for all x, y ∈ X 。

The  following  theorem  concerns  the  feasibility  of pro-

posed gear shifting management system。

Theorem 1: In gear shift operations, the control laws (11)-

(12) make the dynamic system of a DCT driveline system  x

to be constrained as:

uon,FB(iΔT )= kpeon(iΔT )+ 

x(t, x0, u) ∈ A, ∀t ∈ [tc, ts] (17)

= 。  uof f (iΔT ) if u˙on < 0, ∀t ∈ [tc, ts] (12)

uof f ,FF (iΔT )   if u˙on ≥ 0, ∀t ∈ [tc, ts]

x0 = [x1(0), x2(0),。。。, xn(0)]

umin(t) ≤ u(t) ≤ umax(t), ∀t ≥ 0

n

where, uon,FF  and uon,FB  are the feed-forward and feedback

A ¾ {x ∈ R

| u˙on ≥ 0 and  u˙of f  ≤ 0}。

control input of the oncoming clutch, uof f ,FF the feed- forward control input of the off-going clutch, i the sampling instant, and ΔT the sampling period, respectively。 The torque tracking error of the oncoming clutch is defined as eon =

Tcd − Tc。

proportional-integrative (PI) type controller, and the feedforward controller is composed of predetermined torque profiles and low pass filters。 In order to make the control input  differentiable,  First  order  phase  lags  are  applied to

(11) and (12) as

u˙ f ,on + aonuf,on = aonuon (13)

u˙ f ,of f + aof f uf,of f = aof f uof f (14) where, aon  and aof f   are filter design parameters, and   uf,on

and uf,of f  filtered control inputs。

To analysis the characteristic of the proposed method, the following property is required。

Proof: Given control inputs (11) and (12), the resulting trajectories of uon(t) and uof f (t) satisfies the following properties that can be easily obtained from Lemma 1 along with the assumption A1 and  A2。

on(t )(t  − ti)。 (18)

uof f (ts) ≤ uof f (ti)+ ur     (ti)(ts − ti)。 (19)

Hence, uon  is to be a convex function, and uof f  is to be a concave function [11]。 To illustrate more details of this control scheme, the driveline model described in Section II can be adopted。 Since the solid shaft clutch torque Tc1 and the hollow shaft clutch torque Tc2 are connected by kinematic constraints (i。e。 gearbox), both clutch shaft models (4)    and

(5) are combined into

TJ  = ψ1Tc1 + ψ2Tc2 (20)

where

it1if 1

Lemma 1:  Let X  be a convex  set in R。 Suppose that   the

function  f  is  differentiable,  then  its  gradient  6 f  exists at

each point x, y ∈ X 。 Then,  f  is convex such that

f (y) ≥ f (x)+ 6 f (x)T (y − x) (15)

TJ  is the torque that represent the moment of inertia of the

solid and the hollow shaft, and ψ1 and ψ2 the function of the

gear ratio。 Since the control inputs are convex and concave functions, (20) can be rewritten  by

TJ  = αuon + (1 − α)uof f , α ∈ [0, 1]。 (21)

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