Research Article  Open Access
Disturbance Rejection in a OneHalf Semiactive Vehicle Suspension by means of a FuzzyH_{∞} Controller
Abstract
A fuzzyH_{∞} control, improved with weighting functions, has been designed and applied to a novel model of a onehalf semiactive lateral vehicle (OHSLV) suspension. The herein contribution resides in the development and computation of an H_{∞} controller with parallel distributed compensation (PDC) designed from a highly nonlinear system modelled via the Takagi–Sugeno (TS) fuzzy approach. A fuzzyH_{∞} controller is synthesized for an OHSLV TS fuzzy model of a suspension with two magnetorheological (MR) dampers including actuators’ nonlinear dynamics. The realism of results has been improved by considering the MR damper’s behaviours (viscoplasticity, hysteresis, and saturation) and the handling of the phase angle of the sinusoidal disturbance, not included in other reported work. Timedomain tests remark transient time achievements, whereas precise performance criterion indices in the frequency domain are employed to assess the generated outcomes. The proposed solution complies with all performance criteria compared with a benchmark passive average suspension that fails in satisfying most of the performance criteria.
1. Introduction
Ground vehicle suspension systems provide a certain level of passenger comfort and vehicle stability by covering a set of basic functions such as supporting vehicle’s weight, keeping tires in contact with the road, holding an optimal height of the vehicle, and isolating passengers against vibrations from road’s disturbances, among others [1]. From the variety of available suspensions, semiactive solutions have proven their contribution to an acceptable level of simultaneous comfort and stability. Filled with electrorheological (ER) or magnetorheological (MR) fluids [2, 3], they can modify their viscosity from liquid to semisolid in less than ten milliseconds [4]. Due to the advantages with respect to ER dampers, this research employs MR dampers for the suspension [5, 6].
Considering that MR dampers have nonlinear phenomena such as saturation, hysteresis, and dynamics of a fluid going through an orifice [7], obtaining an accurate modelling becomes a critical task when applying them for vehicle suspensions. One of the most employed MR damper nonlinear representations is the Bouc–Wen model, which is handled in this research [4].
The baseline study in vehicle suspensions is the passive onehalf vehicle suspension. The analysis on ride comfort and vehicle stability can be obtained through a halfvehicle model, where roll or pitch dynamics are added to the vertical motion [8]. If the focus of the study is to analyse the effect of road irregularities, road bumps, and potholes on vehicle stability and passenger comfort, studies should include the pitch angle; however, if the aim is the vehicle’s behaviour during road curves, the roll angle is the adequate option [9].
The rationale of this research is to improve the deficiencies exhibited by the passive suspension in the time domain and frequency domain, although working with semiactive suspensions increases the system’s complexity. This study works with a semiactive vehicle suspension model that includes vertical and pitch dynamics through the socalled bicycle model, a fourdegreeoffreedom (4DOF) system also known as the suspension lateral model [10]. This representation merges both front wheels into a single front one, and both rear wheels in the same manner. The lateral model includes vertical motion of the concentrated front and rear wheels, along with pitch and vertical dynamics of the sprung mass, and it is portrayed in Figure 1.
In Figure 1, it is assumed that front and rear suspensions have the same characteristics, and the centre of mass is not exactly in the middle of the vehicle; thus, distances a and b, measured from the centre of gravity (COG), are different. Moreover, front m_{tf} and rear m_{tr} unsprung masses could be different as well. To analyse the pitch angle θ for the sprung mass m_{s}, a moment of inertia I_{y} with respect to horizontal and transversal axes is required. Furthermore, k_{sf} and k_{sr} represent the front and rear suspension springs’ constants, respectively, whereas c_{sf} and c_{sr} are the dampers’ constants. In addition, tires’ stiffness is represented by k_{tf} and k_{tr}. All Z variables describe vertical displacements (Z_{s}, Z_{tf}, and Z_{tr} are related to the system’s masses, whereas Z_{rf} and Z_{rr} illustrate disturbance inputs from the road profile).
The state of the art includes outstanding control proposals for 4DOF onehalf vehicle suspension with vertical and lateral dynamics. Pan and Fan [11] developed a variable universe fuzzy control system with variable damping and stiffness, whose results are compared against passive suspension and common fuzzy controller. Sun and Yang [12] developed an adaptive fuzzy PID controller for a 5DOF suspension with good outcomes in the frequency domain. Adaptive controllers have been derived in their own research direction. Sun et al. [13] proposed an adaptive backstepping control strategy with parameter uncertainties, and their controller design considered actuator’s nonlinearities. Sun et al. [14] introduced a faulttolerant adaptive robust controller able to compensate parameter uncertainties and disturbances. Krauze and Kasprzyk [15] considered road profile information as a preview signal and designed an adaptive feedforward leastmean squares (LMS) algorithm for a semiactive suspension with a Bouc–Wen model and generated results in the frequency domain. Sun et al. [16] developed an adaptive vibration control strategy for nonlinear uncertain suspension and employed a hydraulic cylinder as the actuator. Another research line is state/output feedback and H_{∞} controllers. Kong et al. [17] designed an H_{∞}static output feedback controller (SOFC) based on particle swarm optimization (PSO) and differential evolution (DE) algorithms. Li et al. [18] applied multiobjective control to synthesize an H_{∞}/H_{2} SOFC, whereas Suzuki et al. [19] reported a robust H_{2} control to improve comfort and stability. Wang et al. [20] tested a SOFC based on variable substitution as an optimization algorithm to relax restriction during calculation of a feasible gain matrix.
More recent results have focused on semiactive suspensions with MR dampers. Krauze and Kasprzyk [21] employed the damper’s Spencer model and developed a LQ statefeedback controller to improve two performance criteria in the frequency domain. Mahdi [22] designed an LQRchaos controloriented system based on an optimal Ott–Grebogi–Yorke approach with relevant results in the time domain. Wu and Liu [23] reported an MR semiactive system represented with a piecewise approximation model that modelled nonlinearities with a hyperbolic tangent function. Results were generated in the frequency domain. In addition, Pang et al. [24] developed a fuzzy controller for an experimental MR semiactive suspension based on neural networks and particle swarm optimization.
All these contributions are notable results in frequency and time domains, and their outcomes were compared against a passive or active benchmark suspension. Even though the state of the art provides important improvements in passenger comfort and vehicle stability, for the best of authors’ knowledge, none of them individually reports a solution that considers actuator’s nonlinear dynamics in controller computation, as well as timedomain and frequencydomain tests paired with performance indexes. Moreover, an opportunity area is the lack of study of the phase angle in the profile signal when it is analysed as a sinusoidal input.
This work considers the reported 4DOF onehalf vehicle suspension lateral model in the study of FélixHerrán et al. [25] and the fuzzyH_{∞} controller for a onequarter vehicle developed in the study of FélixHerrán et al. [26], and it includes the disturbance phase angle and computes a fuzzyH_{∞} controller that complies with a set of performance criteria. This article is organized as follows: Section 2 states the performance criteria, and Section 3 provides a brief review of the controloriented onehalf suspension model applied herein. Section 4 reviews some peculiar aspects when employing sinusoidal signals as disturbances from road profile, Section 5 solves the fuzzyH_{∞} problem for the TS nonlinear suspension, Section 6 presents the numerical case study and carries out a comparative analysis between the passive and the proposed semiactive solution. Finally, conclusions and brief information about further research work complete the article in Section 7. It is out of the scope of this report to develop the MR damper’s characteristics and to review the Takagi–Sugeno fuzzy model of a onehalf semiactive vehicle suspension: lateral approach.
2. Performance Criteria
In automotive suspensions, the most important aspects to consider are passenger comfort and vehicle stability. The goal is to have a suspension system that meets both performance indices simultaneously, and therefore, these benchmarks must be considered during the controller design process. Before criteria are presented, it is important to give more information about passenger comfort and vehicle stability.
The comfort in this study refers to the performance index related to the vibrations generated from the road profile. These vibrations travel through the physical elements of the vehicle and are transmitted to passengers, causing annoyance when vibrations are inside a range of frequencies (human body resonant frequencies are around 5 Hz [27] and exceed certain specified amplitude values). For example, human dizziness and motion sickness are stronger around 1 Hz. Even though passenger comfort level is very subjective (persons are different regarding their physical aspects such as anthropological and sensorial capacities), it is necessary to come up with a standard to measure passenger comfort.
Vehicle stability is related to steering wheel changes and disturbances from the environment, e.g., road profile irregularities. In this research, stability is focused on the suspension’s ability to keep the tires in contact with the road surface against external disturbances (the road profile) [28].
2.1. Time Domain
The objective is to keep all the variables of interest inside the physical limits and to reduce, as much as possible, the overshoot and the settling time of chassis displacement, tire displacement, chassis acceleration, suspension deflection, pitch angle, and pitch acceleration when compared with a passive suspension [29–33].
2.2. Frequency Domain
The focus is to evaluate vertical and pitch suspension performances of average city vehicles in terms of some frequencydomain indices. Accepted input signals for these tests are Z_{r} = 0.015 sin ωt(m) for low frequencies and Z_{r} = 0.001 sin ωt(m) for high frequencies [28–30]. The benchmarks are described below:(1)Passenger comfort at low frequencies (0–4 Hz): limit the relation (gain) chassis displacement/road profile to be less than 2.0, nearby the sprung mass resonance frequency for an average city vehicle, i.e., 1.1 Hz.(2)Road holding (0–15 Hz): limit the relation (gain) tire displacement/road profile to be less than 1.8, around the sprung mass resonance frequency, i.e., approximately 10 Hz for a compact city vehicle.(3)Passenger’s comfort at high frequencies (4–30 Hz): maintain chassis rms (rootmean square) acceleration below the limits reported in the study of Wong [28] to ensure a passenger comfort condition for 8 hours.(4)Suspension travel (deflection) within physical limits (0–4 Hz): this is a restriction that must be considered when evaluating vehicle suspensions. Deflection must always remain within the damper’s physical limits to increase its useful life and avoid nonmodelled dynamics when limits are reached.(5)To the best of authors’ knowledge, no specific criterion related to θ has been found in reported work. The aim in this inquiry is to decrease the pitch angle θ, as much as possible, between 0 and 4 Hz, compared against a reference passive suspension, as in [31].(6)Wong [28] displayed a benchmark proposed by the ISO (International Organization of Standardization). The criterion illustrates the maximum allowable rms chassis pitch angle acceleration within a frequency range of interest. Like rms vertical chassis acceleration, the author provides curves that bound the maximum value to ensure passenger comfort during certain time. The goal is to have 8 hours of passenger comfort for pitch acceleration.
Because the performance criteria for vertical and pitch acceleration are expressed in and the generated herein results are presented in , it is required to have an equation to relate both units. To develop this idea, D’Souza and Gang [34] dealt with equations that measure the moment of inertia I for a set of different solids rotating with respect to an arbitrary axis. Considering a bicycle model vehicle’s sprung mass, m_{s}, is viewed as a cylinder of radius r that rotates around a pitch axis y, the following equation holds:
From equation (1), r can be expressed as follows:where I_{y} has previously been defined for the sprung mass. Thus, with the calculated r, angular acceleration is converted into a linear acceleration as presented below:
3. OneHalf Semiactive Suspension Model
The passive dampers have been replaced by magnetorheological elements to achieve different damping forces in real time. The target system is a 4DOF onehalf suspension with two MR dampers, as in Figure 2. Dynamic equations and some other basic aspects of a general bicycle model are developed by Rajamani [35].
From Figure 2, the following state variables were defined: , , , , , , , , , and ; and ; and and . The suspension system is defined in the following equations:where , , and .
The proposed OHSLV system contains two MR dampers, and each damper provides two nonlinearities to the whole system. As a result, to are the four nonlinear terms to model, and they are defined in the following equations:where , , and . In addition, and are defined bywhere is equal to −1 if ζ < 0, 0 if ζ = 0, and +1 if ζ > 0. Explained in the study of Tanaka and Wang [36], 2^{n} linear subsystems are generated from n nonlinearities. Hence, Z_{1} to Z_{4} are replaced with 16 linear subsystems interconnected with fuzzy membership functions (M_{1}, M_{2}, N_{1}, N_{2}, P_{1}, P_{2}, Q_{1}, and Q_{2}) and linking functions (h_{1} to h_{16}). The whole system is fully expressed with the following Takagi–Sugeno fuzzy model where each nonlinear term was represented by two linear systems interconnected via fuzzy membership functions. Further explanations about the MR damper’s composition and its behaviour are extensively explained in [4], whereas the complete development of the halfvehicle semiactive suspension model with the MR dampers is found in [25]:where is the state vector, represents the command input, and is the disturbance signal coming from the road profile. Subindex i refers to the ith linear subsystem, whereas is the controlled output vector that complies with previously stated performance criteria.
For further control purposes, the state vector contains all relevant system’s state variables, and it is defined as and associated with A_{i}, whereas the command input vector [i_{f}i_{r}]^{T} is related to B_{i}, and the disturbance input vector [Z_{rf}Z_{rr}]^{T} is linked to B_{w}.
4. Considering a Realistic Scenario
When working with a onehalf vehicle suspension model, an important aspect is the disturbance input phase angle. The 4DOF onehalf vehicle suspension has two external inputs coming from the road profile (Z_{rf} and Z_{rr}). It could be assumed that in real life, sinusoidal signals representing irregular surfaces have the same amplitude and frequency, but the phase angle among them is different. This is due to vehicle’s length, translational velocity, and uneven road profiles.
The authors of this study consider that phase angle φ must be included in simulation work to consider real conditions. Ignoring angle φ or if both tires are always in phase is unrealistic. For example, Figure 3 portrays a vehicle moving with translational velocity . Signals Z_{rf} and Z_{rr} are always in phase, which means that φ is always zero; seldom true in real suspension systems. Two unrealistic scenarios are illustrated (both disturbances are exactly on top of the sinusoidal signal, or both at the bottom).
A more realistic scenario is depicted in Figure 4, where φ is not zero, i.e., Z_{rf} and Z_{rr} are not in phase.
To estimate phase angle φ some calculations must be performed from previous knowledge of (translational velocity), f (frequency), and distances a and b. Velocity , usually expressed in , must be translated into , by
Moreover, the wavelength λ of a sinusoidal wave can be defined as the spatial period of the wave, i.e., the distance over the wave, before the pattern repeats itself [37]:where f is the wave’s frequency, i.e., number of instances per unit time of a repeating event, whereas T is the signal’s period; that is, the duration of one cycle in a repeating event [38]. In equation (22), λ is defined as follows:
In Figure 5, it can be noted that generates an integer when Z_{rf} and Z_{rr} are in phase; however, as mentioned before, this is an occasional situation. The generic relation between vehicle length (a + b) and wavelength λ is given as follows:where E is an integer and F is a fraction that represents the phase angle φ, as in Figure 5. Equation (24) is the baseline to obtain F:where F is a fraction of one entire cycle of the wave, and it is closely related to the phase angle φ. For that reason, F has a value between 0 and 2π (radians). From equation (25), φ could be measured in degrees and radians as presented in the following equations:
In Figure 5, the vehicle length a + b is constant, and the graphical magnitude of φ is measured from the rear dot closest to the amplitude axis, to the wave’s peak. The herein simulation work calculated φ for each frequency applied value and included it in the input signal, a disturbance coming from the road profile. To the best of authors’ knowledge, this phase angle φ consideration for road profile inputs Z_{rf} and Z_{rr} has not been reported yet.
5. Controller Design
For this research, control design effort focused on stability and frequencydomain performance via a fuzzyH_{∞} controller. Even though the first approach in control design was a static state feedback controller (SSFC) to guarantee stability in closed loop, there were not enough degrees of freedom to design based on performance criteria; hence, a fuzzyH_{∞} controller became an attractive option (the closedloop diagram is presented in Figure 6).
5.1. FuzzyH_{∞} Controller to Achieve Stated Performance Criteria in Frequency Domain
The nonlinear onehalf suspension model is a set of linear subsystems interconnected via fuzzy membership functions [25]. The applied closed loop is as follows.
For the OHSLV, each TS model’s linear subsystem can be represented by a linear timeinvariant (LTI) system, as defined in equations (19) and (20). The applied control approach referred to the H_{∞} norm stated aswhere the right part of the equation represents the largest singular value σ for the transfer function , which is the ratio (controlled output z/input disturbance ) along a frequency range. It is important to mention that an H_{∞} approach seeks to keep norm below a target value, along a frequency of interest. In this research, the H_{∞} controller aims to obtain a fuzzy controller that guarantees stability as well as a closed loop system with a γ disturbance rejection level. The general formulation has been reported in the study of Assawinchaichote et al. [39] and presented in the form of linear matrix inequalities [40] as in the following equations:where N is the number of subsystems, γ > 0, and
Individual controller gains K_{j} were calculated for each linear subsystem in the onehalf vehicle suspension. Moreover, the total fuzzy control law was composed through parallel distributed compensation (PDC) [36, 41]:where the set of must comply with
For the system defined in equations (19) and (20), the controller synthesis employed matrices C_{z}, D_{u}, and D_{w} as follows:
The proposed equations (34) and (35) adhere to the state, command input, and disturbance input vectors described at the end of Section 3. An analysis on equation (34) remarks that the fuzzyH_{∞} approach impacts on the unsprung mass displacement and on the front and rear tires’ displacements.
5.2. Modification to the Control Strategy via Weighting Functions
This work applied weighting functions [42] in the controller gains’ computation to improve the control outcomes. According to Son et al. [43], a valid weighting function must be stable, rational, and represented with minimum phase transfer function, i.e., all poles and zeros must be in the half left plane of the socalled S plane.
To improve comfort and stability performance, three weighting functions were added to control synthesis in equation (29). For lowfrequency performance, i.e., 0.5–4 Hz, a highpass filter was included. For highfrequency enhancement (beyond 7 Hz), two lowpass filters, one per unsprung mass, were considered in the controller’s calculation. These weighting functions (W_{f}) of affine transfer functions changed the norm definition in equation (28) to be as follows:
The modified system was verified for stability and detectability [44]. It is important to highlight that weighting functions were incorporated to obtain a controller; i.e., it is only for the controller design. The next step is to define the structure and numerical values for W_{f}. For this research context, onehalf suspension corresponds to a city vehicle with sprung mass resonant frequency between 1.1 and 1.5 Hz, whereas unsprung masses hold resonant frequencies in a range of 2.3 to 3 Hz. [35]. The second resonant peak for tire displacement is near 10 Hz. With this information, two firstorder highpass filters, with similar cutoff frequency, were proposed, and their transfer function is defined as follows:where , and the first order lowpass filter has the following transfer function:where . The cornerfrequency numerical values and are presented and explained in Section 6. In addition, equations (37) and (38) are transformed into the statespace domain to achieve the following equations:where f_{s} stands for the filtersprung mass, fu1 refers to filterunsprung mass 1, and fu2 symbolizes filterunsprung mass 2. The augmented system is represented as follows:where the desired output vector z_{f} should comply with the performance criteria in Section 2. Equations (43) and (44) hold for each linear subsystem of the TS fuzzy model.
5.3. IllConditioned Matrices and LMIController Computation
It is widely known that a system in the state space can be represented in many ways without altering the system’s eigenvalues and that this representation has an impact on controller calculation [45]. Although from one realization to another one, the state, input, and output matrices have different numerical values, the eigenvalues do not change, and overall behaviour remains the same. Furthermore, the eigenvectors are different among representations, but they are related by means of a transformation matrix. A general transformation matrix T is defined in the following equation:where stands for the modified and the statespace realization is as follows:
For the target system in equations (43) and (44), the transformation matrix T generated the following representation:
When solving a set of LMIs in MATLAB, the convergence to a solution depends on the statespace realization [40]. In this study, during the computation process of K_{j}, it was observed that the calculated gains were very small and a hypothesis was stated. Due to the matrix ill conditioning (directly generated from the fuzzy TS representation), it was difficult to calculate a set of K_{j} that fulfilled the feasibility problem for solving all the LMIs. Authors tested different statespace representations and through the modal canonical transformation [46], larger values of K_{j} were computed to generate a more significant command vector [i_{f}i_{r}]^{T}. The exploration of different statespace representations to solve the LMIs was an important part in the design and computation of the feedback gains.
Due to the transformation, a modified gain K_{t} to work for a modified vector xt was calculated. However, after the gain was computed, the TS fuzzy model was left aside, and the controller was applied to the original nonlinear differential equations system; therefore, the controller gains must be returned to the original system’s coordinates as follows:
In a TS fuzzy model constituted by a set of linear subsystems, several controller gains K_{i} are going to be obtained and equation (49) is employed as follows:where K_{ti} is the computed gain K_{i} for the transformed linear subsystem and T is the transformation matrix.
6. Case Study
Simulation results based on real data support the theoretical work in previous sections. The MR damper Bouc–Wen model numerical data were taken from a realistic damper’s characterization [47], whereas the rest of parameters were adopted from [25]. MR dampers do not respond instantaneously to current changes; they are modelled as firstorder systems with a time constant η = 0.052 ms [4]. The herein research employs a damper’s maximum extension of 2.5 cm. In addition, front and rear suspensions and tire stiffnesses are the same, and dampers c_{sf} and c_{sr} are the referenced damping values (passive case). Furthermore, the centre of gravity is not in the middle of the vehicle’s length; thus, a is different from b. It is worthwhile to remember that distances a and b are measured from the vehicle’s COG to the point where the chassis is in contact with the front and rear suspensions, respectively. Timedomain and frequencydomain tests consider the realistic scenario about the phase angle between Z_{rf} and Z_{rr}, as explained in Section 4. Table 1 lists all numerical values employed in the simulation work.

For the weighting functions, the cutoff frequencies f_{c} were chosen based on the suspension’s resonance peaks and finely tuned to trial and error. One highpass filter with f_{c1} = 5.5 Hz and two lowpass filters with f_{c2} = f_{c3} = 2.3 Hz were selected to decrease the suspension’s resonant peaks. This information was considered during the fuzzyH_{∞} controller synthesis.
Based on equations (47) and (48) and considering the baseline fuzzyH_{∞} LMI formulation in equations (29) and (30), the controller’s gains were calculated. The goal was to minimize the target variables contained in equation (44). It is relevant to remark that even though the illconditioning matrix behaviour was reduced with the applied system’s transformation, the best computed value for γ was equal to 4.0; i.e., this was the smallest γ value that satisfied feasibility LMI problem programmed in MATLAB.
All the simulation work was carried out in MATLAB Simulink. From the MATLAB workspace, the values of the constants were loaded, and the tests were executed in time and frequency domains with the suspension systems developed in Simulink. Figure 7 presents the block diagram about the onehalf semiactive suspension system with the fuzzyH_{∞} controller.
6.1. TimeDomain Tests
This test refers to the system’s response when the disturbance is a road bumplike signal of 4.0 cm high. To describe this behaviour, a cosine function like an inverted bell was employed to represent the road bump, as explained in [32], also applied in [48]. The disturbance is described in equations (51) and (52), also depicted in Figure 8:
Another important issue about the disturbance is that the road bump excites the front tire, and after some time, it affects the rear tire. The time between the front and rear wheels depends on the distance (a + b) as well as the vehicle’s forward velocity, . The road bump signal seeks to comply with the realistic disturbance scenario explained in Section 4; thus, the simulation effort computed φ for each frequency testing value and included it in the input signal.
Timedomain suspension responses are depicted in Figures 9–16. The performance of the fuzzyH_{∞} controller is compared against a passive suspension system with c = 1,000 Ns/m. Simulation time considers ; the required time to measure the variables of interest.
Table 2 extracts the key variables from Figures 9–16 to compare suspensions’ performance via 16 criteria. Mp stands for the maximum value in cm for Z_{s} and suspension deflections and in degrees for the pitch angle displacement. A_{tt} refers to the tire attenuation percentage of the displacement with respect to the road profile. Val is the peak acceleration numerical value registered for COG chassis and pitch angle accelerations. Moreover, the settling time (t_{s}) was compared in quantitative and qualitative manners, depending on whether the steady state was reached in the simulation time or not.

In Table 2, the semiactive suspension improved nine criteria, four indexes had similar results, and three requirements had better results with the passive suspension. Regarding chassis, front and rear suspensions, and pitch suspensions, the improvement was detected at the second transient overshoot and in the settling time as well. About physical limits, the two suspensions reached the maximum dampers’ limits. In addition, both suspensions exhibited a similar performance for tire transmissibility. For COG chassis acceleration, the fuzzyH_{∞} solution improved the settling time. Finally, pitch angle acceleration for the semiactive proposal performed a drawback in the settling time. In general terms, under the testing conditions established herein, the proposed semiactive suspension in closed loop had a better performance than the passive suspension.
6.2. Comparison with Reported Work
As mentioned in the introduction, other authors have also obtained results that improve the passive suspension performance. Although it is complicated to compare the outcomes of this study with other one because disturbance characteristics and numerical suspension’s parameters are different and research efforts focus on different process variables, a modest comparison among recent reported work is accomplished to provide some external validity to this research. From the reported outcomes in the time domain (response to a bumplike disturbance), this inquiry calculated the percentage of achieved improvement with respect to a benchmark passive suspension.
Wang et al. [20] developed an optimized static output feedback controller and reduced by nearly 90% the maximum peak value registered for and . Pang et al. [24] implemented a variable universe fuzzy control with fuzzy neural networks and particle swarm optimization and reduced and by approximately 39%, whereas suspension deflections (front and rear) were reduced by 33%. In addition, Benariba et al. [49] developed a suspension with sliding mode control supplemented with Lyapunov surfaces and managed to reduce the Z_{s} and θ by approximately 40%, while the deflection was reduced by 70%. The above results are compared against those obtained herein. A reduction of 10% was achieved for Z_{s} and θ, and 8% for and . Although the differences between suspension performances are considerable, this study has the advantage of measuring a larger set of process variables in the time domain and including tests in the frequency domain with clearly defined performance indices. It is important to highlight that all performance indexes, time domain and frequency domain, comply with the defined criteria in Section 2.
6.3. Frequency Domain Tests
The signal inputs for frequencydomain simulations were defined in Section 2.2. Frequency results are presented as descriptive functions, also known as pseudobodes [50], and they are illustrated in Figures 17–24.
For suspension deflection, this must be within MR damper’s physical limits (±2.5 cm for the MR dampers in this case study) as portrayed in Figures 18 and 19. Distances a and b are measured from the vehicle’s COG to the point where the chassis is in contact with the front and rear suspensions, respectively.
The authors did not find a reported maximum or upper boundary value for pitch angle θ to state that a comfort index is met; however, the objective is to reduce this angle as much as possible to have more stability; i.e., pitch angle magnitude is inversely proportional to stability [31]. Figure 22 depicts the pitch angle for the passive and semiactive suspensions. The maximum pitch angle with the passive suspension is 1.1°, whereas a top angle of 0.9° is obtained with the semiactive suspension. Hence, for the experiment carried out, the semiactive suspension is 18.2% more stable than passive solution.
Figures 23 and 24 portray and . The wide black line indicates a passenger’s comfort limit. In the former one, the limit refers to the maximum allowable rms vertical chassis acceleration of the chassis, whereas the latter figure includes rms maximum limit for lateral acceleration over pitch axis. In Figure 24, pitch angle acceleration is expressed in m/s^{2}, as result of having applied equation (3). Both chassis and pitch angle accelerations are rootmeansquare values. If accelerations are maintained below the marked boundary, it can be stated that a passenger’s comfort will be preserved for 8 hours as explained in [28].
The results in Figures 17–24 are summarized in Table 3, where the numerical values are the highest possible in the testing range. Compared with a reference passive suspension, the proposed fuzzyH_{∞} solution complies with all criteria.
It is worthwhile to remark the defined performance criteria in Section 2 because they help to highlight the relevance of the obtained results. For low frequencies, the passive suspension was not able to keep the (chassis displacement/road profile) relation below 2.0, and it also could not maintain the gain (tire displacement/road profile) smaller than 1.8 around the sprung masses’ resonance frequencies. In the terms defined by the performance criteria, the chassis was not able to isolate passengers from the road disturbances, nor to guarantee the tires to be always in contact with the floor. The same situation applies for suspension travel at lowfrequencies. The passive suspension failed to keep it inside the physical range and this problem reduces the damper’s useful life. These three difficulties were solved with the proposed solution.
The suspension with MR dampers improved the performance criteria and physical restriction maximum values in almost all cases, except for the rms and ; however, both criteria complied with the comfort criterion reported by Wong [28]. A quantitative comparison between both performances highlights the relevance of the herein outcomes. When comparing the table’s first four performance index maximum values, it is observed that fuzzyH_{∞} improved the performance of the passive suspension between 30% and 35%. Moreover, the pitch angle was enhanced by 18%, and the rear suspension travel was reduced by 53%. The opportunity areas are observed in the rms and , where the passive suspension performed better.
7. Conclusions and Future Work
It is possible to synthesize a competitive fuzzyH_{∞} controller for a onehalf semiactive vehicle suspension with two MR dampers that considers actuator’s dynamics and weighted functions during controller design. Due to all nonlinearities in the control design, the design challenge increases due to additional restrictions, but it contributes to more realistic results. To the best of author’s knowledge, there is not reported work that considers actuator’s dynamics in the controller computation for this type of system.
The proposed control complied with all frequencydomain requirements and increased considerably the passive suspension performance in the time domain Two additional performance criteria, not included in the reviewed work, had been considered. Root mean square vertical and pitch chassis accelerations were measured and compared with benchmarks proposed by the International Organization of Standardization. Although indices of ride comfort at high frequencies were not improved, they were kept below the limit. The developed research work is justifiable due to the enhancement achieved with the obtained semiactive suspension. Moreover, another contribution is the analysis in the disturbance signal phase angle between front and rear tires. This realistic and relevant issue is not mentioned in previous reported work.
This research has some drawbacks. Each time the state vector is sensed, several membership functions are calculated to generate the control signals towards both dampers. This computational load is added to the operations demanded by the parallel distributed compensator (PDC). This could have a negative impact on the implementation stage. Another disadvantage in the complexity is related to controller computation. A 4DOF suspension with 4 nonlinearities generates 16 linear subsystems and 256 linear matrix inequalities. This is a considerable number of restrictions to satisfy as part of the controller’s synthesis.
Currently, authors are exploring modifications to the fuzzyH_{∞} controller formulation. To improve the performance criteria, an H_{2} approach could consider minimal energy aspects. This last strategy requires to deal with greater complexity during controller computation because additional restrictions must be satisfied. In addition, if the control problem is reformulated, γ could pass to the filters’ gains to satisfy the set of LMI while keeping its value less than 1.
Another research line is predictive control. If the length of the vehicle is known, and a disturbance affects the front wheel, after certain time, there will also be an impact on the rear wheel. This information could be employed for control purposes based on an estimated prediction for the back suspension.
A third option is the inclusion of roll angle and/or the yaw dynamics. A current onehalf vehicle could be complemented with the roll angle analysis, which is related to rollover and steering stability criteria. The authors consider that this research work has potential towards the fullvehicle suspension control, i.e., vertical, pitch, roll, and yaw dynamics.
Finally, to assess the computed controller in a physical testing system would complement these results. A test bed with MR dampers, springs, tires, and the rest of mechanical components would help to detect possible adjustments when moving from an ideal simulation environment towards a more complex scheme with other external variables such as data acquisition, computational cost of the control algorithm, signal filtering, and mechanical couplings, among others. Although the necessary infrastructure for this type of experiments is not available, it could be managed with partners from another educational institution.
Data Availability
The required data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors are grateful to Tecnologico de Monterrey, Escuela de Ingeniería y Ciencias, through the GIEERobótica for funding the publication cost of this research.
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Copyright © 2019 L. C. FélixHerrán et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.