Enhancing the trade-off between ride comfort and active actuation requirements via an inerter-based passive-active-combined automotive suspension

Ride comfort is an important indicator to evaluate the dynamic performance of automotive vehicles. One method for improving ride comfort is to incorporate an active actuator into the passive suspension. However, the improvement is closely linked with the required actuation power and force. Increased actuation requirements may lead to higher energy consumption and a larger actuator size. To enhance the trade-off between ride comfort and active actuation requirements, a new design approach for searching for the optimal passive-active-combined suspension across many different mechanical component layouts incorporating an inerter is proposed in this paper. Compared with traditional designs where the suspension is limited to a few special layouts, via this approach the optimal passive part among all network possibilities with pre-determined numbers of each element type (springs, dampers and inerters), and optimal controller parameters of the parallel active actuator can be identified. Considering a quarter-car model, with a benchmark combined active-passive suspension in which the passive part is a spring-damper, the optimal inerter-based suspension can reduce the active force by more than 48% and regenerate 6 W more average power in the active part while achieving the same ride comfort. Note that in this case, two constraints are always considered: the road-holding ability (as indicated by the dynamic tyre load) and suspension travel of the identified suspension will not be worse than those of the benchmark suspension. The optimal trade-off obtained with this approach serves as a powerful tool in the automotive suspension design as it provides guidance on the following three aspects: the optimal ride comfort, the minimum power consumption and the feasibility of specific actuation requirements, where the second point is obtained in conjunction with the power conservation theorem proposed in this paper which proves that the total power consumed by the passive and active parts will not vary with changes in the passive part and active controller parameters.


Introduction
The running automotive will be subject to the excitation from road irregularities, which may cause discomfort to the driver and influence driving safety.In order to improve the ride quality, researchers proposed incorporating an active actuator to supplement the passive suspension and studied various control strategies [1][2][3][4][5], but it is found that the level of improvement is linked to the level of energy consumption and actuator force.These increased actuation requirements may lead to some negative issues, such as reduced reliability, larger actuator dimensions and costs.Therefore, when designing a passive-active-combined suspension, the question of how we might achieve the optimal trade-off between ride comfort and active actuation requirements should be considered.
The traditional approach designing the active part after the passive part has already been fixed cannot reach the full potential of the whole suspension, which will compromise the trade-off between ride comfort and active actuation requirements [6,7].To tackle this, Smith et al. [8] and Corriga [9] proposed to first design an active controller and then synthesise a part of it with passive components to alleviate actuation requirements of the remaining active part.Furthermore, Sun et al. [10] suggested using the integrated passive and active optimisation to tune suspension parameters together.Herber et al. [11,12] investigated more network possibilities of the combined suspension using the graph theory and identified the configuration which achieved the optimal performance considering passenger comfort, handling performance, and control effort.These studies only considered the cases where the passive part only consisted of springs and dampers.
It is reasonable to anticipate that further trade-off improvements can be achieved by integrating an inerter into the passive part of combined suspensions.The inerter is a passive element with the property that the generated force is proportional to the relative acceleration across its two terminals [13].The application of the inerter in passive suspension systems was first investigated in [14] and then various beneficial inerter-based layouts have been identified in the vehicle field [15][16][17][18].The concept of semi-active inerter has also been studied [19][20][21][22][23].For example, Chen et al. [21] demonstrated that a semi-active inerter can improve the force tracking performance of a vehicle suspension.Hu et al. [22] proposed to use semi-active means to realise an ideal skyhook inerter and verified its effectiveness.A hydraulic prototype of the semi-active inerter was developed by Wang et al. [23], which achieved two adjustable inertance.There were also studies combining an inerter-based passive device with an active (or a semi-active) controller.For example, Matamoros-Sanchez and Goodall [24] demonstrated the beneficial effects of the inertance in ride quality enhancement.Liu et al. [25] proposed a hydraulic electric inerter that combined a hydraulic inerter with a controllable linear motor.Ge et al. [26] designed an active tuned inerter damper based on the combination of an active actuator and a flywheel.Chen et al. [27,28] studied the combination of an inerter-based passive part with a semi-active damper, where in [27] the passive part adopted six fixed-structure networks and in [28] the passive part was formulated via network synthesis.While the above-mentioned studies have shown promising results, the answer to the following question is still unknown: to achieve the optimal trade-off between ride comfort and active actuation requirements, what is the optimal combination between a passive part with a given number of springs, dampers and inerters and an active controller?
The authors partially answered this question in [29], but only the combined suspension consisting of a three-element passive network and the skyhook controller was considered.To more fully address this problem, this paper proposes a systematic design approach for inerter-based passive-active-combined automotive suspensions, which allows the identification of the optimal passive configuration among all network possibilities with pre-determined numbers of each element type (springs, dampers, inerters), and optimal controller parameters of the parallel active actuator.The contributions of this approach are twofold: (1) The optimal trade-off between ride comfort and two active actuation requirements (the root-mean-square (r.m.s.) active force and the average power consumption) can be achieved.For example, with the same ride comfort level as the traditional combined suspension, the optimal one can reduce the active force by more than 48% and regenerate 6 W more average power in the active part; (2) The approach can provide guidance on the automotive suspension design in the following three aspects: (2a) the optimum ride comfort with and without constraining active actuation requirements; and reversely, the minimum actuation requirements for a target ride comfort; (2b) the minimum average power consumed by the active part; (2c) the feasibility of a given set of active force and power consumption requirements, and the boundary between feasible and unfeasible regions.Specifically, the design guidance (2b) is obtained in conjunction with the power conservation theorem proposed in this paper which proves that for any active controllers taking the format of the linear state-feedback-based control, the total average power consumed by the passive and active parts will not vary with changes in the passive part and active controller parameters.This conclusion builds on the case in [30] where the suspension force is assumed to be dependent on its terminal relative movement.
This paper is structured as below: in Section 2, the quarter-car model with a passiveactive-combined suspension is introduced, where the active part adopts the linear statefeedback-based controller and the candidate passive layouts are formulated using the structure-immittance technique.Expressions for ride comfort and active actuation requirements in the H 2 -norm form are then presented in Section 3, followed by the power conservation theorem.The nested optimisation facilitated with the linear matrix inequality (LMI) technique is also described.In Section 4, the optimal trade-off between ride comfort and each actuation requirement, i.e. the r.m.s.active force and the average power consumption, is identified.In Section 5, the design guidance for optimal suspensions is graphically demonstrated with contour plots.Finally, conclusions are drawn in Section 6.

Quarter-car model and active control scheme
To demonstrate the advantages of this approach in automotive suspension design, an example quarter-car model is used as shown in Figure 1, with a sprung mass m s , an unsprung mass m u and a tyre modelled as a spring with stiffness k t .In line with other studies about vehicle suspension design (e.g.[14,15]), damping characteristics of the tyre have been neglected.The displacements of the sprung mass and the unsprung mass are denoted as x s and x u respectively, with their positive directions shown in the figure.The displacement of the road profile is presented by x r .The suspension static stiffness together with  other quarter-car parameters including the sprung mass, the unsprung mass, and the tyre stiffness are listed in Table 1.These are typical values for passenger cars [15].As shown in Figure 1, the combined suspension considered in this work consists of two parts: one is a passive strut consisting of a static spring k s and a passive layout represented by the admittance Y p (s).Note that the candidate configurations for Y p (s) will be introduced in Section 2.2.The other part of the combined suspension is an active actuator, where different control schemes according to application requirements can be applied.In order to illustrate the generalizability of the design approach and the power conservation theorem which will be proposed in Section 3.2, we define the following general linear state-feedback-based controller whose generated force can be determined by the linear combination of any states of the quarter car through a transfer function.It is expressed in the Laplace domain as where ûa is the active force, the transfer functions G 1 (s), G 2 (s), and G 3 (s) are controller gains with respect to the states xs , xu , and xr (s is the Laplace parameter and the symbolr epresents that the variable is in the Laplace domain).This general format can cover many widely-used classic active controllers based on state feedback, e.g.skyhook or groundhook controller, PID controller and those based on a full state-feedback control.The equivalent gains for these classic control schemes are listed in Table 2.

Traditional and candidate passive layouts
A parallel-spring-damper is selected as the default passive part of the combined automotive suspension, as shown in Figure 2(a), denoted by L0.This layout has formed the passive part of the automotive suspension in many studies (e.g.[6,31]) and as such we treat it as our baseline reference here.Some previous studies indicated that adding an inerter to the traditional layout L0 can improve the dynamic performance of the system [24,32], so a simple inerter-based L1 as shown in Figure 2(b) is also employed for comparison in this work.
The candidate networks for Y p (s) are formulated with the structure-immittance technique [33].Given any number of springs, dampers and inerters, all network possibilities can be enumerated with this technique, which is characterised by two key steps: (1) Formulates the generic networks covering all possible layouts and (2) Removes redundant elements from the generic networks to obtain candidate network layouts.In this paper, as an example we consider the network possibilities of Y p (s) in the following two cases: (1) two-element networks consisting of one inerter and one damper, denoted as '1b1c' case and (2) three-element networks consisting of one spring, one inerter and one damper, denoted as '1k1b1c' case.
Taking the 1k1b1c case as an example, based on the formulation procedures of the structure-immittance technique demonstrated in [33], two generic networks can be obtained as shown in Figure 3.The force to velocity transfer functions of these generic  networks in the structural-immittance format are and respectively, where b ≥ 0 and c ≥ 0. Note that there are four springs in each generic network and three of them should be removed each time to obtain a new candidate network, i.e. for Equation (2) which corresponds to Figure 3(a), three of the parameters k 1 , k 2 , 1/k 3 , k 4 must equal zero, whereas for Equation (3) which corresponds to Figure 3(b), three of the parameters 1/k 5 , 1/k 6 , 1/k 7 , k 8 must equal zero.These two generic networks along with the constraint that only one spring exists, cover all eight layout possibilities of the 1k1b1c case (enumerated in [33]).It should be noted that the network with a spring directly connecting the sprung mass with the unsprung mass would degrade to a two-element layout, because the spring has to be omitted to keep the static stiffness of the suspension constant.It is worth mentioning that, as detailed in [33], the structure-immittance technique can also be applied to networks with more elements.

Performance measures and nested optimisation
Ride comfort, active actuation requirements and two typical design constraints for an automotive suspension are introduced in Section 3.1.Specially, the power conservation in the combined suspension is investigated in Section 3.2 and then based on this the mathematical expression of the average power consumption index in the H 2 -norm form is derived.Lastly, in Section 3.3 the nested optimisation procedures are described, which will be used to identify the optimal combined suspension in Sections 4 and 5.

Dynamic performance and actuation requirement indices
The design approach proposed in this paper aims to identify the combined suspension providing the optimal trade-off between ride comfort and active actuation requirements.For the considered quarter-car model, the ride comfort index J r is expressed as the root-meansquare (r.m.s.) value of the sprung mass vertical acceleration [15].By using the Parsaval's theorem [34], the calculation of J r in the time domain can be converted to the frequency domain as where H ẋr →ẍ s (jω) is the transfer function from ẋr to ẍs and S ẋr = 4π 2 κV is the power spectral density of the random road velocity, as detailed in [18].In this study, we select the road roughness parameter κ = 5 × 10 −7 m 3 /cycle and the vehicle velocity V = 25 m/s, in line with [15].According to the standard , the ride comfort index J r in Equation ( 4) can be converted to To prevent the saturation of the suspension mechanism and the actuator, there should be a constraint on the suspension travel.In this work, the r.m.s.suspension travel is adopted since under the random road excitation, the r.m.s.value is appropriate to capture the statistical characteristics.This r.m.s.travel constraint also links with the constraint on the maximum suspension travel.According to the Three-sigma rule (as adopted in [24,35]), restricting the r.m.s.suspension travel within the value J s0 is equivalent to the constraint that the probability of the maximum suspension stroke exceeding 3J s0 is kept to only about 0.3%.Using the same formulation procedure as the ride comfort index J r , the r.m.s.suspension travel J s can also be expressed in the H 2 -norm form as The dynamic tyre load is considered as another constraint to ensure that the road-holding ability will not deteriorate.Similarly, the r.m.s.dynamic tyre load index J t can be expressed as Note that in real-life situations the tyre load can be measured with sensors [36] or estimated with model-based observers [37].
Regarding the active actuation requirements, the r.m.s.active force and the average power consumption are considered.The r.m.s.active force index J f defined as can represent the continuous output capability of an active actuator.It is often regarded as an important factor to determine the actuator size for continuous operation [38], so minimising the required r.m.s.active force during the design stage is important.The average power consumption index P a is defined by the average of the instantaneous power consumed by the active part, i.e.
The average power consumption P a is another important metric for active systems as it can represent the power that must be supplied from (or returned to) the electrical supply [39,40].Through a careful design, the power requirement for the active system can potentially be reduced, and power autonomy, or even power absorption might be achieved without sacrificing ride comfort.According to Equation ( 9), a positive P a value means that, on average, the active part consumes power from the external supply, but conversely a negative P a indicates that, on average, the active actuator could generate net positive energy that can be stored in the external supply.In other words, P a > 0 means the active actuator is working in the motor mode while P a < 0 means the active actuator is working in the generator mode.The approach to calculate Equation ( 9) will be introduced in Section 3.2 where the minimum P a will be obtained based on the proposed power conservation theorem.

Power conservation
It will be demonstrated in this subsection that the total average power consumed by the passive and active parts is independent of all suspension and vehicle parameters with the exception of the tyre stiffness.This intriguing finding was initially introduced in a quartercar model where the suspension force û is assumed to be determined by its terminal relative movement (x s − xu ) through an admittance Y(s) [30], i.e. û = sY(s)(x s − xu ).However, this work did not consider the case where the absolute motion of the sprung, unsprung mass or the road profile was used as feedback signals for the active controller.Here, we are showing that any controller defined in the format of the linear state-feedback-based control scheme (Equation ( 1)) is equivalent with an admittance relating the suspension force to its relative movement so the power conservation theorem still holds.
Theorem 3.1: For the quarter-car model shown in Figure 1 where the active actuator adopts the linear state-feedback-based control scheme expressed by Equation (1), the following two conclusions hold.
(I) It is equivalent to the system in Figure 4 with an active controller defined as ûa (II) If it is stable and the road disturbance is a white noise with the velocity profile expressed by S ẋr = 4π 2 κV, the total average power consumed by the combined suspension P total , including the power in the passive part P p and the active part P a , is Please refer to Appendix 1 for the proof of Theorem 3.1.Since the sign of the right hand side of Equation ( 11) is negative, the combined suspension always absorbs power from the road excitation.With the vehicle parameters listed in Table 1, it can be calculated that the total average power consumed by the combined suspension in Figure 1 is −37 W (or the absorbable average power is 37 W).The following example is given to better illustrate Theorem 3.1.It is assumed that the Y p (s) in the passive part of the combined suspension is a parallel-damper-inerter layout, i.e.Y p (s) = c + bs where c is the damping and b is the inertance.The active part adopts a special linear state-feedback-based control scheme -skyhook controller, i.e. u a = −c sky ẋs where c sky is the controller gain.The suspension stiffness together with other vehicle parameters are the same as those in Table 1.The parameters for the passive part of the combined suspension are b = 3 kg and c = 653 N/m.The power variation in the passive part P p , the active part P a and the entire suspension P total with the dimensionless variable c sky /c is plotted in Figure 5.It can be seen that while the power in each part changes with c sky /c, the total power keeps constant.
Based on Theorem 3.1, the average power consumption index P a in Equation ( 9) can be re-expressed in the H 2 -norm form with the equation P a = P total − P p .Specifically, by using the Parsaval's theorem [34], the power consumed by a passive damper is where c is the damping, ẋd1 and ẋd2 denote the terminal velocities of the damper.The '−' sign in Equation (12) denotes that the passive damper is always absorbing power from the road excitation (If there is no power harvesting device, the power absorbed by the damper will finally be dissipated).As the damper is the only dissipation element in the passive part, and in this work the 1b1c and 1k1b1c cases are considered, we can have P p = P damper and then the average power consumed by the active part can be expressed as It can be seen from Equation (13) that the minimum average power consumed by the active part is −2π 2 κVk t (−37 W with vehicle parameters listed in Table 1).
The dynamic performance indices expressed by Equations ( 5), ( 6) and ( 7) together with the actuation requirement indices expressed by Equations ( 8) and ( 13) will be adopted for the optimal design of combined suspensions in Sections 4 and 5.

Nested optimisation
Two-level nested optimisation will be used in this work to search for the optimal combined suspension solution, where the passive and active parts will be optimised in the outer and inner loops, respectively.For each iteration when the passive parameters are fixed in the outer loop, in the inner loop the active part will then be optimised and some existing algorithms, such as the linear matrix inequality (LMI) technique [40][41][42], can be used to design the best active controller.Compared with the simultaneous optimisation, optimal design variables to be searched for in the nested optimisation can be reduced to those of the passive part so that the optimisation could be much more efficient [43].
Regarding the active part expressed by Equation (1), there are three transfer functions, G 1 (s), G 2 (s) and G 3 (s), that need to be optimised.Making use of the first conclusion of Theorem 3.1, the active part can be converted to ûa (s) = sY a (s)(x s (s) − xu (s)) so that only the transfer function Y a (s) will be optimised.Based on this, the combined suspension design becomes optimising the Y p (s) and Y a (s) in Figure 4.The equations of motion for Figure 4 can be expressed in the state-space format as where x = [x u − x r , ẋu , x s − x u , ẋs ] T denotes the states of the quarter car.The actuation force u is the total suspension force consisting of forces generated by both the passive part Y p (s) and the active part Y a (s).w is the external excitation which in this case is the road profile velocity, i.e. w = ẋr .y is the measured output and here it is the terminal relative velocity of the suspension, i.e. y = ẋs − ẋu .z is the vector of output signals, i.e.
The entries of z correspond to system responses for calculating ride comfort, suspension travel and dynamic tyre load, respectively.Note that the active force u a and the terminal relative velocities of the damper (ẋ d1 − ẋd2 ) used for computing active actuation requirements have not been included in the vector z as they cannot be expressed explicitly by the states x and the total suspension force u, but will be introduced after additional states of the passive layout Y p (s) are integrated into the vehicle model.The matrices A, B, B w , C z , E and C are listed in Appendix 2. The combined control including both the passive and active parts becomes û = (Y p (s) + Y a (s))ŷ.Our goal is to design the optimal transfer functions, Y p (s) and Y a (s), that can achieve the best ride comfort with constraints satisfied.
To facilitate using nested optimisation, the passive layout Y p (s) of the combined suspension needs be integrated into the state-space model as expressed in Equation ( 14) and the active force u a should be kept as the only actuation force u in Equation (14).To achieve this, the transfer function Y p (s) of the passive part, which is formulated in Section 2.2, should be converted to the state-space format as follows: where y = ẋs − ẋu is the terminal relative velocity of the suspension, u p is force generated by the passive layout Y p (s), ξ is the vector comprising additional states introduced by Y p (s), and A p , B p , C p , D p are gain matrices with their entries relying on element values of Y p (s).
The element values are denoted by the vector θ p , for example, θ p = c for the traditional layout L0 and θ p = [b c] for the inerter-based layout L1.With additional states of Y p (s), the terminal relative velocity of the damper used for computing the average power consumption index P a can be expressed as where T 1 and T 2 are constant vectors in relevance to the topology of candidate passive layouts.The output vector used for calculating the indices for both the dynamic performance and active actuation requirements becomes z = [z T u a (ẋ d1 − ẋd2 )] T where each entry is denoted by zi , i = 1, 2, 3, 4, 5. Substituting Equation (15) into Equation ( 14) and defining η = [x T ξ T ] T , the quartercar model integrated with the passive layout Y p (s) can be described as (16) where ûa = Y a (s)ŷ and 16) will be used for calculating the performance measures of interest.If suspension travel, dynamic tyre load and r.m.s.active force are constrained to be no greater than J s0 , J t0 and J f 0 , and the average power consumed by the active part is constrained to be no higher than P a0 , the nested optimisation strategy can be summarised as: where θ low and θ up are the lower and upper boundaries for passive element values θ p .The genetic algorithm is employed to optimise θ p in the outer loop Equation (17a) while the LMI technique is employed to design the active controller Y a (s) in the inner loop Equation (17b).Please see Appendix 3 for a detailed description of how the LMI technique is used here.The flowchart of this nested loop is given in Figure 6.The input is an initial population of θ p generated by the genetic algorithm.After substituting θ p into the dynamic equations of the quarter-car model Equation ( 16), the only remaining design variables are the parameters in the active controller Y a (s).Then, the LMI technique is adopted to optimise Y a (s) under this θ p .If the resulted ride comfort J * r (θ p ) is optimal, then both θ p and Y a (s) are output.Otherwise, update θ p via the genetic algorithm and repeat the above steps until the optimal ride comfort is achieved.Note that the output of this nested optimisation is one solution providing the optimal ride comfort J r under four constraints: the suspension travel J s ≤ J s0 , the dynamic tyre load J t ≤ J t0 , the r.m.s.active force J f ≤ J f 0 and the average power consumption P a ≤ P a0 .To obtain the optimal trade-off curve between the ride comfort J r and r.m.s active force J f , the nested optimisation Equation ( 17) needs to be performed with different J f 0 values while keeping J s0 , J t0 and P a0 unchanged.Similarly, to obtain the optimal trade-off curve between the ride comfort J r and average power consumption P a , the nested optimisation needs to be performed with different P a0 values while keeping J s0 , J t0 and J f 0 unchanged.

Optimal trade-off between ride comfort and active actuation requirements
Before employing the proposed approach to obtain the optimal trade-off between ride comfort and each active actuation requirement (the r.m.s.active force and the average power consumption), the benchmark performance is given first.
The optimal ride comfort index J r for the purely passive suspension L0 without any constraints is 0.880 m/s 2 .The corresponding suspension travel and tyre load indices are calculated as J s0 = 0.0104m and J t0 = 541N, respectively.As for the following optimisation, the suspension travel and dynamic tyre load for all combined suspensions are constrained to be no greater than those of the optimal passive suspension L0, i.e.J s ≤ J s0 and J t ≤ J t0 .Note that from here, the labels Li&Active refer to the combined suspension integrating a passive layout Li in parallel with an active actuator.

Ride comfort v.s. r.m.s. active force
The trade-off between ride comfort and r.m.s.active force is first investigated.In order to avoid a huge energy consumption in the active part, for each pareto-optimality point a constraint on the average power consumption P a is considered.As an example, it is restricted to be P a ≤ 0 W in Figure 7(a), which means that averaging over time the active part works as a generator and absorbs energy from the road excitation.In the figure, all crossing points of the curves with J f = 0 N refer to the optimal ride comfort provided by the passive parts alone.When the active control is added, the ride comfort keeps improving with increase of the r.m.s.active force.The red dashed curve is the optimal trade-off achieved by L0&Active, while the orange curve is with L1&Active.By comparing the above two trade-off curves, it can be seen that the benefit of including an inerter in the combined suspension is evident, especially in the range of low r.m.s.active force values.This benefit can be further expanded without even needing to increase the number of elements in the passive part, as demonstrated by the green curve which shows the optimal trade-off among combined suspensions with a 1b1c passive layout.This system is termed as L2&Active and the passive part L2 is shown in Figure 8(a).Compared with L1&Active, the optimal L2&Active further improves the ride comfort in the range of high r.m.s.active force values.The blue curve is the optimal trade-off among combined suspensions where Y p (s) consists of one spring, one damper and one inerter (the 1k1b1c case).Its passive part is shown in Figure 8  L3&Active, when the r.m.s.active force approaches about 100 N the trade-off improvements become minor.This is because a majority of the suspension functionality comes from the active part -employing the optimal passive part will not result in an obvious benefit.
To show that the r.m.s.dynamic tyre load index J t is not deteriorated when designing the combined suspension, as an example three points at J f = 33.1 N in Figure 7(a) are selected: the red point, the green square point and the blue square point.A comparison of the power spectral density (PSD) of the dynamic tyre load for these three cases is shown in Figure 7(b).It can be seen that three PSD curves almost overlap.

Ride comfort v.s. average power consumption
The trade-off between ride comfort and average power consumption of the active part is now investigated.In Figure 9, an example active force constraint is considered, i.e.J f ≤ 33.1N (the J f value at the benchmark point A L0 in Figure 7(a)).The optimal combined suspensions identified are still L2&Active for the 1b1c case and L3&Active for the 1k1b1c case.It can be seen that all curves will not extend to the negative x-axis infinitely.For example, the peak power is −7.88 W for L0&Active, L1&Active and L2&Active, and −8.04 W for L3&Active.Left to the peak power is the unfeasible region where every set of (P a , J r ) is unfeasible.Specifically, the unfeasible region for L3&Active is shaded in the figure.The peak power along with the unfeasible region will be detailedly explained in Section 5 as a design guidance for optimal combined suspensions.Compared with the traditional L0&Active, a significant trade-off improvement is achieved with the identified suspensions L2&Active and L3&Active.For example, considering the same ride comfort level as the benchmark point A L0 , using L2&Active and L3&Active can further absorb the power by 4.95 W and 6.44 W on average, i.e.Point A L0 moves to A f L2 and A f L3 .To distinguish from the power constraint in Figure 7(a), the superscript f is used here to denote the r.m.s.active force constraint J f ≤ 33.1 N. The passive and active parameters of the points A L0 , A f L2 , A f L3 , A  The optimal trade-off between ride comfort J r and average power consumption P a satisfying J f ≤ 33.1 N constraint.

Design guidance for optimal combined suspensions
Up to this point, we have obtained the optimal trade-off between ride comfort and two active actuation requirements separately.In this section, a deeper investigation of the relationship between ride comfort, active force and power consumption is demonstrated via contours plots, and then based on this the design guidance for automotive suspensions is established.

Contour plots of optimal ride comfort
The contour plot of the traditional combined suspension L0&Active is first illustrated in Figure 10, where the grey bold curve is the boundary between feasible and unfeasible regions in terms of two actuation requirements.In the feasible region, the coloured curves are the contour lines indicating different ride comfort levels.The ride comfort value for each contour line is presented in the colour scale to the right of the graph, which ranges from 0.653 m/s 2 to 0.793 m/s 2 with a uniform interval of 0.010 m/s 2 .It can be seen that for the same average power consumption, the increase of r.m.s.active force will improve ride comfort while under the same r.m.s.active force, ride comfort can be improved by increasing the power consumption.It can be seen from the distance between every two contour curves that ride comfort is much more sensitive to the variation of actuation requirements in the range of high ride comfort values than low ride comfort values.The pareto front between two actuation requirements to achieve the optimal ride comfort is presented as the purple bold curve.In the optimal region bounded by this curve, the increase of either actuation requirement will not lead to any improvement of ride comfort.It should be noted that the optimal trade-off between ride comfort and one actuation requirement discussed in Section 4 can be regarded as a cross section of the contour plot at a given value of the other actuation requirement, so the benchmark point A L0 in Figures 7(a) and 9 can be directly mapped to Figure 10.
The performance improvement achieved by using combined suspensions with the optimal 1b1c and 1k1b1c passive parts are illustrated in Figure 11 where the dotted, dashed and solid curves denote the performance of L0&Active, L2&Active and L3&Active, respectively.The coloured and grey curves represent the same as in Figure 10, and the shaded area shows the unfeasible region of L3&Active.The contour curves with the example ride comfort 0.753 m/s 2 and with the optimal ride comfort 0.653 m/s 2 are plotted to show the trade-off improvement of optimal combined suspensions.The significant reduction of the active force and average power consumption achieved by the identified L2&Active and L3&Active can be clearly seen from the zoom-in plot of this figure.As each improvement can be linked with previously-discussed Figures 7(a) and 9, they will not be repeated here.

Design guidance for combined suspensions
Based on the contour plots discussed above, we can establish the design guidance (DG) for optimal combined suspensions which mainly focuses on the following three aspects: (a) DG1: optimal ride comfort J r -the optimum J r without constraining active force J f and power consumption P a requirements is indicated by the optimal region in the contour plots, i.e. the purple area in Figures 10 and 11; and for a given set of actuation requirements (J f , P a ), the optimal J r can be obtained by tracking the contour curve passing through this point.For example, the optimal J r of L0&Active under the constraint of Point B (110 N, -15 W) is 0.663 m/s 2 as shown in Figure 10.With the same constraint, the optimal J r of L3&Active is 0.658 m/s 2 as shown in Figure 11.Reversely, given a target ride comfort level J r , the relation between the minimum J f and corresponding P a can also be obtained by tracking the contour curve with the value of J r ; (b) DG2: the minimum average power consumed by the active actuator P amin (also the maximum power absorption when it works as a generator) -according to the power conservation theorem proposed in Section 3.2, the total power consumed by the combined suspension with parameters listed in Table 1 is −37 W, i.e.P a + P p = −37 W. Since the passive part is always absorbing power P p ≤ 0, the power consumed by the active part satisfies P a ≥ −37 W. The minimum power consumption in the active part is P amin = −37 W which is denoted by the flat part of the grey curves in Figures 10  and 11 -it also means that the maximum power absorption of the active part is 37 W; (c) DG3: the boundary between feasible and unfeasible (J f , P a ) values -it is shown as the grey curves in Figures 10 and 11.On these curves, each point means the possible maximum average power absorbed by the active part for a certain level of r.m.s.active force.Below this curve is the unfeasible region where every set of (J f , P a ) values cannot be achieved.

Conclusions
In this paper, a novel design approach for searching for the optimal passive-activecombined suspension incorporating an inerter has been proposed.A quarter-car model was studied as an example and the combined suspension was composed of a passive part with a pre-determined number of each element type (springs, dampers and inerters) and a linear state-feedback-based active controller.In this case, two constraints are always considered: the road-holding ability and suspension travel of the identified suspension will not be worse than those of the benchmark suspension.Via this approach, the optimal trade-off between ride comfort and active actuation requirements, i.e. r.m.s.active force and average power consumption, was achieved.For example, compared to the traditional combined suspension where the passive part is a spring-damper parallel, the optimal inerter-based suspension can reduce the r.m.s.active force by more than 48% and absorb about 6 W more power on average while achieving the same ride comfort.Note that while in the case study the passive part with one spring, one damper and one inerter is considered, the proposed design approach can also be directly applied to a suspension system with multiple springs/dampers/inerters.It is reasonable to anticipate that with more elements, the trade-off can be further improved.The power conservation of the combined suspension has also been investigated, and it was found that for any linear state-feedback-based controller, the total power consumed by the passive and active parts only relies on the tyre stiffness and will not vary with changes in the passive suspension and active controller parameters.
Based on the optimal trade-off obtained via the proposed design approach and the power conservation theorem, the design guidance for combined suspensions has been given, including the following three aspects: (a) the optimal ride comfort with and without constraining actuation requirements, and the minimum actuation requirements with a target ride comfort; (b) the minimum power consumption of the active part (also the maximum power absorption when it works in the generator mode); and (c) the feasibility of a given set of active force and power consumption requirements.Note that the focus of this paper is to demonstrate the feasibility of the proposed design approach in identifying the optimal network for passive-active-combined suspensions.The physical realisation of the identified suspension network and its experimental validation will be considered as the next step of this research.
The combined suspension parameters of points A L0 , A p L2 , A p L3 , A f L2 and A f L3 are listed in Table A1, where a i and b i (i = 0, 1, 2, . . ., 6) are the coefficients of the active controller Y a (s) expressed in the transfer-function format as

Appendix 3. Active controller design through the LMI approach
Before employing the LMI approach to solve the inner loop of the nested optimisation, Equation (17b) is first converted to the following equation by Equations ( 5), ( 6), ( 7), ( 8) and (13). where The main procedure of the LMI approach is to express all cost functions in the form of linear matrix inequalities and to obtain the controller gain through solving them.For the inner loop Table A1.Passive and active paremeters of identified combined suspensions.
where Czi and Ēi are the ith rows of Cz and Ē, respectively, and * represents an entry that is induced by symmetry.The proof of this can be found in [41] and is not introduced here.The control problem in the inner loop of the nested optimisation, i.e.Equation (C1), is to find the controller gain Y a (s) such that γ 1 of H w→z 1 2 < γ 1 is minimised subject to H w→z i 2 < γ i (i = 2, 3, 4, 5).This requires to minimise γ 1 with the matrix inequalities (C2a), (C2b) and (C2c) satisfied simultaneously for i = 1, 2, 3, 4, 5.Note that the traditional LMI approach adopts the same matrices X, Y, A e , B e , C e , D e when i take different values.Despite the loss of the true global optimality with this condition, it will lead to a convex optimisation problem which can be easily solved by MATLAB LMI Toolbox ® .As the optimisation tool is not the focus of this work, the authors did not investigate less conservative LMI conditions and suggest readers referring to [44,45]  where M and N should be chosen such that In this paper, the order of the controller is assumed to be the same as the system.In this case, M and N are invertible and can be obtained by performing the singular value decomposition of Equation (C4).Then, the gain matrices for Y a (s) are unique and can be explicitly calculated from the following equations:

Figure 1 .
Figure 1.The quarter-car model with a general passive-active-combined suspension.

Figure 2 .
Figure 2. Passive part with (a) the traditional layout L0 and (b) the simple inerter-based layout L1.

Figure 3 .
Figure 3. Two generic networks (a) N1 and (b) N2 for the 1k1b1c case with the condition that only one spring exists for each network.

Figure 4 .
Figure 4.The equivalent quarter-car model with the active controller replaced by a transfer function Y a (s).

Figure 5 .
Figure 5.The average power consumption in the combined suspension.

Figure 6 .
Figure 6.The flowchart showing detailed steps of the nested optimisation.

Figure 7 .
Figure 7. (a) The optimal trade-off between ride comfort J r and r.m.s.active force J f satisfying P a ≤ 0 W constraint.(b) A comparison of the dynamic tyre load PSD of L0&Active (red point in Figure(a)), L2&Active (green square point) and L3&Active (blue square point).

Figure 9 .
Figure9.The optimal trade-off between ride comfort J r and average power consumption P a satisfying J f ≤ 33.1 N constraint.

Figure 10 .
Figure10.The optimal ride comfort J r of the traditional combined suspension L0&Active considering both actuation requirements.

Figure 11 .
Figure 11.The ride comfort improvement of the combined suspensions L2&Active (dashed lines) and L3&Active (solid lines) compared with the traditional suspension L0&Active (dotted lines).

Table 1 .
Parameters of the quarter-car model.

Table 2 .
Equivalent gains for some classic active control schemes.