# A New Approach to the Robust Control Design of Fuzzy Automated Highway Systems

Authors: Qingmin Huang; Zeyu Yang; Jin Huang; Hui Yin
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, a new approach to the decentralized control design for vehicle platooning for uncertain automated highway systems is proposed. The uncertainty in the system, which is nonlinear and (possibly) fast time-varying, is bounded. The bound is assumed to be within a prescribed fuzzy set. A creative transformation is made to the system, which converts a local problem to a global problem. Based on the fuzzy description of the uncertainty bound and the transformation, a class of decentralized control is proposed in which each vehicle only needs the knowledge of its preceding vehicle in the platoon. No acceleration feedback or the information of the leading vehicle is required. Both the vehicle platooning system and the control are deterministic, hence notif-then fuzzy rule-based. The performance of the resulting controlled system is twofold. First, the collision avoidance performance is guaranteed under any safe initial conditions regardless of the value of the uncertainty. Second, the minimization of a fuzzy-based performance index is guaranteed based on an optimal choice of a control design parameter. Numerical simulations are conducted to validate the efficiency of the proposed algorithm.


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## Body

## 1. Introduction

The Automated Highway System (AHS) is a proposed intelligent transportation system (ITS) technology designed to increase capacity and safety with the increasingly severe traffic congestion since the last two decades. The basic idea is by grouping vehicles into platoons at closer spacing under automatic control, which cannot be achieved by human drivers alone, to increase the capacity on highways [1, 2]. With the distances between vehicles becoming smaller, safety (or collision avoidance) is the first and foremost concern, which must be guaranteed for the AHS. From the control design point of view, stability is often the primary system performance. However, a platoon of automated vehicles is an interconnected system in which stability of each component system per se is not sufficient to guarantee the boundedness of the spacing errors for all the vehicles [3], which is directly related to safety.For this reason, the alternative string stability has been proposed, which implies the boundedness of all the states of the interconnected system if the initial states of the vehicles are within some neighborhood of the equilibrium states [4]. Efforts are then devoted to linearize the system through control with required platoon information (acceleration feedback and/or information of the leading vehicle) and to use transfer function analysis to investigate string stability [5–8]. Many important contributions have been made (for a survey, see, for example, [9–12]). However, as it was pointed out in [13], the string stability alone does not provide a warranty for global collision avoidance of the platoon. Rather, the initial conditions should be within a neighborhood around the equilibrium position in order to prevent vehicle collision [14, 15], hence a local performance. Earlier research studies regarding local collision avoidance can be found in [16–18]. The restriction on the initial conditions reduces the range of applications. This promotes a need to further design a controller for the platoon which guarantees global collision avoidance under any safe initial condition (that is, as long as the collision does not occur initially).One salient feature of the paper is that the uncertainty in the system, which may come from parameter variations (the mass variation and the movement of the passengers and/or the stowage) and aerodynamics (the unmeasurable side wind), as well as external disturbances (the rolling resistance and slope of road), is described using fuzzy set theory [19–21], instead of fuzzy if-then inference rules [22, 23]. Neither the platoon model nor the controller is fuzzy if-then rule-based, hence not the Takagi–Sugeno–Kang (TSK) fuzzy system [24, 25] nor the Mamdani-type fuzzy controller [26]. This distinguishes the current work from some others in fuzzy systems and control research. For example, the uncertainty bound may be close to 2. Here, “close to” is a linguistic variable which is associated with a fuzzy set. Earlier works related to fuzzy if-then rule-based approach for platooning can be found in [27, 28].The main contributions of this paper are fourfold. First, a creative state transformation is made on the error dynamics of each vehicle in the platoon. This converts alocal problem to a global problem. Second, fuzzy set instead of fuzzy if-then rules is used to describe the uncertainty bound of the system. A class of decentralized robust controllers is proposed based on the fuzzy description of the uncertainty bound. Each vehicle in the platoon only needs the information of its preceding vehicle for the control. No acceleration feedback or information of the following or leading vehicle or other vehicle is required. The control is proven to render each transformed error dynamics globally practically stable regardless of the uncertainty. Third, it is shown that the global practical stability leads to global collision avoidance and the spacing error will converge to a small region. Fourth, an optimal design problem associated with the fuzzy-based performance index and the control cost is formulated, which enables us to choose an optimal gain for a control parameter. It is proven that the global solution to the optimal problem exists and is unique. Furthermore, the closed-form (that is, the analytic) expression of the optimal solution is shown. This completely solves the problem.


## 2. Vehicle Platooning Model

### 2.1. Platoon Configuration

Consider a platoon of automated vehicles traveling on a straight line of a highway and closely following one another, as shown in Figure1. The position of the leading vehicle and the ith following vehicle in the platoon is denoted by xL∈R and xi∈R with respect to an inertial frame, respectively. The actual space Δi between the ith vehicle and its preceding one is given by(1)Δit=xi−1t−xit−li−1,where li−1 is the length of the i−1th vehicle in the platoon. Henceforth, subscripts L and i are used to label the leader and the ith follower in the platoon, respectively.Figure 1
The position history of the four vehicles in the platoon under robust controller with zero initial conditions.Suppose the desired space between theith and i−1th vehicle is denoted by a constant scalar Δid. Then, the space error is given by(2)eit=Δid−Δit=Δid+xit−xi−1t+li−1.Notice thatΔits=0 means a collision happens at time ts. A definition related to the collision avoidance performance is given as follows.Definition 1 (collision avoidance).
A platoon of controlled vehicles is collision avoidable if given anysafe initial condition (that is, initially noncolliding or eit0<Δid):(3)eit<Δid,for all t>t0.Remark.
It is worth pointing out here that collision avoidance (3) is the foremost performance to be guaranteed for automated vehicles. From the practical design point of view, one should first guarantee that the platoon is collision avoidable.


### 2.2. Platoon Error Dynamics

The longitudinal equation of motion for theith vehicle is determined by the Newton’s second law as(4)Mixit,σit,tx¨it=uit−cixt,x˙it,σit,t×x˙itx˙it−Fixit,x˙it,σit,t,where t∈R is the time, σi∈Σi⊂Rp is the uncertain parameter representing all the uncertainty in the vehicle, and ui∈R is the control input which is from the vehicle propulsive/braking effort. Moreover, we assume Mi⋅, ci⋅, and Fi⋅ are continuous, where Mixi,σi,t∈R represents the vehicle mass, −cix,x˙i,σi,tx˙ix˙i∈R represents the aerodynamic drag force, and −Fixi,x˙i,σi,t∈R represents the rolling resistance force and other external disturbances acting on the vehicle.Differentiating (2) with respect to time, we get the space error velocity and space error dynamics as follows:(5)e˙it=x˙it−x˙i−1t,(6)e¨it=x¨it−x¨i−1t.With (4) in (6) yields (henceforth, arguments of functions are sometimes omitted when no confusion is likely to arise),(7)e¨i=Mi−1ui−cix˙ix˙i−Fi−Mi−1−1ui−1−ci−1x˙i−1x˙i−1−Fi−1.The control objective for vehicle platooning is stated as follows: design the controlui in (4) for the ith vehicle in the platoon with available information (will be elaborated later) such that the resulting controlled platoon is collision avoidable under arbitrary safe initial conditions.


### 2.3. Bounding and Structure Conditions

We decomposeMi, ci, and Fi in (4) as follows:(8)Mixi,σi,t=M¯ixi,t+ΔMixi,σi,t,cixi,x˙i,σi,t=c¯ixi,x˙i,t+Δcixi,x˙i,σi,t,Fixi,x˙i,σi,t=F¯ixi,x˙i,t+ΔFixi,x˙i,σi,t,where M¯i, c¯i, and F¯i denote the “nominal” portions with M¯i>0 (this is always feasible since the nominal portion is the designer’s discretion), while ΔMi, Δci, and Δgi are the uncertain portions. The functions M¯i⋅, ΔMi⋅, c¯i⋅, Δci⋅, F¯i⋅, and ΔFi⋅ are all continuous.Denote(9)Dixi,t≔M¯i−1xi,t,(10)ΔDixi,σi,t≔Mi−1xi,σi,t−M¯i−1xi,t,(11)Eixi,σi,t≔M¯ixi,tMi−1xi,σi,t−1.Hence,ΔDixi,σi,t=Dixi,tEixi,σi,t.Assumption 1.
(i) Let the initial state of the error dynamics of theith vehicle in the platoon, which is uncertain, be represented by qi0=ei0e˙i0T. For each entry of qi0, namely, qi0j, j=1,2, there exists a fuzzy set Xi0j in a universe of discourse Ξij∈R characterized by a membership function μΞij:Ξij⟶0,1. That is,(12)Xi0j=qi0j,μΞijqi0j∣qi0j∈Ξij.where Ξij is known and compact. (ii) For each entry of the vector σi, namely, σij, j=1,2,⋯,p, the function σij⋅ is Lebesgue measurable. (iii) For each σij, there exists a fuzzy set Sij in a universe of discourse Σij∈R characterized by a membership function μij:Σij⟶0,1. That is,(13)Sij=σij,μijσij∣σij∈Σij.where Σij is known and compact.Assumption 2.
(1) There exists a (possibly unknown) parameterρEi∈R such that for all xi,t∈R×R, σi∈Σi,
(14)
min
σ
i
∈
Σ
i
E
i
x
i
,
σ
i
,
t
≥
ρ
E
i
>
−
1.
(2)
The unknown parameterρEi belongs to a known fuzzy number.Remark.
In the special case thatMi=M¯i (i.e., no uncertainty in the ith vehicle mass), we get Ei=0. Hence, one can choose ρEi=0. The assumption imposes the effect of uncertainty on the possible deviation of Mi from M¯i to be within a unidirectional threshold.


### 2.4. State Transformation

We propose the following transformation for the space errorei:(15)zi1=lnΔid−eiΔid,which means that(16)ei=Δid−Δidexpzi1.Differentiating (15) with respect to time, we get(17)z˙i1=−1Δidexp−zi1e˙i,(18)z¨i1=−1Δidexp−zi1−z˙i1e˙i−1Δidexp−zi1e¨i.Let(19)zi2≔zi1−1Δidexp−zi1e˙i,yields(20)z˙i1=−zi1+zi2,z˙i2=z˙i1−1Δidexp−zi1−z˙i1e˙i−1Δidexp−zi1e¨i.With (7) in (20), the transformed space error dynamics is in the form of(21)z˙i1=−zi1+zi2,z˙i2=−zi1+zi2−zi1−zi22,(22)−1Δidexp−zi1Mi−1ui−cix˙ix˙i−Fi−Mi−1−1ui−1−ci−1x˙i−1x˙i−1−Fi−1.Remark.
From (15), we have for all −∞<ei<Δid, the corresponding zi1 is −∞<zi1<∞. Furthermore, zi1=0⟺ei=0. This is in fact the main reason for the transformation from ei,e˙i to zi1,zi2. If both zi1t and zi2t are bounded, then space error eit∈−∞,Δid, hence no collision.Remark.
The statezi2 given by (19) is selected to make the transformed space error dynamics in a lower triangular form so that the backstepping method [29] can be adopted for the control design, which will be explained in the next section.


## 2.1. Platoon Configuration

Consider a platoon of automated vehicles traveling on a straight line of a highway and closely following one another, as shown in Figure1. The position of the leading vehicle and the ith following vehicle in the platoon is denoted by xL∈R and xi∈R with respect to an inertial frame, respectively. The actual space Δi between the ith vehicle and its preceding one is given by(1)Δit=xi−1t−xit−li−1,where li−1 is the length of the i−1th vehicle in the platoon. Henceforth, subscripts L and i are used to label the leader and the ith follower in the platoon, respectively.Figure 1
The position history of the four vehicles in the platoon under robust controller with zero initial conditions.Suppose the desired space between theith and i−1th vehicle is denoted by a constant scalar Δid. Then, the space error is given by(2)eit=Δid−Δit=Δid+xit−xi−1t+li−1.Notice thatΔits=0 means a collision happens at time ts. A definition related to the collision avoidance performance is given as follows.Definition 1 (collision avoidance).
A platoon of controlled vehicles is collision avoidable if given anysafe initial condition (that is, initially noncolliding or eit0<Δid):(3)eit<Δid,for all t>t0.Remark.
It is worth pointing out here that collision avoidance (3) is the foremost performance to be guaranteed for automated vehicles. From the practical design point of view, one should first guarantee that the platoon is collision avoidable.


## 2.2. Platoon Error Dynamics

The longitudinal equation of motion for theith vehicle is determined by the Newton’s second law as(4)Mixit,σit,tx¨it=uit−cixt,x˙it,σit,t×x˙itx˙it−Fixit,x˙it,σit,t,where t∈R is the time, σi∈Σi⊂Rp is the uncertain parameter representing all the uncertainty in the vehicle, and ui∈R is the control input which is from the vehicle propulsive/braking effort. Moreover, we assume Mi⋅, ci⋅, and Fi⋅ are continuous, where Mixi,σi,t∈R represents the vehicle mass, −cix,x˙i,σi,tx˙ix˙i∈R represents the aerodynamic drag force, and −Fixi,x˙i,σi,t∈R represents the rolling resistance force and other external disturbances acting on the vehicle.Differentiating (2) with respect to time, we get the space error velocity and space error dynamics as follows:(5)e˙it=x˙it−x˙i−1t,(6)e¨it=x¨it−x¨i−1t.With (4) in (6) yields (henceforth, arguments of functions are sometimes omitted when no confusion is likely to arise),(7)e¨i=Mi−1ui−cix˙ix˙i−Fi−Mi−1−1ui−1−ci−1x˙i−1x˙i−1−Fi−1.The control objective for vehicle platooning is stated as follows: design the controlui in (4) for the ith vehicle in the platoon with available information (will be elaborated later) such that the resulting controlled platoon is collision avoidable under arbitrary safe initial conditions.


## 2.3. Bounding and Structure Conditions

We decomposeMi, ci, and Fi in (4) as follows:(8)Mixi,σi,t=M¯ixi,t+ΔMixi,σi,t,cixi,x˙i,σi,t=c¯ixi,x˙i,t+Δcixi,x˙i,σi,t,Fixi,x˙i,σi,t=F¯ixi,x˙i,t+ΔFixi,x˙i,σi,t,where M¯i, c¯i, and F¯i denote the “nominal” portions with M¯i>0 (this is always feasible since the nominal portion is the designer’s discretion), while ΔMi, Δci, and Δgi are the uncertain portions. The functions M¯i⋅, ΔMi⋅, c¯i⋅, Δci⋅, F¯i⋅, and ΔFi⋅ are all continuous.Denote(9)Dixi,t≔M¯i−1xi,t,(10)ΔDixi,σi,t≔Mi−1xi,σi,t−M¯i−1xi,t,(11)Eixi,σi,t≔M¯ixi,tMi−1xi,σi,t−1.Hence,ΔDixi,σi,t=Dixi,tEixi,σi,t.Assumption 1.
(i) Let the initial state of the error dynamics of theith vehicle in the platoon, which is uncertain, be represented by qi0=ei0e˙i0T. For each entry of qi0, namely, qi0j, j=1,2, there exists a fuzzy set Xi0j in a universe of discourse Ξij∈R characterized by a membership function μΞij:Ξij⟶0,1. That is,(12)Xi0j=qi0j,μΞijqi0j∣qi0j∈Ξij.where Ξij is known and compact. (ii) For each entry of the vector σi, namely, σij, j=1,2,⋯,p, the function σij⋅ is Lebesgue measurable. (iii) For each σij, there exists a fuzzy set Sij in a universe of discourse Σij∈R characterized by a membership function μij:Σij⟶0,1. That is,(13)Sij=σij,μijσij∣σij∈Σij.where Σij is known and compact.Assumption 2.
(1) There exists a (possibly unknown) parameterρEi∈R such that for all xi,t∈R×R, σi∈Σi,
(14)
min
σ
i
∈
Σ
i
E
i
x
i
,
σ
i
,
t
≥
ρ
E
i
>
−
1.
(2)
The unknown parameterρEi belongs to a known fuzzy number.Remark.
In the special case thatMi=M¯i (i.e., no uncertainty in the ith vehicle mass), we get Ei=0. Hence, one can choose ρEi=0. The assumption imposes the effect of uncertainty on the possible deviation of Mi from M¯i to be within a unidirectional threshold.


## 2.4. State Transformation

We propose the following transformation for the space errorei:(15)zi1=lnΔid−eiΔid,which means that(16)ei=Δid−Δidexpzi1.Differentiating (15) with respect to time, we get(17)z˙i1=−1Δidexp−zi1e˙i,(18)z¨i1=−1Δidexp−zi1−z˙i1e˙i−1Δidexp−zi1e¨i.Let(19)zi2≔zi1−1Δidexp−zi1e˙i,yields(20)z˙i1=−zi1+zi2,z˙i2=z˙i1−1Δidexp−zi1−z˙i1e˙i−1Δidexp−zi1e¨i.With (7) in (20), the transformed space error dynamics is in the form of(21)z˙i1=−zi1+zi2,z˙i2=−zi1+zi2−zi1−zi22,(22)−1Δidexp−zi1Mi−1ui−cix˙ix˙i−Fi−Mi−1−1ui−1−ci−1x˙i−1x˙i−1−Fi−1.Remark.
From (15), we have for all −∞<ei<Δid, the corresponding zi1 is −∞<zi1<∞. Furthermore, zi1=0⟺ei=0. This is in fact the main reason for the transformation from ei,e˙i to zi1,zi2. If both zi1t and zi2t are bounded, then space error eit∈−∞,Δid, hence no collision.Remark.
The statezi2 given by (19) is selected to make the transformed space error dynamics in a lower triangular form so that the backstepping method [29] can be adopted for the control design, which will be explained in the next section.


## 3. Robust Control Design and Performance Analysis

We now propose a robust controller for the transformed system to realize the performance of uniform boundedness and uniform ultimate boundedness. Then, we show that the controlled platoon will have guaranteed collision avoidance in the error space.


### 3.1. Robust Control Design

Let(23)pi1=c¯ix˙ix˙i+F¯i+Di−1Di−1ui−1−c¯i−1x˙i−1x˙i−1−F¯i−1,(24)pi2=−Di−1Δidexpzi1−2zi2+zi1−zi22.Assumption 3.
(1) There exists a possibly unknown parameterαi and a known function Πi⋅:R×R×R×R×R×R⟶R+ such that for all xi,xi−1,x˙i,x˙i−1,ui−1,t∈R×R×R×R×R×R, σi∈Σi,
(25)
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(2)
The unknown parameterαi belongs to a known fuzzy number.Remark.
The fuzzy numbersρEi and αi are relevant to σi. Their associated membership functions can be determined via μi⋅’s, the fuzzy arithmetic and the decomposition theorem [30]. The extreme value of the fuzzy number ρEi can be evaluated since the universes of discourse Σi’s are known.
Let(26)μi≔−1Δidzi2exp−zi1Πi.
We propose the following control:(27)ui=pi1+pi2+pi3,where(28)pi3=−γiDi−1μiΠi,γi>0.Theorem 1.
Letδi≔zi1zi2T∈R2. Subject to Assumptions 1–3, the control (27) renders the transformed system (22) and (21) the following performance:(i)
Uniform boundedness: for anyri>0, there is a diri<∞ such that if δit0≤ri, then δit≤diri for all t>t0(ii)
Uniform ultimate boundedness: for anyri>0 with δit0≤ri, there exists a d¯i>0 such that δit≤d¯i for any d¯i>d¯i as t≥t0+Tid¯i,ri, where Tid¯i,ri<∞Proof.
Consider the Lyapunov function candidate:(29)Vi=12zi12+zi22.
For a given uncertaintyσi⋅, the derivative of Vi along the trajectory of the controlled system is to be evaluated. First, we have(30)V˙i=zi1z˙i1+zi2z˙i2.
Next, in view of (21), the first term on the RHS of (30) is given by(31)zi1z˙i1=−zi12+zi1zi2.
The second term on the RHS of (30) is(32)zi2z˙i2=zi2−zi1+zi2−zi2zi1−zi22−1Δidzi2exp−zi1Mi−1ui−cix˙ix˙i−Fi−Mi−1−1ui−1−ci−1x˙i−1x˙i−1−Fi−1.
By the third term on the RHS of (32), after decomposing Mi−1, ci, and Fi by (9)–(11) and noting that Mi−1=Di+ΔDi, we get(33)−1Δidzi2exp−zi1Mi−1ui−cix˙ix˙i−Fi−Mi−1−1ui−1−ci−1x˙i−1x˙i−1−Fi−1=−1Δidzi2exp−zi1Di+ΔDipi1+pi2+pi3−c¯i+Δcix˙ix˙i−F¯i+ΔFi−Di−1+ΔDi−1×ui−1−c¯i−1+Δci−1x˙i−1x˙i−1−F¯i−1+ΔFi−1=−1Δidzi2exp−zi1Dipi1−c¯ix˙ix˙i−F¯i−Di−1ui−1−c¯i−1x˙i−1x˙i−1−F¯i−1+Dipi2+ΔDipi1+pi2−cix˙ix˙i−Fi−ΔDi−1ui−1−ci−1x˙i−1x˙i−1−Fi−1+Di−Δcix˙ix˙i−ΔFi−Di−1−Δci−1x˙i−1x˙i−1−ΔFi−1+Di+ΔDipi3.
By (23), it can be shown that(34)Dipi1−c¯ix˙ix˙i−F¯i−Di−1ui−1−c¯i−1x˙i−1x˙i−1−F¯i−1=0.
Next, by (24),(35)−1Δidzi2exp−zi1Dipi2=1Δidzi2exp−zi1DiDi−1Δidexpzi1−2zi2+zi1−zi22=−2zi22+zi2zi1−zi22.
By Assumption 3, we obtain(36)−1Δidzi2exp−zi1ΔDipi1+pi2−cix˙ix˙i−Fi−ΔDi−1ui−1−ci−1x˙i−1x˙i−1−Fi−1+Di−Δcix˙ix˙i−ΔFi−Di−1−Δci−1x˙i−1x˙i−1−ΔFi−1≤1+ρEi1Δidzi2exp−zi1αiΠi≤1+ρEiαiμi.
By (28), Assumption 2, and (26),(37)−1Δidzi2exp−zi1Di+ΔDipi3≤−1Δidzi2exp−zi1Di+DiEi−γiDi−1μiΠi=−γiDiDi−11+ρEi−1Δidzi2exp−zi1Πiμi=−γi1+ρEiμi2.
With (33)–(37) in (32), we have(38)zi2z˙i2≤zi2−zi1+zi2−zi2zi1−zi22−2zi22+zi2zi1−zi22+1+ρEiαiμi−γi1+ρEiμi2=−zi1zi2−zi22+1+ρEiαiμi−γi1+ρEiμi2.
With (31) and (38) in (30), we obtain(39)V˙i≤−zi12−zi22+1+ρEiαiμi−γi1+ρEiμi2.
Since(40)1+ρEiαiμi−γi1+ρEiμi2≤1+ρEiαi24γi,we have (noting that δi2=zi12+zi22)(41)V˙i≤−δi2+ɛiγi,where εi≔1+ρEiαi2/4. This means that V˙i is negative definite for all δi such that(42)δi>ɛiγi.
Since all universes of discourseΣij’s are compact (hence, closed and bounded), εi is bounded. Noticing that γi is crisp, we conclude that V˙i is a negative definite for sufficiently large δi. Therefore, upon invoking the standard arguments as in [31], we conclude the solution to the transformed error dynamics of the ith vehicle is uniform boundedness with(43)diri=Riifri≤Ri,riifri>Ri,Ri=εiγi.
Furthermore, uniform ultimate boundedness also follows with(44)d¯i=Ri,Tid¯i,ri=0,ifri≤d¯i,ri2−d¯i2d¯i2−Ri2,otherwise,Q.E.D.Remark.
Notice the control given by (23) only needs the information of the preceding vehicle. No acceleration feedback or information of the leader or other followers are required. One may argue that the control given by (23) needs the information of M¯i−1,c¯i−1,F¯i−1,andui−1 besides xi−1 and x˙i−1 of the preceding vehicle, which increases the burden of communication. However, a straightforward modification to the control will show that without the information of M¯i−1,c¯i−1, F¯i−1,andui−1 (noting that Mi−1−1ui−1−ci−1x˙i−1x˙i−1−Fi−1 is the acceleration of the i−1th vehicle), the control still works as long as this unknown acceleration is bounded by a known function Π˜ixi,xi−1,x˙i,x˙i−1,t.Remark.
In the special case when the system is without uncertainty, that isΔDi≡0,Δci≡0,andΔFi≡0, we may choose Πi=0 and hence pi3=0. The control ui=pi1+pi2 will render V˙i≤−δi2. Therefore, we conclude that δi⟶0 as t⟶∞. Notice that zi1=0⟺ei=0. Hence, ei⟶0 as t⟶∞. Furthermore, if we choose pi2=0, then V˙i=0. This means if ei=0 initially, then ei=0 for all t≥t0.Remark.
Notice that the third term in the control scheme (27) is for compensating the uncertainty and γi is the control gain. From (43) and (44), it can be concluded that the gain γi can be used to manipulate the size of the uniform ultimate boundedness region. In particular, the larger the gain, the smaller the size. This stands for a trade-off between the performance and the cost. As a result, it may be interested in seeking an optimal choice of γi for a compromise, which will be elaborated later.


### 3.2. Guaranteed Collision Avoidance

Theorem 2.
Consider a platoon of vehicles with theith vehicle described by (4). Suppose that Assumptions 1–3 are met. The control (27) for the ith following vehicle guarantees the collision avoidance of the platoon under arbitrary safe initial condition.Proof.
For theith follower in the platoon, if an initial condition eit0e˙it0T and a desired space Δid are specified, the corresponding initial condition in the transformed state of the same vehicle is given by(45)zi1t0=lnΔid−eit0Δid,(46)zi2t0=zi1t0−1Δidexp−zi1t0e˙it0.
Suppose control (27) is proposed, and hence the transformed state of the ith follower is uniform bounded. By the definition of uniform boundedness in Theorem 1 with (43), we conclude that for any ri>0, there is a diri<∞ such that if(47)δit0=zi12t0+zi22t0≤ri,then(48)δit≤diri,where(49)diri≔εiγi,ifri≤εiγi,ri,ifri>εiγi,for all t≥t0. From (48), we have(50)zi12t+zi22t≤diri,which means that in the worst case(51)zi1t=lnΔid−eitΔid≤diri.
From (51), we get(52)Δid−eitΔid≤expdiri,(53)Δid−eitΔid≥exp−diri.
Therefore,(54)eit≥1−expdiriΔid,(55)eit≤1−exp−diriΔid.
Since0<diri<∞, from (54), we get(56)eit>−∞,and from (55), we have(57)eit<Δid,for all t>t0. Consequently, by Definition 1, we conclude that the platoon is collision avoidable. Q.E.D.


## 3.1. Robust Control Design

Let(23)pi1=c¯ix˙ix˙i+F¯i+Di−1Di−1ui−1−c¯i−1x˙i−1x˙i−1−F¯i−1,(24)pi2=−Di−1Δidexpzi1−2zi2+zi1−zi22.Assumption 3.
(1) There exists a possibly unknown parameterαi and a known function Πi⋅:R×R×R×R×R×R⟶R+ such that for all xi,xi−1,x˙i,x˙i−1,ui−1,t∈R×R×R×R×R×R, σi∈Σi,
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The unknown parameterαi belongs to a known fuzzy number.Remark.
The fuzzy numbersρEi and αi are relevant to σi. Their associated membership functions can be determined via μi⋅’s, the fuzzy arithmetic and the decomposition theorem [30]. The extreme value of the fuzzy number ρEi can be evaluated since the universes of discourse Σi’s are known.
Let(26)μi≔−1Δidzi2exp−zi1Πi.
We propose the following control:(27)ui=pi1+pi2+pi3,where(28)pi3=−γiDi−1μiΠi,γi>0.Theorem 1.
Letδi≔zi1zi2T∈R2. Subject to Assumptions 1–3, the control (27) renders the transformed system (22) and (21) the following performance:(i)
Uniform boundedness: for anyri>0, there is a diri<∞ such that if δit0≤ri, then δit≤diri for all t>t0(ii)
Uniform ultimate boundedness: for anyri>0 with δit0≤ri, there exists a d¯i>0 such that δit≤d¯i for any d¯i>d¯i as t≥t0+Tid¯i,ri, where Tid¯i,ri<∞Proof.
Consider the Lyapunov function candidate:(29)Vi=12zi12+zi22.
For a given uncertaintyσi⋅, the derivative of Vi along the trajectory of the controlled system is to be evaluated. First, we have(30)V˙i=zi1z˙i1+zi2z˙i2.
Next, in view of (21), the first term on the RHS of (30) is given by(31)zi1z˙i1=−zi12+zi1zi2.
The second term on the RHS of (30) is(32)zi2z˙i2=zi2−zi1+zi2−zi2zi1−zi22−1Δidzi2exp−zi1Mi−1ui−cix˙ix˙i−Fi−Mi−1−1ui−1−ci−1x˙i−1x˙i−1−Fi−1.
By the third term on the RHS of (32), after decomposing Mi−1, ci, and Fi by (9)–(11) and noting that Mi−1=Di+ΔDi, we get(33)−1Δidzi2exp−zi1Mi−1ui−cix˙ix˙i−Fi−Mi−1−1ui−1−ci−1x˙i−1x˙i−1−Fi−1=−1Δidzi2exp−zi1Di+ΔDipi1+pi2+pi3−c¯i+Δcix˙ix˙i−F¯i+ΔFi−Di−1+ΔDi−1×ui−1−c¯i−1+Δci−1x˙i−1x˙i−1−F¯i−1+ΔFi−1=−1Δidzi2exp−zi1Dipi1−c¯ix˙ix˙i−F¯i−Di−1ui−1−c¯i−1x˙i−1x˙i−1−F¯i−1+Dipi2+ΔDipi1+pi2−cix˙ix˙i−Fi−ΔDi−1ui−1−ci−1x˙i−1x˙i−1−Fi−1+Di−Δcix˙ix˙i−ΔFi−Di−1−Δci−1x˙i−1x˙i−1−ΔFi−1+Di+ΔDipi3.
By (23), it can be shown that(34)Dipi1−c¯ix˙ix˙i−F¯i−Di−1ui−1−c¯i−1x˙i−1x˙i−1−F¯i−1=0.
Next, by (24),(35)−1Δidzi2exp−zi1Dipi2=1Δidzi2exp−zi1DiDi−1Δidexpzi1−2zi2+zi1−zi22=−2zi22+zi2zi1−zi22.
By Assumption 3, we obtain(36)−1Δidzi2exp−zi1ΔDipi1+pi2−cix˙ix˙i−Fi−ΔDi−1ui−1−ci−1x˙i−1x˙i−1−Fi−1+Di−Δcix˙ix˙i−ΔFi−Di−1−Δci−1x˙i−1x˙i−1−ΔFi−1≤1+ρEi1Δidzi2exp−zi1αiΠi≤1+ρEiαiμi.
By (28), Assumption 2, and (26),(37)−1Δidzi2exp−zi1Di+ΔDipi3≤−1Δidzi2exp−zi1Di+DiEi−γiDi−1μiΠi=−γiDiDi−11+ρEi−1Δidzi2exp−zi1Πiμi=−γi1+ρEiμi2.
With (33)–(37) in (32), we have(38)zi2z˙i2≤zi2−zi1+zi2−zi2zi1−zi22−2zi22+zi2zi1−zi22+1+ρEiαiμi−γi1+ρEiμi2=−zi1zi2−zi22+1+ρEiαiμi−γi1+ρEiμi2.
With (31) and (38) in (30), we obtain(39)V˙i≤−zi12−zi22+1+ρEiαiμi−γi1+ρEiμi2.
Since(40)1+ρEiαiμi−γi1+ρEiμi2≤1+ρEiαi24γi,we have (noting that δi2=zi12+zi22)(41)V˙i≤−δi2+ɛiγi,where εi≔1+ρEiαi2/4. This means that V˙i is negative definite for all δi such that(42)δi>ɛiγi.
Since all universes of discourseΣij’s are compact (hence, closed and bounded), εi is bounded. Noticing that γi is crisp, we conclude that V˙i is a negative definite for sufficiently large δi. Therefore, upon invoking the standard arguments as in [31], we conclude the solution to the transformed error dynamics of the ith vehicle is uniform boundedness with(43)diri=Riifri≤Ri,riifri>Ri,Ri=εiγi.
Furthermore, uniform ultimate boundedness also follows with(44)d¯i=Ri,Tid¯i,ri=0,ifri≤d¯i,ri2−d¯i2d¯i2−Ri2,otherwise,Q.E.D.Remark.
Notice the control given by (23) only needs the information of the preceding vehicle. No acceleration feedback or information of the leader or other followers are required. One may argue that the control given by (23) needs the information of M¯i−1,c¯i−1,F¯i−1,andui−1 besides xi−1 and x˙i−1 of the preceding vehicle, which increases the burden of communication. However, a straightforward modification to the control will show that without the information of M¯i−1,c¯i−1, F¯i−1,andui−1 (noting that Mi−1−1ui−1−ci−1x˙i−1x˙i−1−Fi−1 is the acceleration of the i−1th vehicle), the control still works as long as this unknown acceleration is bounded by a known function Π˜ixi,xi−1,x˙i,x˙i−1,t.Remark.
In the special case when the system is without uncertainty, that isΔDi≡0,Δci≡0,andΔFi≡0, we may choose Πi=0 and hence pi3=0. The control ui=pi1+pi2 will render V˙i≤−δi2. Therefore, we conclude that δi⟶0 as t⟶∞. Notice that zi1=0⟺ei=0. Hence, ei⟶0 as t⟶∞. Furthermore, if we choose pi2=0, then V˙i=0. This means if ei=0 initially, then ei=0 for all t≥t0.Remark.
Notice that the third term in the control scheme (27) is for compensating the uncertainty and γi is the control gain. From (43) and (44), it can be concluded that the gain γi can be used to manipulate the size of the uniform ultimate boundedness region. In particular, the larger the gain, the smaller the size. This stands for a trade-off between the performance and the cost. As a result, it may be interested in seeking an optimal choice of γi for a compromise, which will be elaborated later.


## 3.2. Guaranteed Collision Avoidance

Theorem 2.
Consider a platoon of vehicles with theith vehicle described by (4). Suppose that Assumptions 1–3 are met. The control (27) for the ith following vehicle guarantees the collision avoidance of the platoon under arbitrary safe initial condition.Proof.
For theith follower in the platoon, if an initial condition eit0e˙it0T and a desired space Δid are specified, the corresponding initial condition in the transformed state of the same vehicle is given by(45)zi1t0=lnΔid−eit0Δid,(46)zi2t0=zi1t0−1Δidexp−zi1t0e˙it0.
Suppose control (27) is proposed, and hence the transformed state of the ith follower is uniform bounded. By the definition of uniform boundedness in Theorem 1 with (43), we conclude that for any ri>0, there is a diri<∞ such that if(47)δit0=zi12t0+zi22t0≤ri,then(48)δit≤diri,where(49)diri≔εiγi,ifri≤εiγi,ri,ifri>εiγi,for all t≥t0. From (48), we have(50)zi12t+zi22t≤diri,which means that in the worst case(51)zi1t=lnΔid−eitΔid≤diri.
From (51), we get(52)Δid−eitΔid≤expdiri,(53)Δid−eitΔid≥exp−diri.
Therefore,(54)eit≥1−expdiriΔid,(55)eit≤1−exp−diriΔid.
Since0<diri<∞, from (54), we get(56)eit>−∞,and from (55), we have(57)eit<Δid,for all t>t0. Consequently, by Definition 1, we conclude that the platoon is collision avoidable. Q.E.D.


## 4. Optimal Robust Control

### 4.1. Fuzzy-Based Performance Index

In the previous analysis, we know that the system performance can be guaranteed by a deterministic control design. The size of the uniform ultimate boundedness region decreases asγi increases. Furthermore, the size approaches zero when γi approaches infinity. Therefore, both the performance and control effort are affected by the control gain γi. From the practical design point of view, one may be interested in seeking an optimal choice of γi for a compromise among various conflicting criteria.We first explore more on the deterministic performance of the uncertain system. From (41), we know that(58)V˙it≤−δi2+εiγi,≤−2Vit+εiγi,where Vit0=1/2zi12t0+zi22t0. This is a differential inequality [32], whose analysis can be made according to [33]. The following is needed for our analysis of (58).Definition 2 (see [32]).
Ifωψ,t is a scalar function of the scalars ψ and t in some open connected set D, we say a function ψt, t0≤t≤t¯ and t¯≥t0 is a solution of the differential inequality(59)ψ˙t≤ωψt,t,on t0,t¯ if ψt is continuous on t0,t¯ and its derivative on t0,t¯ satisfies (59).Theorem 3 (see [32]).
Letωϕ,t be continuous on an open connected set D∈R2 such that the initial value problem for the scalar equation(60)ϕ˙t=ωϕt,t,ϕt0=ϕ0,has a unique solution. If ϕt is a solution of (60) on t0≤t≤t¯ and ψt is a solution of (59) on t0≤t≤t¯ with ψt0≤ϕt0, then ψt≤ϕt for t0≤t≤t¯.Instead of exploring the solution of the differential inequality, which is often nonunique and not available, the theorem suggests that it may be feasible to study the upper bound of the solution. The reasoning is, however, based on that the solution of (60) is unique.Theorem 4 (see [33]).
Consider the differential inequality (59) and the differential equation (60). Suppose that for some constant L>0, the function ω⋅ satisfies the Lipschitz condition(61)ωv1,t−ωv2,t≤v1−v2,for all points v1,t,v2,t∈D. Then, any function ψt that satisfies the differential inequality (59) for t0≤t<t¯ satisfies also the inequality(62)ψt≤ϕt,for t0≤t≤t¯.We consider the differential equation(63)r˙it=−2rit+εiγi,rit0=Vi0=Vit0.The RHS satisfies the global Lipschitz condition withL=2. We proceed with solving the differential equation (63). This results in(64)rit=Vi0−εi2γiexp−2t−t0+εi2γi.Therefore,(65)Vit≤rit,or(66)Vit≤Vi0−εi2γiexp−2t−t0+εi2γi,for all t≥t0. By the same argument, we also have, for any ts and any τ≥ts,(67)Viτ≤Vis−εi2γiexp−2τ−ts+εi2γi,where Vis=Vits=1/2zi12ts+zi22ts. The time ts is when the control scheme (27) starts to be executed. It does not need to be t0.SinceViτ=1/2δiτ2, the RHS of (67) provides an upper bound of 1/2δiτ2. This in turn leads to an upper bound of δiτ. For each τ≥ts, let(68)ηiγi,τ,ts=Vis−εi2γiexp−2τ−ts,(69)ηi∞γi=εi2γi.Notice that for eachγi, ts, ηiγi,τ,ts⟶0 as τ⟶∞.Definition 3.
Consider a fuzzy set:(70)N=ν,μNνν∈N.
For any functionf:N⟶R, the D-operation Dfν is given by(71)Dfν=∫NfνμNνdν∫NμNνdν.Remark.
In a sense, theD-operation Dfν takes an average value of fν over μNν. In the special case that fν=ν, this is reduced to the well-known center-of-gravity defuzzification method (see, e.g., [34]). If N is crisp (i.e., μNν=1 for all ν∈N), then Dfν=fν.Lemma 1.
For any crisp constanta∈R,(72)Dafν=aDfν.Proof.
By Definition 3,(73)Dafν=∫NafνμNνdν∫NμNνdν=a∫NfνμNνdν∫NμNνdν=aDfν,Q.E.D.
We now propose the following performance index. For anyts, let(74)Jiγi,ts≔D∫ts∞ηi2γi,τ,tsdτ+Γi1Dηi∞2γi+Γi1γi2≕Ji1γi,ts+Γi1Ji2γi+Γi2Ji3γi,where Γi1>0,Γi2>0. The performance index consists of three parts. The first part Ji1γi,ts may be interpreted as the value of the overall transient performance (via the integration) from time ts. The second part Ji2γi may be interpreted as the value of the steady state performance. The third part Ji3γi may be interpreted as the value of the control effort. Both Γi1 and Γi2 are weighting factors. The weighting of J1 is normalized to be unity.
For anyts, given Δi, Γi1, and Γi2, our design problem is to choose γi>0 in (28) such that the performance index Jiγi,ts is minimized.Remark.
A standard LQG (i.e., linear-quadratic- Gaussian) problem in stochastic control is to minimize a performance index which is the average (via the expectation value operation in probability) of the overall state and control accumulation. The current approach may be viewed, loosely speaking, as a parallel, though not equivalent, in fuzzy dynamical systems. However, one cannot be too careful in distinguishing the differences. For example, the Gaussian probability distribution implies that the uncertainty is unbounded (although a higher bound is predicted by a lower probability). In the current consideration, the uncertainty bound is always finite. Also, the standard LQG does not take parameter uncertainty into account.
One can show that(75)∫ts∞ηi2γi,τ,tsdτ=Vis−εi2γi2∫ts∞exp−4τ−tsdτ=Vis−εi2γi2−14exp−4τ−tsts∞=14Vis−εi2γi2.
Taking theD-operation yields,(76)D∫ts∞ηi2γi,τ,tsdτ=D14Vis−εi2γi2=14DVis2−14γiDVisεi+116γi2Dεi2.
The last equality is due to Lemma1. Next, we analyze the cost Ji2γi. Again, by Lemma 1,(77)Dηi∞2γi=Dεi24γi2=14γi2Dεi2.
With (76) and (77) in (74), we obtain(78)Jiγi,ts=14DVis2−14γiDVisεi+116γi2Dεi2+Γi114γi2Dεi2+Γi2γi2≕λ1−λ2γi+λ3γi2+Γi1λ4γi2+Γi2γi2,where λ1=1/4DVis2, λ2=1/4DVisεi, λ3=1/16Dεi2, and λ4=1/4Dεi2.


### 4.2. Formulation of the Optimal Design Problem

The optimal design problem is then equivalent to the following constrained optimization problem. For anyts,(79)minγiJiγi,ts,subject toγi>0.For anyts, taking the first order derivative of Ji with respect to γi,(80)∂Ji∂γi=λ2γi2−2λ3γi3−2Γi1λ4γi3+2Γi2γi4=1γi3λ2γi−2λ3−2Γi1λ4+2Γi2γi4.That∂Ji/∂γi=0 leads to(81)λ2γi−2λ3−2Γi1λ4+2Γi2γi4=0,or(82)λ2γi+2Γi2γi4=2λ3+Γi1λ4,which is a quartic equation.Theorem 5.
SupposeDεi≠0. For any ts, given λ1, λ2, λ3, and λ4, the solution γi>0 to (82) always exists and is unique, which globally minimizes the performance index (79).Proof.
Letfγi≔λ2γi+2Γi2γi4. Then, f0=0 and f⋅ is continuous in γi. In view of λ2>0 and Γi2>0, we conclude that f⋅ is strictly increasing in γi. Since Dεi≠0, we have Dεi>0, Dεi2>0, λ3>0, λ4>0, and, therefore, 2λ3+Γi1λ4>0. Consequently, the solution γi>0 to (82) always exists and is unique. For the unique solution γi>0 that solves (82),(83)∂2Ji∂γi2=−3γi4λ2γi2λ3−2Γi1λ4+2Γi2γi4+1γi3λ2+8Γi2γi3=1γi3λ2+8Γi2γi3>0.
Therefore, the positive solutionγi>0 of the quartic equation (82) solves the constrained minimization problem (68). Q.E.D.Remark.
In the special case that the fuzzy setsSi are crisp, DVis=Vis, Dεi=εi, etc. The current setting still applies. The optimal design can also be found by solving (82), the solution of which will be given in the next section.


### 4.3. Solution to the Optimal Problem

The solutions of the quartic equation (82) depend on the following cubic resolvent:(84)z3+−4r¯z−q¯2=0,where(85)r¯=−1Γi2λ3+Γi1λ4,q¯=λ22Γi2.Letp1^=−4r¯ and p2^=−q¯2. The discriminant D^ of the cubic resolvent is(86)D^=p1^33+p2^22.SinceΓi2>0, λ3>0, Γi1>0, and λ4>0, we have r¯<0, thus p1^>0, so that D^>0. The solutions of the cubic resolvent are given by(87)z1=u^+v^,(88)z2=−u^+v^2+u^−v^i32,(89)z3=−u^+v^2−u^−v^i32,where(90)u^=−p2^2+D^1/3,v^=−p2^2−D^1/3.The cubic resolvent has one real solution and two complex conjugate solutions. This in turn implies that the quartic solution has two real solutions and one pair of complex conjugate solutions. The maximum real solution, which is positive and is therefore the optimal solution to the constrained optimization problem, of the quartic equation is given by(91)γiopt=12z1+z2+z3.Withz1, z2, and z3 into (91), the positive solution of the quartic equation is given by(92)γiopt=12u^+v^+7u^2+7v^2−10u^v^cosψ2,where(93)ψ=tan−13/2u^−v^−1/2u^+v^.With (82), the cost J in (78) can be rewritten as(94)Ji=λ1−λ2γi+λ3γi2+Γi1λ4γi2+Γi2γi2=λ1−1γi2λ2γi+2Γi2γi4+λ3γi2+Γi1λ4γi2+3Γi2γi2=λ1−1γi22λ3+Γi1λ4+λ3γi2+Γi1λ4γi2+3Γi2γi2=λ1−λ3γi2−Γi1λ4γi2+3Γi2γi2=λ1−1γi2λ3+Γi1λ4+3Γi2γi4.With (92), the minimum cost is given by(95)Jimin=λ1−4z1+z2+z32×λ3+Γi1λ4−316Γi2z1+z2+z34.Remark.
Combining the results of Sections 2–4, the robust control scheme (27) using the optimal design of γi>0 renders the solution of the transformed error dynamic system uniformly bounded and uniformly ultimately bounded. In addition, the performance index Ji given by (78) is globally minimized. Furthermore, the closed-form (i.e., analytic) solution is shown. The optimization problem is completely solved.
The optimal design procedure is summarized as follows:Step 1: for givenΔid, the control terms p1 in (23) and p2 in (24) are obtained.Step 2: obtain the fuzzy descriptions ofαi and ρEi and the bounding function Π⋅. The control p3 is given by (28) with γi undetermined.Step 3: calculateλ1,2,3,4 based on the α-cuts of the membership functions, the fuzzy arithmetic, and the decomposition theorem.Step 4: for givenΓi1>0andΓi2>0, calculate the optimal gain γiopt in (92). The resulting minimal performance index can be obtained by (95).Step 5: the robust control scheme is given by (27).


## 4.1. Fuzzy-Based Performance Index

In the previous analysis, we know that the system performance can be guaranteed by a deterministic control design. The size of the uniform ultimate boundedness region decreases asγi increases. Furthermore, the size approaches zero when γi approaches infinity. Therefore, both the performance and control effort are affected by the control gain γi. From the practical design point of view, one may be interested in seeking an optimal choice of γi for a compromise among various conflicting criteria.We first explore more on the deterministic performance of the uncertain system. From (41), we know that(58)V˙it≤−δi2+εiγi,≤−2Vit+εiγi,where Vit0=1/2zi12t0+zi22t0. This is a differential inequality [32], whose analysis can be made according to [33]. The following is needed for our analysis of (58).Definition 2 (see [32]).
Ifωψ,t is a scalar function of the scalars ψ and t in some open connected set D, we say a function ψt, t0≤t≤t¯ and t¯≥t0 is a solution of the differential inequality(59)ψ˙t≤ωψt,t,on t0,t¯ if ψt is continuous on t0,t¯ and its derivative on t0,t¯ satisfies (59).Theorem 3 (see [32]).
Letωϕ,t be continuous on an open connected set D∈R2 such that the initial value problem for the scalar equation(60)ϕ˙t=ωϕt,t,ϕt0=ϕ0,has a unique solution. If ϕt is a solution of (60) on t0≤t≤t¯ and ψt is a solution of (59) on t0≤t≤t¯ with ψt0≤ϕt0, then ψt≤ϕt for t0≤t≤t¯.Instead of exploring the solution of the differential inequality, which is often nonunique and not available, the theorem suggests that it may be feasible to study the upper bound of the solution. The reasoning is, however, based on that the solution of (60) is unique.Theorem 4 (see [33]).
Consider the differential inequality (59) and the differential equation (60). Suppose that for some constant L>0, the function ω⋅ satisfies the Lipschitz condition(61)ωv1,t−ωv2,t≤v1−v2,for all points v1,t,v2,t∈D. Then, any function ψt that satisfies the differential inequality (59) for t0≤t<t¯ satisfies also the inequality(62)ψt≤ϕt,for t0≤t≤t¯.We consider the differential equation(63)r˙it=−2rit+εiγi,rit0=Vi0=Vit0.The RHS satisfies the global Lipschitz condition withL=2. We proceed with solving the differential equation (63). This results in(64)rit=Vi0−εi2γiexp−2t−t0+εi2γi.Therefore,(65)Vit≤rit,or(66)Vit≤Vi0−εi2γiexp−2t−t0+εi2γi,for all t≥t0. By the same argument, we also have, for any ts and any τ≥ts,(67)Viτ≤Vis−εi2γiexp−2τ−ts+εi2γi,where Vis=Vits=1/2zi12ts+zi22ts. The time ts is when the control scheme (27) starts to be executed. It does not need to be t0.SinceViτ=1/2δiτ2, the RHS of (67) provides an upper bound of 1/2δiτ2. This in turn leads to an upper bound of δiτ. For each τ≥ts, let(68)ηiγi,τ,ts=Vis−εi2γiexp−2τ−ts,(69)ηi∞γi=εi2γi.Notice that for eachγi, ts, ηiγi,τ,ts⟶0 as τ⟶∞.Definition 3.
Consider a fuzzy set:(70)N=ν,μNνν∈N.
For any functionf:N⟶R, the D-operation Dfν is given by(71)Dfν=∫NfνμNνdν∫NμNνdν.Remark.
In a sense, theD-operation Dfν takes an average value of fν over μNν. In the special case that fν=ν, this is reduced to the well-known center-of-gravity defuzzification method (see, e.g., [34]). If N is crisp (i.e., μNν=1 for all ν∈N), then Dfν=fν.Lemma 1.
For any crisp constanta∈R,(72)Dafν=aDfν.Proof.
By Definition 3,(73)Dafν=∫NafνμNνdν∫NμNνdν=a∫NfνμNνdν∫NμNνdν=aDfν,Q.E.D.
We now propose the following performance index. For anyts, let(74)Jiγi,ts≔D∫ts∞ηi2γi,τ,tsdτ+Γi1Dηi∞2γi+Γi1γi2≕Ji1γi,ts+Γi1Ji2γi+Γi2Ji3γi,where Γi1>0,Γi2>0. The performance index consists of three parts. The first part Ji1γi,ts may be interpreted as the value of the overall transient performance (via the integration) from time ts. The second part Ji2γi may be interpreted as the value of the steady state performance. The third part Ji3γi may be interpreted as the value of the control effort. Both Γi1 and Γi2 are weighting factors. The weighting of J1 is normalized to be unity.
For anyts, given Δi, Γi1, and Γi2, our design problem is to choose γi>0 in (28) such that the performance index Jiγi,ts is minimized.Remark.
A standard LQG (i.e., linear-quadratic- Gaussian) problem in stochastic control is to minimize a performance index which is the average (via the expectation value operation in probability) of the overall state and control accumulation. The current approach may be viewed, loosely speaking, as a parallel, though not equivalent, in fuzzy dynamical systems. However, one cannot be too careful in distinguishing the differences. For example, the Gaussian probability distribution implies that the uncertainty is unbounded (although a higher bound is predicted by a lower probability). In the current consideration, the uncertainty bound is always finite. Also, the standard LQG does not take parameter uncertainty into account.
One can show that(75)∫ts∞ηi2γi,τ,tsdτ=Vis−εi2γi2∫ts∞exp−4τ−tsdτ=Vis−εi2γi2−14exp−4τ−tsts∞=14Vis−εi2γi2.
Taking theD-operation yields,(76)D∫ts∞ηi2γi,τ,tsdτ=D14Vis−εi2γi2=14DVis2−14γiDVisεi+116γi2Dεi2.
The last equality is due to Lemma1. Next, we analyze the cost Ji2γi. Again, by Lemma 1,(77)Dηi∞2γi=Dεi24γi2=14γi2Dεi2.
With (76) and (77) in (74), we obtain(78)Jiγi,ts=14DVis2−14γiDVisεi+116γi2Dεi2+Γi114γi2Dεi2+Γi2γi2≕λ1−λ2γi+λ3γi2+Γi1λ4γi2+Γi2γi2,where λ1=1/4DVis2, λ2=1/4DVisεi, λ3=1/16Dεi2, and λ4=1/4Dεi2.


## 4.2. Formulation of the Optimal Design Problem

The optimal design problem is then equivalent to the following constrained optimization problem. For anyts,(79)minγiJiγi,ts,subject toγi>0.For anyts, taking the first order derivative of Ji with respect to γi,(80)∂Ji∂γi=λ2γi2−2λ3γi3−2Γi1λ4γi3+2Γi2γi4=1γi3λ2γi−2λ3−2Γi1λ4+2Γi2γi4.That∂Ji/∂γi=0 leads to(81)λ2γi−2λ3−2Γi1λ4+2Γi2γi4=0,or(82)λ2γi+2Γi2γi4=2λ3+Γi1λ4,which is a quartic equation.Theorem 5.
SupposeDεi≠0. For any ts, given λ1, λ2, λ3, and λ4, the solution γi>0 to (82) always exists and is unique, which globally minimizes the performance index (79).Proof.
Letfγi≔λ2γi+2Γi2γi4. Then, f0=0 and f⋅ is continuous in γi. In view of λ2>0 and Γi2>0, we conclude that f⋅ is strictly increasing in γi. Since Dεi≠0, we have Dεi>0, Dεi2>0, λ3>0, λ4>0, and, therefore, 2λ3+Γi1λ4>0. Consequently, the solution γi>0 to (82) always exists and is unique. For the unique solution γi>0 that solves (82),(83)∂2Ji∂γi2=−3γi4λ2γi2λ3−2Γi1λ4+2Γi2γi4+1γi3λ2+8Γi2γi3=1γi3λ2+8Γi2γi3>0.
Therefore, the positive solutionγi>0 of the quartic equation (82) solves the constrained minimization problem (68). Q.E.D.Remark.
In the special case that the fuzzy setsSi are crisp, DVis=Vis, Dεi=εi, etc. The current setting still applies. The optimal design can also be found by solving (82), the solution of which will be given in the next section.


## 4.3. Solution to the Optimal Problem

The solutions of the quartic equation (82) depend on the following cubic resolvent:(84)z3+−4r¯z−q¯2=0,where(85)r¯=−1Γi2λ3+Γi1λ4,q¯=λ22Γi2.Letp1^=−4r¯ and p2^=−q¯2. The discriminant D^ of the cubic resolvent is(86)D^=p1^33+p2^22.SinceΓi2>0, λ3>0, Γi1>0, and λ4>0, we have r¯<0, thus p1^>0, so that D^>0. The solutions of the cubic resolvent are given by(87)z1=u^+v^,(88)z2=−u^+v^2+u^−v^i32,(89)z3=−u^+v^2−u^−v^i32,where(90)u^=−p2^2+D^1/3,v^=−p2^2−D^1/3.The cubic resolvent has one real solution and two complex conjugate solutions. This in turn implies that the quartic solution has two real solutions and one pair of complex conjugate solutions. The maximum real solution, which is positive and is therefore the optimal solution to the constrained optimization problem, of the quartic equation is given by(91)γiopt=12z1+z2+z3.Withz1, z2, and z3 into (91), the positive solution of the quartic equation is given by(92)γiopt=12u^+v^+7u^2+7v^2−10u^v^cosψ2,where(93)ψ=tan−13/2u^−v^−1/2u^+v^.With (82), the cost J in (78) can be rewritten as(94)Ji=λ1−λ2γi+λ3γi2+Γi1λ4γi2+Γi2γi2=λ1−1γi2λ2γi+2Γi2γi4+λ3γi2+Γi1λ4γi2+3Γi2γi2=λ1−1γi22λ3+Γi1λ4+λ3γi2+Γi1λ4γi2+3Γi2γi2=λ1−λ3γi2−Γi1λ4γi2+3Γi2γi2=λ1−1γi2λ3+Γi1λ4+3Γi2γi4.With (92), the minimum cost is given by(95)Jimin=λ1−4z1+z2+z32×λ3+Γi1λ4−316Γi2z1+z2+z34.Remark.
Combining the results of Sections 2–4, the robust control scheme (27) using the optimal design of γi>0 renders the solution of the transformed error dynamic system uniformly bounded and uniformly ultimately bounded. In addition, the performance index Ji given by (78) is globally minimized. Furthermore, the closed-form (i.e., analytic) solution is shown. The optimization problem is completely solved.
The optimal design procedure is summarized as follows:Step 1: for givenΔid, the control terms p1 in (23) and p2 in (24) are obtained.Step 2: obtain the fuzzy descriptions ofαi and ρEi and the bounding function Π⋅. The control p3 is given by (28) with γi undetermined.Step 3: calculateλ1,2,3,4 based on the α-cuts of the membership functions, the fuzzy arithmetic, and the decomposition theorem.Step 4: for givenΓi1>0andΓi2>0, calculate the optimal gain γiopt in (92). The resulting minimal performance index can be obtained by (95).Step 5: the robust control scheme is given by (27).


## 5. Simulation Results

In this section, numerical simulations will be conducted to examine the behavior of a vehicle platoon traveling on a highway under the robust control law (27) developed in previous sections. Consider there are totally four vehicles in a platoon, that is, one leader with three followers. The following parameters are chosen for numerical simulations. The nominal vehicle masses (kg) M¯L=M¯1=M¯2=M¯3=1000, the nominal aerodynamic coefficients c¯L=c¯1=c¯2=c¯3=0.3, the nominal resistance forces (N) F¯L=F¯1=F¯2=F¯3=100, and the vehicle lengths lL=l1=l2=l3=5m. The uncertainties in each vehicle which are (possibly) time-varying are given by ΔML=35sin0.1t,ΔM1=32cos0.5t,ΔM2=40cost, ΔM3=−25cos0.1t, ΔcL=0.02, Δc1=0.01, Δc2=−0.03, Δc3=−0.02, ΔFL=180sin0.5t, ΔF1=200sint, ΔF2=190sint−π/6, and ΔF3=160sint−π/6. Notice that all uncertainties are bounded. Suppose the parameters ρE1, ρE2, and ρE3 in Assumption 2 are all “close to 0”, where the associated membership functions are given by (all triangular)(96)μρE1,2,3ν=103ν+1,−0.3≤ν≤0,−103ν+1,0≤ν≤0.3,and the known functions in Assumption 3 are given by(97)Π1,2,3=α1,2,30.1e˙i2+0.1ei2+0.6,with the uncertain parameters α1,2,3 all “close to 1,” where the associated membership functions are given by (all triangular)(98)μα1,2,3ν=2ν−1,12≤ν≤1,−2ν+3,1≤ν≤32.Hence, Assumptions 2 and 3 are met.The leading vehicle is under the control as follows:(99)u¯L≔c¯Lx˙Lx˙L+F¯L,(100)uL=u¯L+2500sin0.2πt−5,if5<t≤10,u¯L−1500sin0.2πt−15,if15<t≤20,u¯L,otherwise.For comparison purpose, we consider two types of control for the followers. The first is the robust controller (27):(101)ui=pi1+pi2+pi3,i=1,2,3.The second is the PD controller:(102)ui=−ki1ei−ki2e˙i,i=1,2,3,where ki1 and ki2 are the proportional and derivative control gains.Suppose the desired space is given byΔ1d=Δ2d=Δ3d=5m. For the PD controller, we choose ki1=220andki2=500,i=1,2,3. Two cases of initial conditions are considered (notice that all initial conditions are crisp): first, the zero initial condition case with initial positions xts=xt0=x0=100908070T and initial velocities x˙ts=x˙t0=x˙0=10101010T. We find that the position errors e=000T and velocity errors e˙=000T. Recalling that Vis=1/2zi12ts+zi22ts, we have Vis=0,i=1,2,3,4, By using the fuzzy arithmetic and the decomposition theorem, we obtain λ1=0, λ2=0, λ3=0.0039, and λ4=0.0156, and the quartic equation (82) is given by(103)2Γi2γi4=20.0039+0.0156Γi2.We choose five sets of weighting factorsΓ1 and Γ2 for demonstration. Their values and the corresponding γiopt and Jmin are summarized in Table 1.Table 1
Weighting factors/optimal gain/minimum cost.
(Γi1,Γi2)
Γ
i
1
/
Γ
i
2
γ
i
opt
J
i
min
(100, 1)
100
1.1187
2.5031
(10, 1)
10
0.6326
0.8004
(1, 1)
1
0.3738
0.2795
(1, 10)
0.1
0.2102
0.8839
(1, 100)
0.01
0.1182
2.7951Next, the critical case with initial positionsxts=xt0=x0=100948882T and initial velocities x˙ts=x˙t0=x˙0=10121416T. Similarly, it can be found that ets=444T and e˙ts=222T (these mean the vehicles are only 1m apart and they are approaching initially). Recalling that Vis=1/2zi12ts+zi22ts, we have Vis=1.3747,i=1,2,3,4. By using the fuzzy arithmetic and the decomposition theorem, we obtain λ1=0.4725, λ2=0.0859, λ3=0.0039, and λ4=0.0156. By following (84) to (95), the quartic equation (82) is given by(104)0.0859γi+2Γi2γi4=20.0039+0.0156Γi2.Again, five sets of weighting factorsΓi1 and Γi2 are chosen for demonstration. Their values and the corresponding γiopt and Jimin are summarized in Table 2.Table 2
Weighting factors/optimal gain/minimum cost.
(Γi1,Γi2)
Γ
i
1
/
Γ
i
2
γ
i
opt
J
i
min
(100, 1)
100
2.5118
6.9956
(10, 1)
10
0.8280
1.2878
(1, 1)
1
0.3627
0.5156
(1, 10)
0.1
0.1133
1.3640
(1, 100)
0.01
0.0357
13.522With these parameters obtained, numerical simulations are conducted. The final results are shown in Figures2–13. Figures 2–9 show the performance of the platoon under zero initial conditions. Figures 2–5 show the position and velocity histories along x-direction of each vehicle in the platoon under the PD controller and robust controller (γiopt=0.3738, when using Γi1=Γi2=1) with zero initial condition, respectively. All trajectories are smooth and no abrupt changes occurred. Figures 6 and 7 show the space error histories for the vehicles under the PD controller and robust controller, respectively. It is shown that the space errors of the followers under PD control oscillate and the maximum magnitude increases. The collisions occur (Δ2,3>5m) for the second and third vehicles at around t=18. This is not surprising since this control does not guarantee stability. On the other hand, the space errors of the followers under robust control always stay in a region smaller than 0.3 m and hence no collisions.Figure 2
The position history of the four vehicles in the platoon under the PD controller with zero initial conditions.Figure 3
The position history of the four vehicles in the platoon under robust controller with zero initial conditions.Figure 4
The velocity history of the four vehicles in the platoon under the PD controller with zero initial conditions.Figure 5
The velocity history of the four vehicles in the platoon under the robust controller with zero initial conditions.Figure 6
The space error history of the four vehicles in the platoon under the PD controller with zero initial conditions.Figure 7
The space error history of the four vehicles in the platoon under the robust controller with zero initial conditions.Figure 8
The control force history of the four vehicles in the platoon under the PD controller with zero initial conditions.Figure 9
The control force history of the four vehicles in the platoon under the robust controller with zero initial conditions.Figure 10
The space error history of the four vehicles in the platoon under the PD controller with critical initial conditions.Figure 11
The control force history of the four vehicles in the platoon under the robust controller with critical initial conditions.Figure 12
Comparison of the space error of the first follower under differentγ1min.Figure 13
Comparison of the control effort of the first follower under differentγ1min.It is interesting to see that in Figure7, the third vehicle is the one with least upper bound of the space error, which means the space error is well suppressed under the robust controller. Figures 8 and 9 show the control effort history under two different controllers. It appears that the magnitude of the robust controller is smaller than that of the PD controller.Figures10–13 show the performance of the platoon under critical initial conditions. Figure 10 shows the space error histories for each vehicle under the robust controller. Despite initially being very close to each other, which means a very dangerous proximity, the space error eit for i=1,2,3, under robust control is always under 5m (i.e., eit<Δid) and hence no collision. Moreover, it shows that the space error of each follower enters a small region (less than 0.3) in less than 5s and stays there, thereafter, regardless of the acceleration changes of the leader at the time periods 5<t≤10 and 15<t≤20. Figure 11 shows the control effort history under critical initial conditions. It can be shown that after the initial position and velocity error is suppressed (after 5s), the effort of the robust controller is smaller than that of the PD controller. For comparison, Figures 12 and 13 show the space error history and its corresponding control effort of the first follower under five γiopt, respectively. It appears that a higher optimal gain γiopt renders a faster settling of the space error ei at the beginning. This is because more weighting was assigned to the transient-state performance J11.


## 6. Conclusion

A new approach is proposed to the decentralized control design for vehicle platooning for uncertain automated highway systems. The uncertainty in the system is nonlinear and (possibly) fast time-varying. The only information about the uncertainty is that it is assumed to be within a prescribed fuzzy set. Based on a creative transformation, a class of decentralized control is proposed for the vehicle platoon. Each vehicle in the platoon only needs the knowledge of the preceding vehicle. No acceleration feedback or the information of the lead vehicle is required. The resulting controlled platoon is global collision avoidance. The resulting performance of the controlled platoon is two folds: one deterministic and one fuzzy. The deterministic performance assures that under the worst case we still have collision avoidance under arbitrary safe initial conditions; this assures the bottom line, while the fuzzy information allows us further consider optimization problem. A control design parameter is selected to minimize a fuzzy-based performance index. Both the analytic forms (i.e., closed forms) of the control design parameter and the resulting minimum performance index are obtained. The optimization problem is completely solved. The simulation results compare the performance of the system under the proposed control with PD control. It shows that even under a very critical initial state, where collision appears to be imminent, the robust control was able to pull out and avoid collision.



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