Optimal Vibration Control for Half-Car Suspension on In-Vehicle Networks in Delta Domain

and Applied Analysis 3 Actuator Suspension Sensor ECU A/D D/A Communication Communication v(t) u(t − τ


Introduction
In the past years, communication networks have been applied greatly and widely into the advanced vehicle systems, such as electronic control units (ECUs), sensors, and actuators which are all connected over the high-speed in-vehicle networks (IVNs), for example, Local Interconnect Network, Controller Area Network, Media Oriented Systems Transport [1][2][3], and so forth.However, in this kind of IVNs, two important issues emerge for the controller design problems.One is the network-induced delay issue.Over communication networks, the time delays generated between sensor-controller, controller-actuator, ECU computation delay, and so forth are unavoidably encountered.As well known, even a small time delay can make the systems disastrously unstable or generate oscillations [4][5][6].So, this issue should be taken into account when we design a controller for a system over networks.The next issue is the system modeling problem.Actually, signals processed by microprocessors are digital, and most of those produced by sensors or put into actuators are analogs.Thus, the continuous-time plant is combined with a discrete-time controller, where A/D and D/A converts are used to combine these two different signals.Therefore, a sampled-data system is more appropriate for the reality of networked-control system.In previous studies, there are two main approaches to get the sampled-data systems, those are, indirect approach of continuous-time domain and direct approach of discrete-time domain.However, the former is only suitable for simple control algorithms, and the latter has two drawbacks: one is that the discrete-time model is unable to approach to its corresponding continuous-time one as the sampling frequency increases; and the other is that the discretized system can cause oscillations and unstable phenomenon as the sampling frequency increases.
Consequentially, we present two strategies to deal with these above-mentioned issues.First, we introduce the delta operator approach to build a sampled-data model for the networked-control system due to its accurate approximation

Problem Statement
2.1.System Modeling.Consider a four-degree-of-freedom half-car model (refered to in [15]) where the suspension motion is determined by the following dynamic equations: through Newton-Euler method.In this half-car suspension model, the sprung mass   and unsprung one   are separated by spring, damper, and actuator, which are placed in parallel.The tire of the vehicle is modeled as a spring.Vertical motion   () and pitch motion () of the sprung mass are considered, as well as the vertical motion of the front unsprung mass   () and the rear one   ().
Consider it working on an IVNs-based environment, as depicted in Figure 1, where , , and   denote the system state, control input, and measured output, respectively; V is the road excitation input; ,  are constant ECU-actuator delay and sensor-ECU delay, respectively, (actuator delay and sensor delay for short) which are always assumed to be known and constant in such IVNs environment.
With the purpose of replacing (1) into the state-space representation, define the state, control, and disturbance vectors as the controlled output vector as and the measured output vector as Together with the associated dynamic equations and the definitions ( 2)-(4), the system (1) thus is rewritten in the state-space representation as with () ∈ R 8 , () ∈ R 2 ,   () ∈ R 4 , and   () ∈ R 6 as the state vector, control output, measurement vector, and controlled output, respectively, , ,  1 ,  2 , , and  as the real constant matrices of appropriate dimensions, and () ∈ C([−, 0]; R 8 ) as the initial state vector.
Consequently, the sampled-data system form of the system (6) can be got by applying the sampled-data controller where there is a zero-order holder with {  } as the sampling times, (  ) as the state on time of   = , and  as a constant data controller gain, denoting the actuator delay  = ℎ 1  +  1  and the sensor delay  = ℎ 2  +  2  with ℎ  ∈ N and 0 ≤   < 1 ( = 1, 2).Notice that the delays are expressed in either integers or nonintegers.Then, the sampled-data system of (1) can be described by where ℎ = ℎ 1 + ℎ 2 and  =  1 +  2 .Here, the triple (, ,  1 ) is assumed to be completely controllable and observable.
Further, the sampled-data system (8) will be converted to delta domain and without actuator delays.There are two situations that should be considered concerning the nonintegral part  of the time delay:  ∈ [0, 1] and  ∈ [1,2].However, the derivation procedures of these two situations are similar.So, for the sake of simplicity, the derivation procedure of the former situation will be presented in what follows.
Introducing the delta operator as and letting  = ( + 1 −  2 ) and  0 = ( −  2 ) discretize the sampled-data system (8) in the delta-domain form Noting that the system (10) is the delta-domain sampleddata system with actuator delays, for the convenience of calculation, it will be transformed without actuator delays using the model transformation approach.From the first difference equation in (10), it follows the analytical expression of state response where e  d, Consequently, define a new state vector as Equation ( 10) yields the analytical expression of state response for the transformed system which produces its state equation in delta-domain form Through the same way, defining the new measured and controlled outputs with , the output equations of the transformed system are got as follows Then, the state equation ( 15) and the output equation ( 17) consist the transformed equivalent sampled-data system in delta domain without actuator delays: Furthermore, the road excitation should be described by an exosystem in delta domain in order to employ the statespace representation designing the OVC.

Road Disturbance Modeling.
According to ISO 2631 standards, the road displacement power spectral density (PSD) is usually approximately represented in the formulation of with Ω as the spatial frequency,   as the road roughness constant, and  as the sort of the road as shown in Table 1.Due to the low-pass-filter characteristic of vehicle tires and suspension, the road displacement   () ( = , ) can be approximately simulated by a finite Fourier series sum where   = 2  /10 3 √/10 are the amplitudes,   =  0 are the frequencies with the time frequency internal  0 = 2V 0 /,   are the random phases which follow a uniform distribution in [0, 2), V 0 is a constant horizontal velocity,  is the given road segment length, and positive integer  limits the considered frequency band.Letting the disturbance state vector the road velocity V() = [ ẋ  (), ẋ  ()]  then is described by the exosystem ẇ () =  () , with in which 0 and I  represent the zero matrix and the order identity matrix, respectively.Using delta operator (9), exosystem ( 22) is transformed into the delta-domain form where  = (e  − )/.Hence, both the original system and the exosystem are transformed into the delta-domain sampled-data systems so that we would design the OVC for them.

Problem Formulation.
The principal variables for the evaluation of the suspension system are sprung mass acceleration ẍ  and φ determining the ride comfort, suspension deflection   −   indicating the limit of vehicle body motion, and tire deflection   −  ensuring the road holding ability.The purpose is to reduce the acceleration of vehicle body and decrease the dynamic tire forces for improving the road holding ability and the stability of vehicles facing road excitation.
In practice, control  and controlled output   are unable synchronously zero so that the general infinite-horizon performance index is not convergent.In this case, an average infinite-horizon performance index could be chosen as with  =   2  0  2 ,  0 = diag{  } ( = 1, 2, . . ., 6) as positive semidefinite matrices and  = diag{  } ( = 1, 2) as a positive definite matrix, assuming that    =  with  an arbitrary matrix and the triple (, , ) completely controllable and observable.
Remark 1.When  ∈ [1,2], the delta-domain sampled-data system is described by with The derivation procedure of this situation is similar to that of  ∈ [0, 1].It is omitted for the simplification reason.

Sampled-Data OVC Design
Theorem 2. Consider the optimal vibration control problem described by the time-delay sampled-data system (10) under disturbance (24) respecting the average performance index (25).The optimal vibration control law is existent and unique and given by where  is the unique positive definite solution of Riccati equation: 1 is the unique solution of Stein equation: Proof.According to Pontryagin's minimum principle, the optimal control problem combined by the system (18) and the performance index (25) results in the two-point boundary value problem in delta-operator form: which yields with the optimal control law From (31) or (32a) and (32b) it is clear that  and  are the linear relationship, so denote the costate vector Consequently, on one hand, together with (34) and (32a), it gives On the other hand, from (34) and (32b), the following equation holds: Substituting ( 35) into (36) yields Due to ( − 2 ) and ( − 2 ) arbitrarily satisfying (37), it results in Riccati equation (29) and Stein equation (30).Further, the uniqueness is to be proved.According to linear optimal regulator theory, there exists a unique positive definite solution  for Riccati equation ( 26) and the closedloop system of (18) is asymptotically stable, which implies that matrix is Hurwitz, that is,   for  = 1, 2, . . ., .Therefore, from (39) and ( 40), the following inequality holds: As a result, Stein equation (30) has the unique solution matrix  1 [16].The uniqueness of  and  1 leads to the uniqueness of OVC (28).This ends the proof.

Physical Realization of OVC.
The optimal control law  * [  ] (28) contains the physically unrealizable disturbance state ( − 2 ) and some unmeasurable variables in ( − 2 ) for economical or practical reasons.In this case, a reduced-order observer can be constructed to reconstruct theses states.Defining the augmented vector ]  yields the following augmented system in delta domain combined by ( 18) and (24): The pair ( Ã, C) can be proved to be completely observable.

Choosing an arbitrary matrix 𝐻 such that Γ = [ C𝑇 𝐻 𝑇 ]
is nonsingular and defining Let the nonsingular transformation be as follows: where ( − 2 ) =  1 ( − 2 ).The new state equations follow in the delta-domain form: Defining a new variable with  the gain matrix to be selected, ( 46) and (45) give  Noting that and substituting (47) into (48) yield Corresponding to (49), the reduced-order observer is constructed: where ψ( − 2 ) and ζ( − 2 ) are the state and output of the observer, respectively.Denoting observer errors ψ( and (49), the error state equation is got: Hence, the error equation gain  should be regulated to make (51) asymptotically stable, such that lim  → ∞ ( − 2 ) = 0.
The observable pair ( Ã, C) results in the observable pair ( CÃ Π 2 ,  ÃΠ 2 ).Consequentially, the gain  is enabled to make the eigenvalues of ( −  C) ÃΠ 2 assigned in the lefthalf complex plane, which means that error equation ( 51) is asymptotically stable.Therefore, the reconstructed states can replace the unavailable ones.Through this mean, states in OVC (28) are replaced by the augmented state which gives the dynamical control law with

Simulation Examples
In this section, a half-car suspension model will be employed to carry out the simulations.We will take two cases of the simulations: firstly, to demonstrate the closed-loop matrices of the continues time, discrete time, and the delta domain taking different sampling period  in order to verify the deltadomain matrix enables approximating to the continues-time   (i)   is the continuous-time closed-loop matrix:   (56c) Comparing with the continuous-time closed-loop matrix (54), from (55a) to (55c) we can see the discretized matrices approach to the unit matrix as the sampling period  decreases; from (56a) to (56c) we can see that the deltadomain matrices approach to the original continuous-time matrix (54) as  decreases.Evidently, the delta-domain approach is more appropriate for the high-sampling IVNs system.
Case 2. Setting sampling period  = 0.2 s and comparing with the responses of the accelerations, deflections of suspension and tire, and control input of OLS shown in Figures 2, 3 System responses in Figures 2-4 show that they diverged when without control (of OLS).On the contrary, in Figures 5-8, when the suspension was controlled under OVC, the suspension responses were stabilized.Moreover, they achieved relatively low magnitude and satisfied the desired requirement.

Conclusions
This paper has presented the OVC design for sampleddata system with time delays with its application to a half-car suspension using delta-domain approach.Through this approach, the built model provides more realistic and appropriate property.The delay compensators guarantee the closed-loop stability and requested performance.The simulation has demonstrated that the designed controller can efficiently make the system performance achieve the desired goal and the design approach proposed in this study is effective and feasible.

Table 1 :
Road grades and PSDs.