, particularly the comfort and balanced approaches, significantly reduce vibratory energy in the mid-frequency range (10–50 rad/s). At higher frequencies, the handling-oriented strategy exhibits slightly higher acceleration levels, emphasizing improved road-holding capability over absolute ride comfort.

Overall, Figure 11 demonstrates that comfort-focused control strategies are most effective in reducing displacement and acceleration for improved ride smoothness, while handling-focused strategies prioritize stability and road contact, especially at higher frequencies. The balanced strategy offers a compromise between comfort and handling, highlighting the importance of tuning suspension control to meet specific performance objectives.

Time Domain H∞ control approach

To evaluate the performance of the proposed H∞ and μ-synthesis controllers, simulations were performed in MATLAB/Simulink using a nonlinear quarter-car model under two road excitations: a single bump and an ISO Class E random road profile. These profiles were chosen because they represent complementary and widely accepted conditions for suspension performance assessment.

The single bump profile, with a height of 0.0525 m and a width of 1.8 m, represented a deterministic, high-amplitude transient disturbance suitable for evaluating the system’s ability to suppress peak acceleration and body displacement. This height corresponds to standard vehicle body excitation tests and remains below the ISO 8608 Class D and E thresholds for harsh road conditions.

The ISO Class E random road profile, generated using a filtered white noise method based on ISO 8608 power spectral density (PSD) parameters, simulated broadband stochastic road irregularities typical of severely deteriorated surfaces.

Together, these two profiles captured both transient and steady-state vibration characteristics, providing a comprehensive evaluation of ride comfort and road-holding performance. Additional road classifications were excluded to maintain computational efficiency but could be explored in future studies.

Simulation time was set to 5 s, and sampling frequency was 1 kHz. Initial conditions were zero, and disturbances were applied at t = 0.5 s. The system was evaluated under both nominal and uncertain parameters (±20% mass and damping), using the same initial conditions for all benchmarked controllers to ensure fairness.

Active suspension systems offer notable improvements over passive designs across key performance metrics, as depicted in Figure 12. In terms of suspension deflection under road disturbances (Figure 12(a)), the passive system exhibits the largest displacement (peak: 0.078 m), reflecting limited vibration isolation. Active control strategies significantly reduce suspension travel, with peak displacements of 0.031 m (active-comfort), 0.024 m (balanced), and 0.020 m (active-handling), thereby enhancing ride quality and minimizing mechanical wear.

Figure 12.

Time domain H∞ control responses. (a) suspension deflection, (b) vertical acceleration, (c) relative displacement, (d) actuator force.

The body acceleration response (Figure 12(b)), a critical indicator of ride comfort, further demonstrates the benefits of active control. The passive system results in substantial oscillations (+3.95 m/s², −5.6 m/s²), whereas the active strategies achieve notably smoother responses: active-comfort (+2.296 m/s², −3.4 m/s²), balanced (±2.078 m/s²), and active-handling (±1.816 m/s²). These improvements contribute to reduced passenger fatigue and more stable conditions for sensitive autonomous vehicle systems. Similarly, the relative displacement between the sprung mass and wheel (Figure 12(c) shows considerable oscillations for the passive system (±0.144 m). Active control strategies significantly limit this motion, with peak values of +0.112 m/−0.080 m (active-comfort), ±0.1067 m (balanced), and ±0.100 m (active-handling), leading to reduced component stress and enhanced ride stability. However, these performance gains require additional control effort, as indicated in Figure 12(d). While the passive system operates without actuator force (0 kN), the active strategies demand increasing levels of control force: ±0.491 kN (active-comfort), ±0.657 kN (balanced), and ±1.3 kN (active-handling). This highlights the trade-off between dynamic performance improvements and increased energy consumption, necessitating careful optimization in practical implementations.

In general, Figure 15 highlights the advantages of active suspension over passive systems, showing significant reductions in suspension deflection, body acceleration, and relative displacement, while improving ride comfort and stability. However, active systems require varying levels of actuator force, with handling-oriented strategies demanding the highest energy, illustrating the balance between performance gains and energy efficiency.

A time domain robust Mu control approach

The comparative performance of passive and robust active suspension systems under transient road disturbances is presented in Figure 13(a)–(d). In Figure 13(a), the passive suspension amplifies the road disturbance, reaching a peak displacement of 0.035 m (67% of the 0.0525 m input bump) with delayed settling time, while the robust active controller reduces the peak by 26% (0.026 m) and achieves faster stabilization. This enhancement in vibration isolation contributes directly to improved ride comfort and system stability. Figure 13(b) illustrated that the robust controller effectively managed vertical accelerations, which are critical for passenger comfort. The passive suspension exhibited a peak acceleration of 1.426 m/s², whereas the robust controller reduced this to 1.3325 m/s². Moreover, the passive system-maintained oscillations of approximately 1.2 m/s² beyond 0.6 s, while the active system demonstrated faster damping, stabilizing around 0.5–0.6 m/s².

Figure 13.

Time domain robust mu control responses. (a) suspension working space, (b) vertical acceleration, (c) relative displacement, (d) actuator force.

The minor oscillations that appeared after the main transient response were mainly attributed to resonance effects near the suspension system’s natural frequency, approximately 10 rad/s, and to the actuator’s dynamic limitations. The actuator was modeled as a first-order system with finite bandwidth, introducing phase lag and reducing damping near the resonance region. Although the proposed μ-synthesis controller effectively reduced the peak acceleration, the remaining oscillations reflected an inherent trade-off between ride comfort and actuator constraints considered during the weighting function design. These oscillations could have been further minimized by slightly increasing the low-frequency damping weight in the generalized plant, incorporating a notch or lead compensator around the dominant resonance frequency, or using an actuator with a higher dynamic response. Additional simulations revealed that halving the actuator time constant reduced the post-transient oscillations by about 15%, confirming that actuator bandwidth significantly influenced residual vibrations. Consequently, the observed oscillations were interpreted not as a loss of robustness but as a practical outcome of controller design compromises.

The relative displacement response shown in Figure 13(c) mirrors this trend. While both systems initially dip to −0.0476 m, the passive suspension rebounds to a higher peak (0.035 m) compared to the robust system (0.0278 m), indicating reduced tire contact and compromised handling in the passive design.

The performance trade-off is depicted in Figure 13(d), where the robust system requires actuator forces up to ±0.18 kN to actively control suspension dynamics, whereas the passive system operates without active force input. Despite the additional control effort, the robust suspension maintains a balance between dynamic performance gains and practical energy expenditure, making it a promising solution for enhancing ride comfort, road-holding, and system adaptability in modern vehicles.

Overall all Figure 13(a)–(d) demonstrate that the robust active suspension significantly outperforms the passive system in reducing suspension deflection, vertical acceleration, and rebound motion under transient road disturbances, while requiring moderate actuator forces. These results highlight the robust controller’s ability to enhance comfort and stability with efficient control effort.

Validation approach

To validate the effectiveness of the proposed H∞ and μ-synthesis control strategy, simulation results are benchmarked against three commonly used control approaches:

(1) Passive Suspension: A conventional passive suspension system was modeled using fixed spring and damper parameters, serving as a baseline for performance comparison. No active control was applied, and its response relied solely on the mechanical characteristics of the suspension components.

(2) Fuzzy PID Control: A hybrid controller combining fuzzy logic with a PID structure was developed. The fuzzy module adjusted PID gains dynamically based on suspension deflection and acceleration inputs, enhancing adaptability under varying road conditions. Triangular membership functions and an expert-defined rule base were used to improve ride comfort and control effort balance.

(3) Sliding Mode Control (SMC): A robust nonlinear controller was designed based on a sliding surface defined using suspension deflection and velocity. The control law ensured insensitivity to parameter variations and external disturbances, with a boundary layer introduced to mitigate chattering effects.

(4) Linear Quadratic Regulator (LQR): An optimal state-feedback controller was formulated by minimizing a quadratic cost function that penalized suspension deflection and control effort. The weighting matrices were tuned to balance ride comfort and actuator usage while maintaining system stability across both nominal and uncertain conditions.

(5) Bioinspired Control: A bioinspired algorithm, mimicking reflexive control mechanisms found in biological systems, was implemented. The controller utilized adaptive gain modulation and sensory feedback to respond rapidly to road disturbances, aiming to enhance comfort and road-holding through biologically motivated strategies.

• All controllers were evaluated under consistent simulation settings, including:

• Identical road disturbances (bump and ISO Class E),

• The same quarter-car parameters (both nominal and with ±20% uncertainty),

• Uniform actuator saturation and sensor noise constraints.

This standardized framework ensured an objective and direct comparison of performance across all control strategies.

Control strategy evaluation

The proposed robust control strategy, integrating H∞ control and μ-synthesis, demonstrates superior performance in both ride comfort and road handling compared to existing methods, as confirmed by simulations and comparative analysis. As shown in Figure 14 (Vertical Acceleration Comparison), the controller reduces peak body acceleration by 54% (from 3.95 m/s² to 1.816 m/s²), outperforming Fuzzy PID (35% reduction) and Sliding Mode Control (38%) from recent literature.

Figure 14.

Vertical acceleration compression.

Figure 15 (Suspension Travel Frequency Response) reveals a 35% improvement in resonance attenuation at the critical 10 rad/s frequency, significantly surpassing the performance of LQR and bioinspired approaches. Additionally, Figure 16 (Robustness to Mass Variations) demonstrates the controller’s stability under ±20% mass variations, maintaining consistent performance, whereas conventional methods experience degradation under similar conditions.

Figure 15.

Suspension travel frequency response.

Figure 16.

Robustness to mass variations.

These findings are corroborated by Table 2, which illustrates the proposed method’s advantages in energy efficiency (30% savings) and the effective handling of structured uncertainty, addressing critical limitations of previous approaches.^6,22 Table 3 further validates the practical feasibility of the controller, confirming that vertical acceleration remains below the ISO 2631-1 standard threshold of 2.5 m/s² for comfort, supporting its real-world applicability.

Table 2.

Performance comparison of active suspension control strategies.

Study/source	Control method	Vertical accel. reduction	Suspension travel reduction	Robustness to uncertainties	Energy savings	Robustness under ±20% uncertainty
Proposed method (2024)	H∞ + μ-synthesis	54% (3.95 to 1.816 m/s²)	35% (0.078 to 0.020 m)	Explicit handling of mass, stiffness, and time delay variations, 1/60s	30% (vs. conventional active)	Moderate
⁶	Fuzzy PID	35%	25%	Limited adaptation to parameter changes	15%	Moderate
⁷	Fuzzy-LQR (PSO-optimized)	42%	28%	Moderate robustness to load changes	18%	Moderate
²	Nonlinear sliding mode	38%	22%	Handles nonlinearities but not structured uncertainties	12%	Moderate
²³	Robust H∞ (electromagnetic actuator)	40%	28%	Focuses on actuator dynamics only	22%	Good
²⁴	Continuous sliding mode	36%	22%	Good for disturbances, weak on parameter variations	20%	Good
²⁵	Robust H∞ (semi-active)	45%	30%	Conservative bounds reduce performance	25%	Very good
²⁶	Bioinspired robust control	48%	32%	Adaptive but not systematic for all uncertainties	35%	Very good

Table 3.

Validation against industry standards.

Feature	Proposed system performance	Industry standard (ISO/UNECE)	Comparison
Vertical acceleration	1.816 m/s²	Comfortable: 1.0–2.5 m/s² (ISO 2631-1)	Within the ideal range
Suspension travel	0.020 m	Design limit: 0.025–0.030 m	Better than typical
Tire load variation	26.7% reduction	Typical improvement: 15%–20%	Superior
Actuator force	±1.3 kN	Commercial hydraulic actuator range	Feasible

The energy-saving claim (∼30%) is based on the total actuator energy consumption, calculated using the integral of the actuator power over time:

E_{a c t} = \int_{0}^{T} F_{a} (t) \cdot u_{a} (t) d t

(49)

where Fa(t) is the actuator force,

u_{a} (t) = {\dot{x}}_{b} (t) - {\dot{x}}_{w} (t)

is the relative velocity across the actuator.

This metric quantifies the net mechanical energy input required for active control. The proposed controller demonstrates reduced Eact compared to conventional H∞, LQR, and fuzzy PID controllers over identical operating conditions.

The fairness of the comparative results illustrated in Figures 14–16 was maintained by ensuring identical modeling conditions and simulation parameters for all controllers. The same quarter-car model, actuator dynamics, sensor noise characteristics, and structured uncertainty bounds of ±20% were consistently applied throughout the analysis. For the deterministic bump road, the excitation timing and amplitude were kept identical, whereas for the ISO Class E random road, identical random seeds and power spectral density (PSD) parameters were used to guarantee equivalent input conditions across all control strategies.

All benchmark controllers namely, passive, fuzzy PID, sliding mode, LQR, bioinspired, and the proposed H∞–μ controller were tuned using a unified two-stage process comprising coarse stabilization followed by fine performance adjustment, with identical evaluation indices applied to each. Additionally, a Monte Carlo analysis consisting of 100 trials with parameter variations within ±20% was performed for every controller. The mean and standard deviation of performance metrics, including RMS body acceleration, RMS suspension travel, and actuator energy consumption, were calculated. The mean responses are plotted in Figures 14–16, while shaded regions denoted ±1 standard deviation. These systematic procedures ensured that all comparisons were objective, statistically reliable, and fully reproducible across different control methodologies.

To further evaluate robustness, the controllers were tested under ISO Class E road excitation, representing a high-disturbance environment. Under this condition, the proposed μ-synthesis controller successfully maintained suspension deflection within 0.03 m and vertical acceleration below the ISO 2631-1 comfort threshold of 1.2 m/s² RMS. These results demonstrated that the proposed design achieved superior disturbance rejection and ride comfort compared to conventional control strategies.

The μ-synthesis controller outperformed all baselines in terms of ride comfort (body acceleration or vertical acceleration), handling (suspension travel), and energy efficiency, while maintaining robust performance under structured uncertainty. These results align with and surpass those reported in recent studies [2, 6, 7, and 24-27], highlighting the novelty and strength of the proposed method.

The proposed robust control strategy outperforms existing methods in ride comfort and road handling. It achieves a 54% reduction in peak body acceleration, a 35% improvement in resonance attenuation, and maintains stability under mass variations. The method also offers 30% energy savings and ensures compliance with ISO 2631-1 comfort standards, proving its practical feasibility for real-world use.

Conclusion

This study developed and evaluated a robust control strategy for a quarter-car active suspension system using a combined H-infinity and μ-synthesis approach. The controller was designed to address key challenges in suspension dynamics, including structured uncertainties such as actuator time delays of up to 1/60 s. A representative quarter-car model of a mid-size sedan was used, with parameters including a sprung mass of 600 kg, unsprung mass of 60 kg, suspension stiffness of 18,000 N/m, and tire stiffness of 150,000 N/m.

Simulation studies conducted in MATLAB/Simulink under various road excitations, including bump inputs and ISO Class E road profiles, demonstrated that the proposed robust control scheme significantly outperforms conventional controllers such as PID, fuzzy logic, and disturbance compensation-based robust (DCBR) methods. Specifically, the H∞ and μ-synthesis-based controller achieved a 54% reduction in vertical body acceleration, a 35% decrease in suspension travel, and a 26.7% improvement in tire load variation control, resulting in enhanced ride comfort, road-holding, and overall handling performance. Additionally, the controller exhibited strong robustness under structured parameter uncertainties and road disturbances modeled as sinusoidal inputs with amplitudes of 0.01–0.03 m and frequencies ranging from 1 to 10 Hz.

These results confirm the potential of robust control strategies to overcome the limitations of classical and intelligent control approaches, particularly in managing real-world uncertainties and maintaining consistent performance. The dynamic adaptability and improved energy efficiency of the proposed strategy make it highly suitable for modern applications in autonomous and electric vehicles, where stability and performance must be ensured under varying load and terrain conditions.

• However, despite the promising simulation outcomes, several practical considerations must be addressed to advance real-world implementation:

• Real-time implement ability: μ-synthesis controllers often yield high-order transfer functions that may pose computational challenges for embedded automotive systems. Applying model order reduction techniques is essential to ensure compatibility with real-time control hardware.

• Sensor requirements: The current design assumes full-state feedback, requiring measurements of body and wheel displacements and velocities. In practical applications, this would necessitate additional sensors or the integration of state observers (e.g., Kalman filters), potentially increasing system cost and complexity.

• Scalability and integration: While the proposed control architecture was validated using a quarter-car model, scaling it to half- and full-vehicle systems introduces cross-coupled dynamics and increased degrees of freedom. Coordinated control across multiple suspension units must be carefully addressed.

Future research will focus on enhancing the practical applicability of the proposed control strategy. Reduced-order controllers will be developed for real-time implementation on embedded systems, and advanced state estimation techniques such as Kalman filters will be integrated to address sensing limitations. The control framework will be extended to half- and full-vehicle models to capture dynamic coupling, and hardware-in-the-loop (HIL) testing will be conducted for real-time validation. Additionally, energy consumption analysis will be performed to evaluate comfort–efficiency trade-offs, particularly for electric vehicles.

Despite strong simulation performance, several limitations were identified. The study used a linearized quarter-car model, neglecting nonlinearities like suspension geometry, tire stiffness variation, and actuator saturation. Full-state feedback was assumed, which may not be practical without state observers, and the computational feasibility of the reduced-order controller was not experimentally verified. Actuator losses were also not considered.

Future work should therefore extend the controller to nonlinear and full-vehicle models, validate it experimentally using HIL setups, incorporate observer-based estimation, and explore adaptive or gain-scheduled μ-synthesis designs. Multi-objective optimization considering comfort, handling, and energy efficiency is also recommended to enhance real-world applicability.

Footnotes

Acknowledgments

The authors would like to express their sincere gratitude to the Department of Motor Vehicle Engineering, College of Engineering, Ethiopian Defence University, for providing the facilities and support necessary for this research. Special thanks are extended to colleagues who contributed through valuable discussions and technical assistance during the study.

ORCID iDs

Yilkal Misganaw

Tatek Mamo

Kumlachew Yeneneh

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

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

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