Noninteracting Control via Static Measurement Feedback for ...
774
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 4, APRIL 1999
where y^(t; ) is the output of the neural control scheme, Jtrack (c ) is a cost function for the tracking error, defined on a given specific reference input d(t) and tf is a finite time horizon. The matrix inequality constraint can be related to Lemma 1 as well as to Lemma 2, respectively, for imposing global asymptotic stability or I/O stability with a fixed disturbance attenuation level 3 : Such methods have been successfully applied for the discrete-time recurrent neural networks using NLq theory in [19].
VII. CONCLUSION In this paper absolute stability and dissipativity of continuous-time recurrent neural networks with two hidden layers have been studied. These types of models occur when one considers nonlinear models and controllers that are parameterized by multilayer perceptrons with one hidden layer. For the autonomous case a classical Lur’e system representation and Lur’e system with multilayer perceptron nonlinearity is given. Sufficient conditions for absolute stability and dissipativity have been derived from a Lur’e–Postnikov Lyapunov function and a storage function of the same form. The criteria are expressed as matrix inequalities. They can be employed in order to impose closed-loop stability in Narendra’s dynamic backpropagation procedure and for nonlinear H1 control.
[17] E. Polak and Y. Wardi, “Nondifferentiable optimization algorithm for designing control systems having singular value inequalities,” Automatica, vol. 18, no. 3, pp. 267–283, 1982. [18] E. D. Sontag and H. Sussmann, “Complete controllability of continuoustime recurrent neural networks,” Syst. Contr. Lett., vol. 30, pp. 177–183, 1997. [19] J. A. K. Suykens, J. P. L. Vandewalle, and B. L. R. De Moor, Artificial Neural Networks for Modeling and Control of Non-Linear Systems. Boston, MA: Kluwer, 1995. [20] J. A. K. Suykens, B. De Moor, and J. Vandewalle, “Nonlinear system identification using neural state space models, applicable to robust control design,” Int. J. Contr., vol. 62, no. 1, pp. 129–152, 1995. , “NLq theory: A neural control framework with global asymptotic [21] stability criteria,” Neural Networks, vol. 10, no. 4, pp. 615–637, 1997. [22] A. J. van der Schaft, “A state-space approach to nonlinear 1 control,” Syst. Contr. Lett., vol. 16, pp. 1–8, 1991. , “ 2 -gain analysis of nonlinear systems and nonlinear state [23] feedback 1 control,” IEEE Trans. Automat. Contr., vol. 37, pp. 770–784, 1992. [24] H. Verrelst, K. Van Acker, J. Suykens, B. Motmans, B. De Moor, and J. Vandewalle, “Application of NLq neural control theory to a ball and beam system,” European J. Contr., vol. 4, no. 2, pp. 148–157, 1998. [25] M. Vidyasagar, Nonlinear Systems Analysis. Englewood Cliffs, NJ: Prentice-Hall, 1993. [26] J. C. Willems, “Dissipative dynamical systems I: General theory. II: Linear systems with quadratic supply rates,” Archive for Rational Mechanics and Analysis, vol. 45, pp. 321–343, 1972. [27] C.-F. Yung, Y.-P. Lin, and F.-B. Yeh, “A family of nonlinear 1 output feedback controllers,” IEEE Trans. Automat. Contr., vol. 41, pp. 232–236, 1996. [28] J. M. Zurada, Introduction to Artificial Neural Systems. West, 1992.
H
L H
H
REFERENCES [1] F. Albertini and E. D. Sontag, “For neural networks, function determines form,” Neural Networks, vol. 6, pp. 975–990, 1993. [2] , “State observability in recurrent neural networks,” Syst. Contr. Lett., vol. 22, pp. 235–244, 1994. [3] J. A. Ball, J. W. Helton, and M. L. Walker, “ 1 control for nonlinear systems with output feedback,” IEEE Trans. Automat. Contr., vol. 38, pp. 546–559, 1993. [4] A. R. Barron, “Universal approximation bounds for superposition of a sigmoidal function,” IEEE Trans. Inform. Theory, vol. 39, no. 3, pp. 930–945, 1993. [5] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, Studies in Applied Mathematics, vol. 15. Philadelphia, PA: SIAM, 1994. [6] S. Haykin, Neural Networks: A Comprehensive Foundation. Englewood Cliffs, NJ: Macmillan, 1994. [7] D. J. Hill and P. J. Moylan, “Connections between finite-gain and asymptotic stability,” IEEE Trans. Automat. Contr., vol. AC-25, no. 5, pp. 931–936, 1980. [8] , “The stability of nonlinear dissipative systems,” IEEE Trans. Automat. Contr., vol. AC-21, pp. 708–711, 1976. [9] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, U.K.: Cambridge Univ. Press, 1985. [10] K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforward networks are universal approximators,” Neural Networks, vol. 2, pp. 359–366, 1989. [11] A. Isidori and A. Astolfi, “Disturbance attenuation and 1 control via measurement feedback in nonlinear systems,” IEEE Trans. Automat. Contr., vol. 37, pp. 1283–1293, 1992. [12] A. Isidori and W. Kang, “ 1 control via measurement feedback for general nonlinear systems,” IEEE Trans. Automat. Contr., vol. 40, pp. 466–472, 1995. [13] H. K. Khalil, Nonlinear Systems. New York: Macmillan, 1992. [14] M. Leshno, V. Y. Lin, A. Pinkus, and S. Schocken, “Multilayer feedforward networks with a nonpolynomial activation function can approximate any function,” Neural Networks, vol. 6, pp. 861–867, 1993. [15] K. S. Narendra and K. Parthasarathy, “Gradient methods for the optimization of dynamical systems containing neural networks,” IEEE Trans. Neural Networks, vol. 2, no. 2, pp. 252–262, 1991. [16] K. S. Narendra and J. H. Taylor, Frequency Domain Criteria for Absolute Stability. New York: Academic, 1973.
H
Noninteracting Control via Static Measurement Feedback for Nonlinear Systems with Relative Degree S. Battilotti
Abstract— In this paper the authors give a necessary and sufficient geometric condition for achieving noninteraction via static measurement feedback for nonlinear systems with vector relative degree. Their analysis relies on the theory of connections and as a result gives systematic procedures for constructing a decoupling feedback law. Index Terms—Measurement feedback, noninteracting control.
I. THE CLASS OF SYSTEMS AND CONTROL LAWS Let us consider the affine nonlinear systems of the form m
x_ = f (x) +
H
gj (x)uj j =1
yi = hi (x);
H
z = k(x)
i = 1; 1 1 1 ; m;
(1)
where x 2 M; a smooth (Hausdorff) manifold, the ui ’s are input functions from a suitable function space (e.g., measurable -valued functions defined on closed intervals of the form [0; T ]); the yi ’s are the -valued output functions and z 2 s is the vector of Manuscript received September 23, 1997. Recommended by Associate Editor, A. J. van der Schaft. The author is with the Dipartimento di Informatica e Sistemistica, 00184 Roma, Italy. Publisher Item Identifier S 0018-9286(99)02087-5.
0018–9286/99$10.00 1999 IEEE
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 4, APRIL 1999
variables which are available for feedback, and f; g1 1 1 1 ; gm are smooth vector fields on some open subset U M with g1 ; 1 1 1 ; gm linearly independent on U and f : Moreover, h1 ; 1 1 1 ; hm and k are smooth functions defined on U ; with hi and k : Denote by h the vector h1 1 1 1 hm T and by g the matrix g1 1 1 1 gm : We will consider static measurement-feedback laws of the form
(0) = 0 )
(
(0) = 0
(0) = 0 ( )
u = (k(x)) + (k(x))v (2) with smooth functions and defined on some open subset Z s ; (k(x)) 2 G L( ; m); where G L( ; m) is the general linear group of m 2 m invertible matrices over ; and v the new input vector. II. MOTIVATIONS
()=
PROBLEM STATEMENT
AND
Assume that k x x; i.e., the state vector is available for feedback. Moreover, if not otherwise stated, assume that U M: Let G be the distribution spanned by g1 ; 1 1 1 ; gm : A distribution on M is said to be weakly f; g -invariant at x 2 M; if there exists a neighborhood U0 of x such that on U0
=
( )
[f; 1] [g; 1]
+1 + 1: (f; g)-invariant
1
G
(4)
G
(5)
1 is said to be globally weakly (f; g)-invariant at each x :
if it is weakly
2 M
If (4) and (5) are not satisfied, one can try to modify the behavior of (1) through (2) in such a way that these properties are achieved for the closed-loop system. A distribution on M is said to be f; g -invariant at x 2 M; if there exists a static state feedback law (2), defined in a neighborhood U0 of x; such that
1
( )
[f + g ; 1] 1 [g ; 1] 1:
(6)
(7)
If the feedback law is defined on ; then 1 is said to be globally (f; g)-invariant. Weak (f; g )-invariance was first introduced for linear systems in M
[3] and, independently, in [4], for nonlinear systems affine in the input in [5] and for general nonlinear systems in [6]. If (1) is a linear system and is a subspace of n ; (5) and (7) are trivially satisfied, local and global definitions coincide, and weak f; g -invariance is equivalent to f; g invariance. For nonlinear system (1), weak f; g -invariance is implied by f; g -invariance but the converse is not true in general. A powerful lemma for studying f; g -invariance is given by the Quaker lemma [1]–[6]. The main difficulty in extending the Quaker lemma to a global setting hides behind using arguments heavily based on partial differential equations (PDE’s). The key observation, which gives a deep insight into global obstructions, has been first noted in [7], followed by [8], where it is shown that weak f; g -invariance can be interpreted geometrically saying that a subbundle of the normal bundle of is “invariant under parallel transport” along leaves of : Now, assume that only k x is available for feedback. A distribution on M is said to be weakly f; g; k -invariant at x 2 M; if it is weakly f; g -invariant at x 2 M and there exists a neighborhood U0 of x such that
1
( )
( )
( )
( )
( )
( )
1
1
1
()
(
( )
[f; 1 [g; 1
)
(8) ker dk] 1 ker dk] 1: (9) 1 is said to be globally weakly (f; g; k)-invariant if it is weakly (f; g; k)-invariant at each x : A distribution 1 on is said to be (f; g; k)–invariant at x ; \
\
2 M
M
775
1
is said to be globally If the feedback law is defined on M; then f; g; k -invariant. Weak f; g; k invariance was first introduced in [3] and, independently, in [4] for linear systems and in [5] for nonlinear systems, affine in the input. If (1) is a linear system and is spanned by a set of constant vectors of n ; local and global definitions coincide and weak f; g; k -invariance is equivalent to f; g; k invariance [3], [4]. For nonlinear systems (1), weak f; g; k -invariance is implied by f; g; k -invariance but the converse is not true in general. Necessary and sufficient (existence) conditions for local f; g; h invariance are given in [9]. As is well known, weak f; g -invariance is a natural tool for solving many control problems, such as disturbance decoupling and noninteracting control, as long as the state x is available for feedback and local solutions are sought (see [2] and [10] for an exhaustive discussion). For global f; g invariance, additional assumptions to weak f; g -invariance must be imposed (see above). If only z is available for feedback, weak f; g; k invariance still does the job for linear systems [3], [4]. However, in a nonlinear setting, one needs additional assumptions to fully characterize (global) f; g; k invariance in terms of weak f; g; k invariance. In general, these extra assumptions are tailored to guarantee that a state-feedback law (2), which renders a given distribution invariant, can be “expressed” as a measurement feedback law (2). In this paper, exploring the route of f; g; k invariance, we will focus our attention on the problem of rendering (1) noninteractive via static measurement feedback (2). We say that the system (1), (2) is noninteractive if each output is influenced (or “controlled”) only by one input. Moreover, we consider the class of nonlinear systems (1) which have uniform vector relative degree on M [10], i.e., the same vector relative degree at each x 2 M (see [12] for motivations). Noninteracting Control via Static Measurement Feedback Laws (NSM): Find, if possible, a static feedback law (2), defined on a neighborhood U of x ; such that the system (1), (2) is noninteractive and has uniform vector relative degree on U : When U M; we will refer to the noninteracting control problem by global noninteracting control via static measurement (GNSM). When the functions f; g; h; and k are analytic, a stronger problem can be formulated (strong input–output decoupling; see [10]). The local noninteracting control problem via static output feedback (i.e., k h) was first solved in [12] under an additional assumption. This assumption was removed in [13], where a necessary and sufficient condition for solving NSM is given. Unfortunately, although the global validity of this condition is equivalent to GNSM being solvable, no global constructive method is pointed out in [13] for checking this condition. In this paper, we give a necessary and sufficient (checkable) condition for solving GNSM, using a complete different approach, advocated in [8] for global f; g invariance and leading naturally to a global solution of the noninteracting control problem. We point out the obstruction lying between linear and nonlinear f; g; k -invariance. Our proof gives a constructive procedure for obtaining the decoupling feedback law and has, in local coordinates, a very simple interpretation. Moreover, our condition recovers the ones given in [13] and in [9] for local f; g; k invariance. One of the contributions of this paper is, in our opinion, to give (global) constructive tools which can also be used to solve measurement feedback problems different from noninteracting control such as either disturbance decoupling or model matching.
(
)
(
)
1
(
(
)
(
)
)
(
)
(
)
( )
( )
( )
(
)
(
)
(
(
)
1 )
= 0
=
=
( )
(
)
(
)
2 M
if there exists a static measurement feedback law (2), defined in neighborhood U0 of x; such that
[f + g( k); 1] 1 [g( k); 1] 1:
(10)
(11)
III. THEORETICAL BACKGROUND AND DEFINITIONS • We assume that concepts, as bundles, sections, etc., are familiar to the reader. We refer to standard textbooks such as [14] and [15]. By T M we will denote the tangent bundle on M
( )
776
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 4, APRIL 1999
1( )
and by the same symbol we will denote both distributions and their associated subbundles. Let C M be the set of smooth functions on M and X M be the set of smooth vector fields on M: • A smooth section X of a vector bundle over M is a smooth mapping X M ! such that X idM ; where idM is the identity on M: A vector field on M is a section of T M : If q is the dimension of and if there exist q everywhere independent sections (frame of ), then is said to be trivial. • A smooth connection r on T M is a mapping X; Y 7! 1 M 2 X 1 M ! X 1 M ; which rX Y; with r X 1 is C M -linear w.r.t. X; -linear w.r.t. Y and such that rX fY LX f Y f rX Y; f 2 C 1 M : A smooth connection r on a vector bundle D ! M is a mapping X; Y 7! rX Y; with r X 1 M 2 D ! D; which is C 1 M -linear w.r.t. X; -linear w.r.t. Y and such that
1( )
:
1 =
1
1
:
( ) ) +
1 1 ( ) ( )
( ) ( ) ( )
r
2
M
Q
Q
M
M
M Q
M
Q
Q
M
M
r
r
2
Q
Q
M
r
!
M
Q
Q
r
IV. NECESSARY AND SUFFICIENT CONDITION FOR NONINTERACTING CONTROL VIA STATIC MEASUREMENT FEEDBACK 3
Let Ri be the maximal controllability distribution for (1) contained in j 6=i dhj ; supposed to exist on all M: This distribution can be obtained through a constructive algorithm (see [2]). We will assume that at each step of this algorithm we obtain a nonsingular distribution on M: Under this assumption, such an algorithm ends in a finite number of steps. More simply, we will say that R3i is regularly computable on M: m 3 Let R3 j 6=i Rj and let R be the strong accessii=1 bility distribution or, equivalently, the smallest distribution which is invariant under f and g and contains G (see [2] for constructive 3 algorithms). Moreover, let Di R3i \ G and Ei j 6=i Rj : 3 A1) R and Ri are regularly computable on M for i dk and dk Ei have constant ; 1 1 1 ; m; Ei ; R3 ; dimension on M: Moreover, dk Ei is involutive. A2) System (1) has uniform vector relative degree on M: A3) R has constant dimension n on M: The distribution dk Ei is not involutive, in general. However, if, in particular, k x h x and under Assumptions A1) and A2), it is easy to show that dk Ei is indeed involutive (see [12, Sec. 6.1]). In the framework of local noninteracting control with internal stability (see the end of this section), A3) is a standard assumption [4], [16] and in the linear case amounts to require that the systems be controllable [4]. If internal stability is not required, Assumption A3) is not needed but for simplifying the formulas involved. Before stating the main result of this section, we will discuss some basic facts.
ker
=
; (6
)
=
1
ker
ker + ( )= ( ) ker +
=6
ker + ker +
=
( )
( )
( )
( = +( + )
)
1
( ) ( )=( : ( ( ) : ( ) ( ) X (fY ) = (LX f )Y + f X Y; f C 1 ( ): = ( ) The subbundle 1 determines the quotient bundle T ( )=1; having x = (Tx )=1x as fiber over x: If a Riemaniann metric is introduced on ; can be identified with a subbundle 0 of T ( ) (normal bundle of 1; denoted by 1? ), ( having as fiber over x the subspace 0x = 1? Tx : As x a consequence, any section Y of T ( ) decomposes uniquely as Y1 + YQ : According to this decomposition, one can define X Y = X YQ = [X; YQ ]Q ; with X 1; Y is a section of and YQ is the unique vector field in 0 which projects onto Y at each x by the natural projection Tx ( ) Tx ( )=1: is shown to define a connection on and to be independent ( ( )) = ( of the choice of 0 : If X YQ = 0; we say that Y (or YQ ; (6 ) once Y and YQ are identified as above) is parallel along leaves ( ) of 1: Such a connection is commonly referred to as the Bott connection. ( ( )) r
•
( )
1
For each i 2 f ; 1 1 1 ; mg; let decompose T M as follows. If a Riemaniann metric is introduced on M; the quotient bundle Qi T M =Ei can be identified with a subbundle Q0i of T M : Thus, any vector field X on M can be uniquely decomposed as XQ XE : Moreover, under Assumptions A1) and A2), Ei \ G \ Di ([16] and [12]). This implies that Q0i can be chosen in such a way to include Di : In particular, Q0i Di Di Ei ? : In what follows, by i 0 r we will denote the Bott connection such that rX Y X; Y 0 Q : Given any distribution or vector field X on M; we will denote i or its projection into Q0i (or Qi ; once Qi is identified with Q0i ) by i X ; respectively. The projection of X onto Di will be also denoted by XD : By definition, for each i 2 f ; 1 1 1 ; mg the distribution Ei is globally weakly f; g -invariant and, under Assumption A1), it is also involutive (see [2] or [12] for a detailed proof). Thus, through each p 2 M there is a maximal connected integral manifold LEp ; the leaf of Ei passing through p: As pointed out in [8], global weak f; g -invariance of Ei can be interpreted from a geometric point of view by saying that the projection of G into Qi is invariant under parallel transport along leaves (of Ei ). This can be stated equivalently as follows: if W ; is a Frobenius chart and is followed by projection into q (q is the codimension of Ei ), then, at each point x of any leaf of Ei intersected with W ; G Ei x projects to the same subspace of T ~ (x) q : Let Fi denote the folation of Ei : Assume that M=Fi is a smooth manifold and let F M ! M=Fi be the projection map. At each p 2 M one can assign a subspace of T (p) M=Fi by i Di F p f F 3p X jX 2 Di p g: Since G D1 8 1 1 1 8 Dm and j2I R3j \ G j 2I Dj ; with I f ; 1 1 1 ; mg ([16]), global weak f; g invariance of Ei implies that the projection of Di onto Qi is invariant under parallel transport along leaves. Thus, i Di F p does not depend on the choice of p on the leaf of Ei and Di i is a smooth distribution on M=Fi (see also [8]). Given any distribution or vector field X on M; we will denote (when this does make sense) its projection onto T M=Fi by i or X i ; respectively. If there exists a complementary subbundle Hi to Di i in T M=Fi ; one can pull back Hi to obtain a complementary subbundle Ni to Di in Qi : With Q0i chosen as above, the subbundle Ni is isomorphic to ? under the projection T M ! Q the subbundle Ni0 Di Ei i and, thus, by construction Ni0 is invariant under parallel transport along leaves. Denoting by Gi the distribution spanned by gi ; it can be easily seen that Assumption A2) implies that for each i 2 f ; 1 1 1 ; mg and in a neighborhood U0 of each point p 2 M there exists ki 2 f ; 1 1 1 ; mg such that Gk i Di : For achieving a global result and according to our previous discussion, we assume the following. A4) After possibly renumbering the inputs and for each i 2 i Di : Moreover, M=Fi ; i ; 1 1 1 ; m; are f ; 1 1 1 ; mg; Gi smooth (not necessarily Hausdorff) manifolds, Di i is trivial and there exists a complementary subbundle Hi to Di i in
=
+
=[
] 1
(
)
1
)
~
)
( + )
:
)
()
=6
= 1
1
^
(
)
(
=( + )
1^ )
( )
1
1
=
=
1
=0
=1
T (M=Fi ): When k(x) = x; Assumption A4) is exactly the one invoked in [8] (Proposition 5.1). If M=Fi is also Hausdorff, a Riemaniann metric
can be introduced and the existence of a complementary subbundle i to Di i in T M=Fi is automatically guaranteed. Let K be the foliation of dk and Ki be the foliation of dk \ Ei : Let M=Ki be a smooth manifold, {K M=Ki ! M be the inclusion map, and K M ! M=Ki be the projection map. Note that, since dk is involutive, X; dk dk for all X2 dk \ Ei : Thus, the distribution Si f K 3p X jX 2 dk p g is a smooth distribution on M=Ki : H
(
)
(ker )
(ker ) (ker )( )
ker
ker
:
:
[ ker ] ker = ( )
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 4, APRIL 1999
Moreover, by L1 p we will denote the leaf, passing through p; of an involutive and nonsingular distribution : Our last assumption is the following and its role and interpretation will be clear in the proof of the main theorem. 1 E A5) M=K; M=Ki ; and {0 K Lp are smooth manifolds and there exists a smooth manifold Oi M=Ki ; transversal to each {0K1 LEp in M=Ki ; intersecting each {0K1 LEp only once and such that Si p Tp Oi : Our main result is the following. Theorem 1: Assume A1)–A5). GNSM is solvable if and only if for each i 2 f ; 1 1 1 ; mg and for each p 2 M there exists a neighborhood Up of p such thati for any frame Z1 ; 1 1 1 ; Zd ; 1 1 1 ; Zl of dk; defined on Up ; with Zj parallel for all j d and Zd+1 ; 1 1 1 ; Zl a frame of dk \ Ei • for j d and for all X 2 Ei i Zj ; gs i D s ;111;m (12) ji gs i i rX Zj ; fD D ji rX fD (13)
1
()
1
ker
(ker )
[ [
] = ] =
=1
with ji constant along leaves of Ei in U0 • for j d s ;111;m riZ gs i i i rZ f :
+1
=0 =0
=1
)
(
)
on
=1
Di = spanfWi g (W1 1 1 1 Wm ) = g( k) [f + g( k); Ei ] Ei [g( k); Ei ] Ei
+
)
:
gi
= ii Wi :
] ]
(21)
(16c)
j 6= i and for some ji 2 C 1 (M); constant ker dk: Thus, (12) holds for all s = 1; 1 1 1 ; m:
Finally, let us prove (13). For the vector field uniquely as
along leaves of
f
decomposes
+ fD + fN : (22) Since Ei is globally weakly (f; g ) invariant and Ni0 is invariant under f = fE
parallel transport along leaves, it must be
[fN ; Ei ] Ei (see [8]). From (22) and (23) riX f i riX fD
=
(23)
= cXi0 Wi
C 1 (M)
(24)
for some cXi0 2 and for all X 2 Ei : By (17), (24) and since riX Wi cXi0 Wi for all X 2 Ei ; LX i for some i 2 C 1 M ; constant along riX fD i Wi leaves of dk; and for all X 2 Ei or equivalently
(
=0 +~ )=0 ker
( ~ + ( )
~
)
LX ~i = 0cXi0 LZ ~i =0
=
(25)
A necessary condition for (24) is
LZ cXi0 =
(16d)
(17)
=0 = i:
=
(16b)
By global weak (f; g )-invariance of Ei ; riX gi i = cX;i gi i for X 2 Ei and for some cX;i 2 C 1 (M); and by (16d) and (17) [ ii01 cX;i + LX ( ii01)]gi i = riX Wi = 0 for X 2 Ei : Thus, since ii is constant along leaves of ker dk LX ( ii01 ) = 0 ii01 cX;i LZ ( ii01 ) =0 (18) for X 2 Ei and Zj 2 ker dk: A necessary condition for (18) is that LZ cX;i = bjs cX ;i (19) s
]] [[
]
0 = [Zj ; riX gi i ]i 0 riX [Zj ; gi i ]i 0 ri[Z ;X]gi i
(16a)
As a consequence of (17a), (17b) and A4), there exists ii 2 everywhere on M and constant along leaves of i
]] [ [
0=[ [
[
it follows that
M: This, together with A4), implies (14) and (15).
C 1 (M) nonzero ker dk such that
ker + + ] = ( )
( ) [
for
1
)
(ker )
ker (ker ) = +
(15)
=
(
]=6
(14)
(
[
where Zj ; X s bjs Xs and X1 ; 1 1 1 ; Xk is a frame of Ei (see [2, Th. 6.2.3]). Let Z1 ; 1 1 1 ; Zd ; 1 1 1 ; Zl be any local frame of dk; defined in Up ; with Zj i parallel for j d and Zd+1 ; 1 1 1 ; Zl a frame of dk \ Ei (this frame always exists, since dk \ Ei has constant dimension on M and dk Ei is involutive). Moreover, let us decompose T M as Ei Q0i ; with Q0i Di Ni0 ; and let i Zj ; gi i D (20) ji gi for j d and some ji 2 C 1 M : Since Zj i is parallel and Ni0 is invariant under parallel transport along leaves, i.e., Ni0 ; Ei Ni0 Ei ; from the Jacobi identity Zj ; X; gi i 0 X; Zj ; gi i 0 Zj ; X ; gi i
for all X 2 Ei and j d: As a consequence of (17)–(21), LX ji for all X 2 Ei : This, together with (20), implies (12) with s Moreover, from (17a) and (17b) gj i ji Wi
Remark: Conditions (12) and (13) are trivially satisfied for linear systems and (14) and (15) are nothing but (8) and (9) (i.e., weak f; g; k invariance). Thus, system (12), (13) is the nonlinear obstruction to f; g; k invariance of each distribution R3i : Moreover, as far as a local solution of noninteracting control via static measurement feedback is sought, the condition G i i Di (after possibly renumbering the inputs) is guaranteed by Assumption A2) while the existence of a complementary subbundle Hi to Di i in T M=Fi and A5) are automatically satisfied. It is easy to show that (12)–(15) recover the local condition given in [13] and [9]. Proof (Only If): Fix i 2 f ; 1 1 1 ; mg: Since GNSM is solvable, following the proof of [12, Proposition 3.1], one can prove that there exist everywhere nonzero vector fields W1 ; 1 1 1 ; Wm and a static feedback law (2), both defined on M and with k M ! GL ; m ; such that for each i ; 1 1 1 ; m
(
777
where
s
bjs cX i0
(26)
[Zj ; X ] = 6s bjs Xs (see [2, Th. 6.2.3]). Moreover, [Zj ; Wi ]D = ji Wi :
(27)
Using (20), with gi i replaced with fD ; and (26) and (27), one obtains (13). 0i ; with (If): First, we prove (12). Let decompose T as i 0i 0 i i : From A4) one obtains gi i ii Wi (28)
(M) E + Q
Q = D +N
=~
for some ii 2 C 1 M ; nonzero everywhere on M; and for some parallel Wi which spans Di : Indeed, G i i is invariant under parallel transport along leaves and Di i is a smooth distribution on M=Fi : Since F 3p Di ! T (p) M=Fi is one-to-one with image equal to Di i F p and by triviality of Di i ; one can define F 3p 01 Yi F p ; where Yi is a smooth, never Wi p
~
( )
( ) : ( ( ( )) ~ ( ) = (( ) ) ( ( ( )))
)
778
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 4, APRIL 1999
vanishing vector field of Di i : This automatically defines a function
ii 2 C (M); which is nonzero on M and satisfies (28) (see [8]). ~ i is parallel and spans Di : Moreover, W Let Z1 ; 1 1 1 ; Zd ; 1 1 1 ; Zl be any frame, defined in a neighborhood Up of p 2 M; with Zj i parallel for j d and Zd+1 ; 1 1 1 ; Zl a frame of (ker dk) \ Ei : Along any integral curve s(t) of Zd+1 ; 1 1 1 ; Zl ; entirely contained in Up
Let ~ii
1
riZ W~ i = d dtii gi i + ii riZ gi i
(29)
LZ ii = 0
(30)
d + 1 and with Zj (s(t)) = s_ (t): Since W~ i is parallel and Zj 2 Ei for j d + 1; it follows from (14) and (29) that for
in
j
Up and for all j d + 1:
i
On the other hand, since Zj is parallel for j constant along leaves of Ei ; one can find a parallel Di and is defined on M; such that for j d
[Zj ; Wi ]D =
d and ji is Wi ; which spans
ji Wi :
(31)
~ i as above. Let M=K be the projection M=Ki ! Oi Indeed, let W and let E = {O M=K K ; where {O : Oi ! M is the inclusion map. We will show that the vector field Wi = ~ i ( ii E )01 satisfies (30). From Jacobi identity W 0 = [Zj ; [X; W~ i ]] 0 [X; [Zj ; W~ i ]] 0 [[Zj ; X ]; W~ i ] ~i X 2 Ei ; since Zj i is parallel for all j r and W
for parallel, it follows that
is also
[Zj ; W~ i ]D = ~ji W~ i
(32)
for j d; with ~ji constant along leaves of Ei : As a consequence of (12), (28), and (32)
LZ ii = (0
ji
+ ~ji ) ii :
(33)
E ) = (0
Indeed, choose local coordinates 3T (x)) such that
+ ~ji )( ii E ): (34) T (x) = (0T (x); 1T (x); 2T (x); ji
Ei = span @@ 2 ; @@ 3
ker dk = span @@ ; @@ : 1 3 Locally around p; E is the map T 01 (0 ; 1 ; 0; 0): Note that, since Zj i is parallel for j d; the projection of Zj onto span(@=@0 ); (@=@1 )g depends only on 0 and 1 : For similar reasons, the functions ji and ~ji depend only on 0 and 1 : Moreover, by (30), ii is independent of 3 : Since Ei \ (ker dk) = spanf(@=@3 )g; (34) follows from (33). ~ i ( ii E )01 : From (32), it follows that (31) holds with Wi = W Indeed, by (32) and (34)
Wi ~ji ( ii E ) = [Zj ; Wi ]( ii = [Zj ; Wi ]( ii
E ) + Wi LZ ( ii E ) E ) + Wi ( ~ji 0 ji )( ii E ) which, since ii E is nonzero on M; implies (31).
(35)
and
LZ ~ii = 0; in
j d+1
(36)
Up [by (30)]. From (31) and (35), for j d ji Wi
= [Zj ; Wi ]D = (LZ ~ii )gi i + ~ii [Zj ; gi i ]D = (LZ ~ii )gi i + ji Wi
which implies
LZ ~ii = 0
jd
(37)
on Up : Repeating the above arguments for a neighborhood Up of each point p 2 M; from (36) and (37) one obtains LX ~ii = 0 on M and for all X 2 ker dk: Since G i i = Di ; then gs i = si Wi for some si 2 C 1 (M) and for all s 6= i: As a consequence of (12), (14), and (31) ji si Wi
= [Zj ; si Wi ]D =
ji si Wi
+ (LZ si )Wi
which proves that
LZ si = 0
(38)
for all s 6= i and for all j: Since M=K is a smooth manifold, as a consequence of (36)–(38), there exist functions ^ii 2 C 1 (M=K) and ^ji 2 C 1 (M=K) such that ~ii = ^ii K and ji = ^ji K ; where K : M ! M=K is the projection map. Moreover, since Ei \ G = 6j 6=i Dj ; Di \ (6j6=i Dj ) = 0 and since G = D1 81 1 18Dm and ~ii is nonzero on M; there exists a smooth function k: M ! GL( ; m) such that
k) = (W1 . . . Wm ) Di = spfWi g riX g( k)i = 0
From (33), since Zj i is parallel for all j d and ji 0 ~ji is constant along leaves, it is easy to realize (in local coordinates) that
ii E 2 C 1 (M) and satisfies
LZ ( ii
= ii ( ii E )01 : Thus Wi = ~ii gi i
g(
(39)
for all X 2 Ei : Finally, using similar arguments to those above (a detailed proof is omitted for lack of space), one proves the existence of k: M ! m such that
riX f + g( k)i = 0 2 Ei : In particular, if we decompose f as in fD = !i Wi ; defining !~ i = !i 0 !i E ~ = 0( k)(~!1 1 1 1 !~ m )T ; one has for all X and write
(40) (22) and
[X; f + g ~]D = [X; fD 0 Wi !~ i ]D = [X; Wi (!i E )]D = 0 for all X 2 Ei : This implies (40) with ~ = k:
From (39) and (40) and by direct inspection in coordinates
x1 ; 1 1 1 ; xm+1 of the closed-loop system x_ = f + g( k) + g( k)v; y = h(x); with v being the new input vector, we
conclude that the closed-loop system is noninteractive (see also [11, Proposition 3.3]). Moreover, since k: M ! GL(m; ); the closed-loop system has also uniform vector relative degree on M: We want to remark that also the stability issue can be discussed and results similar to those contained in [12, Ch. 4] can be derived.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 4, APRIL 1999
We conclude with a simple example. Let us study the noninteracting control problem for the following system:
x_ 1 = 0ex x23 + ex u1 x_ 2 = u2 x_ 3 = x1 + x2 + u1 + u2 yi = xi i = 1; 2; z = x3
f
g; ker dz \ E1
= span (@=@x1 ); (@=@x3 )
Optimal Average Cost Manufacturing Flow Controllers: Convexity and Differentiability Michael H. Veatch and Michael C. Caramanis
with x = (x1 ; x2 ; x3 )T : We have E1 = span(@=@x2 ); (@=@x3 )g;
E2
779
f
g
= span (@=@x2 )
and ker dz \ E2 = spanf(@=@x1 )g: Assumptions A1)–A3) are trivially satisfied. Moreover, D1 = spanfex (@=@x1 ) + (@=@x3 )g and D2 = spanf(@=@x2 ) + (@=@x3 )g: Moreover, Di i is trivial and since M=Fi = we can choose Hi = 0: Thus, Assumption A4) is satisfied. Finally, since M=Ki = 2 choose Oi = fxi 2 g so that also A5) is satisfied. If ker dz = spanf(@=@x1 ); (@=@x2 )g; it is easy to check that (12)–(15) hold true. Thus, Theorem 1 applies. Following the proof of Theorem 1, we can easily derive the decoupling controller. Indeed, we have E (x) = (x1 ; 0; 0)T and E (x) = (0; x2 ; 0)T : Moreover, !1 (x) = 0x23 ; !1 E (x) = 0; !2 (x) = 0 and !2 E (x) = 0; so that u1 = 0!1 (x) + !1 E (x) + v1 = x23 + v1 and u2 = v2 is the desired decoupling control law. REFERENCES [1] S. N. Singh, “A modified algorithm for invertibility in nonlinear systems,” IEEE Trans. Automat. Contr., vol. 26, pp. 595–598, 1981. [2] A. Isidori, Nonlinear Control Systems, 3rd ed. New York: Springer Verlag, 1994. [3] G. Basile and G. Marro, “Controlled and conditioned invariant subspaces in linear systems theory,” J. Optimization Theory and Appl., vol. 3, pp. 306–315, 1969. [4] W. M. Wohnam, Linear Multivariable Control: A Geometric Approach, 2nd ed. New York: Springer Verlag, 1979. [5] A. Isidori, A. J. Krener, C. G. Giorgi, and S. Monaco, “Nonlinear decoupling via feedback: A differential geometric approach,” IEEE Trans. Automat. Contr., vol. 26, pp. 331–345, 1981. [6] H. Nijmeijer and A. Van der Schaft, “Controlled invariance of nonlinear systems,” IEEE Trans. Automat. Contr., vol. 27, pp. 904–914, 1982. [7] C. I. Byrnes and A. Krener, “On the existence of globally (f ; g )invariant distributions,” in Proc. Differential Geometric Control Theory Conf., Cambridge, MA, 1983. [8] W. P. Dayawansa, D. Cheng, W. B. Boothby, and T. J. Tarn, “Global (f; g )-invariance of nonlinear systems,” SIAM J. Contr. Optim., vol. 26, pp. 1119–1131, 1988. [9] H. Nijmeijer and A. J. Van Der Schaft, “Controlled invariance by static output feedback for nonlinear systems,” Syst. Contr. Lett., vol. 2, pp. 122–129, 1982. [10] C. Byrnes and A. Isidori, “Asymptotic stabilization of minimum phase systems,” IEEE Trans. Automat. Contr., vol. 36, pp. 1122–1137, 1991. [11] H. Nijmeijer and A. J. Van Der Schaft, Nonlinear Dynamical Control Systems. New York: Springer Verlag, 1990. [12] S. Battilotti, Noninteracting Control with Stability for Nonlinear Systems, Lecture Notes in Control and Information Sciences, vol. 196. Berlin, Germany: Springer Verlag, 1994. [13] H. J. Huijberts, L. Colpier, and P. Moreau, “Nonlinear input-output decoupling by static output feedback,” in Proc. European Control Conf., Rome, Italy, 1995, pp. 1057–1061. [14] W. B. Boothby, An Introduction to Differentiable Manifolds and Riemaniann Geometry. New York: Academic, 1975. [15] R. Bott, Lectures on Characteristic Classes of Foliations, Lecture Notes in Mathematics, vol. 279. New York: Springer Verlag, 1972. [16] H. Nijmeijer and J. M. Schumacher, “The regular local noninteracting control problem,” SIAM J. Contr. Optim., vol. 8, pp. 1232–1245, 1986.
Abstract— The authors consider the control of a production facility consisting of a single workstation with multiple failure modes and part types using a continuous flow control model. Technical issues concerning the convexity and differentiability of the differential cost function are investigated. It is proven that under an optimal control policy the differential cost is C 1 on attractive control switching boundaries. Index Terms— Average cost minimization, differentiability, manufacturing flow control, value function.
I. INTRODUCTION Manufacturing systems subject to discrete disturbances (failures, setup changes, and the like) have been studied extensively using a fluid model approximation, where surplus or backlog of production is represented by a continuous variable (see [5] for justification). The goal is to control production with a state feedback policy that minimizes the average cost of production surplus and backlog under a constant demand rate and stochastic production capacity. Little is known about the structure of the optimal policy for systems involving more than one part type; see Srivatsan and Dallery [12] Perkins and Srikant [8] and Veatch and Caramanis [14] for some recent exceptions. Instead, algorithms have been developed to compute a reasonable control policy using infinitesimal perturbation analysis or direct computation of average cost [2], [6], [7]. However, some of these algorithms rely on properties of the differential cost functions that have not been rigorously proven. Sethi et al. [10] prove the existence of the potential cost function that is closely related to the differential cost. This paper investigates the continuity of the differential cost function’s derivative on control switching surfaces, which are hypersurfaces in the state space that form the boundaries between state space regions characterized by a constant optimal control. We show that the differential cost is, at least in some cases, continuously differentiable, justifying the assumption made in some previous papers and supporting the quadratic approximation used in [2]. Convexity of the differential cost is also established. II. THE FLOW CONTROL MODEL We consider the flow control model of Liberopoulos and Caramanis [6], which generalizes the multiple unreliable machine model of [2]. The system state is (x(t); (t)), where x = (x1 ; 1 1 1 ; xn ); xi is the continuous production surplus of part type i, and is the discrete machine state. When xi (t) > 0 there is a surplus and when xi (t) < 0 there is a shortage and demand is backlogged. The machine state is governed by a continuous-time irreducible Markov chain on a finite state space E . Let Q = [q ]; ; 2 E be the generator, i.e., q is the transition rate from state to state and q = 0 6= q . Manuscript received April 30, 1997. Recommended by Associate Editor, E. K. P. Chong. This work was supported by National Science Foundation under Grant DDM-9215368. M. H. Veatch is with the Department of Mathematics, Gordon College, Wenham, MA 01984 USA. M. C. Caramanis is with the Department of Manufacturing Engineering, Boston University, Boston, MA 02215 USA (e-mail: [email protected]). Publisher Item Identifier S 0018-9286(99)02093-0.
0018–9286/99$10.00 1999 IEEE
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 4, APRIL 1999
where y^(t; ) is the output of the neural control scheme, Jtrack (c ) is a cost function for the tracking error, defined on a given specific reference input d(t) and tf is a finite time horizon. The matrix inequality constraint can be related to Lemma 1 as well as to Lemma 2, respectively, for imposing global asymptotic stability or I/O stability with a fixed disturbance attenuation level 3 : Such methods have been successfully applied for the discrete-time recurrent neural networks using NLq theory in [19].
VII. CONCLUSION In this paper absolute stability and dissipativity of continuous-time recurrent neural networks with two hidden layers have been studied. These types of models occur when one considers nonlinear models and controllers that are parameterized by multilayer perceptrons with one hidden layer. For the autonomous case a classical Lur’e system representation and Lur’e system with multilayer perceptron nonlinearity is given. Sufficient conditions for absolute stability and dissipativity have been derived from a Lur’e–Postnikov Lyapunov function and a storage function of the same form. The criteria are expressed as matrix inequalities. They can be employed in order to impose closed-loop stability in Narendra’s dynamic backpropagation procedure and for nonlinear H1 control.
[17] E. Polak and Y. Wardi, “Nondifferentiable optimization algorithm for designing control systems having singular value inequalities,” Automatica, vol. 18, no. 3, pp. 267–283, 1982. [18] E. D. Sontag and H. Sussmann, “Complete controllability of continuoustime recurrent neural networks,” Syst. Contr. Lett., vol. 30, pp. 177–183, 1997. [19] J. A. K. Suykens, J. P. L. Vandewalle, and B. L. R. De Moor, Artificial Neural Networks for Modeling and Control of Non-Linear Systems. Boston, MA: Kluwer, 1995. [20] J. A. K. Suykens, B. De Moor, and J. Vandewalle, “Nonlinear system identification using neural state space models, applicable to robust control design,” Int. J. Contr., vol. 62, no. 1, pp. 129–152, 1995. , “NLq theory: A neural control framework with global asymptotic [21] stability criteria,” Neural Networks, vol. 10, no. 4, pp. 615–637, 1997. [22] A. J. van der Schaft, “A state-space approach to nonlinear 1 control,” Syst. Contr. Lett., vol. 16, pp. 1–8, 1991. , “ 2 -gain analysis of nonlinear systems and nonlinear state [23] feedback 1 control,” IEEE Trans. Automat. Contr., vol. 37, pp. 770–784, 1992. [24] H. Verrelst, K. Van Acker, J. Suykens, B. Motmans, B. De Moor, and J. Vandewalle, “Application of NLq neural control theory to a ball and beam system,” European J. Contr., vol. 4, no. 2, pp. 148–157, 1998. [25] M. Vidyasagar, Nonlinear Systems Analysis. Englewood Cliffs, NJ: Prentice-Hall, 1993. [26] J. C. Willems, “Dissipative dynamical systems I: General theory. II: Linear systems with quadratic supply rates,” Archive for Rational Mechanics and Analysis, vol. 45, pp. 321–343, 1972. [27] C.-F. Yung, Y.-P. Lin, and F.-B. Yeh, “A family of nonlinear 1 output feedback controllers,” IEEE Trans. Automat. Contr., vol. 41, pp. 232–236, 1996. [28] J. M. Zurada, Introduction to Artificial Neural Systems. West, 1992.
H
L H
H
REFERENCES [1] F. Albertini and E. D. Sontag, “For neural networks, function determines form,” Neural Networks, vol. 6, pp. 975–990, 1993. [2] , “State observability in recurrent neural networks,” Syst. Contr. Lett., vol. 22, pp. 235–244, 1994. [3] J. A. Ball, J. W. Helton, and M. L. Walker, “ 1 control for nonlinear systems with output feedback,” IEEE Trans. Automat. Contr., vol. 38, pp. 546–559, 1993. [4] A. R. Barron, “Universal approximation bounds for superposition of a sigmoidal function,” IEEE Trans. Inform. Theory, vol. 39, no. 3, pp. 930–945, 1993. [5] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, Studies in Applied Mathematics, vol. 15. Philadelphia, PA: SIAM, 1994. [6] S. Haykin, Neural Networks: A Comprehensive Foundation. Englewood Cliffs, NJ: Macmillan, 1994. [7] D. J. Hill and P. J. Moylan, “Connections between finite-gain and asymptotic stability,” IEEE Trans. Automat. Contr., vol. AC-25, no. 5, pp. 931–936, 1980. [8] , “The stability of nonlinear dissipative systems,” IEEE Trans. Automat. Contr., vol. AC-21, pp. 708–711, 1976. [9] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, U.K.: Cambridge Univ. Press, 1985. [10] K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforward networks are universal approximators,” Neural Networks, vol. 2, pp. 359–366, 1989. [11] A. Isidori and A. Astolfi, “Disturbance attenuation and 1 control via measurement feedback in nonlinear systems,” IEEE Trans. Automat. Contr., vol. 37, pp. 1283–1293, 1992. [12] A. Isidori and W. Kang, “ 1 control via measurement feedback for general nonlinear systems,” IEEE Trans. Automat. Contr., vol. 40, pp. 466–472, 1995. [13] H. K. Khalil, Nonlinear Systems. New York: Macmillan, 1992. [14] M. Leshno, V. Y. Lin, A. Pinkus, and S. Schocken, “Multilayer feedforward networks with a nonpolynomial activation function can approximate any function,” Neural Networks, vol. 6, pp. 861–867, 1993. [15] K. S. Narendra and K. Parthasarathy, “Gradient methods for the optimization of dynamical systems containing neural networks,” IEEE Trans. Neural Networks, vol. 2, no. 2, pp. 252–262, 1991. [16] K. S. Narendra and J. H. Taylor, Frequency Domain Criteria for Absolute Stability. New York: Academic, 1973.
H
Noninteracting Control via Static Measurement Feedback for Nonlinear Systems with Relative Degree S. Battilotti
Abstract— In this paper the authors give a necessary and sufficient geometric condition for achieving noninteraction via static measurement feedback for nonlinear systems with vector relative degree. Their analysis relies on the theory of connections and as a result gives systematic procedures for constructing a decoupling feedback law. Index Terms—Measurement feedback, noninteracting control.
I. THE CLASS OF SYSTEMS AND CONTROL LAWS Let us consider the affine nonlinear systems of the form m
x_ = f (x) +
H
gj (x)uj j =1
yi = hi (x);
H
z = k(x)
i = 1; 1 1 1 ; m;
(1)
where x 2 M; a smooth (Hausdorff) manifold, the ui ’s are input functions from a suitable function space (e.g., measurable -valued functions defined on closed intervals of the form [0; T ]); the yi ’s are the -valued output functions and z 2 s is the vector of Manuscript received September 23, 1997. Recommended by Associate Editor, A. J. van der Schaft. The author is with the Dipartimento di Informatica e Sistemistica, 00184 Roma, Italy. Publisher Item Identifier S 0018-9286(99)02087-5.
0018–9286/99$10.00 1999 IEEE
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 4, APRIL 1999
variables which are available for feedback, and f; g1 1 1 1 ; gm are smooth vector fields on some open subset U M with g1 ; 1 1 1 ; gm linearly independent on U and f : Moreover, h1 ; 1 1 1 ; hm and k are smooth functions defined on U ; with hi and k : Denote by h the vector h1 1 1 1 hm T and by g the matrix g1 1 1 1 gm : We will consider static measurement-feedback laws of the form
(0) = 0 )
(
(0) = 0
(0) = 0 ( )
u = (k(x)) + (k(x))v (2) with smooth functions and defined on some open subset Z s ; (k(x)) 2 G L( ; m); where G L( ; m) is the general linear group of m 2 m invertible matrices over ; and v the new input vector. II. MOTIVATIONS
()=
PROBLEM STATEMENT
AND
Assume that k x x; i.e., the state vector is available for feedback. Moreover, if not otherwise stated, assume that U M: Let G be the distribution spanned by g1 ; 1 1 1 ; gm : A distribution on M is said to be weakly f; g -invariant at x 2 M; if there exists a neighborhood U0 of x such that on U0
=
( )
[f; 1] [g; 1]
+1 + 1: (f; g)-invariant
1
G
(4)
G
(5)
1 is said to be globally weakly (f; g)-invariant at each x :
if it is weakly
2 M
If (4) and (5) are not satisfied, one can try to modify the behavior of (1) through (2) in such a way that these properties are achieved for the closed-loop system. A distribution on M is said to be f; g -invariant at x 2 M; if there exists a static state feedback law (2), defined in a neighborhood U0 of x; such that
1
( )
[f + g ; 1] 1 [g ; 1] 1:
(6)
(7)
If the feedback law is defined on ; then 1 is said to be globally (f; g)-invariant. Weak (f; g )-invariance was first introduced for linear systems in M
[3] and, independently, in [4], for nonlinear systems affine in the input in [5] and for general nonlinear systems in [6]. If (1) is a linear system and is a subspace of n ; (5) and (7) are trivially satisfied, local and global definitions coincide, and weak f; g -invariance is equivalent to f; g invariance. For nonlinear system (1), weak f; g -invariance is implied by f; g -invariance but the converse is not true in general. A powerful lemma for studying f; g -invariance is given by the Quaker lemma [1]–[6]. The main difficulty in extending the Quaker lemma to a global setting hides behind using arguments heavily based on partial differential equations (PDE’s). The key observation, which gives a deep insight into global obstructions, has been first noted in [7], followed by [8], where it is shown that weak f; g -invariance can be interpreted geometrically saying that a subbundle of the normal bundle of is “invariant under parallel transport” along leaves of : Now, assume that only k x is available for feedback. A distribution on M is said to be weakly f; g; k -invariant at x 2 M; if it is weakly f; g -invariant at x 2 M and there exists a neighborhood U0 of x such that
1
( )
( )
( )
( )
( )
( )
1
1
1
()
(
( )
[f; 1 [g; 1
)
(8) ker dk] 1 ker dk] 1: (9) 1 is said to be globally weakly (f; g; k)-invariant if it is weakly (f; g; k)-invariant at each x : A distribution 1 on is said to be (f; g; k)–invariant at x ; \
\
2 M
M
775
1
is said to be globally If the feedback law is defined on M; then f; g; k -invariant. Weak f; g; k invariance was first introduced in [3] and, independently, in [4] for linear systems and in [5] for nonlinear systems, affine in the input. If (1) is a linear system and is spanned by a set of constant vectors of n ; local and global definitions coincide and weak f; g; k -invariance is equivalent to f; g; k invariance [3], [4]. For nonlinear systems (1), weak f; g; k -invariance is implied by f; g; k -invariance but the converse is not true in general. Necessary and sufficient (existence) conditions for local f; g; h invariance are given in [9]. As is well known, weak f; g -invariance is a natural tool for solving many control problems, such as disturbance decoupling and noninteracting control, as long as the state x is available for feedback and local solutions are sought (see [2] and [10] for an exhaustive discussion). For global f; g invariance, additional assumptions to weak f; g -invariance must be imposed (see above). If only z is available for feedback, weak f; g; k invariance still does the job for linear systems [3], [4]. However, in a nonlinear setting, one needs additional assumptions to fully characterize (global) f; g; k invariance in terms of weak f; g; k invariance. In general, these extra assumptions are tailored to guarantee that a state-feedback law (2), which renders a given distribution invariant, can be “expressed” as a measurement feedback law (2). In this paper, exploring the route of f; g; k invariance, we will focus our attention on the problem of rendering (1) noninteractive via static measurement feedback (2). We say that the system (1), (2) is noninteractive if each output is influenced (or “controlled”) only by one input. Moreover, we consider the class of nonlinear systems (1) which have uniform vector relative degree on M [10], i.e., the same vector relative degree at each x 2 M (see [12] for motivations). Noninteracting Control via Static Measurement Feedback Laws (NSM): Find, if possible, a static feedback law (2), defined on a neighborhood U of x ; such that the system (1), (2) is noninteractive and has uniform vector relative degree on U : When U M; we will refer to the noninteracting control problem by global noninteracting control via static measurement (GNSM). When the functions f; g; h; and k are analytic, a stronger problem can be formulated (strong input–output decoupling; see [10]). The local noninteracting control problem via static output feedback (i.e., k h) was first solved in [12] under an additional assumption. This assumption was removed in [13], where a necessary and sufficient condition for solving NSM is given. Unfortunately, although the global validity of this condition is equivalent to GNSM being solvable, no global constructive method is pointed out in [13] for checking this condition. In this paper, we give a necessary and sufficient (checkable) condition for solving GNSM, using a complete different approach, advocated in [8] for global f; g invariance and leading naturally to a global solution of the noninteracting control problem. We point out the obstruction lying between linear and nonlinear f; g; k -invariance. Our proof gives a constructive procedure for obtaining the decoupling feedback law and has, in local coordinates, a very simple interpretation. Moreover, our condition recovers the ones given in [13] and in [9] for local f; g; k invariance. One of the contributions of this paper is, in our opinion, to give (global) constructive tools which can also be used to solve measurement feedback problems different from noninteracting control such as either disturbance decoupling or model matching.
(
)
(
)
1
(
(
)
(
)
)
(
)
(
)
( )
( )
( )
(
)
(
)
(
(
)
1 )
= 0
=
=
( )
(
)
(
)
2 M
if there exists a static measurement feedback law (2), defined in neighborhood U0 of x; such that
[f + g( k); 1] 1 [g( k); 1] 1:
(10)
(11)
III. THEORETICAL BACKGROUND AND DEFINITIONS • We assume that concepts, as bundles, sections, etc., are familiar to the reader. We refer to standard textbooks such as [14] and [15]. By T M we will denote the tangent bundle on M
( )
776
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 4, APRIL 1999
1( )
and by the same symbol we will denote both distributions and their associated subbundles. Let C M be the set of smooth functions on M and X M be the set of smooth vector fields on M: • A smooth section X of a vector bundle over M is a smooth mapping X M ! such that X idM ; where idM is the identity on M: A vector field on M is a section of T M : If q is the dimension of and if there exist q everywhere independent sections (frame of ), then is said to be trivial. • A smooth connection r on T M is a mapping X; Y 7! 1 M 2 X 1 M ! X 1 M ; which rX Y; with r X 1 is C M -linear w.r.t. X; -linear w.r.t. Y and such that rX fY LX f Y f rX Y; f 2 C 1 M : A smooth connection r on a vector bundle D ! M is a mapping X; Y 7! rX Y; with r X 1 M 2 D ! D; which is C 1 M -linear w.r.t. X; -linear w.r.t. Y and such that
1( )
:
1 =
1
1
:
( ) ) +
1 1 ( ) ( )
( ) ( ) ( )
r
2
M
Q
Q
M
M
M Q
M
Q
Q
M
M
r
r
2
Q
Q
M
r
!
M
Q
Q
r
IV. NECESSARY AND SUFFICIENT CONDITION FOR NONINTERACTING CONTROL VIA STATIC MEASUREMENT FEEDBACK 3
Let Ri be the maximal controllability distribution for (1) contained in j 6=i dhj ; supposed to exist on all M: This distribution can be obtained through a constructive algorithm (see [2]). We will assume that at each step of this algorithm we obtain a nonsingular distribution on M: Under this assumption, such an algorithm ends in a finite number of steps. More simply, we will say that R3i is regularly computable on M: m 3 Let R3 j 6=i Rj and let R be the strong accessii=1 bility distribution or, equivalently, the smallest distribution which is invariant under f and g and contains G (see [2] for constructive 3 algorithms). Moreover, let Di R3i \ G and Ei j 6=i Rj : 3 A1) R and Ri are regularly computable on M for i dk and dk Ei have constant ; 1 1 1 ; m; Ei ; R3 ; dimension on M: Moreover, dk Ei is involutive. A2) System (1) has uniform vector relative degree on M: A3) R has constant dimension n on M: The distribution dk Ei is not involutive, in general. However, if, in particular, k x h x and under Assumptions A1) and A2), it is easy to show that dk Ei is indeed involutive (see [12, Sec. 6.1]). In the framework of local noninteracting control with internal stability (see the end of this section), A3) is a standard assumption [4], [16] and in the linear case amounts to require that the systems be controllable [4]. If internal stability is not required, Assumption A3) is not needed but for simplifying the formulas involved. Before stating the main result of this section, we will discuss some basic facts.
ker
=
; (6
)
=
1
ker
ker + ( )= ( ) ker +
=6
ker + ker +
=
( )
( )
( )
( = +( + )
)
1
( ) ( )=( : ( ( ) : ( ) ( ) X (fY ) = (LX f )Y + f X Y; f C 1 ( ): = ( ) The subbundle 1 determines the quotient bundle T ( )=1; having x = (Tx )=1x as fiber over x: If a Riemaniann metric is introduced on ; can be identified with a subbundle 0 of T ( ) (normal bundle of 1; denoted by 1? ), ( having as fiber over x the subspace 0x = 1? Tx : As x a consequence, any section Y of T ( ) decomposes uniquely as Y1 + YQ : According to this decomposition, one can define X Y = X YQ = [X; YQ ]Q ; with X 1; Y is a section of and YQ is the unique vector field in 0 which projects onto Y at each x by the natural projection Tx ( ) Tx ( )=1: is shown to define a connection on and to be independent ( ( )) = ( of the choice of 0 : If X YQ = 0; we say that Y (or YQ ; (6 ) once Y and YQ are identified as above) is parallel along leaves ( ) of 1: Such a connection is commonly referred to as the Bott connection. ( ( )) r
•
( )
1
For each i 2 f ; 1 1 1 ; mg; let decompose T M as follows. If a Riemaniann metric is introduced on M; the quotient bundle Qi T M =Ei can be identified with a subbundle Q0i of T M : Thus, any vector field X on M can be uniquely decomposed as XQ XE : Moreover, under Assumptions A1) and A2), Ei \ G \ Di ([16] and [12]). This implies that Q0i can be chosen in such a way to include Di : In particular, Q0i Di Di Ei ? : In what follows, by i 0 r we will denote the Bott connection such that rX Y X; Y 0 Q : Given any distribution or vector field X on M; we will denote i or its projection into Q0i (or Qi ; once Qi is identified with Q0i ) by i X ; respectively. The projection of X onto Di will be also denoted by XD : By definition, for each i 2 f ; 1 1 1 ; mg the distribution Ei is globally weakly f; g -invariant and, under Assumption A1), it is also involutive (see [2] or [12] for a detailed proof). Thus, through each p 2 M there is a maximal connected integral manifold LEp ; the leaf of Ei passing through p: As pointed out in [8], global weak f; g -invariance of Ei can be interpreted from a geometric point of view by saying that the projection of G into Qi is invariant under parallel transport along leaves (of Ei ). This can be stated equivalently as follows: if W ; is a Frobenius chart and is followed by projection into q (q is the codimension of Ei ), then, at each point x of any leaf of Ei intersected with W ; G Ei x projects to the same subspace of T ~ (x) q : Let Fi denote the folation of Ei : Assume that M=Fi is a smooth manifold and let F M ! M=Fi be the projection map. At each p 2 M one can assign a subspace of T (p) M=Fi by i Di F p f F 3p X jX 2 Di p g: Since G D1 8 1 1 1 8 Dm and j2I R3j \ G j 2I Dj ; with I f ; 1 1 1 ; mg ([16]), global weak f; g invariance of Ei implies that the projection of Di onto Qi is invariant under parallel transport along leaves. Thus, i Di F p does not depend on the choice of p on the leaf of Ei and Di i is a smooth distribution on M=Fi (see also [8]). Given any distribution or vector field X on M; we will denote (when this does make sense) its projection onto T M=Fi by i or X i ; respectively. If there exists a complementary subbundle Hi to Di i in T M=Fi ; one can pull back Hi to obtain a complementary subbundle Ni to Di in Qi : With Q0i chosen as above, the subbundle Ni is isomorphic to ? under the projection T M ! Q the subbundle Ni0 Di Ei i and, thus, by construction Ni0 is invariant under parallel transport along leaves. Denoting by Gi the distribution spanned by gi ; it can be easily seen that Assumption A2) implies that for each i 2 f ; 1 1 1 ; mg and in a neighborhood U0 of each point p 2 M there exists ki 2 f ; 1 1 1 ; mg such that Gk i Di : For achieving a global result and according to our previous discussion, we assume the following. A4) After possibly renumbering the inputs and for each i 2 i Di : Moreover, M=Fi ; i ; 1 1 1 ; m; are f ; 1 1 1 ; mg; Gi smooth (not necessarily Hausdorff) manifolds, Di i is trivial and there exists a complementary subbundle Hi to Di i in
=
+
=[
] 1
(
)
1
)
~
)
( + )
:
)
()
=6
= 1
1
^
(
)
(
=( + )
1^ )
( )
1
1
=
=
1
=0
=1
T (M=Fi ): When k(x) = x; Assumption A4) is exactly the one invoked in [8] (Proposition 5.1). If M=Fi is also Hausdorff, a Riemaniann metric
can be introduced and the existence of a complementary subbundle i to Di i in T M=Fi is automatically guaranteed. Let K be the foliation of dk and Ki be the foliation of dk \ Ei : Let M=Ki be a smooth manifold, {K M=Ki ! M be the inclusion map, and K M ! M=Ki be the projection map. Note that, since dk is involutive, X; dk dk for all X2 dk \ Ei : Thus, the distribution Si f K 3p X jX 2 dk p g is a smooth distribution on M=Ki : H
(
)
(ker )
(ker ) (ker )( )
ker
ker
:
:
[ ker ] ker = ( )
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 4, APRIL 1999
Moreover, by L1 p we will denote the leaf, passing through p; of an involutive and nonsingular distribution : Our last assumption is the following and its role and interpretation will be clear in the proof of the main theorem. 1 E A5) M=K; M=Ki ; and {0 K Lp are smooth manifolds and there exists a smooth manifold Oi M=Ki ; transversal to each {0K1 LEp in M=Ki ; intersecting each {0K1 LEp only once and such that Si p Tp Oi : Our main result is the following. Theorem 1: Assume A1)–A5). GNSM is solvable if and only if for each i 2 f ; 1 1 1 ; mg and for each p 2 M there exists a neighborhood Up of p such thati for any frame Z1 ; 1 1 1 ; Zd ; 1 1 1 ; Zl of dk; defined on Up ; with Zj parallel for all j d and Zd+1 ; 1 1 1 ; Zl a frame of dk \ Ei • for j d and for all X 2 Ei i Zj ; gs i D s ;111;m (12) ji gs i i rX Zj ; fD D ji rX fD (13)
1
()
1
ker
(ker )
[ [
] = ] =
=1
with ji constant along leaves of Ei in U0 • for j d s ;111;m riZ gs i i i rZ f :
+1
=0 =0
=1
)
(
)
on
=1
Di = spanfWi g (W1 1 1 1 Wm ) = g( k) [f + g( k); Ei ] Ei [g( k); Ei ] Ei
+
)
:
gi
= ii Wi :
] ]
(21)
(16c)
j 6= i and for some ji 2 C 1 (M); constant ker dk: Thus, (12) holds for all s = 1; 1 1 1 ; m:
Finally, let us prove (13). For the vector field uniquely as
along leaves of
f
decomposes
+ fD + fN : (22) Since Ei is globally weakly (f; g ) invariant and Ni0 is invariant under f = fE
parallel transport along leaves, it must be
[fN ; Ei ] Ei (see [8]). From (22) and (23) riX f i riX fD
=
(23)
= cXi0 Wi
C 1 (M)
(24)
for some cXi0 2 and for all X 2 Ei : By (17), (24) and since riX Wi cXi0 Wi for all X 2 Ei ; LX i for some i 2 C 1 M ; constant along riX fD i Wi leaves of dk; and for all X 2 Ei or equivalently
(
=0 +~ )=0 ker
( ~ + ( )
~
)
LX ~i = 0cXi0 LZ ~i =0
=
(25)
A necessary condition for (24) is
LZ cXi0 =
(16d)
(17)
=0 = i:
=
(16b)
By global weak (f; g )-invariance of Ei ; riX gi i = cX;i gi i for X 2 Ei and for some cX;i 2 C 1 (M); and by (16d) and (17) [ ii01 cX;i + LX ( ii01)]gi i = riX Wi = 0 for X 2 Ei : Thus, since ii is constant along leaves of ker dk LX ( ii01 ) = 0 ii01 cX;i LZ ( ii01 ) =0 (18) for X 2 Ei and Zj 2 ker dk: A necessary condition for (18) is that LZ cX;i = bjs cX ;i (19) s
]] [[
]
0 = [Zj ; riX gi i ]i 0 riX [Zj ; gi i ]i 0 ri[Z ;X]gi i
(16a)
As a consequence of (17a), (17b) and A4), there exists ii 2 everywhere on M and constant along leaves of i
]] [ [
0=[ [
[
it follows that
M: This, together with A4), implies (14) and (15).
C 1 (M) nonzero ker dk such that
ker + + ] = ( )
( ) [
for
1
)
(ker )
ker (ker ) = +
(15)
=
(
]=6
(14)
(
[
where Zj ; X s bjs Xs and X1 ; 1 1 1 ; Xk is a frame of Ei (see [2, Th. 6.2.3]). Let Z1 ; 1 1 1 ; Zd ; 1 1 1 ; Zl be any local frame of dk; defined in Up ; with Zj i parallel for j d and Zd+1 ; 1 1 1 ; Zl a frame of dk \ Ei (this frame always exists, since dk \ Ei has constant dimension on M and dk Ei is involutive). Moreover, let us decompose T M as Ei Q0i ; with Q0i Di Ni0 ; and let i Zj ; gi i D (20) ji gi for j d and some ji 2 C 1 M : Since Zj i is parallel and Ni0 is invariant under parallel transport along leaves, i.e., Ni0 ; Ei Ni0 Ei ; from the Jacobi identity Zj ; X; gi i 0 X; Zj ; gi i 0 Zj ; X ; gi i
for all X 2 Ei and j d: As a consequence of (17)–(21), LX ji for all X 2 Ei : This, together with (20), implies (12) with s Moreover, from (17a) and (17b) gj i ji Wi
Remark: Conditions (12) and (13) are trivially satisfied for linear systems and (14) and (15) are nothing but (8) and (9) (i.e., weak f; g; k invariance). Thus, system (12), (13) is the nonlinear obstruction to f; g; k invariance of each distribution R3i : Moreover, as far as a local solution of noninteracting control via static measurement feedback is sought, the condition G i i Di (after possibly renumbering the inputs) is guaranteed by Assumption A2) while the existence of a complementary subbundle Hi to Di i in T M=Fi and A5) are automatically satisfied. It is easy to show that (12)–(15) recover the local condition given in [13] and [9]. Proof (Only If): Fix i 2 f ; 1 1 1 ; mg: Since GNSM is solvable, following the proof of [12, Proposition 3.1], one can prove that there exist everywhere nonzero vector fields W1 ; 1 1 1 ; Wm and a static feedback law (2), both defined on M and with k M ! GL ; m ; such that for each i ; 1 1 1 ; m
(
777
where
s
bjs cX i0
(26)
[Zj ; X ] = 6s bjs Xs (see [2, Th. 6.2.3]). Moreover, [Zj ; Wi ]D = ji Wi :
(27)
Using (20), with gi i replaced with fD ; and (26) and (27), one obtains (13). 0i ; with (If): First, we prove (12). Let decompose T as i 0i 0 i i : From A4) one obtains gi i ii Wi (28)
(M) E + Q
Q = D +N
=~
for some ii 2 C 1 M ; nonzero everywhere on M; and for some parallel Wi which spans Di : Indeed, G i i is invariant under parallel transport along leaves and Di i is a smooth distribution on M=Fi : Since F 3p Di ! T (p) M=Fi is one-to-one with image equal to Di i F p and by triviality of Di i ; one can define F 3p 01 Yi F p ; where Yi is a smooth, never Wi p
~
( )
( ) : ( ( ( )) ~ ( ) = (( ) ) ( ( ( )))
)
778
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 4, APRIL 1999
vanishing vector field of Di i : This automatically defines a function
ii 2 C (M); which is nonzero on M and satisfies (28) (see [8]). ~ i is parallel and spans Di : Moreover, W Let Z1 ; 1 1 1 ; Zd ; 1 1 1 ; Zl be any frame, defined in a neighborhood Up of p 2 M; with Zj i parallel for j d and Zd+1 ; 1 1 1 ; Zl a frame of (ker dk) \ Ei : Along any integral curve s(t) of Zd+1 ; 1 1 1 ; Zl ; entirely contained in Up
Let ~ii
1
riZ W~ i = d dtii gi i + ii riZ gi i
(29)
LZ ii = 0
(30)
d + 1 and with Zj (s(t)) = s_ (t): Since W~ i is parallel and Zj 2 Ei for j d + 1; it follows from (14) and (29) that for
in
j
Up and for all j d + 1:
i
On the other hand, since Zj is parallel for j constant along leaves of Ei ; one can find a parallel Di and is defined on M; such that for j d
[Zj ; Wi ]D =
d and ji is Wi ; which spans
ji Wi :
(31)
~ i as above. Let M=K be the projection M=Ki ! Oi Indeed, let W and let E = {O M=K K ; where {O : Oi ! M is the inclusion map. We will show that the vector field Wi = ~ i ( ii E )01 satisfies (30). From Jacobi identity W 0 = [Zj ; [X; W~ i ]] 0 [X; [Zj ; W~ i ]] 0 [[Zj ; X ]; W~ i ] ~i X 2 Ei ; since Zj i is parallel for all j r and W
for parallel, it follows that
is also
[Zj ; W~ i ]D = ~ji W~ i
(32)
for j d; with ~ji constant along leaves of Ei : As a consequence of (12), (28), and (32)
LZ ii = (0
ji
+ ~ji ) ii :
(33)
E ) = (0
Indeed, choose local coordinates 3T (x)) such that
+ ~ji )( ii E ): (34) T (x) = (0T (x); 1T (x); 2T (x); ji
Ei = span @@ 2 ; @@ 3
ker dk = span @@ ; @@ : 1 3 Locally around p; E is the map T 01 (0 ; 1 ; 0; 0): Note that, since Zj i is parallel for j d; the projection of Zj onto span(@=@0 ); (@=@1 )g depends only on 0 and 1 : For similar reasons, the functions ji and ~ji depend only on 0 and 1 : Moreover, by (30), ii is independent of 3 : Since Ei \ (ker dk) = spanf(@=@3 )g; (34) follows from (33). ~ i ( ii E )01 : From (32), it follows that (31) holds with Wi = W Indeed, by (32) and (34)
Wi ~ji ( ii E ) = [Zj ; Wi ]( ii = [Zj ; Wi ]( ii
E ) + Wi LZ ( ii E ) E ) + Wi ( ~ji 0 ji )( ii E ) which, since ii E is nonzero on M; implies (31).
(35)
and
LZ ~ii = 0; in
j d+1
(36)
Up [by (30)]. From (31) and (35), for j d ji Wi
= [Zj ; Wi ]D = (LZ ~ii )gi i + ~ii [Zj ; gi i ]D = (LZ ~ii )gi i + ji Wi
which implies
LZ ~ii = 0
jd
(37)
on Up : Repeating the above arguments for a neighborhood Up of each point p 2 M; from (36) and (37) one obtains LX ~ii = 0 on M and for all X 2 ker dk: Since G i i = Di ; then gs i = si Wi for some si 2 C 1 (M) and for all s 6= i: As a consequence of (12), (14), and (31) ji si Wi
= [Zj ; si Wi ]D =
ji si Wi
+ (LZ si )Wi
which proves that
LZ si = 0
(38)
for all s 6= i and for all j: Since M=K is a smooth manifold, as a consequence of (36)–(38), there exist functions ^ii 2 C 1 (M=K) and ^ji 2 C 1 (M=K) such that ~ii = ^ii K and ji = ^ji K ; where K : M ! M=K is the projection map. Moreover, since Ei \ G = 6j 6=i Dj ; Di \ (6j6=i Dj ) = 0 and since G = D1 81 1 18Dm and ~ii is nonzero on M; there exists a smooth function k: M ! GL( ; m) such that
k) = (W1 . . . Wm ) Di = spfWi g riX g( k)i = 0
From (33), since Zj i is parallel for all j d and ji 0 ~ji is constant along leaves, it is easy to realize (in local coordinates) that
ii E 2 C 1 (M) and satisfies
LZ ( ii
= ii ( ii E )01 : Thus Wi = ~ii gi i
g(
(39)
for all X 2 Ei : Finally, using similar arguments to those above (a detailed proof is omitted for lack of space), one proves the existence of k: M ! m such that
riX f + g( k)i = 0 2 Ei : In particular, if we decompose f as in fD = !i Wi ; defining !~ i = !i 0 !i E ~ = 0( k)(~!1 1 1 1 !~ m )T ; one has for all X and write
(40) (22) and
[X; f + g ~]D = [X; fD 0 Wi !~ i ]D = [X; Wi (!i E )]D = 0 for all X 2 Ei : This implies (40) with ~ = k:
From (39) and (40) and by direct inspection in coordinates
x1 ; 1 1 1 ; xm+1 of the closed-loop system x_ = f + g( k) + g( k)v; y = h(x); with v being the new input vector, we
conclude that the closed-loop system is noninteractive (see also [11, Proposition 3.3]). Moreover, since k: M ! GL(m; ); the closed-loop system has also uniform vector relative degree on M: We want to remark that also the stability issue can be discussed and results similar to those contained in [12, Ch. 4] can be derived.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 4, APRIL 1999
We conclude with a simple example. Let us study the noninteracting control problem for the following system:
x_ 1 = 0ex x23 + ex u1 x_ 2 = u2 x_ 3 = x1 + x2 + u1 + u2 yi = xi i = 1; 2; z = x3
f
g; ker dz \ E1
= span (@=@x1 ); (@=@x3 )
Optimal Average Cost Manufacturing Flow Controllers: Convexity and Differentiability Michael H. Veatch and Michael C. Caramanis
with x = (x1 ; x2 ; x3 )T : We have E1 = span(@=@x2 ); (@=@x3 )g;
E2
779
f
g
= span (@=@x2 )
and ker dz \ E2 = spanf(@=@x1 )g: Assumptions A1)–A3) are trivially satisfied. Moreover, D1 = spanfex (@=@x1 ) + (@=@x3 )g and D2 = spanf(@=@x2 ) + (@=@x3 )g: Moreover, Di i is trivial and since M=Fi = we can choose Hi = 0: Thus, Assumption A4) is satisfied. Finally, since M=Ki = 2 choose Oi = fxi 2 g so that also A5) is satisfied. If ker dz = spanf(@=@x1 ); (@=@x2 )g; it is easy to check that (12)–(15) hold true. Thus, Theorem 1 applies. Following the proof of Theorem 1, we can easily derive the decoupling controller. Indeed, we have E (x) = (x1 ; 0; 0)T and E (x) = (0; x2 ; 0)T : Moreover, !1 (x) = 0x23 ; !1 E (x) = 0; !2 (x) = 0 and !2 E (x) = 0; so that u1 = 0!1 (x) + !1 E (x) + v1 = x23 + v1 and u2 = v2 is the desired decoupling control law. REFERENCES [1] S. N. Singh, “A modified algorithm for invertibility in nonlinear systems,” IEEE Trans. Automat. Contr., vol. 26, pp. 595–598, 1981. [2] A. Isidori, Nonlinear Control Systems, 3rd ed. New York: Springer Verlag, 1994. [3] G. Basile and G. Marro, “Controlled and conditioned invariant subspaces in linear systems theory,” J. Optimization Theory and Appl., vol. 3, pp. 306–315, 1969. [4] W. M. Wohnam, Linear Multivariable Control: A Geometric Approach, 2nd ed. New York: Springer Verlag, 1979. [5] A. Isidori, A. J. Krener, C. G. Giorgi, and S. Monaco, “Nonlinear decoupling via feedback: A differential geometric approach,” IEEE Trans. Automat. Contr., vol. 26, pp. 331–345, 1981. [6] H. Nijmeijer and A. Van der Schaft, “Controlled invariance of nonlinear systems,” IEEE Trans. Automat. Contr., vol. 27, pp. 904–914, 1982. [7] C. I. Byrnes and A. Krener, “On the existence of globally (f ; g )invariant distributions,” in Proc. Differential Geometric Control Theory Conf., Cambridge, MA, 1983. [8] W. P. Dayawansa, D. Cheng, W. B. Boothby, and T. J. Tarn, “Global (f; g )-invariance of nonlinear systems,” SIAM J. Contr. Optim., vol. 26, pp. 1119–1131, 1988. [9] H. Nijmeijer and A. J. Van Der Schaft, “Controlled invariance by static output feedback for nonlinear systems,” Syst. Contr. Lett., vol. 2, pp. 122–129, 1982. [10] C. Byrnes and A. Isidori, “Asymptotic stabilization of minimum phase systems,” IEEE Trans. Automat. Contr., vol. 36, pp. 1122–1137, 1991. [11] H. Nijmeijer and A. J. Van Der Schaft, Nonlinear Dynamical Control Systems. New York: Springer Verlag, 1990. [12] S. Battilotti, Noninteracting Control with Stability for Nonlinear Systems, Lecture Notes in Control and Information Sciences, vol. 196. Berlin, Germany: Springer Verlag, 1994. [13] H. J. Huijberts, L. Colpier, and P. Moreau, “Nonlinear input-output decoupling by static output feedback,” in Proc. European Control Conf., Rome, Italy, 1995, pp. 1057–1061. [14] W. B. Boothby, An Introduction to Differentiable Manifolds and Riemaniann Geometry. New York: Academic, 1975. [15] R. Bott, Lectures on Characteristic Classes of Foliations, Lecture Notes in Mathematics, vol. 279. New York: Springer Verlag, 1972. [16] H. Nijmeijer and J. M. Schumacher, “The regular local noninteracting control problem,” SIAM J. Contr. Optim., vol. 8, pp. 1232–1245, 1986.
Abstract— The authors consider the control of a production facility consisting of a single workstation with multiple failure modes and part types using a continuous flow control model. Technical issues concerning the convexity and differentiability of the differential cost function are investigated. It is proven that under an optimal control policy the differential cost is C 1 on attractive control switching boundaries. Index Terms— Average cost minimization, differentiability, manufacturing flow control, value function.
I. INTRODUCTION Manufacturing systems subject to discrete disturbances (failures, setup changes, and the like) have been studied extensively using a fluid model approximation, where surplus or backlog of production is represented by a continuous variable (see [5] for justification). The goal is to control production with a state feedback policy that minimizes the average cost of production surplus and backlog under a constant demand rate and stochastic production capacity. Little is known about the structure of the optimal policy for systems involving more than one part type; see Srivatsan and Dallery [12] Perkins and Srikant [8] and Veatch and Caramanis [14] for some recent exceptions. Instead, algorithms have been developed to compute a reasonable control policy using infinitesimal perturbation analysis or direct computation of average cost [2], [6], [7]. However, some of these algorithms rely on properties of the differential cost functions that have not been rigorously proven. Sethi et al. [10] prove the existence of the potential cost function that is closely related to the differential cost. This paper investigates the continuity of the differential cost function’s derivative on control switching surfaces, which are hypersurfaces in the state space that form the boundaries between state space regions characterized by a constant optimal control. We show that the differential cost is, at least in some cases, continuously differentiable, justifying the assumption made in some previous papers and supporting the quadratic approximation used in [2]. Convexity of the differential cost is also established. II. THE FLOW CONTROL MODEL We consider the flow control model of Liberopoulos and Caramanis [6], which generalizes the multiple unreliable machine model of [2]. The system state is (x(t); (t)), where x = (x1 ; 1 1 1 ; xn ); xi is the continuous production surplus of part type i, and is the discrete machine state. When xi (t) > 0 there is a surplus and when xi (t) < 0 there is a shortage and demand is backlogged. The machine state is governed by a continuous-time irreducible Markov chain on a finite state space E . Let Q = [q ]; ; 2 E be the generator, i.e., q is the transition rate from state to state and q = 0 6= q . Manuscript received April 30, 1997. Recommended by Associate Editor, E. K. P. Chong. This work was supported by National Science Foundation under Grant DDM-9215368. M. H. Veatch is with the Department of Mathematics, Gordon College, Wenham, MA 01984 USA. M. C. Caramanis is with the Department of Manufacturing Engineering, Boston University, Boston, MA 02215 USA (e-mail: [email protected]). Publisher Item Identifier S 0018-9286(99)02093-0.
0018–9286/99$10.00 1999 IEEE
