Constrained Dynamic Output-Feedback Robust H∞ Control of Active Inerter-Based Half-Car Suspension System With Parameter Uncertainties

In this study, a multi-objective dynamic output-feedback robust ${H_{2}}/{H_{\infty} }$ controller for an active inerter-based half-car suspension system in the presence of parameter uncertainty and external disturbance has been investigated. Its main goal is to improve the inherent trade-offs between ride quality, handling performance, and suspension travel and to guarantee the allowable level for suspension stroke and input constraint. Inerters have been widely used to suppress undesirable vibrations in various mechanical structures. The advantage of inerters is that the realized equivalent mass ratio (inertance over primary structure mass) is greater than its actual mass ratio, leading to higher performance for the same effective mass. First, the dynamics and state space of the active inerter-based suspension system for a half-car model with parameter uncertainties have been achieved. To meet the specified objectives, and guarantee the prescribed disturbance attenuation level of the closed-loop system, the Lyapunov stability function and linear matrix inequality (LMI) technique have been used to fulfill the proposed approach. In the case of feasibility, sufficient LMI conditions by solving a convex optimization problem afford the stabilizing gain of the controller. According to numerical simulations, the active inerter-based suspension system in the presence of parameter uncertainties and external disturbance performs much better than both a passive suspension with inerter and active suspension without inerter.


I. INTRODUCTION
The main objective of vehicle suspension systems development is to reduce the acceleration of the car body and passengers while ensuring good contact between tires and the road. In addition, the suspension travel must be restricted to the permissible working space. These objectives (ride comfort, road holding, and suspension travel) can be in opposition to each other, and the design problem is to find a compromise between them [1], [2]. With regard to the control structure of the suspension systems, three main categories have been developed to achieve the required performance of the vehicle The associate editor coordinating the review of this manuscript and approving it for publication was Zhiguang Feng . suspension systems, namely passive, semi-active, and active suspension systems [1]. In many investigations, the active suspension system has been indicated as an effective method for improving suspension performance [1], [3]. Nowadays, research on the improvement of suspension performance is focused on two aspects: first, the rational and precise design of advanced vehicle suspension, and second, the search for optimal control methods.
In order to avoid the consequences of vibration, many techniques have been introduced such as isolating systems from vibration, controlling systems, redesigning systems to change their natural frequencies, using tuned mass dampers, and many more [4]. Tuned mass dampers (TMDs) are widely employed to suppress unwanted vibrations in various mechanical structures, such as preventing damage to buildings due to seismic excitation, suppressing bridge vibrations, achieving the best characteristics of cutting processes, reducing floor or balcony vibrations, achieving stable rotations of rotors, stabilizing drill strings, etc [4], [5]. The classical TMD consists of a mass on a linear spring, and it is well known that the classical TMD is particularly effective in reducing the response of the main structure in principal resonance, but at other frequencies (even near the resonance frequency) it increases the amplitude of the system's motion [5]. Therefore, we must always consider whether we want to most effectively damp vibrations at a particular frequency or whether we want to achieve tolerable damping characteristics over a wide range of vibration frequencies. This problem is capable to be minimized by novel TMDs containing inerters or magnetorheological dampers, which are intensively developed nowadays [6]. Inerter is a device with two free-moving terminals whose generated force is proportional to the relative acceleration of its terminals (idealized model) [7]. The proportional constant is called inertance with the unit kilogram. Inerter possesses the effect of mass amplification and would provide much greater inertia compared to its own mass, thus increasing the inertia of the entire dynamic system rather than increasing the mass [4], [5]. Due to its mechanical properties, it is therefore an efficient structure for damping vibrations. On the other hand, the main motivation for the proposal of inerter lies in the incompleteness of the force-current analogy between mechanical and electrical systems. The introduction of the inerter completes the analogy between the mechanical network spring-damper-inerter and the electrical network inductor-resistor-capacitor. Therefore, the systematic methods for the synthesis of passive electrical networks can be directly applied to the development of inerter-based mechanical networks [4].
The rack-and-pinion, ball-screw, and hydraulic (or fluid) inerters are the three most commonly used inerters. Depending on whether a flywheel is used in the realization, they can be divided into two categories, namely flywheel-based inerters and non-flywheel inerters. When the inertance is fixed, the inerter is passive; when the inertance can be adjusted, the inerter is semi-active [4], [8].
The inerter is employed as a passive element in the majority of applications, in the sense that the inertance cannot be adjusted by online control actions. Then the performance of the system has been investigated passively or actively using the controller. In [7], analytical solutions for some inerter-based suspension structures were derived based on a quarter-car model, and the performance benefits of using inerters in vehicle suspensions were analytically demonstrated. In [9], several performance requirements, including ride comfort, suspension deflection, and tire deflection, were analytically investigated. As a result, the analytical solutions for six suspension configurations were taken, and it was derived that the performance indexes of complex networks are better than the simpler ones. In [10], the nonlinearities of inerter and their influence on suspension performance were studied. In [11], a mechatronic network structure combining a ball-screw inerter with a permanent magnet electric machine was proposed. One of the main advantages of this mechatronic structure is that the system impedance can be realized by combining mechanical and electrical networks. As a result, the higher order system impedance can be easily realized without taking up much space. In [12], eight inerter-based networks were combined with sky-hook controlled and ground-hook controlled actuators to verify the performance benefits of inerter.
In [13], the active inerter-based suspension has employed a controllable actuator to generate the required force. Although it uses the most energy, it offers higher dynamic performance as compared to passive and semi-active suspensions. To solve the problem of energy requirement due to the force-generating actuators in the active suspension systems, an inerter-based electromagnetic device was presented in [14] and implemented in the vehicle suspension system. The proposed device not only improves the performance of the suspension, but also generates an amount of electrical energy that can be used by other parts of the vehicle, especially the energy required to operate the actuator.
To find a compromise between the conflicting performances of the vehicle suspension system, many approaches have been proposed based on various control techniques, such as fuzzy logic and neural network control [1], [15], [16], adaptive control [15], [16], [17], model predictive control [18], sliding mode control [17], [19], H ∞ control [2], [3], [20], [21], etc. In particular, the application of H ∞ control of the active vehicle suspension system has been intensively investigated in the context of robustness and damping of road disturbances. Furthermore, it has been recognized that it is not only an effective way to compromise between conflicting performance requirements, but also to optimize either a weighted single objective function with hard constraints or a multi-objective function [1], [21].
Most problems that engineers encounter in practical applications involve some degree of uncertainty, including parameter uncertainties and model uncertainties. Therefore, system uncertainties should always be considered when designing a control system for stability and performance [22]. Changes in vehicle inertial properties, such as vehicle sprung mass (due to the number of passengers in the vehicle, the load it is carrying, or the aerodynamic forces), have direct effects on vehicle ride comfort, handling, and braking performances. In addition, uncertainty with regard to stiffness may be caused by a variety of factors, including variability in manufacturing processes and quality control, uncertainty in material properties and element dimensions, etc. There is difficulty in choosing a fixed inerter to satisfy the vehicle performances at the sprung mass natural frequency without deteriorating significantly at the unsprung mass natural frequency [23]. Accordingly, uncertainty in the inertance of inerter should be considered 46052 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
when evaluating the performance of the inerter-based vehicle suspension system.
On the other hand, It should be noted that the state-feedback control method requires all information about the state variables. In practical applications, such an assumption is usually unrealistic, since online measurement of all state information is not only difficult to achieve (such as tire deflection) but also increases the cost and complexity of the implementation [1], [2]. Consequently, a practical alternative would be to design a controller based on output-feedback, an effective and practical control scheme that employs only a portion of the measured states for active vehicle suspension systems.
To solve the aforementioned problems, a parameterdependent control technique could be applied to realize robust control of vehicle suspension systems regardless of changes in vehicle parameters. The linear matrix inequality (LMI) approach is a practical and powerful tool for dealing with output-feedback control and system uncertainties [22]. In [24], a non-fragile static output-feedback controller for quarter-car active suspension systems was presented, taking into account the input time-delay. The resulting constraints were obtained in the form of bilinear matrix inequalities (BMI), which require iterative methods to be resolved. In [20], a static output-feedback H ∞ controller for an active quarter-car suspension system with input time-delay and parameter uncertainty was investigated. In [21] a constrained dynamic output-feedback H ∞ controller for an active quarter-car suspension system was designed, where the uncertainties of the parameters were not considered. In [2] a dynamic output-feedback H ∞ controller for a quarter-car active suspension system was designed to deal with actuator time delays. In [22] convex optimization method was employed to develop a robust fixed-order nonfragile dynamic output-feedback controller for the nonlinear active suspension system of a quarter-car model with uncertainty. Considering all these works, we intend to investigate the active inerter-based half-car suspension system by considering all factors, including external disturbance, parametric uncertainty, structural limitations, input constraint, and inaccessibility of all state variables. It is worth noting that this work is not a simple application of an existing method on active suspension systems, but that the theoretical findings are also novel and nontrivial.
In this paper, the active inerter-based half-car suspension system is investigated based on the parallel-connected configuration, since this configuration is simple and spacesaving [8]. The H ∞ (energy-to-energy) control is employed to optimize the ride comfort of the suspension system, which measures the accelerations of the body, including both the heaving and pitching movements. In addition, the tire deflection and suspension deflection are constrained by their peak values in the time domain. Hence, the hard limits of the suspension system are considered using a generalized H 2 (GH 2 ) norm (energy-to-peak) in the form of an LMI. Moreover, generalized H 2 (GH 2 ) norm is utilized to reduce the gain of the controller, which results in avoiding measurement noise amplification and saturation of the actuator [25]. Sufficient stability conditions and performance criteria are derived in the form of LMIs employing the direct Lyapunov method.
The main contributions of this work can be summarized as follows: • In this paper, we purpose to design a multi-objective dynamic output-feedback robust H 2 /H ∞ controller for the active inerter-based suspension system for a half-car model with parameter uncertainties. This approach will allow us to achieve the main goal of finding a compromise between the basic performance requirements for advanced vehicle suspension systems, including road holding, ride comfort, suspension deflection, and energy consumption.
• For the first time, we have presented the state space of the active vehicle suspension system for the half-car model with the presence of inerter in its dynamics and evaluate the performance of this system with parameter uncertainties and external disturbance.
• The hard limits of the suspension and tire deflection are considered utilizing a generalized H 2 (GH 2 ) norm in the form of an LMI. It ensures that the energy of road disturbance is suppressed in the controlled output (consisting of the hard limits of the suspension system), thus satisfying the physical constraints of the suspension system.
• In the real implementation of the active suspension system, high gain of the controller can lead to major problems such as noise amplification and saturation of the actuator. To avoid such problems, an LMI is introduced to guarantee the maximum possible control force of the actuator.
• The stability conditions are derived as linear matrix inequalities (LMIs) and therefore the stabilization gain of the system is obtained by solving the convex optimization problem.
The subsequent parts of this paper are structured into four sections. The description of the active inerter-based suspension system for a half-car model is presented in Section II. Section III contains the problem formulation for constrained dynamic output-feedback robust H ∞ control based on the solvability of LMIs for the active inerter-based suspension system with parameter uncertainties. In Section IV, the proposed controller is applied to the active inerter-based half-car suspension system for performance evaluation. Finally, the conclusion of our findings is given in Section V.
Notation: The following nomenclature will be utilized throughout this paper. In a symmetric block matrix or complex matrix expressions, an asterisk ( * ) indicates a term that is induced by symmetry. The notation P > 0 (≥ 0) is used to denote that P is a real symmetric and positive definite (semi-definite) matrix. R n stands for the n-dimensional Euclidean space and the superscript T denotes matrix transposition. I and 0 are utilized to indicate the identity and zero matrices with appropriate dimensions, and diag {· · · } stands for a block-diagonal matrix. Let ∥•∥ symbolize the induced norm for matrices and the Euclidean norm for vectors. ∥•∥ L 2 represents the L 2 norm of a signal defined as ∥v(t)∥ 2 L 2 = ∞ 0 ∥v(s)∥ 2 ds. Moreover, ∥•∥ L ∞ denotes the L ∞ norm of a signal defined as ∥v(t)∥ 2 L ∞ = sup t≥0 ∥v(s)∥ 2 ds.

II. ACTIVE INERTER-BASED HALF-CAR SUSPENSION SYSTEM MODELLING
Since the spring force depends on the displacement and the damper force depends on the velocity, the idea of the inerter is to act against accelerations. Accordingly, the inerter is connected in parallel to the spring and damper between the wheel and the chassis. The main function of the inerter is to counteract the vibrations coming from the tire, thus improving the contact between the wheel and the ground [23]. In this regard, the parallel inerter has been shown to reduce the overall suspension force by removing the spring force, resulting in improved handling [8]. The half-car model of the active suspension system equipped with inerter, as shown in Fig.1, can be reduced to 4DOF system considering the vertical and angular dynamics [3]. The model is assembled by one sprung mass (car body) that is connected to two unsprung masses (representing the front and rear wheels) and includes heave and pitch modes of vibrations. The two front and rear unsprung masses are free to move vertically and are confronted with the road disturbance input. In Fig.1, m s represents the mass of the car body, m uf and m ur are the unsprung masses of the front and rear tires, respectively. b sf and b sr denote the inertance of the inerter for the front and rear axles of suspensions, respectively; c sf and c sr represent the damping coefficients of suspension elements for the front and rear suspensions, respectively. k sf and k sr represent the stiffnesses of the suspension for the front and rear suspensions, respectively. Likewise, k tf and k tr are the front and rear tire stiffnesses, respectively; u f (t) and u r (t) denote the front and rear actuator force inputs, respectively. ϕ(t) is the pitch angle, z c (t) is the displacement of the center of gravity, l ϕ is the pitch moment of inertia about the center of mass, l 1 is the distance between the front axle and the center of gravity, and l 2 is the distance between the rear axle and FIGURE 1. Active inerter-based half-car suspension system. the center of gravity. z sf (t) and z sr (t) represent the vertical displacements of the front and rear body, respectively; z uf (t) and z ur (t) denote the vertical displacements of the front and rear unsprung masses, respectively. z rf (t) and z rr (t) denote the road disturbance inputs to the front and rear wheels, respectively. It is assumed that the tires are in contact with the road at all times, and the characteristics of the suspension elements are linear.
We assume that the exact value of the sprung mass (m s (t)) and inertance of the inerters (b sf (t), b sr (t)) are not known, but their maximum and minimum values are available. Also we assume that the pitch angle ϕ(t) is small enough. Accordingly, the displacements of the sprung mass can be established by Assuming linear dampers, springs, and inerters, differential equations of motion can be calculated by applying Newton's second law as follows It is noteworthy that the equations of motion for the passive system can be obtained by letting u f (t) = u r (t) = 0. Defining eight state variables as follows where x 1 (t) is the suspension deflection of the front car body, x 2 (t) denotes the vertical velocity of the front car body, x 3 (t) represents the suspension deflection of the rear car body, x 4 (t) indicates the vertical velocity of the rear car body, x 5 (t) is the tire deflection of the front car body, x 6 (t) denotes the vertical velocity of the front wheel, x 7 (t) represents the tire deflection of the rear car body, and x 8 (t) is the vertical velocity of the rear wheel. Accordingly, by defining active inerter-based half-car suspension system can be represented by the following state-space equatioṅ where The equations for α i (t), β i (t), κ i (t), and ρ i (t) can be found in Appendix A. As mentioned earlier, ride comfort, suspension deflection, and road-holding ability are the three most important performance criteria to consider when developing controllers for vehicle suspension systems. Therefore, the suspension outputs are divided into two categories including the optimization output and the constraint output which are explained in the following subsections.

A. RIDE COMFORT
Indeed, minimization of the vertical acceleration sensed by the rider is the paramount assignment of the suspension system, leading to ride comfort and lesser depreciation. In other words, ride comfort refers to the general sensation of noise, vibration, and motion inside a driven vehicle, and it impacts the comfort, safety, and health of the passengers. Hence, both the heave accelerationz c (t) and the pitch accelerationφ(t) of the half-car suspension system are chosen as the first control output vector, that is where q 1 and q 2 are weighting constants and, normally, q 1 is selected as 1 and q 2 = q 1 √ l 1 l 2 [3]. The ride comfort performance of the suspension system is optimized by the concept of the H ∞ control (energy-to-energy) to measure the body accelerations. As a performance measure, the H ∞ norm (or L 2 -gain) leads to organize active suspension enacting adequately in a wide range of shock and vibration environments and we intend to decrease the H ∞ norm of the suspension system from the road disturbance v(t) to the controlled output z 1 (t) due to reducing the heave and pitch vibrations of the vehicle.z c (t) andφ(t) are given by Eqs. (2) and (3), respectively, and the controlled output z 1 (t) can be described as follows where The equations for υ i (t), and µ i (t) are given in Appendix A.

B. SUSPENSION DEFLECTION LIMITATION
Vehicle suspension must be capable of supporting the static weight of the vehicle. Accordingly, in order to avoid ride comfort deterioration and mechanical structural damage, the active suspension controllers should be qualified to preclude the suspension from hitting its travel limit. Consequently, the suspension deflection requires to be limited within a suitable range induced by the constraint of the mechanical structure [2], which is defined as (10) where z f max and z r max are the maximum suspension travel hard limits, under any road disturbance inputs and vehicle running conditions for the front and rear tires, respectively. The suspension deflection space does not need to be minimal, but only its peak value should be limited. As regards L ∞ norm of a signal in the time domain, it actually represents its peak value, that is The L ∞ norm of the suspension deflection output under the energy-bounded road disturbance input can be optimized according to That is, v(t) ∈ L 2 0 ∞ , to perceive the hard requirement for the suspension deflection. Actually, this is a generalized H 2 (GH 2 ) or energy-to-peak optimization problem [3].

C. ROAD HOLDING ABILITY
In practical vehicle systems, during maneuvers such as accelerating, deaccelerating, or cornering, there are many forces acting on the wheel that can lift it off the ground and leads to losing control of the vehicle, in either drive or steering senses. Hence, the dynamic tire load should not exceed the static tire load to ensure firm uninterrupted contact of both front and rear wheels to the road [3], according to where F f and F r are static tire loads of front and rear wheels, respectively. They can be calculated by Such as suspension deflection, here we have another peak value optimization problem that can be dealt with in the same manner. Consequently, the tire load and the hard constraints on the suspension deflection are defined as the second control output, that is, Therefore, z 2 (t) can be described as follows where In the next section, we present the main results. That is, proper conditions are proposed for the closed-loop system in the presence of parametric uncertainties such that, 1. The system without external disturbance is asymptotically stable.

III. FORMULATION OF THE CONSTRAINED DYNAMIC OUTPUT-FEEDBACK ROBUST H ∞ CONTROL PROBLEM
To discuss the main results, we consider a more general system, which can be defined by the following state-space equationsẋ where ∈ R q is the external disturbance vector, and since we are considering an output-feedback control paradigm for the system defined in Eq. (17), it is assumed that p < n. In the same way, matrices A(t), B(t), D(t), C 1 (t), D 12 (t), C 2 (t) and C 3 (t), are all uncertain matrices with appropriate dimensions.
where (t) represents the uncertainties. In addition, the uncertainties are assumed to be structurally bounded, i.e.
where E(t) T E(t) ≤ I; moreover, M and N are appropriately dimensioned matrices. Next, we design the full order dynamic output-feedback controller for the system in Eq. (17) as follows: wherex(t) ∈ R n is the state vector of the dynamic controller.Â,B, andĈ are appropriately dimensioned matrices of the controller to be obtained. Let us definex(t) = [ x(t) Tx (t) T ] T , then applying the controller in Eq. (20) to the system in Eq. (17), the closed-loop system is obtained as followsẋ Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
In this section, we will solve the problem of the constrained dynamic output-feedback H ∞ controller for the system with parameter uncertainties. Theorem 1 presents the conditions under which the uncertain closed-loop system without external disturbance becomes asymptotically stable, and in the presence of external disturbance a prescribed disturbance attenuation level is achieved. This can be accomplished by minimizing the multi-objective robust H 2 /H ∞ criterion of the closed-loop system under the external disturbance v(t) to the controlled outputs z 1 (t), and z 2 (t) via a suitable quadratic Lyapunov function.
Assumption 1: In this paper, it is assumed that the external disturbance signal v(t) is square-integrable, that is 26]): Let Q, and F be real matrices of appropriate dimensions with F satisfying F T F ≤ I. Then, for any scalar ε > 0, we have Lemma 2 (Schur Complement [27]): Given constant matrices 1 , 2 and 3 satisfying 1 = T 1 and 2 > 0 , then Theorem 1: Assuming positive constants γ 1 , γ 2 , and δ i for i = 1, 2, 3, the linear uncertain system (Eq. (17)) with dynamic output-feedback controller (Eq. (20)) in the absence of external disturbance is asymptotically stable and in the presence of external disturbance satisfies ∥z 1 (t)∥ 2 L 2 < γ 2 1 ∥v(t)∥ 2 L 2 , subject to ∥z 2 (t)∥ 2 L ∞ < γ 2 2 ∥v(t)∥ 2 L 2 for all v(t) ∈ L 2 0 ∞ , if there exist symmetric positive definite matrix p > 0, symmetric matrices R T = R and S T = S, the matrix E, gain matrices L and K, and ε j > 0 for j = 1, . . . , 12, such that the following LMIs hold Proof: The Lyapunov function is chosen as follows: and p = p T > 0 is the matrix to be chosen. The derivative of V (t) is taken aṡ Now, consider the following index Assuming zero initial condition, that is, x(t) = φ(t) = 0, we have V (t)| t=0 = 0. Then, for any non-zero v(t) ∈ L 2 0 ∞ , there holds, It is supposed that ζ = x(t) T v(t) T T , and Assuming the zero-disturbance input (v(t) = 0); if Eq. (32) is negative-definite ( 1 < 0), thenV (t) < 0 and the asymptotic stability of system in Eq. (21) is guaranteed.
where M, N, S, R, T and U satisfy the following equalities, since they result from the fact that pp −1 = I 2n .
In addition, define 1 and 2 as follows Noting Eqs. (36) and (37), it can easily be verified that p 1 = 2 . It follows that which implies that the matrices 1 and 2 in Eq. (37) are square invertible. It is established that the matrix p can be constructed as p = 2 −1 1 , and it follows from Eqs. (25) and (38) that p > 0. Inserting T 2Ā (t) Pre-and post-multiplying Eq. (39) by diag ( 1 , I, I) and its transpose (Congruent transformation), respectively, we obtain 1 Inserting matrices from Eq. (21), and 1 , 2 from Eq. (37) in Eq. (40), then the calculation of multiplying of matrices, we have wherê VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
Noting Eq. (18), we can separate Eq. (44) to the certain and uncertain parts, that is

By using Eq (19), it is easily obtained that
If we substitute Eq. (46) into the uncertain part of Eq. (45) and then use lemma 1, we can find the upper bound for each element as fallows where¯ is the same in Eq. (24). Eventually, by applying lemma 2 to each i with i = 1, . . . , 16, we can construct an LMI in the form of Eq. (24). Having a feasible solution for the conditions in Eqs. (24)- (25), and ε i > 0 with i = 1, . . . , 10, guarantees 1 < 0, which further implies that J ∞ < 0 in Eq. (31), and therefore Integrating both sides of this inequality from zero to any At the same time, it can be derived that x T (t)px(t) < γ 2 1 ∥v(t)∥ 2 2 + V (0). Furthermore, employing the Schur complement, the feasibility of the following inequality p. Under zero initial conditions, we have V (0) = 0 and V (∞) ≥ 0. Afterwards, it can be easily determined from Eq. (21) and Eq. (49) that for all t ≥ 0, Pre-and post-multiplying Eq. (49) by diag ( 1 , I) and its transpose (Congruent transformation), respectively, we obtain Noting Eq. (18), we can separate Eq. (51) to the certain and uncertain parts, that is If we substitute Eq. (46) into the uncertain part of Eq. (52) and then use lemma 1, we can find the upper bound for each VOLUME 11, 2023 46059 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. element as fallows By adding LMI parts of Eqs. (53)-(54) to the constant part of Eq. (52), and by applying lemma 2 to each 17 and 18 , we can construct an LMI in the form of Eq. (26). Therefore, having a feasible solution for the conditions in Eq. (26) guarantees ∥z 2 (t)∥ 2 L ∞ < γ 2 2 ∥v(t)∥ 2 L 2 . The control signal is generated by the hydraulic actuator and is limited due to its saturation. Remember that the GH 2 norm is defined as an L 2 −L ∞ induced norm (or 'energy to peak' norm). It guarantees that the L ∞ -norm (its max) does not exceed a certain maximum. Therefore, if the following matrix inequality holds, we can conclude that |u(t)| ≤ u max whereC = 0Ĉ . Pre-and post-multiplying Eq. (55) by diag ( 1 , I) and its transpose (Congruent transformation), respectively, we can construct an LMI in the form of Eq. (27). Therefore, the proof of Theorem 1 is completed. The active vehicle suspension system without inerter and parameter uncertainty can be defined by the following state-space equations [3]: Corollary 1 can be used to achieve the parameters of the constrained dynamic output-feedback robust H ∞ controller for the linear system without parameter uncertainties.
Remark 2: It should be noted that in order to find the non-singular matrix M in Eq. (43), we considered the matrix N to be matrix I.
Remark 3: Since the half-car suspension is a MIMO system, when solving LMIs (27) or (60), it should be solved separately for each inputs considering u f max and u r max .

IV. APPLICATION TO ACTIVE INERTER-BASED HALF-CAR SUSPENSION SYSTEM
In this section, to illustrate the effectiveness of the proposed controller in Section III, we will apply it to the active inerterbased half-car suspension system described in Section II. The parameters of the inerter-based half-car suspension system are listed in Table 1. Note that in our simulations we assume that m s , b sf , and b sr are uncertain. We use a variable mass profile (14% mass uncertainty) and a continuous uniform random distribution for the inertance of inerter (8% inertance uncertainty) to generate them, as shown in Fig. 2. The design constraints and the design parameters for the controller are shown in Table 2.   Considering that we can only measure x 1 , x 2 , x 3 , x 4 (we cannot measure tire deflection), the dynamic output-feedback for the controller is defined as follows By setting γ 1 = 8 , γ 2 = 12 , δ 1 = δ 2 = δ 3 = 0.1 and solving the convex optimization problem formulated in Theorem 1 employing the YALMIP toolbox [28], the parameters of the dynamic output feedback controller are determined using Eq. (42), as given in Appendix C.
And for brevity, we will indicate the proposed controller as Controller I hereafter. To assess the performance of the proposed controller, acquired results will be compared to the results obtained in corollary 1 to the constrained dynamic output-feedback robust H ∞ control for active vehicle suspension without inerter and parameter uncertainty, and will be denoted as Controller II for brevity. The parameters of the dynamic output feedback Controller II are determined with design parametersC 3 = C 3 ,γ 1 = 7 , andγ 2 = 9 , as given in Appendix C. Remark 4: Given the definition of the GH 2 norm, prior knowledge of road disturbances may be beneficial in determining an appropriate value for γ 2 . For a given energy of road disturbance, a larger γ 2 value allows for more variation of constrained outputs. Consideration of the normalized constrained outputs indicates that the γ 2 value is less than the inverse of the worst-case disturbance energy. In this study, it is considered as a parameter for the design of the controller.
According to ISO 2361, improving ride comfort is equivalent to minimizing the vertical and horizontal accelerations of the vehicle body. The human body is more sensitive to vibrations in the frequency range of 4Hz to 8Hz in the vertical direction and 1Hz to 2Hz in the horizontal direction [29]. Therefore, we first focus on the frequency responses from the ground velocity to the heave and pitch accelerations for the passive and closed-loop systems using the constrained dynamic output-feedback robust H ∞ controllers. From Fig. 3, we can see that the desired controller I and the controller II can provide the lower value of the H ∞ norm over the frequency range of 1Hz -8Hz.
Performance of the half-car suspension system is capable to be assessed by examining six response quantities, that is, the sprung mass heave accelerationz c (t), the pitch accelerationφ(t), the suspension deflection x 1 (t) and x 3 (t) for the front and rear wheels, respectively, and the tire deflection x 5 (t) and x 7 (t) for the front and rear wheels, respectively. In the following subsections, we will utilize Shock (Bump) and Vibration (Rough Road) road profiles to evaluate the performance of the half-car suspension system with respect to vehicle handling, ride comfort, energy consumption, and working space of the suspension.

A. BUMP RESPONSE
Here the bumps or potholes with relatively short duration and high intensity confronted in a smooth surface was considered to reveal the transient response characteristic, which is given by Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.  where a and l are the height and the length of the bump, respectively. We choose a = 0.1 m, l = 2 m and the vehicle forward velocity as v 0 = 18 km/h. Here, the road profile z rr (t) for the rear wheel is assumed to be same as for the front wheel profile z rf (t) but with a time-delay of (l 1 + l 2 )/v 0 . The response of the active half-car suspension system with inerter by using Controller I and without inerter by using Controller II, and passive suspension with inerter are compared in Fig. 4. Fig. 4 displays the heave acceleration, pitch acceleration, suspension deflection, and tire deflection for the front and rear wheels, respectively. The control efforts for the front and rear wheels of the active controllers are also plotted in Fig. 5. It can be seen from Fig. 4 that the Controllers I and II compared to the passive suspension system acquire better responses. It is confirmed by the simulation results that bump response quantities for heave and pitch accelerations of active inerter-based suspension system are better than active suspension without inerter. It should be noted that Controller I cannot show any significant improvement over Controller II for suspension deflection. On the other hand, the required control effort for Controller I is less than for Controller II, which is shown in Fig. 5.
To qualitatively evaluate the control efforts of two active control methods, their energy consumption is calculated through the following L 2 norm value: whereT = 3 s denotes the simulation time. Energy consumption of two controllers is shown in Table 3. It can be seen from this Table, active suspension system with inerter has a superior performance compared to the active suspension without inerter and low gain of the Controller I lead to the less required energy consumption. As mentioned in the introduction, the advantages of using the inerters are not only to improve the performance of the vehicle suspension system in terms of ride comfort, suspension travel, road holding, and energy consumption, but they can also generate the amount of electrical energy required to operate the actuator.

B. RANDOM RESPONSE
Generally, it is capable to assume random vibrations as road disturbances, which are consistent and typically specified as the random process. The ground displacement power spectral density (PSD) is defined as follows where 0 = 1/2π stands for reference spatial frequency and is a spatial frequency. The value of S g ( 0 ) denotes a measure for the roughness coefficient of the road. n 1 and n 2 represent the road roughness constants.
In particular, if the vehicle is presumed to travel with a constant horizontal speed v 0 over a given road, it is capable to simulate the force resulting from the road irregularities by the following series where s n = 2s g (n ) , = 2π/L, and L is the length of the road segment considered. The amplitudes s n of the excitation harmonics are assessed from the road spectra selected. Additionally, the value of the fundamental temporal frequency ω 0 is determined from ω 0 = 2π L v 0 . While the phases ϕ n are treated as random variables, following a uniform distribution in the interval [0, 2π).
According to ISO2631 standards, road class D (poor quality) S g ( 0 ) = 256 × 10 −6 m 3 , and road class E (very poor quality) S g ( 0 ) = 1024 × 10 −6 m 3 , are selected as a typical road profile. In this paper, n 1 = 2, n 2 = 1.5, L = 100, N f = 200 and the horizontal speed v 0 = 36 m/s, are utilized to generate the random road profiles as shown in Fig. 6.
The random response of the half-car suspension system for two road class profiles are compared in Fig. 7 and Fig. 8. These figures display the heave acceleration, pitch acceleration, suspension deflection, and tire deflection for the front and rear wheels, respectively. It can be seen from Fig. 7, and Fig. 8 that the Controllers I and II compared to the passive suspension system acquire better responses. It is confirmed by the simulation results that random response quantities for all performance requirements of active inerter-based suspension system are better than active suspension without inerter.
To assess the probabilistic properties of the random response, the Monte Carlo simulation is utilized. Therefore, taking into account the random variable ϕ n of the excitation applied, the performance index of the Root Mean Square  (RMS) is determined by the expected values: For sprung mass heave acceleration J 1 , pitch acceleration J 2 , suspension deflection for the front J 3 and rear J 4 , tire deflection for the front J 5 and rear J 6 , have been considered as the RMS values; whereT = L/v 0 is the temporal measurement period. For calculating RMS values, we have considered T = 5 in Eqs. (66)-(71) and the simulation has been run randomly 100 times.
To validate the effectiveness of controller I in dealing with the active inerter-based suspension system, the RMS ratios J I I i (t)/J I i (t), J P i (t)/J I i (t), i = 1, 2, 3, 4, 5, 6, are calculated, where J I i (t) denotes the RMS value of the active suspension system with inerter by using Controller I, J I I i (t) denotes the RMS value of the active suspension system without inerter by using Controller II, and J P i (t) is the RMS value of the passive suspension system with inerter.
Tables 4 and 5 represent the results of RMS ratios for Controller I, Controller II, and passive suspension system VOLUME 11, 2023 46063 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.  for the poor (class D) and very poor (class E) quality road profiles. The control efforts for the front and rear wheels of the active controllers are also shown in Tables 4 and 5.
It can be seen from Tables 4, and 5 that the RMS ratios of the active inerter-based suspension system with Controller I is always less than the passive suspension system (the response ratio is more than 1). On the other hand, it can be seen that heave acceleration, pitch acceleration, and tire deflection for the front and rear wheels with Controller I acquire better response compared to Controller II.
In addition, the required control effort for Controller I is less than for Controller II.

V. CONCLUSION
In this paper, the performance of the active inerter-based half-car suspension system in the presence of parameter uncertainties and external disturbance has been investigated. A constrained dynamic output-feedback robust H ∞ controller has developed to optimize the L 2 gain of the heave and pitch accelerations of the suspension system to enhance the ride comfort performance. Whereas, the generalized H 2 (GH 2 ) norm has been utilized to insert the hard limits of the suspension and tire deflections to guarantee that these performance criteria do not exceed their pre-specified maximum values. Furthermore, to restrict the gain of the controller, L 2 − L ∞ norm of input has minimized. Finally, to validate the effectiveness of the proposed approach, it has been applied to the active inerter-based half-car suspension system to minimize the influence of parameter uncertainties and road disturbance on the suspension system performance. It has been observed that for all performance requirements, the active inerter-based suspension system achieves better response compared to both the active suspension without inerter and passive suspension with inerter. The proposed method is expected to pave the way for the application of theoretical findings to practical vehicle suspension systems.    Most of his work is devoted to mathematical modeling, measurement and control in fluid power systems, including pneumatic and hydraulic servo drives and servo valves controlling flow or pressure and speed or force. Experimental verification of the assumptions introduced in the theoretical work is one of the most important elements of his scientific work. The application of modern control systems based on microcontrollers is also a part of his work. Currently, he is teaching students and is cooperating with industry in research and development projects. He is a member of the Polish Academy of Sciences, the Machine Building Committee, and the AGH Scientific Discipline Council for Automation, Electronics, Electrical Engineering, and Space Technologies. He is the coauthor of three books and more than 120 articles and holds six patents. His distinguished merit has contributed to dynamics and control of structures, specializing in experimental research. He has committed to help university students to develop their skills in signal analysis, system identifications, measurement systems, and parts of automation control. He has involved in didactic activity at three levels of study, such as, engineer, master's, and doctoral degree. He is an Associate Editor of Journal of Low Frequency Noise, Vibration and Active Control.