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Feedback equivalence of control systems
Systems & Control Letters 8 (1987) 463-465 North-Holland
463
Feedback equivalence of control systems R.B. G A R D N E R
*
Mathematics Department, University of North Carolina, Chapel Hill, NC 27514, USA
W.F. SHADWICK
**
Pure Mathematics Department, Universityof Waterloo, Waterloo, Ontario, Canada N2L 3G1
Received 30 September 1986 Revised 9 October 1986 Abstract: Cartan's method of equivalence is applied to the problem of equivalence of 2-state systems with scalar control under feedback. The differential invariants produced by the method completely characterize equivalence classes. The phenomenon of linearizability is associated with the presence of infinite Lie pseudogroups. The generic non-linearizable case has a geometricallydefined variational problem which yields a time optimal closed loop control. Kewvords: Feedback, Cartan equivalence method.
Section 1 The problem of feedback equivalence of control systems has been considered by many authors (e.g. [1,6-10,12]). In this note we indicate some results from a new approach to the local problem of equivalence, in the case of a 2-state system with scalar control, dx d"'~ = f ( x ,
u),
x ~ R 2, u ~ R.
(1.1)
We consider feedback as given by the local coordinate transformations [=t, 2 = q0(x),
our solution to this problem. The first is that linearizability may be characterized in terms of differential inoariants - functions made up from f ( x , u) and its partial derivatives, which Cartan's procedure constructs, and is accompanied b y the presence of infinite Lie pseudogroups. The second is that, in the non-linearizable case, the form of the structure equations for the equivalence problem allows us to recognize the presence of an intrinsically defined variational problem. It turns out that the solutions of this problem provide closed loop time optimal controls for (1.1). The first of these features appears to extend to the general case of systems with n states and p controls. The second extends to an intrinsically defined subclass of systems with n states and n - 1 controls. As our solution is given in terms of differential invariants, one may program tests, involving only differentiation, which give necessary and sufficient conditions for equivalence, using symbolic manipulation routines. This approach provides the four equivalence classes of linearizable systems (1.1) and their normal forms [13]. For systems which are not linearizable we construct 3 canonically associated 1-forms to1, to2, to3 whose exterior derivatives yield three differential invariants in the structure equations dt~i----- ~ ½cjikt~J A tok, i,j
i=1,2,3.
These functions m a y be used to determine necessary and sufficient conditions for equivalence of two non-linearizable systems by a process which involves only differentiation.
(1.2)
u), and apply Cartan's method of equivalence [2,3] to characterize equivalence classes of systems (1.1) under (1.2). There are two novel and important features of Research supported by NSERC grant A7895 ** and NSF grant DMS 8505434 *
Section 2 In order to describe the solution, it is necessary to cast the system (1.1) in terms of exterior differential equations and to characterize the coordinate transformations (1.2) in terms of conditions on their Jacobians. It is clear that the system (1.1) may be replaced
0167-6911/87/$3.50 © 1987, Elsevier Science Publishers'B.V. (North-Holland)
464
R.B. Gardner, I¥.F. Shadwick / F e e d b a c k equivalence of control systems
without loss of generality, we assume
by the vector of 1-forms
= d x - f ( x , u) dt
G=([ A
on R 4 with coordinates t, x 1, x 2, u. The condition for a second system A=
to be equivalent to (1.1) under a coordinate transformation is that there be a diffeomorphism • : R 4 --->R 4 such that ~*(d2-fdt)
=
a(dx-f
(2.1a)
(2.1b)
= =
.),
and this is an over-determined equivalence problem. In order to cast this problem in the form described in [3] it is necessary to perform a 0-th order reduction [2,4]. Let U0, VoC R 4 be open sets in which f ¢ 0 and f 4 : 0 respectively. If A o and A-0 are such that Aof= t (1, 0) = Aof, then necessary and sufficient conditions for a diffeomorphism q~:U0 ~ V0 to satisfy (2.1) are given by
(1 o A'-od2 = dfi I
0 0
A B
d Aodx ~ du
(2.2)
where
1 ax)
A ---- 0
a 2 E GL(2, R)
and
C 4: O.
As the 1-forms dt and d [ are invariant and decouple from the rest of the problem, it suffices to consider the conditions given in terms of A o d x -and du. Following [3] we now construct the 1forms to := t (to1, to2, to3) defined by
CJt
b2)
du
/
•
The procedure described in [3] now yields first order invariants (2.4a) where
and, if
t*~=t, o
and B = (b z
A : = f -,i , ,t _ f zf,,2
If • has the form (1.2) then
•
[lo"]
dt)
for a ~ GL(2, R).
2.
and
g] ~GL(3,")
a2
d-Z~ = / ( 2 , fi) dt
ft ~ 0,
J
which is defined on U × G, where U c R 3 is an open set on which f = (f~,)~ 0 and on which,
m 2 ~ O,
a = -a2
b=
_
~'-,x' _
1
(2.4b)
second order invariants +
k A },.~2
-/'1
'
2 t 1 2 (f,,f.,,-f,,f,,u)"
(2.5a) (2.5b)
(a2)2A 3
The various branches of the problem are analyzed by using coframes adapted to conditions satisfied by these invariants. Tl-ie cases in which transitive pseudogroups arise are given in Table 1. These are the four cases in which the system (1.1) is feedback linearizable. We note that the condition f . .2f , . ,1, - f . , , 1f . . . 2 = 0
(2.6)
is sufficient for an equation to fall into one of Cases I through IV and hence characterizes linearizable systems. The remaining ease to consider is the one in which b ~ 0. In this case one obtains a co-frame (to1, to2, to3) in which all of the group parameters
Table 1 Case
Characterization
Normal form
I
rn 1 = 0 , m 2 = 0
dxl dx2 = 0 dt = 1 , dt
II
m I q" 0, m 2 = 0
d x l = u 1, d x 2 dt --d'7- = 0
III
m2~0, a=0, b=0
d x l = U 1, d x 2 = x 2 dt
IV
ra2 q" O' a -~ O' b = O
dxl dx2 =xk dt = ul' " - ~ -
R.B. Gardner, W..F. Shadwick / Feedback equivalence of control systems
al, a2, b a, b 2 and C are assigned as functions of u) and its partial derivatives. The structure equations are
465
References
f(x,
d6o1 = e6o3 A 6o2, d6o2 = 6o3 A 6ol + 16o3 A 6o2,
(2.7)
d6o3 = J6o 3 A 6oz - K6ol A 6o2,
where e = + 1. These are the structure equations of the equivalence problem for a regular Lagrangian in one independent and one dependent variable [3] from which we may conclude that there is an intrinsic variational problem associated to the non-linearizable case. The integrand for this problem is the 1-form 6ol which is defined by
2 i1 6ol=--~dxl+L~dx 2. Along solutions of the system (2.1), d x a = f l dt and d x 2 _ f 2 dt so 6ol __ dt and hence
Stot
6oa = t
to.
(solutions)
Thus the variational problem 8f 6ol = 0 J(solutions) is the time optimal control problem. Euler-Lagrange equations are given by
The
6o2 = 0 , 6o3 ~ 0,
and the solution of these equations yields a dosed loop time optimal control u(x ~, x2).
[1] R. Brockett, Feedback invariants for nonlinear systems, IFAC Congress, Helsinki, Vol. 2 (1978) pp. 1115-1120. [2] P. Brunowski, A classification of linear controllable systems, Kibernetica Cislo 3, Rocnik 6/190 (1970) pp. 173-187. [3] E. Cartan, Les sous-groupes des groupes continues de transformations, Ann. Ec. Norm. 25 (1908) 57-194 (Oeuvres Completes II, pp. 719-856). [4] R.B. Gardner, Differential geometric methods interfacing control theory, in: R. Brockett, R. Millman, H. Sussmann, Eds., Differential Geometric Control Theory, Progress in Mathematics, Vol. 27 (Birkhauser, Boston, M_A, 1983) pp. 117-180. [5] R.B. Gardner and W.F. Shadwick, Overdetermined equivalence problems and apphcations to feedback linearization, in: Proceedings of NSF Differential Geometry Conference, San Antonio, TX (1986). [6] R. Hermann, Global invariants for feedback equivalence and Cauchy characteristics of nonlinear control systems, Part 1, Preprint (1986). [7] L.R. Hunt and R. Su, Linear equivalents of nonlinear time-varying systems, in: Proc. International Symposium on Mathematical Theory of Networks and Systems, Santa Monica, CA (1981) pp. 119-123. [8] B. Jakubczyk and W. Respondek, On linearization of control systems, Bull Acad. Polon. Sci. 28 (1980) 517-522. [9] A. Krener, On the equivalence of control systems and the linearization of non-linear systems, SIAM J. Control 11 (1973) 670-676. [10] A. Krener, Normal forms for linear and non-linear systems, Preprint (1986). [11] W.F. Shadwick, Seminar Lecture Notes on Cartan's Method of Equivalence (1985). [12] R. Su, G. Meyer and L. Hunt, Robustness in nonlinear control, in: R. Brockett, R. Millman, H. Sussmann, Eds., Differential Geometric Control Theory, Progress in Mathematics, Vol. 27 (Birkhauser, Boston, MA, 1983) pp. 316-337. [13] R. Su, On the linear equivalents of nonlinear systems, Systems Control Left. 2 (1981) 48-52.
463
Feedback equivalence of control systems R.B. G A R D N E R
*
Mathematics Department, University of North Carolina, Chapel Hill, NC 27514, USA
W.F. SHADWICK
**
Pure Mathematics Department, Universityof Waterloo, Waterloo, Ontario, Canada N2L 3G1
Received 30 September 1986 Revised 9 October 1986 Abstract: Cartan's method of equivalence is applied to the problem of equivalence of 2-state systems with scalar control under feedback. The differential invariants produced by the method completely characterize equivalence classes. The phenomenon of linearizability is associated with the presence of infinite Lie pseudogroups. The generic non-linearizable case has a geometricallydefined variational problem which yields a time optimal closed loop control. Kewvords: Feedback, Cartan equivalence method.
Section 1 The problem of feedback equivalence of control systems has been considered by many authors (e.g. [1,6-10,12]). In this note we indicate some results from a new approach to the local problem of equivalence, in the case of a 2-state system with scalar control, dx d"'~ = f ( x ,
u),
x ~ R 2, u ~ R.
(1.1)
We consider feedback as given by the local coordinate transformations [=t, 2 = q0(x),
our solution to this problem. The first is that linearizability may be characterized in terms of differential inoariants - functions made up from f ( x , u) and its partial derivatives, which Cartan's procedure constructs, and is accompanied b y the presence of infinite Lie pseudogroups. The second is that, in the non-linearizable case, the form of the structure equations for the equivalence problem allows us to recognize the presence of an intrinsically defined variational problem. It turns out that the solutions of this problem provide closed loop time optimal controls for (1.1). The first of these features appears to extend to the general case of systems with n states and p controls. The second extends to an intrinsically defined subclass of systems with n states and n - 1 controls. As our solution is given in terms of differential invariants, one may program tests, involving only differentiation, which give necessary and sufficient conditions for equivalence, using symbolic manipulation routines. This approach provides the four equivalence classes of linearizable systems (1.1) and their normal forms [13]. For systems which are not linearizable we construct 3 canonically associated 1-forms to1, to2, to3 whose exterior derivatives yield three differential invariants in the structure equations dt~i----- ~ ½cjikt~J A tok, i,j
i=1,2,3.
These functions m a y be used to determine necessary and sufficient conditions for equivalence of two non-linearizable systems by a process which involves only differentiation.
(1.2)
u), and apply Cartan's method of equivalence [2,3] to characterize equivalence classes of systems (1.1) under (1.2). There are two novel and important features of Research supported by NSERC grant A7895 ** and NSF grant DMS 8505434 *
Section 2 In order to describe the solution, it is necessary to cast the system (1.1) in terms of exterior differential equations and to characterize the coordinate transformations (1.2) in terms of conditions on their Jacobians. It is clear that the system (1.1) may be replaced
0167-6911/87/$3.50 © 1987, Elsevier Science Publishers'B.V. (North-Holland)
464
R.B. Gardner, I¥.F. Shadwick / F e e d b a c k equivalence of control systems
without loss of generality, we assume
by the vector of 1-forms
= d x - f ( x , u) dt
G=([ A
on R 4 with coordinates t, x 1, x 2, u. The condition for a second system A=
to be equivalent to (1.1) under a coordinate transformation is that there be a diffeomorphism • : R 4 --->R 4 such that ~*(d2-fdt)
=
a(dx-f
(2.1a)
(2.1b)
= =
.),
and this is an over-determined equivalence problem. In order to cast this problem in the form described in [3] it is necessary to perform a 0-th order reduction [2,4]. Let U0, VoC R 4 be open sets in which f ¢ 0 and f 4 : 0 respectively. If A o and A-0 are such that Aof= t (1, 0) = Aof, then necessary and sufficient conditions for a diffeomorphism q~:U0 ~ V0 to satisfy (2.1) are given by
(1 o A'-od2 = dfi I
0 0
A B
d Aodx ~ du
(2.2)
where
1 ax)
A ---- 0
a 2 E GL(2, R)
and
C 4: O.
As the 1-forms dt and d [ are invariant and decouple from the rest of the problem, it suffices to consider the conditions given in terms of A o d x -and du. Following [3] we now construct the 1forms to := t (to1, to2, to3) defined by
CJt
b2)
du
/
•
The procedure described in [3] now yields first order invariants (2.4a) where
and, if
t*~=t, o
and B = (b z
A : = f -,i , ,t _ f zf,,2
If • has the form (1.2) then
•
[lo"]
dt)
for a ~ GL(2, R).
2.
and
g] ~GL(3,")
a2
d-Z~ = / ( 2 , fi) dt
ft ~ 0,
J
which is defined on U × G, where U c R 3 is an open set on which f = (f~,)~ 0 and on which,
m 2 ~ O,
a = -a2
b=
_
~'-,x' _
1
(2.4b)
second order invariants +
k A },.~2
-/'1
'
2 t 1 2 (f,,f.,,-f,,f,,u)"
(2.5a) (2.5b)
(a2)2A 3
The various branches of the problem are analyzed by using coframes adapted to conditions satisfied by these invariants. Tl-ie cases in which transitive pseudogroups arise are given in Table 1. These are the four cases in which the system (1.1) is feedback linearizable. We note that the condition f . .2f , . ,1, - f . , , 1f . . . 2 = 0
(2.6)
is sufficient for an equation to fall into one of Cases I through IV and hence characterizes linearizable systems. The remaining ease to consider is the one in which b ~ 0. In this case one obtains a co-frame (to1, to2, to3) in which all of the group parameters
Table 1 Case
Characterization
Normal form
I
rn 1 = 0 , m 2 = 0
dxl dx2 = 0 dt = 1 , dt
II
m I q" 0, m 2 = 0
d x l = u 1, d x 2 dt --d'7- = 0
III
m2~0, a=0, b=0
d x l = U 1, d x 2 = x 2 dt
IV
ra2 q" O' a -~ O' b = O
dxl dx2 =xk dt = ul' " - ~ -
R.B. Gardner, W..F. Shadwick / Feedback equivalence of control systems
al, a2, b a, b 2 and C are assigned as functions of u) and its partial derivatives. The structure equations are
465
References
f(x,
d6o1 = e6o3 A 6o2, d6o2 = 6o3 A 6ol + 16o3 A 6o2,
(2.7)
d6o3 = J6o 3 A 6oz - K6ol A 6o2,
where e = + 1. These are the structure equations of the equivalence problem for a regular Lagrangian in one independent and one dependent variable [3] from which we may conclude that there is an intrinsic variational problem associated to the non-linearizable case. The integrand for this problem is the 1-form 6ol which is defined by
2 i1 6ol=--~dxl+L~dx 2. Along solutions of the system (2.1), d x a = f l dt and d x 2 _ f 2 dt so 6ol __ dt and hence
Stot
6oa = t
to.
(solutions)
Thus the variational problem 8f 6ol = 0 J(solutions) is the time optimal control problem. Euler-Lagrange equations are given by
The
6o2 = 0 , 6o3 ~ 0,
and the solution of these equations yields a dosed loop time optimal control u(x ~, x2).
[1] R. Brockett, Feedback invariants for nonlinear systems, IFAC Congress, Helsinki, Vol. 2 (1978) pp. 1115-1120. [2] P. Brunowski, A classification of linear controllable systems, Kibernetica Cislo 3, Rocnik 6/190 (1970) pp. 173-187. [3] E. Cartan, Les sous-groupes des groupes continues de transformations, Ann. Ec. Norm. 25 (1908) 57-194 (Oeuvres Completes II, pp. 719-856). [4] R.B. Gardner, Differential geometric methods interfacing control theory, in: R. Brockett, R. Millman, H. Sussmann, Eds., Differential Geometric Control Theory, Progress in Mathematics, Vol. 27 (Birkhauser, Boston, M_A, 1983) pp. 117-180. [5] R.B. Gardner and W.F. Shadwick, Overdetermined equivalence problems and apphcations to feedback linearization, in: Proceedings of NSF Differential Geometry Conference, San Antonio, TX (1986). [6] R. Hermann, Global invariants for feedback equivalence and Cauchy characteristics of nonlinear control systems, Part 1, Preprint (1986). [7] L.R. Hunt and R. Su, Linear equivalents of nonlinear time-varying systems, in: Proc. International Symposium on Mathematical Theory of Networks and Systems, Santa Monica, CA (1981) pp. 119-123. [8] B. Jakubczyk and W. Respondek, On linearization of control systems, Bull Acad. Polon. Sci. 28 (1980) 517-522. [9] A. Krener, On the equivalence of control systems and the linearization of non-linear systems, SIAM J. Control 11 (1973) 670-676. [10] A. Krener, Normal forms for linear and non-linear systems, Preprint (1986). [11] W.F. Shadwick, Seminar Lecture Notes on Cartan's Method of Equivalence (1985). [12] R. Su, G. Meyer and L. Hunt, Robustness in nonlinear control, in: R. Brockett, R. Millman, H. Sussmann, Eds., Differential Geometric Control Theory, Progress in Mathematics, Vol. 27 (Birkhauser, Boston, MA, 1983) pp. 316-337. [13] R. Su, On the linear equivalents of nonlinear systems, Systems Control Left. 2 (1981) 48-52.