Application of feedback linearisation to the tracking and almost disturbance decoupling control of multi-input multi-output nonlinear system C.-C. Chen and Y.-F. Lin Abstract: The tracking and almost disturbance decoupling problem of multi-input multi-output nonlinear systems based on the feedback linearisation approach are studied. The main contribution of this study is to construct a controller, under appropriate conditions, such that the resulting closed-loop system is valid for any initial condition and bounded tracking signal with the following characteristics: input-to-state stability with respect to disturbance inputs and almost disturbance decoupling, that is, the influence of disturbances on the L 2 norm of the output tracking error can be arbitrarily attenuated by changing some adjustable parameters. One example, which cannot be solved by the first paper of the almost disturbance decoupling problem on account of requiring some sufficient conditions that the nonlinearities multiplying the disturbances satisfy structural triangular conditions, is proposed to exploit the fact that the tracking and the almost disturbance decoupling performances are easily achieved by the proposed approach. To demonstrate the practical applicability, a famous half-car active suspension system has been investigated. 1 Introduction Two well-known tasks of stabilisation and tracking problem are important topics in the field of control. Tracking problem is generally more complicated than stabilisation problem for nonlinear control systems. Many approaches for nonlinear systems have been introduced including feedback linearisation, variable structure control (sliding mode control), backstepping, regulation control, nonlinear H 1 control, internal model principle and H 1 adaptive fuzzy control. Recently, variable structure controls are introduced to deal with nonlinear systems [1]. However, chattering behaviour that may create unmodelled high fre- quency due to the discontinuous switching and imperfect implementation and even drive system to instability is inevitable for variable structure control scheme. Backstepping has been a powerful tool for synthesising con- troller for a class of nonlinear systems. However, a disad- vantage with the backstepping approach is the explosion of complexity which is caused by the complicated repeated differentiations of some nonlinear functions [2, 3]. An output tracking approach is to utilise the scheme of the output regulation control [4] in which the outputs are assumed to be excited by an exosystem. However, the non- linear regulation problem requires solving the difficult sol- ution of partial-differential algebraic equation. Another problem of the output regulation control is that the exosys- tem states need to be switched to describe changes in the output and this will create transient tracking errors [5]. In general, the nonlinear H 1 control has to solve the Hamilton–Jacobi equation, which is a difficult nonlinear partial-differential equation [6–8]. Only for some particular nonlinear systems, we can derive a closed-form solution [9]. The control approach based on internal model principle converts the tracking problem to nonlinear output regulation problem. This approach depends on solving a first-order partial-differential equation of the centre manifold [4]. For some special nonlinear systems and desired trajectories, the asymptotic solutions of this equation via ordinary differ- ential equations have been developed [10, 11]. Recently, H 1 adaptive fuzzy control has been proposed to systemati- cally deal with nonlinear systems [12]. The drawback with H 1 adaptive fuzzy control is that the complex parameter update law makes this approach impractical. During the past decade, significant progress has been made in the research of control approaches for nonlinear systems based on the feedback linearisation theory [1, 13–15]. Moreover, feedback linearisation approach has been applied successfully to address many real controls. These include the control of electromagnetic suspension system [16], pendulum system [17], spacecraft [18], electrohydrau- lic servosystem [19], car-pole system [20] and bank-to-turn missile system [21]. The almost disturbance decoupling problem, which is the design of a controller which attenuates the effect of the disturbance on the output terminal to an arbitrary degree of accuracy, was originally developed for linear and nonlinear control systems in [22, 23], respectively. Henceforward, the problem has attracted considerable attention and many significant results have been developed for both linear and nonlinear control systems [24–26]. The almost disturbance decoupling problem of nonlinear single- input/single-output (SISO) systems was investigated in [23] by state feedback and solved in terms of sufficient conditions for systems with nonlinearities which are not globally Lipschitz and disturbances appearing linearly but possibly multiplying nonlinearities. The resulting state # IEE, 2006 IEE Proceedings online no. 20050025 doi:10.1049/ip-cta:20050025 Paper first received 25th January and in revised form 22nd November 2005 The authors are with the Department of Electrical Engineering, National Formosa University, 64, Wun-Hwa Road, Huwei, Yunlin, Taiwan 632, Republic of China E-mail: [email protected]IEE Proc.-Control Theory Appl., Vol. 153, No. 3, May 2006 331
Transcript
Application of feedback linearisation to the trackingand almost disturbance decoupling control ofmulti-input multi-output nonlinear system
C.-C. Chen and Y.-F. Lin
Abstract: The tracking and almost disturbance decoupling problem of multi-input multi-outputnonlinear systems based on the feedback linearisation approach are studied. The main contributionof this study is to construct a controller, under appropriate conditions, such that the resultingclosed-loop system is valid for any initial condition and bounded tracking signal with the followingcharacteristics: input-to-state stability with respect to disturbance inputs and almost disturbancedecoupling, that is, the influence of disturbances on the L2 norm of the output tracking error canbe arbitrarily attenuated by changing some adjustable parameters. One example, which cannotbe solved by the first paper of the almost disturbance decoupling problem on account of requiringsome sufficient conditions that the nonlinearities multiplying the disturbances satisfy structuraltriangular conditions, is proposed to exploit the fact that the tracking and the almost disturbancedecoupling performances are easily achieved by the proposed approach. To demonstrate thepractical applicability, a famous half-car active suspension system has been investigated.
1 Introduction
Two well-known tasks of stabilisation and tracking problemare important topics in the field of control. Trackingproblem is generally more complicated than stabilisationproblem for nonlinear control systems. Many approachesfor nonlinear systems have been introduced includingfeedback linearisation, variable structure control (slidingmode control), backstepping, regulation control, nonlinearH1 control, internal model principle and H1 adaptivefuzzy control. Recently, variable structure controls areintroduced to deal with nonlinear systems [1]. However,chattering behaviour that may create unmodelled high fre-quency due to the discontinuous switching and imperfectimplementation and even drive system to instability isinevitable for variable structure control scheme.Backstepping has been a powerful tool for synthesising con-troller for a class of nonlinear systems. However, a disad-vantage with the backstepping approach is the explosionof complexity which is caused by the complicated repeateddifferentiations of some nonlinear functions [2, 3]. Anoutput tracking approach is to utilise the scheme of theoutput regulation control [4] in which the outputs areassumed to be excited by an exosystem. However, the non-linear regulation problem requires solving the difficult sol-ution of partial-differential algebraic equation. Anotherproblem of the output regulation control is that the exosys-tem states need to be switched to describe changes inthe output and this will create transient tracking errors [5].
In general, the nonlinear H1 control has to solve theHamilton–Jacobi equation, which is a difficult nonlinearpartial-differential equation [6–8]. Only for some particularnonlinear systems, we can derive a closed-form solution [9].The control approach based on internal model principleconverts the tracking problem to nonlinear output regulationproblem. This approach depends on solving a first-orderpartial-differential equation of the centre manifold [4]. Forsome special nonlinear systems and desired trajectories,the asymptotic solutions of this equation via ordinary differ-ential equations have been developed [10, 11]. Recently,H1 adaptive fuzzy control has been proposed to systemati-cally deal with nonlinear systems [12]. The drawback withH1 adaptive fuzzy control is that the complex parameterupdate law makes this approach impractical. During thepast decade, significant progress has been made in theresearch of control approaches for nonlinear systemsbased on the feedback linearisation theory [1, 13–15].Moreover, feedback linearisation approach has beenapplied successfully to address many real controls. Theseinclude the control of electromagnetic suspension system[16], pendulum system [17], spacecraft [18], electrohydrau-lic servosystem [19], car-pole system [20] and bank-to-turnmissile system [21].
The almost disturbance decoupling problem, which is thedesign of a controller which attenuates the effect of thedisturbance on the output terminal to an arbitrary degreeof accuracy, was originally developed for linear andnonlinear control systems in [22, 23], respectively.Henceforward, the problem has attracted considerableattention and many significant results have been developedfor both linear and nonlinear control systems [24–26]. Thealmost disturbance decoupling problem of nonlinear single-input/single-output (SISO) systems was investigated in[23] by state feedback and solved in terms of sufficientconditions for systems with nonlinearities which are notglobally Lipschitz and disturbances appearing linearly butpossibly multiplying nonlinearities. The resulting state
# IEE, 2006
IEE Proceedings online no. 20050025
doi:10.1049/ip-cta:20050025
Paper first received 25th January and in revised form 22nd November 2005
The authors are with the Department of Electrical Engineering, NationalFormosa University, 64, Wun-Hwa Road, Huwei, Yunlin, Taiwan 632,Republic of China
IEE Proc.-Control Theory Appl., Vol. 153, No. 3, May 2006 331
feedback control is constructed following a singular pertur-bation approach. The sufficient conditions in [23] requirethat the nonlinearities multiplying the disturbances satisfystructural triangular conditions. Marino et al. [23] showsthat, for the following nonlinear SISO system, the almostdisturbance decoupling problem may not be solvable
_x1ðtÞ ¼ tan�1 x2 þ uðtÞ
_x2ðtÞ ¼ u
y ¼ x1
where u and y denote the input and output, respectively, andu is the disturbance. On the contrary, the sufficient con-ditions given in [23] (in particular the structural conditionson nonlinearities multiplying disturbances) are not necess-ary in this study where a nonlinear state feedback controlis explicitly designed which easily solves the almost dis-turbance decoupling problem. Moreover, to exploit the sig-nificant applicability, this paper also has successfullyderived tracking controller with almost disturbance decou-pling for a famous half-car active suspension system.Throughout the paper, the notation k.k denotes the usualEuclidean norm or the corresponding induced matrix norm.
2 Tracking and almost disturbance decouplingcontroller design
For convenience of demonstration, we start by recallingsome differential geometry definitions [13] as follows.
Definition 1: A mapping f: U ! Rn, where U is a subsetof Rn, is said to be a vector field on U.
Definition 2: Let l: U ! R and f: U ! Rn. The Lie deriva-tive of l with respect to f, written as Lf
l, is defined by
Llf ¼@l
@xf ðxÞ
Definition 3: Let f and g be two vector fields on U , Rn.The Lie bracket of f and g, written as [ f, g], is defined by
½ f ; g�ðxÞ ¼@g
@xf ðxÞ �
@f
@xgðxÞ
Definition 4: A function f: U ! Rn is a diffeomorphism onU if exists a function f21(x) such that f21(f(x)) ¼ x forall x [ U, and both f(x) and f21(x) are continuouslydifferentiable.
Definition 5: Let f1, f2, . . . , fd be vector fields on U , Rn.At any fixed point x [ U, these vector fields span a vectorspace
DðxÞ ¼ spanf f 1ðxÞ; f 2ðxÞ; . . . ; f dðxÞg
We will refer to this assignment by
D ¼ spanf f 1; f 2; . . . ; f dg
which we call a distribution.
Definition 6: A distribution D is involutive if
f 1 [ D and f 2 [ D)½ f 1; f 2� [ D
Definition 7: Let D be a non-singular d-dimensional distri-bution on U , Rn, generated by f1, f2, . . . , fd. Then, D issaid to be completely integrable if for each x0 [ U, thereexists a neighbourhood U0 of x0 and n2 d real-valued
smooth functions l1(x), l2(x), . . . , ln2d (x) such that l1(x),l2(x), . . . , ln2d(x) satisfy the partial-differential equations
@lj@x
f iðxÞ ¼ 0; 1 � i � d; 1 � j � n� d
Now we consider the following nonlinear control systemwith uncertainties and disturbances to design the desiredtracking controller
_x1
_x2
..
.
_xn
266664
377775 ¼
f 1ðx1; x2; . . . ; xnÞ
f 2ðx1; x2; . . . ; xnÞ
..
.
f nðx1; x2; . . . ; xnÞ
266664
377775
þ ½ g1ðx1; x2; . . . ; xnÞ g2ðx1; x2; . . . ; xnÞ
� � � gmðx1; x2; . . . ; xnÞ�
u1ðx1; x2; . . . ; xnÞ
u2ðx1; x2; . . . ; xnÞ
..
.
umðx1; x2; . . . ; xnÞ
266664
377775
þXpj¼1
q�j uj ð1aÞ
y1ðx1; x2; . . . ; xnÞ
y2ðx1; x2; . . . ; xnÞ
..
.
ymðx1; x2; . . . ; xnÞ
26664
37775 ¼
h1ðx1; x2; . . . ; xnÞ
h2ðx1; x2; . . . ; xnÞ
..
.
hmðx1; x2; . . . ; xnÞ
26664
37775 ð1bÞ
that is
_X ðtÞ ¼ f ðX ðtÞÞ þ gðX ðtÞÞuþXpj¼1
q�j uj
yðtÞ ¼ hðX ðtÞÞ
where X(t) U [x1(t) x2(t) � � � xn(t)]T [ <n is the
state vector, u U [u1 u2 � � � um]T [ <m is the input
vector, y U [y1 y2 � � � ym]T [ <m is the output vector,
u U [u1(t) u2(t) � � � up(t)]T is a bounded time-varying
disturbances vector, f U [ f1 f2 � � � fn]T [ <n,
g U [g1 g2 � � � gm] [ <n�m, gi [ <n, i ¼ 1, 2, . . . , nand h U [h1 h2 � � � hm]
T [ <m are smooth vectorfields. The nominal system is then defined as follows
_X ðtÞ ¼ f ðX ðtÞÞ þ gðX ðtÞÞu ð2aÞ
yðtÞ ¼ hðX ðtÞÞ ð2bÞ
The nominal system of the form (2) is assumed to have thevector relative degree fr1, r2, . . . , rmg, that is, the followingconditions are satisfied for all X [ <n
(i)
LgjLkf hiðX Þ ¼ 0 ð3Þ
for all 1 � i � m, 1 � j � m, k , ri2 1, where the operatorL is the Lie derivative and r1þ r2þ � � � þ rm ¼ r.
IEE Proc.-Control Theory Appl., Vol. 153, No. 3, May 2006332
(ii) the m � m matrix
A :¼
Lg1Lr1�1f h1ðX Þ � � � LgmL
r1�1f h1ðX Þ
Lg1Lr2�1f h2ðX Þ � � � LgmL
r2�1f h2ðX Þ
..
. ...
Lg1Lrm�1f hmðX Þ � � � LgmL
rm�1f hmðX Þ
26666664
37777775
ð4Þ
is non-singular.
The desired output trajectory ydi , 1 � i � m and its first ri
derivatives are all uniformly bounded and
yid; yið1Þ
d ; . . . ; yiðri Þ
d
h i��� ��� � Bid; 1 � i � m ð5Þ
where Bdi is some positive constant. Under the assumption
of well-defined vector relative degree, it has been shownthat the mapping
f: <n!<
nð6Þ
defined as
ji :¼
j i1
j i2
..
.
j iri
2666664
3777775:¼
fi1
fi2
..
.
firi
2666664
3777775
:¼
L0f hiðX Þ
L1f hiðX Þ
..
.
Lri�1f hiðX Þ
2666664
3777775;
i ¼ 1; 2; . . . ;m ð7Þ
fkðX ðtÞÞ :¼ hkðtÞ; k ¼ r þ 1; r þ 2; . . . ; n ð8Þ
and satisfying
LgjfkðX ðtÞÞ ¼ 0; k ¼ r þ 1; r þ 2; . . . ; n; 1 � j � m
ð9Þ
is a diffeomorphism onto image, if
(i) the distribution
G :¼ spanfg1; g2; . . . ; gmg ð10Þ
is involutive.
(ii) the vector fields
Y kj ; 1 � j � m; 1 � k � rj ð11Þ
are complete, where
Y kj :¼ ð�1Þk�1adk�1
~f~gj; 1 � j � m; 1 � k � rj ð12Þ
~f ðX Þ :¼ f ðX Þ � gðX ÞA�1ðX ÞbðX Þ ð13Þ
bðX Þ :¼
Lr1f h1ðX Þ
Lr2f h2ðX Þ
..
.
Lrmf hmðX Þ
2666664
3777775
ð14Þ
~g :¼ ½ ~g1 ~g2 � � � ~gm� :¼ gðX ÞA�1ðX Þ ð15Þ
adkf g :¼ b f adk�1f gc ð16Þ
½f g� :¼@g
@Xf ðX Þ �
@f
@XgðX Þ ð17Þ
For the sake of convenience, define the trajectory error to be
eij :¼ j ij � y
iðj�1Þd ; i ¼ 1; 2; . . . ;m; j ¼ 1; 2; . . . ; ri
ð18Þ
ei :¼�ei1 ei2 � � � eiri
�T[ <
ri ð19Þ
and the trajectory error to be multiplied with some adjusta-ble positive constant 1
Assumption 1: For all t � 0, h [ <n2r and j [ <r, thereexists a positive constant M such that the followinginequality holds
kq22ðt;h; �e Þ � q22ðt;h; 0Þk � Mk�ek ð33Þ
IEE Proc.-Control Theory Appl., Vol. 153, No. 3, May 2006 333
where q22(t, h, e ) U q(j, h), that is, the function q22(t, h, �e)is obtained by replacing the variable j with �e(t) for the vari-able q(j, h).For the sake of stating precisely the investigated problem,
define
dij :¼ LgjLri�1f hiðX Þ; 1 � i � m; 1 � j � m ð34Þ
ci :¼ Lrif hiðX Þ; 1 � i � m ð35Þ
and
ei :¼ ai1e
i1 þ ai
2ei2 þ � � � þ ai
rieiri ; 1 � i � m ð36Þ
Definition 8 [1]: A continuous function a: [0, a) ! [0,1)is said to belong to class K if it is strictly increasing anda(0) ¼ 0.
Definition 9 [1]: A continuous function b: [0, a) �[0,1) ! [0,1) is said to belong to class KL if, for eachfixed s, the mapping b(r, s) belongs to class K withrespect to r and, for each fixed r, the mapping b(r, s) isdecreasing with respect to s and b(r, s) ! 0 as s ! 1.
Definition 10 [1]: Consider the system x ¼ f (t, x, u ), wheref: [0,1) � <n
� <n! <n is piecewise continuous in t and
locally Lipschitz in x and u. This system is said to be input-to-state stable if there exists a class KL function b, a class Kfunction g and positive constants k1 and k2 such that for anyinitial state x(t0) with kx(t0)k , k1 and any bounded inputu(t) with supt�t0
ku (t)k , k2, the state exists and satisfies
kxðtÞk � bðkxðt0Þk; t � t0Þ þ g supt0�t�t
ku ðtÞk
� �ð37Þ
for all t � t0 � 0. Now we formulate the tracking problemwith almost disturbance decoupling as follows.
Definition 11 [24]: The tracking problem with almost dis-turbance decoupling is said to be globally solvable bythe state feedback controller u for the transformed-errorsystem by a global diffeomorphism (6), if the controller uenjoys the following properties:
1. It is input-to-state stable with respect to disturbanceinputs.
2. For any initial value �xe0 U [�e(t0) h(t0)]T, for any t � t0
and for any t0 � 0
j yðtÞ � ydðtÞj � b11ðkxðt0Þk; t � t0Þ
þ1ffiffiffiffiffiffiffib22
p b33
�supt0�t�t
kuðtÞk�
ð38Þ
and
ðtt0
½ yðtÞ � ydðtÞ�2 dt �
1
b44
b55ðk�xe0kÞ
þ
ðtt0
b33ðku ðtÞk2Þ dt
ð39Þ
where b22, b44 are some positive constants, b33, b55 areclass K functions and b11 is a class KL function.
Theorem 1: Suppose that there exists a continuously differ-entiable function V: <n2r
! <þ such that the following
three inequalities hold for all h [ <n2r
ðaÞ v1khk2 � V ðhÞ � v2khk
2; v1; v2 . 0 ð40aÞ
ðbÞ rtV þ ðrhV ÞTq22ðt;h; 0Þ � �2axV ðhÞ; ax . 0
ð40bÞ
ðcÞ krhVk � 43khk; 43 . 0 ð40cÞ
then the tracking problem with almost disturbance decou-pling is globally solvable by the controller defined by
u ¼ A�1f�bþ vg ð41Þ
b :¼ ½Lr1f h1 Lr2f h2 � � � Lrmf hm�
Tð42Þ
v :¼ ½v1 v2 � � � vm�T
ð43Þ
vi :¼ yiðriÞ
d � 1�riai1½L
0f hiðX Þ � yid �
�11�riai2½L
1f hiðX Þ � yi
ð1Þ
d � � � � �
�1�1airi½L
ri�1f hiðX Þ � yi
ðri�1Þ
d �; 1 � i � m ð44Þ
Moreover, the influence of disturbances on the L2 norm ofthe tracking error can be arbitrarily attenuated by increasingthe following adjustable parameter NN2 . 1
Hð1Þ :¼H11 H12
H12 H22
" #
:¼
2ax �v23
v1
kfhk2
�1ffiffiffiffiffiffiffiffikð1Þ
p w3Mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2w1l
�min
p" #
26664
�1ffiffiffiffiffiffiffiffikð1Þ
p w3Mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2w1l
�min
p" #
1
1l�max
�2kð1Þkf1
jk2kP1k2
12lminðP1Þ� � � �
�2kð1Þkfm
j k2kPmk2
12lminðPmÞ
37777777775
ð45aÞ
asð1Þ :¼H11 þ H22 � ½ðH11 � H22Þ
2þ 4H2
12�1=2
4ð45bÞ
N :¼ 2asð1Þ ð45cÞ
N1 :¼mþ 1
4
�supt0�t�t
kuðtÞk�2
ð45dÞ
N2 :¼ min v1;kð1Þ
2l�min
� �ð45eÞ
fijð1Þ :¼
1@
@Xhiq
�1 � � � 1
@
@Xhiq
�p
..
. ...
1ri@
@XLri�1
fhiq
�1 � � � 1ri
@
@XLri�1
fhiq
�q
266664
377775;
1 � i � m ð45f Þ
fhð1Þ :¼
@
@Xfrþ1q
�1 � � �
@
@Xfrþ1q
�p
..
. ...
@
@Xfnq
�1 � � �
@
@Xfnq
�p
266664
377775 ð45gÞ
IEE Proc.-Control Theory Appl., Vol. 153, No. 3, May 2006334
where H is positive definite matrix and k(1): <þ! <þ is
any continuous function satisfies
lim1!0
kð1Þ ¼ 0 and lim1!0
1
kð1Þ¼ 0 ð45hÞ
Moreover, the output tracking error of system (1) is expo-nentially attracted into a sphere Br, r ¼
p(N1/NN2), with
an exponential rate of convergence
1
2
NN2
Dmax
�N1
Dmaxr2
� �:¼
1
2a� ð45iÞ
where
Dmax ¼ max v2;k
2l�max
� �ð45jÞ
Proof: Applying the co-ordinate transformation (6) yields
_j1
1 ¼@f1
1
@X
dX
dt
¼@h1@X
f þ g � uþXpj¼1
q�j uj
" #
¼@h1@X
f þ@h1@X
g1u1 þ � � � þ@h1@X
gmum þXpj¼1
@h1@X
q�j u j
¼@h1@X
f þXpj¼1
@h1@X
q�j uj ¼ j 12 þ
Xpj¼1
@h1@X
q�j u j ð46Þ
..
.
_j 1r1�1 ¼
@f1r1�1
@X
dX
dt
¼@Lr1�2
f h1
@Xf þ g � uþ
Xpj¼1
q�j uj
" #
¼@Lr1�2
f h1
@Xf þ
@Lr1�2f h1
@Xg1u1 þ � � �
þ@Lr1�2
f h1
@Xgmum þ
Xpj¼1
@Lr1�2f h1
@Xq�j u j
¼@Lr1�2
f h1
@Xf þ
Xpj¼1
@Lr1�2f h1
@Xq�j uj
¼ Lr1�1f h1 þ
Xpj¼1
@Lr1�2f h1
@Xq�j uj ð47Þ
_j1
r1¼
@f1r1
@X
dX
dt¼
@Lr1�1f h1
@Xf þ g � uþ
Xpj¼1
q�j uj
" #
¼@Lr1�1
f h1
@Xf þ
@Lr1�1f h1
@Xg1u1 þ � � �
þ@Lr1�1
f h1
@Xgmum þ
Xpj¼1
@Lr1�1f h1
@Xq�j uj
¼ Lr1f h1 þ Lg1Lr1�1f h1u1 þ � � � þ LgmL
r1�1f h1um
þXpj¼1
@Lr1�1f h1
@Xq�j u j
¼ c1 þ d11u1 þ � � � þ d1mum þXpj¼1
@Lr1�1f h1
@Xq�j u j
ð48Þ
..
.
_j m1 ¼
@fm1
@X
dX
dt¼
@hm@X
f þ g � uþXpj¼1
q�j uj
" #
¼@hm@X
f þ@hm@X
g1u1 þ � � � þ@hm@X
gmum þXpj¼1
@hm@X
q�j uj
¼ L1f hm þXpj¼1
@h1@X
q�j u j ¼ j m2 þ
Xpj¼1
@hm@X
q�j uj ð49Þ
..
.
_j mrm�1 ¼
@fmrm�1
@X
dX
dt¼
@Lrm�2f hm
@Xf þ g � uþ
Xpj¼1
q�j uj
" #
¼@Lrm�2
f hm
@Xf þ
@Lrm�2f hm
@Xg1u1 þ � � �
þ@Lrm�2
f hm
@Xgmum þ
Xpj¼1
@Lrm�2f hm
@Xq�j uj
¼ Lrm�1f hm þ
Xpj¼1
@Lrm�2f hm
@Xq�j uj
¼ jmrmþXpj¼1
@Lrm�2f hm
@Xq�j uj ð50Þ
_j mrm
¼@fm
rm
@X
dX
dt¼
@Lrm�1f hm
@Xf þ g � uþ
Xpj¼1
q�j uj
" #
¼@Lrm�1
f hm
@Xf þ
@Lrm�1f hm
@Xg1u1 þ � � �
þ@Lrm�1
f hm
@Xgmum þ
Xpj¼1
@Lrm�1f hm
@Xq�j uj
¼ Lrmf hm þ Lg1L
rm�1f hmu1 þ � � � þ LgmL
rm�1f hmum
þXpj¼1
@Lrm�1f hm
@Xq�j uj
¼ cm þ dm1u1 þ � � � þ dmmum þXpj¼1
@Lrm�1f hm
@Xq�j uj
ð51Þ
_hkðtÞ ¼@fk
@X
dX
dt¼
@fk
@Xf þ g � uþ
Xpj¼1
q�j uj
" #
IEE Proc.-Control Theory Appl., Vol. 153, No. 3, May 2006 335
¼@fk
@Xf þ
@fk
@Xg1u1 þ � � � þ
@fk
@Xgmum
þXpj¼1
@fk
@Xq�j uj ð52Þ
¼ Lffk þXpj¼1
@fk
@Xq�j uj
¼ qk þXpj¼1
@fk
@Xq�j uj; k ¼ r þ 1; r þ 2; . . . ; n
As
ciðj ðtÞ;hðtÞÞ :¼ Lrif hiðX ðtÞÞ; 1 � i � m ð53Þ
dij :¼ LgjLri�1f hiðX Þ; 1 � i � m; 1 � j � m ð54Þ
qkðj ðtÞ;hðtÞÞ ¼ LffkðX Þ; k ¼ r þ 1; r þ 2; . . . ; n ð55Þ
the dynamic equations of system (1) in the new co-ordinatesare as follows
So that statement (38) is proved and then the trackingproblem with almost disturbance decoupling is globallysolved. Finally, we will prove that the sphere Br is aglobal attractor for the output tracking error of system (1).
From (78a) and (45d), we obtain
_L � �NN2 kytotalk2
�þ N1 ð85Þ
For kytotalk . r, we have L , 0. Hence any sphere definedby
Br :¼�eh
: k�ek2 þ khk2 � r
� �ð86Þ
is a global final attractor for the tracking error system of thenonlinear control systems (1). Furthermore, it is easyroutine to see that for y � Br, we have
that is the convergence rate towards the sphere Br is equal toa�/2. This completes our proof. A
3 Illustrative example
Consider the half-car active suspension system with disturb-ances shown in Fig. 1. From [29], the dynamic equations aregiven as follows:
_x1 ¼ x2
_x2 ¼1
ms
� Bf þ Brð Þx2 þ aBf � bBrð Þx4 cos x3½
�kf x5 þ Bfx6 � krx7 þ Brx8 þ ff þ frð Þ�
_x3 ¼ x4
_x4 ¼1
JyaBf � bBrð Þx2 cos x3½
� a2Bf þ b2Br
�x4 cos
2 x3 þ akfx5 cos x3
� aBfx6 cos x3 � bkrx7 cos x3
þ bBrx8 cos x3 þ �aff þ bfrð Þ cos x3�
_x5 ¼ x2 � ax4 cos x3 � x6
_x6 ¼1
muf
�Ktfx1 þ Bfx2 þ aKtf sin x3 � aBfx4 cos x3½
þ kf þ Ktfð Þx5 � Bfx6 þ Ktf zrf � ff �
IEE Proc.-Control Theory Appl., Vol. 153, No. 3, May 2006338
_x7 ¼ x2 þ bx4 cos x3 � x8
_x8 ¼1
mur
�Ktrx1 þ Brx2 � bKtr sin x3½ þ bBrx4 cos x3
þ kr þ Ktrð Þx7 � Brx8 þ Ktrzrr � fr� ð89aÞ
y1 ¼ x1 þ x2 :¼ h1
y2 ¼ x3 þ x4 :¼ h2 ð89bÞ
where x1 ¼ z is the displacement of the centre of gravity,x2 ¼ _z is the payload velocity, ms is the mass of the carbody, Bf and Br are the front and rear damping coefficients,a is the distance between front axle and centre of gravity, bis the distance between rear axle and centre of gravity,x3 ¼ u is the pitch angle, x4 ¼ u is the pitch velocity, kfand kr are the front and rear spring coefficients, zsf and zsrare the front and rear body displacement, zuf and zur arefront and rear wheel displacements, x5 ¼ zsf2 zuf thefront wheel suspension travel, x6 ¼ zuf is the front unsprungmass velocity, x7 ¼ zsr2 zur is rear wheel suspension travel,ff ¼ u1 and fr ¼ u2 are the front and rear force inputs, Jy isits centroidal moment of inertia, muf and mur are theunsprung masses on the front and rear wheels, Ktf and Ktr
are the front and rear tire spring coefficients, zrf and zrr arethe front and rear terrain height disturbances and x8 ¼ zuris the rear unsprung mass velocity. The following physicalparameters are chosen in our simulation: ms ¼ 575 kg,Bf ¼ Br ¼ 1000 N/m/s, a ¼ 1.38 m, b ¼ 1.36 m, Jy ¼769 kg/m2, muf ¼ mur ¼ 60 kg, Ktf ¼ Ktr ¼ 190 000 N/m,kf ¼ kr ¼ 16812 N/m, zrf ¼ zrr ¼ mr(12 cos 8pt) andmr ¼ 0.05 m. Hence the mathematical model can berewritten as
_x1 ¼ x2
_x2 ¼ �3:478x2 þ 0:035x4 cos x3 � 29:238x5
þ 1:739x6 � 29:238x7 þ 1:739x8
þ ð0:0017u1 þ 0:0017u2Þ
_x3 ¼ x4
_x4 ¼ 3:563x2 cos x3 � 4:881x4 cos2 x3 þ 30:17x5 cos x3
� 1:794x6 cos x3 � 29:732x7 cos x3
þ 1:768x8 cos x3 þ �0:0018u1 þ 0:00176u2ð Þ cos x3
_x5 ¼ x2 � 1:38x4 cos x3 � x6
_x6 ¼ �3166:667x1 þ 16:667x2 þ 4370 sin x3
� 23x4 cos x3 þ 3446:867x5 � 16:667x6
þ 158:33ð1� cos 8ptÞ � 0:0167u1
_x7 ¼ x2 þ 1:36x4 cos x3 � x8
_x8 ¼ �3166:667x1 þ 16:667x2 � 4306:667 sin x3
þ 22:667x4 cos x3 þ 3446:067x7
� 16:667x8 þ 158:33ð1� cos 8ptÞ � 0:0167u2
ð90aÞ
y1 ¼ x1 þ x2 :¼ h1
y2 ¼ x3 þ x4 :¼ h2 ð90bÞ
Now we will show how to explicitly construct a controllerthat tracks the desired signals yd
1 ¼ yd2¼0 and attenuates
the disturbance’s effect on the output terminal to an arbi-trary degree of accuracy. Let us arbitrarily choosea11 ¼ a1
2 ¼ 0.06, Ac1 ¼ Ac
2 ¼ 20.06, P1 ¼ P2 ¼ 25/3 andl�min ¼ l�max ¼ 25/3. From (41), we obtain the desiredtracking controllers
u1 ¼ �1381:25x4 cos x3 þ 16977:16x5 � 1009:63x6
þ 150:69x7 � 9:07x8 þ 1546:97x2 � 174:48x1
þ 168:54x3ðcos x3Þ�1
þ 449:44x4ðcos x3Þ�1
ð91Þ
u2 ¼ 1360:66x4 cos x3 þ 220:63x5 � 13:244x6 þ 17047:1x7
� 1013:81x8 � 442:33x2 � 178:44x1
� 168:54x3ðcos x3Þ�1
� 449:44x4ðcos x3Þ�1
ð92Þ
It can be verified that the relative conditions of theorem 1are satisfied with 1 ¼ 0.1, Bd
1 ¼ Bd2 ¼ 0, M ¼
p3,
v1 ¼ v2 ¼ 1, ax ¼ 1, v3 ¼ 2 and k ¼p1. Hence the track-
ing controllers will steer the output tracking errors of theclosed-loop system, starting from any initial value, to beasymptotically attenuated to zero by virtue of theorem1. The complete trajectories of the outputs are depicted inFigs. 2a and b with 1 ¼ 0.1. It is easy to prove that, fromthe item H22 of (45a), the feedback-controlled systemwith the tracking controllers achieves better tracking per-formance with smaller parameter 1. This fact can beobserved from Figs. 3a and 3b with 1 ¼ 0.03.
4 Comparative example to existing approaches
Marino et al. [23] exploits the fact that for nonlinear single-input single-output system the almost disturbance decou-pling problem cannot be solved, as the following example
Fig. 1 Half-car active suspension system
IEE Proc.-Control Theory Appl., Vol. 153, No. 3, May 2006 339
shows
_x1ðtÞ
_x2ðtÞ
" #¼
tan�1ðx2Þ
0
" #þ
0
1
" #uþ
1
0
" #uðtÞ ð93aÞ
yðtÞ ¼ x1ðtÞ :¼ hðX ðtÞÞ ð93bÞ
where u and y denote the input and output, respectively,u(t) :¼ 0.5 sin t is the disturbance. On the contrary, thisproblem can be easily solved via the proposed approachin this paper. Following the same procedures shown in thedemonstrated example, the tracking problem with almostdisturbance decoupling problem can be solvable by thestate feedback controller u defined as
u ¼ ð1þ x22Þ½� sin t � ð0:03Þ�2ðx1 � sin tÞ
� ð0:03Þ�1ðtan�1 x2 � cos tÞ� ð94Þ
The output trajectory of feedback-controlled system for (93)is depicted in Fig. 4.
It is worth noting that the sufficient conditions given in[23] (in particular the structural conditions on nonlinearitiesmultiplying disturbances) are not necessary in this studywhere a nonlinear state feedback control is explicitlydesigned which solves the almost disturbance decouplingproblem. For instance, the almost disturbance decouplingproblem is solvable for the system (93) by a nonlinearstate feedback control, according to our proposed approach,while the sufficient conditions given in [23] fail whenapplied to the system (93). The design techniques in thisstudy are also entirely different than those in [23] becausethe singular perturbation tools are not used.
5 Conclusion
In this paper, we have constructed a feedback control algor-ithm which globally solves the tracking problem withalmost disturbance decoupling for multi-input multi-output nonlinear systems. The discussion and practicalapplication of feedback linearisation of nonlinear controlsystems by parameterised co-ordinate transformation havebeen presented. One comparative example is proposed toshow the significant contribution of this paper withrespect to those existing approach. Moreover, a practicalexample of half-car active suspension system demonstratesthe applicability of the proposed differential geometryapproach and the composite Lyapunov approach.Simulation results exploit the fact that the proposedmethodology is successfully applied to feedback
Fig. 2 Output trajectory
a x1 of the half-car active suspension system with 1 ¼ 0.1b x3 of the half-car active suspension system with 1 ¼ 0.1
Fig. 4 Output trajectory of feedback-controlled system for (93)
Fig. 3 Output trajectory
a x1 of the half-car active suspension system with 1 ¼ 0.03b x3 of the half-car active suspension system with 1 ¼ 0.03
IEE Proc.-Control Theory Appl., Vol. 153, No. 3, May 2006340
linearisation problem and achieves the desired tracking andalmost disturbance decoupling performances of the con-trolled system.
6 Acknowledgment
The author greatly appreciates the Editor and anonymousreviewers for their valuable comments to improve thequality of this paper.
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