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On the pseudo-linearization problem for nonlinear systems
Systems & Control Letters 12 (1989) 161-167 North-Holland
161
On the pseudo-linearization problem for nonlinear systems * Jianliang W A N G and Wilson J. R U G H Department of Electrical and Computer Engineering, The Johns Hopkins University, Baltimore, MD 21218, U.S.A. Received 18 March 1988 Revised 3 October 1988
Abstract: Based on well-known results for linear systems, an alternate treatment of the pseudo-linearization problem of Reboulet and Champetier is given. Sufficient conditions for pseudo-linearization are obtained in the general case, and these are shown to be satisfied under mild hypotheses by systems with one or two inputs. Advantages of the new approach include the simplicity of the derivation, and a more explicit representation for a pseudo-linearizing transformation.
Keywords: pseudo-linearization, nonlinear systems, nonlinear control laws
1. Introduction
Pseudo-linearization of a nonlinear system involves computing a state feedback law and invertible state variable change such that the closed-loop system described in terms of the new variables has a family of linearizations that is independent of the closed-loop constant operating point. This notion was introduced by Reboulet and Champetier, and treated from a differential-geometric viewpoint in [2,3,6,7]. We describe here an approach to pseudo-linearization based on the linearization family of the nonlinear system, using a linear transformation due to Luenberger, [5], a linear feedback law due to Ackermann, [1], and results on linearization families in [8,9]. This leads to an alternative derivation of the results in [3,6] for one- and two-input systems, though in the general multi-input case only a sufficient condition is obtained. In the course of the derivation, explicit representations are obtained for a feedback law and a variable change that accomplish pseudo-linearization. We adopt a local viewpoint, though when particular examples are considered the problem often can be solved in a nonlocal fashion. For simplicity, all functions and their derivatives that appear in the sequel are assumed to be continuous. Multi-input nonlinear systems of the form
~(t) = f ( x ( / ) ,
u(t)),
x(t) ~ R", u(t) ~ R", t__ 0,
(a)
will be considered. It is assumed that (1) has a constant operating point family {[x(a), u(a)], a ~ F}, where F is an open neighborhood of 0 ~ R " and, for convenience, x(0)= 0, u(0)= 0. That is, f(x(a), u(a))= 0 for a ~ F. Typically the constant operating point family is parameterized by constant values of the input components and/or state components. Since local results are presented, in order to avoid repetitive statements we implicitly permit the shrinking of F when required by various local arguments. Linearizing (1) about its constant operating point family yields the parameterized linearization family d [x(t) - x ( a ) ] dt
=g(a)[x(t)
f ( ~ ) = ~ ( x ( ~ ) , u(,~)),
-x(a)] +
a ( ~ ) --
G(a)[u(t)
-u(a)]
(x(,~), u(~)),
~ ~ r.
(2)
* Research sponsored by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR-87-0101. 0167-6911/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)
J. Wang, W.J. Rugh /Pseudo-linearizationfor nonlinear svsterns
162
Fig. 1. Nonlinear system structure for pseudo-linearization.
It will be assumed that for each a ~ F, rank G ( a ) = m,
~X
rank-o--~-a( a ) = m,
r a n k [ G ( a ) , F ( a ) G ( a ) . . . . . F " - ' ( a ) G ( a ) ] =n.
(3)
Finally we assume that the controllability indices corresponding to (2), denoted by k 1. . . . . k,,, are constant for all a ~ F. It will become clear that the assumptions of controllability and constant controllability indices are required for the pseudo-linearization problem to be meaningful. The rank assumption on the operating point state will be used in the sequel to determine a from knowledge of x(a). Also, for use in the sequel we note that, since (2) is a linearization family [8],
F(a)
(a)+G(a)-~a(a)=O
forall a ~ F
(4)
2. Pseudo-linearization The pseudo-linearization problem for (1) involves finding an invertible state variable change z ( t ) = P(x(t)), z(t) ~ R n, and a feedback law u(t) = Pn+l(x(t), w(t)), w ( t ) ~ R m, with [OPn+l/Ow](O, 0) nonsingular, such that the closed-loop system shown in Figure 1, when linearized about its constant operating point family {[ w(a), z(a)], a ~ F}, has a constant-parameter linearization family described in the Brunovsky canonical form with controllability indices k 1. . . . . k,,. That is, the closed-loop linearization family is required to take the form
d---[z(t)-z(a)] =F[z(t)-z(a)] +G[w(t)-w(a)] dt P = block diagonal { if1,..
(~ = block diagonal { gl,.
0 0
---
a~_F,
0 0
~Rk, ~1.
(5)
More or less implicit in this problem statement is the requirement that w(a) and z(a) be consistent with the variable change and the requirement that (5) be a linearization family for the closed-loop nonlinear system. Specifically, the closed-loop constant operating point family must satisfy
z(a)=P(x(a)),
u(a)=P~+,(x(a), w(a)),
a¢F,
(6)
163
J. Wang,W.J.Rugh/Pseudo-linearizationfornonlinearsystems
and, for convenience, we will require P ( 0 ) = 0 and P,,+I(0, 0 ) = 0. Also, since (5) is required to be a linearization family it must satisfy a condition analogous to (4), namely
a~ (~) = 0,
. ~ r.
(7)
Direct substitution using the form of ff and G shows that (7) implies, for all a = F,
w(.) = o,
z,(~)
= 0,
i =
(8)
k l + "'" + k , ,
1+2,...,n.
Differentiating the first condition in (6) and combining with (8)gives the requirement (writing transposes to save space)
[[~Zl ]T~_~ (,~) 3P . . . .
[ ~Zkl+l ]T
, o ..... o[-~-~--(~)
, o . . . . . o .....
[ ~Zk~+...+k., ] T ,+1 ~
(,~)
, o ..... o
]T
ax
= -~x ( X( a))~--da (a)
(9)
To solve the pseudo-linearization problem, we first show how to choose K ( a ) ~ R rex', invertible
M(a) ~ R mxm, and invertible Q(a) ~ R "xn such that, for all a ~ F,
Q(a)[F(a) + G(a)K(a)] Q-~(a) = if,
Q(a)G(a)M(a) = G.
(10)
Secondly, we will show how to compute Pn+~(x, w) such that, for all a ~ F, e.+,(x(~),
w(,~)) = ,,(,~),
a , Ox ' . + l (x(,~), w(,~)) = K(,~),
a Pa~ .+l (x(.),
w(.)) = M(.), (11)
where w(a) ~ 0. Finally, we will determine conditions under which (9) can be satisfied with [OP/Ox](x(a)) = Q(a), and show how to compute P(x) such that, for all a ~ F,
P(x(a)) = z(a),
OP -~-x ( x ( a ) ) --- Q(a)
(12)
In order to specify in an explicit fashion the matrices Q(a), K(a) and M(a) that satisfy (10), we will make use of developments leading to the controller form in [5] and the so-called minimum-time deadbeat control law for discrete-time systems in [1]. Denoting the columns of G(a) by g~(a) ..... g.,(a), by hypothesis
C(a) = [ g l ( a ) , r(a)gl(a ) ..... Fk'-l(a)g,(a) ..... g.,(a), r(a)gm(a ) ..... rk"-ag.,(a)]
(13)
is invertible at each a ~ F. Partitioning C-l(a) by rows as
el,(a)
e,k,(") C-'(~) =
(14)
em,(~) em~°,(")
J. Wang,W.J,Rugh / Pseudo-linearizationfor nonlinearsystems
164 we let
elk,(a) elk,(a)F(a) e,k,(a)Fk'-l(a)
e,k,(a)F k' '(a) Q(a) =
a(~)
(15)
emk,.(a)F~,,,-2(a)
~o,(~) emk~(a)F(a) e,nk.,(a)Fk,"-l(a) and
[ e,k~(a)Fk'(a) K(a)-=
-M(a)]
!
(16)
[e,,k,,,(a)Fk"(a) The calculations to verify invertibility of Q(a) and M ( a ) and to show that (10) is satisfied are standard, though tedious unless carried out for a particular case. Now we need to show that the parameterized linear control law
.(t) -.(~)
= K(~)[x(t)
- x(~)] + M(~)[w(t)
- ~(~)]
(17)
with w(a)= 0 is a feedback linearization family, that is, the linearization of a nonlinear feedback law of the form u -=P,+l(x, w) about the closed-loop operating point family. But this follows from Corollary 3.1 in [9], since, using (4),
e'k'(a)Fk'(a) ] K(a) -~a
+ M(a)~aa (a) = -M(a)
~a emkm(a)Fk'(a)
)~(~)
= -M(~)
e,,k,,(a)Fk'-a(a) [ elk,(a)rk'-l(a) = M(a)/
:
G(a)~a(a)=--~a(a ) .
[emkm¢O))km-'(o)
(18)
Indeed, an explicit representation for a nonlinear control law corresponding to (17) can be found as follows [9]. Select m components of x(a) to form :~(a) such that rank[b~/3a](a) = m, and denote by Y the same selection of rn components from x. Then a nonlinear feedback law that satisfies (11) is ~o+l(X, w) = . ( ~ - ' ( ~ ) )
+ K(~-'(~))[x - x(~-'(2))] + M(£-'(~))w.
(19)
J. Wang, W.J. Rugh / Pseudo-linearizationfor nonlinear systems
165
Finally, we must consider conditions under which there exists z(a) such that (9) is satisfied, where [ O P / O x ] ( x ( a ) ) = Q ( a ) is specified in (15). It is straightforward to verify that if k , > 2 , then eik,(a)FJ(a)gk(a) = 0 for j = 0, 1. . . . . k, - 2; k = 1. . . . . m. Thus, for a ~ F,
e,k,(a)FJ+l(a)~a(a)=-eik,(a)FY(a)G(a)~(a)=O,
j=0,1 ..... k,-2.
(20)
Therefore (9) is satisfied if zl(a), Zk, +l(a) ..... Zk~ + ... +k,,, ,+ 1(a) satisfy the following set of m, uncoupled, total differential equations:
oa (a)=elk'(a)-~--da ( a ) ' OZkl+ ..- + k , , , _ l + l
3a
Oa
(a)=ezk'-(a)-~--da (a) . . . . .
Ox (a) = e,,k,,,(a)-~--da (a ).
(21)
If z(a) is such that (9) is satisfied, then a nonlinear variable change that satisfies (12) is
P ( x ) = z( £ - ' ( Y ) ) + Q( £ - l ( Y ) ) [ x - x ( £ - l ( Y ) ) ]
(22)
where .~ and Y are as in (19). [8] Writing the i-th entry of a row vector h as [h]~, and invoking the standard existence condition for solutions to total differential equations we obtain the following sufficient condition for pseudo-linearization. [4] Theorem. Suppose we are given a nonlinear system (1) with linearization family (2) such that (3) is satisfied, and the controllability indices are constant for a ~ F, where I" is a sufficiently small, open neighborhood of 0 ~ R m. Then the nonlinear system is pseudo-linearizable if for all a ~ 1",
Oar ejk '
(a) q = ~ a q
ejk,(a)-~a(a) r'
j=l,...,m;l
(23)
Furthermore, if these conditions are satisfied, then a pseudo-linearizing transformation is specified by (19) and (22). For the case m = 1, (23) is vacuous and pseudo-linearization is always possible under our basic assumptions. Unfortunately, for m > 1 the sufficient condition provided by this theorem is rather restrictive due to the specification of a particular K(a), Q(a), and M ( a ) out of the many that satisfy (10). This can be relaxed somewhat by introducing integrating factors as follows. (See also [7].) In place of (21), we consider
0zl (a) = r l ( a ) e , k , ( a ) Ox
....
OZk~+ ... +k,,._,+l OX Oa (a) = rm(a) emk,,,(a) ~-~-a(a)
(24)
where rl(a ) ..... r,,(a) are as yet arbitrary, continuous, real-valued functions that are nonzero for a ~ F. These functions can be introduced by modifying the definitions of Q ( a ) and M ( a ) (while leaving K ( a ) fixed as in (16)) according to (~(a) = block diagonal(ri(a)tk,×k,, i = 1 . . . . . m }. Q ( a ) , { 1 ~ t ( a ) = M(cQ. diagonal ~ ,
} i = 1..... m ,
(25)
where Ik,xk ' is the ki x ki identity matrix. With Q(a) and M(a) replaced by Q(a) and )Q(a), (10) remains satisfied and (17) is a feedback linearization family for
Pn+l(x, w) = u ( i - ' ( Y ) )
+ K(Y-l(Y))[x
- x(Y-l(Y))] + )l~l(£-'(Y))w
(26)
J. Wang, W.J. Rugh / Pseudo-linearization for nonlinear systems
166
where $ and Y are as before. Also, the equations that determine the nonzero components of z(a) are precisely those in (24). Therefore, if rl(a ) ..... rm(a ) can be found such that the appropriate existence conditions for (24) are satisfied, then the nonlinear variable change P( x ) = z( ~ - l ( Y)) + (~(J?-l(Y))[x - x ( ~ - 1 ( 2 ) ) ]
(27)
together with (26) accomplish pseudo-linearization. Indeed, for the case m = 2 we will establish that such z(a) always exists. Corollary. Suppose we are gioen a nonlinear system (1) with linearization family (2) such that m = 1 or m = 2. (3) is satisfied, and the controllability indices are constant for c~~ I', where F is a sufficiently small, open neighborhood of 0 ~ R m. Then the nonBnear system is pseudo-linearizable by a transformation of the form specified in (26) and (27). Proof. For the case m = 1, we take rl(a ) = 1, and (24) becomes dzl dx a a ( a ) = e l , ( a ) ~ - a (a)
(28)
With the initial condition zl(0) = 0, we have zl(a) = f0~e l , ( o ) ~--~-a dx (o) do
(29)
and a pseudo-linearizing transformation can be constructed as in (26) and (27) (or equivalently, (19) and (22)). In the case m = 2, the existence conditions for (24) become 3al rJ(a)eJk'(a)
= ~
rJ((~)%k'(a)~-~a~ (a) '
j = 1, 2.
(30)
Performing the indicated differentiations gives two, uncoupled, linear partial differential equations 05 05 aj(a)~Ta (a)+bj(a)~--~a2(a)+5(a)rj(a)=0
,
j=l,2,
(31 t
where aj(a)=ejG(a)~-~xaz(a),
b,(a)=-
e
jG (. ) -.~ lO(Xa ) ,
O%G, , Ox Oejk, OX cj(a) = - ~ ((x)-3-~2(a) - - ~ - a z ( a ) - ~ a (a),
j = 1, 2.
(32)
Since [-bj(ct), aj(ot)], j = 1, 2, are rows 1 and k 1 + 1 of (9), a simple rank argument shows that aj(a) and bj(a) do not vanish simultaneously on F. Thus (31) can be solved for any initial condition rj(0), j = 1, 2 [10]. This implies that there exist zl(a) and zj,,+l(a) that satisfy (24),and the proof is complete.
3. Conclusions For systems with one or two inputs, our approach to pseudo-linearization appears to have two main advantages over the original treatment in [3,6]. These are the simplicity of the derivation based on standard linear theory, and a more explicit representation for a pseudo-linearizing transformation. The major disadvantage of the approach is that it is unclear whether the general m > 2 results derived by differential-geometric techniques in [2,3] can be obtained via the approach used here. The difficulty seems to reside in the nonuniqueness of the transformation to the Brunovsky form in (5).
J. Wang, W.J. Rugh /Pseudo-linearizationfor nonlinear systems
167
References [1] J. Ackermann, Sampled-Data Control Systems (Springer-Verlag, Berlin, 1985). [2] C. Champetier, P. Mouyon and J.F. Magni, Pseudo-linearization of nonlinear systems by dynamic precompensation, Proceedings of the 24th IEEE Conference on Decision and Control, Fort Lauderdale, FL (1985) 1371-1372. [3] C. Champetier, P. Mouyon and C. Reboulet, Pseudo-linearization of multi-input nonlinear systems, Proceedings of the 23rd IEEE Conference on Decision and Control, Las Vegas, NV (1984) 96-97. [4] P. Hartmann, Ordinary Differential Equations, Second Edition (Birkhauser, Boston, MA, 1982). [5] D.G. Luenberger, Canonical forms for linear multivariable systems, IEEE Trans. Automat. Control 12 (1967) 290-293. [6] C. Reboulet and C. Champetier, A new method for linearizing nonlinear systems: the pseudo-linearization, Internat. J. Control 40 (1984) 631-638. [7] C. Reboulet, P. Mouyon and C. Champetier, About the local linearization of nonlinear systems, in: M. Fliess, M. Hazewinkel, Eds., Algebraic and Geometric Methods in Nonlinear Control Theory (Reidel, Dordrecht, 1986) 311-322. [8] J. Wang and W.J. Rugh, Parameterized linear systems and linearization families for nonlinear systems, IEEE Trans. Circuits and Systems 34 (1987) 650-657. [9] J. Wang and W.J. Rugh, Feedback linearization families for nonlinear systems, 1EEE Trans. Automat. Control 32 (1987) 935 -940. [10] E.C. Zachmanoglou and D.W. Thoe, Introduction to Partial Differential Equations with Applications (Williams and Wilkens, Baltimore, MD, 1976).
161
On the pseudo-linearization problem for nonlinear systems * Jianliang W A N G and Wilson J. R U G H Department of Electrical and Computer Engineering, The Johns Hopkins University, Baltimore, MD 21218, U.S.A. Received 18 March 1988 Revised 3 October 1988
Abstract: Based on well-known results for linear systems, an alternate treatment of the pseudo-linearization problem of Reboulet and Champetier is given. Sufficient conditions for pseudo-linearization are obtained in the general case, and these are shown to be satisfied under mild hypotheses by systems with one or two inputs. Advantages of the new approach include the simplicity of the derivation, and a more explicit representation for a pseudo-linearizing transformation.
Keywords: pseudo-linearization, nonlinear systems, nonlinear control laws
1. Introduction
Pseudo-linearization of a nonlinear system involves computing a state feedback law and invertible state variable change such that the closed-loop system described in terms of the new variables has a family of linearizations that is independent of the closed-loop constant operating point. This notion was introduced by Reboulet and Champetier, and treated from a differential-geometric viewpoint in [2,3,6,7]. We describe here an approach to pseudo-linearization based on the linearization family of the nonlinear system, using a linear transformation due to Luenberger, [5], a linear feedback law due to Ackermann, [1], and results on linearization families in [8,9]. This leads to an alternative derivation of the results in [3,6] for one- and two-input systems, though in the general multi-input case only a sufficient condition is obtained. In the course of the derivation, explicit representations are obtained for a feedback law and a variable change that accomplish pseudo-linearization. We adopt a local viewpoint, though when particular examples are considered the problem often can be solved in a nonlocal fashion. For simplicity, all functions and their derivatives that appear in the sequel are assumed to be continuous. Multi-input nonlinear systems of the form
~(t) = f ( x ( / ) ,
u(t)),
x(t) ~ R", u(t) ~ R", t__ 0,
(a)
will be considered. It is assumed that (1) has a constant operating point family {[x(a), u(a)], a ~ F}, where F is an open neighborhood of 0 ~ R " and, for convenience, x(0)= 0, u(0)= 0. That is, f(x(a), u(a))= 0 for a ~ F. Typically the constant operating point family is parameterized by constant values of the input components and/or state components. Since local results are presented, in order to avoid repetitive statements we implicitly permit the shrinking of F when required by various local arguments. Linearizing (1) about its constant operating point family yields the parameterized linearization family d [x(t) - x ( a ) ] dt
=g(a)[x(t)
f ( ~ ) = ~ ( x ( ~ ) , u(,~)),
-x(a)] +
a ( ~ ) --
G(a)[u(t)
-u(a)]
(x(,~), u(~)),
~ ~ r.
(2)
* Research sponsored by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR-87-0101. 0167-6911/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)
J. Wang, W.J. Rugh /Pseudo-linearizationfor nonlinear svsterns
162
Fig. 1. Nonlinear system structure for pseudo-linearization.
It will be assumed that for each a ~ F, rank G ( a ) = m,
~X
rank-o--~-a( a ) = m,
r a n k [ G ( a ) , F ( a ) G ( a ) . . . . . F " - ' ( a ) G ( a ) ] =n.
(3)
Finally we assume that the controllability indices corresponding to (2), denoted by k 1. . . . . k,,, are constant for all a ~ F. It will become clear that the assumptions of controllability and constant controllability indices are required for the pseudo-linearization problem to be meaningful. The rank assumption on the operating point state will be used in the sequel to determine a from knowledge of x(a). Also, for use in the sequel we note that, since (2) is a linearization family [8],
F(a)
(a)+G(a)-~a(a)=O
forall a ~ F
(4)
2. Pseudo-linearization The pseudo-linearization problem for (1) involves finding an invertible state variable change z ( t ) = P(x(t)), z(t) ~ R n, and a feedback law u(t) = Pn+l(x(t), w(t)), w ( t ) ~ R m, with [OPn+l/Ow](O, 0) nonsingular, such that the closed-loop system shown in Figure 1, when linearized about its constant operating point family {[ w(a), z(a)], a ~ F}, has a constant-parameter linearization family described in the Brunovsky canonical form with controllability indices k 1. . . . . k,,. That is, the closed-loop linearization family is required to take the form
d---[z(t)-z(a)] =F[z(t)-z(a)] +G[w(t)-w(a)] dt P = block diagonal { if1,..
(~ = block diagonal { gl,.
0 0
---
a~_F,
0 0
~Rk, ~1.
(5)
More or less implicit in this problem statement is the requirement that w(a) and z(a) be consistent with the variable change and the requirement that (5) be a linearization family for the closed-loop nonlinear system. Specifically, the closed-loop constant operating point family must satisfy
z(a)=P(x(a)),
u(a)=P~+,(x(a), w(a)),
a¢F,
(6)
163
J. Wang,W.J.Rugh/Pseudo-linearizationfornonlinearsystems
and, for convenience, we will require P ( 0 ) = 0 and P,,+I(0, 0 ) = 0. Also, since (5) is required to be a linearization family it must satisfy a condition analogous to (4), namely
a~ (~) = 0,
. ~ r.
(7)
Direct substitution using the form of ff and G shows that (7) implies, for all a = F,
w(.) = o,
z,(~)
= 0,
i =
(8)
k l + "'" + k , ,
1+2,...,n.
Differentiating the first condition in (6) and combining with (8)gives the requirement (writing transposes to save space)
[[~Zl ]T~_~ (,~) 3P . . . .
[ ~Zkl+l ]T
, o ..... o[-~-~--(~)
, o . . . . . o .....
[ ~Zk~+...+k., ] T ,+1 ~
(,~)
, o ..... o
]T
ax
= -~x ( X( a))~--da (a)
(9)
To solve the pseudo-linearization problem, we first show how to choose K ( a ) ~ R rex', invertible
M(a) ~ R mxm, and invertible Q(a) ~ R "xn such that, for all a ~ F,
Q(a)[F(a) + G(a)K(a)] Q-~(a) = if,
Q(a)G(a)M(a) = G.
(10)
Secondly, we will show how to compute Pn+~(x, w) such that, for all a ~ F, e.+,(x(~),
w(,~)) = ,,(,~),
a , Ox ' . + l (x(,~), w(,~)) = K(,~),
a Pa~ .+l (x(.),
w(.)) = M(.), (11)
where w(a) ~ 0. Finally, we will determine conditions under which (9) can be satisfied with [OP/Ox](x(a)) = Q(a), and show how to compute P(x) such that, for all a ~ F,
P(x(a)) = z(a),
OP -~-x ( x ( a ) ) --- Q(a)
(12)
In order to specify in an explicit fashion the matrices Q(a), K(a) and M(a) that satisfy (10), we will make use of developments leading to the controller form in [5] and the so-called minimum-time deadbeat control law for discrete-time systems in [1]. Denoting the columns of G(a) by g~(a) ..... g.,(a), by hypothesis
C(a) = [ g l ( a ) , r(a)gl(a ) ..... Fk'-l(a)g,(a) ..... g.,(a), r(a)gm(a ) ..... rk"-ag.,(a)]
(13)
is invertible at each a ~ F. Partitioning C-l(a) by rows as
el,(a)
e,k,(") C-'(~) =
(14)
em,(~) em~°,(")
J. Wang,W.J,Rugh / Pseudo-linearizationfor nonlinearsystems
164 we let
elk,(a) elk,(a)F(a) e,k,(a)Fk'-l(a)
e,k,(a)F k' '(a) Q(a) =
a(~)
(15)
emk,.(a)F~,,,-2(a)
~o,(~) emk~(a)F(a) e,nk.,(a)Fk,"-l(a) and
[ e,k~(a)Fk'(a) K(a)-=
-M(a)]
!
(16)
[e,,k,,,(a)Fk"(a) The calculations to verify invertibility of Q(a) and M ( a ) and to show that (10) is satisfied are standard, though tedious unless carried out for a particular case. Now we need to show that the parameterized linear control law
.(t) -.(~)
= K(~)[x(t)
- x(~)] + M(~)[w(t)
- ~(~)]
(17)
with w(a)= 0 is a feedback linearization family, that is, the linearization of a nonlinear feedback law of the form u -=P,+l(x, w) about the closed-loop operating point family. But this follows from Corollary 3.1 in [9], since, using (4),
e'k'(a)Fk'(a) ] K(a) -~a
+ M(a)~aa (a) = -M(a)
~a emkm(a)Fk'(a)
)~(~)
= -M(~)
e,,k,,(a)Fk'-a(a) [ elk,(a)rk'-l(a) = M(a)/
:
G(a)~a(a)=--~a(a ) .
[emkm¢O))km-'(o)
(18)
Indeed, an explicit representation for a nonlinear control law corresponding to (17) can be found as follows [9]. Select m components of x(a) to form :~(a) such that rank[b~/3a](a) = m, and denote by Y the same selection of rn components from x. Then a nonlinear feedback law that satisfies (11) is ~o+l(X, w) = . ( ~ - ' ( ~ ) )
+ K(~-'(~))[x - x(~-'(2))] + M(£-'(~))w.
(19)
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Finally, we must consider conditions under which there exists z(a) such that (9) is satisfied, where [ O P / O x ] ( x ( a ) ) = Q ( a ) is specified in (15). It is straightforward to verify that if k , > 2 , then eik,(a)FJ(a)gk(a) = 0 for j = 0, 1. . . . . k, - 2; k = 1. . . . . m. Thus, for a ~ F,
e,k,(a)FJ+l(a)~a(a)=-eik,(a)FY(a)G(a)~(a)=O,
j=0,1 ..... k,-2.
(20)
Therefore (9) is satisfied if zl(a), Zk, +l(a) ..... Zk~ + ... +k,,, ,+ 1(a) satisfy the following set of m, uncoupled, total differential equations:
oa (a)=elk'(a)-~--da ( a ) ' OZkl+ ..- + k , , , _ l + l
3a
Oa
(a)=ezk'-(a)-~--da (a) . . . . .
Ox (a) = e,,k,,,(a)-~--da (a ).
(21)
If z(a) is such that (9) is satisfied, then a nonlinear variable change that satisfies (12) is
P ( x ) = z( £ - ' ( Y ) ) + Q( £ - l ( Y ) ) [ x - x ( £ - l ( Y ) ) ]
(22)
where .~ and Y are as in (19). [8] Writing the i-th entry of a row vector h as [h]~, and invoking the standard existence condition for solutions to total differential equations we obtain the following sufficient condition for pseudo-linearization. [4] Theorem. Suppose we are given a nonlinear system (1) with linearization family (2) such that (3) is satisfied, and the controllability indices are constant for a ~ F, where I" is a sufficiently small, open neighborhood of 0 ~ R m. Then the nonlinear system is pseudo-linearizable if for all a ~ 1",
Oar ejk '
(a) q = ~ a q
ejk,(a)-~a(a) r'
j=l,...,m;l
(23)
Furthermore, if these conditions are satisfied, then a pseudo-linearizing transformation is specified by (19) and (22). For the case m = 1, (23) is vacuous and pseudo-linearization is always possible under our basic assumptions. Unfortunately, for m > 1 the sufficient condition provided by this theorem is rather restrictive due to the specification of a particular K(a), Q(a), and M ( a ) out of the many that satisfy (10). This can be relaxed somewhat by introducing integrating factors as follows. (See also [7].) In place of (21), we consider
0zl (a) = r l ( a ) e , k , ( a ) Ox
....
OZk~+ ... +k,,._,+l OX Oa (a) = rm(a) emk,,,(a) ~-~-a(a)
(24)
where rl(a ) ..... r,,(a) are as yet arbitrary, continuous, real-valued functions that are nonzero for a ~ F. These functions can be introduced by modifying the definitions of Q ( a ) and M ( a ) (while leaving K ( a ) fixed as in (16)) according to (~(a) = block diagonal(ri(a)tk,×k,, i = 1 . . . . . m }. Q ( a ) , { 1 ~ t ( a ) = M(cQ. diagonal ~ ,
} i = 1..... m ,
(25)
where Ik,xk ' is the ki x ki identity matrix. With Q(a) and M(a) replaced by Q(a) and )Q(a), (10) remains satisfied and (17) is a feedback linearization family for
Pn+l(x, w) = u ( i - ' ( Y ) )
+ K(Y-l(Y))[x
- x(Y-l(Y))] + )l~l(£-'(Y))w
(26)
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where $ and Y are as before. Also, the equations that determine the nonzero components of z(a) are precisely those in (24). Therefore, if rl(a ) ..... rm(a ) can be found such that the appropriate existence conditions for (24) are satisfied, then the nonlinear variable change P( x ) = z( ~ - l ( Y)) + (~(J?-l(Y))[x - x ( ~ - 1 ( 2 ) ) ]
(27)
together with (26) accomplish pseudo-linearization. Indeed, for the case m = 2 we will establish that such z(a) always exists. Corollary. Suppose we are gioen a nonlinear system (1) with linearization family (2) such that m = 1 or m = 2. (3) is satisfied, and the controllability indices are constant for c~~ I', where F is a sufficiently small, open neighborhood of 0 ~ R m. Then the nonBnear system is pseudo-linearizable by a transformation of the form specified in (26) and (27). Proof. For the case m = 1, we take rl(a ) = 1, and (24) becomes dzl dx a a ( a ) = e l , ( a ) ~ - a (a)
(28)
With the initial condition zl(0) = 0, we have zl(a) = f0~e l , ( o ) ~--~-a dx (o) do
(29)
and a pseudo-linearizing transformation can be constructed as in (26) and (27) (or equivalently, (19) and (22)). In the case m = 2, the existence conditions for (24) become 3al rJ(a)eJk'(a)
= ~
rJ((~)%k'(a)~-~a~ (a) '
j = 1, 2.
(30)
Performing the indicated differentiations gives two, uncoupled, linear partial differential equations 05 05 aj(a)~Ta (a)+bj(a)~--~a2(a)+5(a)rj(a)=0
,
j=l,2,
(31 t
where aj(a)=ejG(a)~-~xaz(a),
b,(a)=-
e
jG (. ) -.~ lO(Xa ) ,
O%G, , Ox Oejk, OX cj(a) = - ~ ((x)-3-~2(a) - - ~ - a z ( a ) - ~ a (a),
j = 1, 2.
(32)
Since [-bj(ct), aj(ot)], j = 1, 2, are rows 1 and k 1 + 1 of (9), a simple rank argument shows that aj(a) and bj(a) do not vanish simultaneously on F. Thus (31) can be solved for any initial condition rj(0), j = 1, 2 [10]. This implies that there exist zl(a) and zj,,+l(a) that satisfy (24),and the proof is complete.
3. Conclusions For systems with one or two inputs, our approach to pseudo-linearization appears to have two main advantages over the original treatment in [3,6]. These are the simplicity of the derivation based on standard linear theory, and a more explicit representation for a pseudo-linearizing transformation. The major disadvantage of the approach is that it is unclear whether the general m > 2 results derived by differential-geometric techniques in [2,3] can be obtained via the approach used here. The difficulty seems to reside in the nonuniqueness of the transformation to the Brunovsky form in (5).
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References [1] J. Ackermann, Sampled-Data Control Systems (Springer-Verlag, Berlin, 1985). [2] C. Champetier, P. Mouyon and J.F. Magni, Pseudo-linearization of nonlinear systems by dynamic precompensation, Proceedings of the 24th IEEE Conference on Decision and Control, Fort Lauderdale, FL (1985) 1371-1372. [3] C. Champetier, P. Mouyon and C. Reboulet, Pseudo-linearization of multi-input nonlinear systems, Proceedings of the 23rd IEEE Conference on Decision and Control, Las Vegas, NV (1984) 96-97. [4] P. Hartmann, Ordinary Differential Equations, Second Edition (Birkhauser, Boston, MA, 1982). [5] D.G. Luenberger, Canonical forms for linear multivariable systems, IEEE Trans. Automat. Control 12 (1967) 290-293. [6] C. Reboulet and C. Champetier, A new method for linearizing nonlinear systems: the pseudo-linearization, Internat. J. Control 40 (1984) 631-638. [7] C. Reboulet, P. Mouyon and C. Champetier, About the local linearization of nonlinear systems, in: M. Fliess, M. Hazewinkel, Eds., Algebraic and Geometric Methods in Nonlinear Control Theory (Reidel, Dordrecht, 1986) 311-322. [8] J. Wang and W.J. Rugh, Parameterized linear systems and linearization families for nonlinear systems, IEEE Trans. Circuits and Systems 34 (1987) 650-657. [9] J. Wang and W.J. Rugh, Feedback linearization families for nonlinear systems, 1EEE Trans. Automat. Control 32 (1987) 935 -940. [10] E.C. Zachmanoglou and D.W. Thoe, Introduction to Partial Differential Equations with Applications (Williams and Wilkens, Baltimore, MD, 1976).