A note on input-to-state stabilization for nonlinear sampled-data systems
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 7, JULY 2002
[10] J. C. Geromel, P. L. D. Peres, and J. Bernoussou, “On a convex parameter space method for linear control design of uncertain systems,” SIAM J. Control Optimiz., vol. 29, no. 2, pp. 381–402, 1991. [11] F. Blanchini, “The gain scheduling and the robust state feedback stabilization problems,” IEEE Trans. Automat. Contr., vol. 45, pp. 2061–2070, Nov. 2000. [12] M. M. Seron, G. A. De Doná, and G. C. Goodwin, “Global analytical model predictive control under input constraints,” in Proc. 39th IEEE Conf. Decision Control, Sydney, Australia, Dec. 2000. [13] T. Parisini and R. Zoppoli, “A receding-horizon regulator for nonlinear systems and a neural approximation,” Automatica, vol. 31, pp. 1443–1451, 1995. [14] B. D. O. Anderson and J. B. Moore, “Detectability and stabilizability of time-varying discrete-time linear systems,” SIAM J. Optimiz., vol. 5, pp. 20–32, 1981. [15] K. Takaba, “Robust servomechanism with preview action for polytopic uncertain systems,” Int. J. Robust Nonlinear Control, vol. 10, pp. 101–111, 2000. [16] H. O. Wang, K. Tanaka, and F. Griffin, “An approach to fuzzy control of nonlinear systems: Stability and design issues,” IEEE Trans. Fuzzy Syst., vol. 4, pp. 14–23, Feb. 1996.
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for the design of input-to-state stabilizing (ISS) controllers based on approximate discrete-time plant models (for more details on ISS see [6], [15], [13], and [14]). In particular, we provide sufficient conditions on the continuous-time plant model, the controller and the approximate discrete-time model, which guarantee that if the controller input-to-state stabilizes the approximate discrete-time plant model it would also input-to-state stabilize the exact discrete-time plant model. Our results apply to dynamic controllers and our approach benefits from the results on numerical integration schemes in [16], [3], and [4]. Related results were investigated in [8], on changes of supply rates for ISS discrete-time systems. II. PRELIMINARIES
A Note on Input-to-State Stabilization for Nonlinear Sampled-Data Systems
Sets of real and natural numbers (including 0) are denoted, respectively, as and . For a given function w: 0 ! n , we use the following notation: wT [k] := w(t); t 2 [kT ; (k + 1)T ] where k 2 and T > 0, and wT [k] is zero elsewhere (in other words wT [k] is a piece of function w(t) in the k th sampling interval [kT ; (k + 1)T ]); and w(k) is the value of the function w(1) at t = kT , k 2 . We denote the norms kwT [k]k1 = sup 2[kT;(k+1)T ] jw( )j and kwk1 := sup 0 jw ( )j and in the case when w (1) is a measurable function (in the Lebesgue sense) we use the essential supremum in the definitions. If kwk1 < 1, then we write w 2 L1 . Consider a continuous-time nonlinear plant with disturbances
Dragan Neˇsic´ and Dina S. Laila
x_ (t) = f (x(t); u(t); w (t))
Abstract—We provide a framework for the design of stabilizing controllers via approximate discrete-time models for sampled-data nonlinear systems with disturbances. In particular, we present sufficient conditions under which a discrete-time controller that input-to-state stabilizes an approximate discrete-time model of a nonlinear plant with disturbances would also input-to-state stabilize (in an appropriate sense) the exact discrete-time plant model. Index Terms—Input-to-state stability, nonlinear, sampled-data.
I. INTRODUCTION A stumbling block in controller design for nonlinear sampled-data control systems is the absence of a good model for the design. Indeed, even if the continuous-time plant model is known, we can not in general compute the exact discrete-time model of the plant since this requires an explicit analytic solution of a nonlinear differential equation. This has motivated research on controller design via approximate discrete-time models for sampled-data nonlinear systems [1], [2], [7]. A drawback of these early results was their limited applicability: they investigate a particular class of plant models, a particular approximate discrete-time plant model (usually Euler) and a particular controller. A more general framework for stabilization of disturbance-free sampled-data nonlinear systems via their approximate discrete-time models that is applicable to general plant models, controllers and approximate discrete-time models was first presented in [10], [11]. In this note, we generalize results in [11] by i) considering sampled-data nonlinear systems with disturbances, and ii) providing a framework Manuscript received April 26, 2001; revised November 19, 2001 and January 8, 2002. Recommended by Associate Editor Z. Lin. This work was supported by the Australian Research Council under the Large Grants Scheme. The authors are with the Department of Electrical and Electronic Engineering, The University of Melbourne, Parkville 3010, Victoria, Australia (e-mail: [email protected]; [email protected]). Publisher Item Identifier 10.1109/TAC.2002.800663.
(1)
where x 2 n , u 2 m and w 2 p are respectively the state, control input and exogenous disturbance. It is assumed that f is locally Lipschitz and f (0; 0; 0) = 0. We will consider two cases: w(1) are measurable functions (in the Lebesgue sense), and w(1) are continuously differentiable functions. We will always make precise which case we consider. The control is taken to be a piecewise constant signal u(t) = u(kT ) =: u(k); 8 t 2 [kT ; (k +1)T ), k 2 , where T > 0 is the sampling period. Also, we assume that some combination (output) or all of the states (x(k) := x(kT )) are available at sampling instant kT , k 2 . The exact discrete-time model for the plant (1), which describes the plant behavior at sampling instants kT , is obtained by integrating the initial value problem x_ (t) = f (x(t); u(k); w(t))
(2)
with given wT [k], u(k) and x0 = x(k), over the sampling interval [kT ; (k + 1)T ]. If we denote by x(t) the solution of the initial value problem (2) at time t with given x0 = x(k), u(k) and wT [k], then the exact discrete-time model of (1) can be written as x(k + 1) =x(k) +
(k+1)T
f (x( ); u(k); w ( ))d kT e =:FT (x(k ); u(k ); wT [k ]):
(3)
We refer to (3) as a functional difference equation since it depends on
wT [k]. We emphasize that FTe is not known in most cases. Indeed, in order to compute FTe we have to solve the initial value problem (2) analytically and this is usually impossible since f in (1) is nonlinear.
Hence, we will use an approximate discrete-time model of the plant to design a controller. Different approximate discrete-time models can be obtained using different methods. For example, we may first assume that the disturbances w(1) are constant during sampling intervals, w(t) = w(k); 8 t 2 [kT ; (k + 1)T ] and then use a classical Runge–Kutta numerical integration scheme (such as Euler) for the
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 7, JULY 2002
initial value problem (2). In this case, the approximate discrete-time model can be written as xk FTa x k ; u k ; w k : (4)
( + 1) = ( ( ) ( ) ( ))
We refer to the approximate model (4) as an ordinary difference equation since FTa does not depend on wT k but on w k . For instance, the Euler approximate model is x k x k Tf x k ;u k ;w k . Recently, numerical integration schemes for systems with measurable disturbances were considered in [3] and [4]. Using these numerical integration techniques, we can obtain an approximate discrete-time model
by: for all T with jwj
2 (0; T 3), all x~ 2
1
2
n with
we have
jx~j 1
1
and all w
1 [VT (FT (~x; w)) 0 VT (~x)] 0 (jx~j) + ~(jwj) +
T
3
1
2
p
(10)
[] () and the function VT is called an ISS-Lyapunov function for the family ( +1) = ( )+ ( ( ) ( ) ( )) FT .
x(k + 1) = FTa (x(k); u(k); wT [k])
(5)
which is in general a functional difference equation. For instance, the simplest such approximate discrete-time model, which is analogous to Euler model, has the following form (k+1)T f x k ; u k ; w s ds (see [3]). Since xk xk kT we will consider semiglobal ISS (see Definition II.2), we will think of FTe and FTa as being defined globally for all small T , even though the initial value problem (2) may exhibit finite escape times [11, p. 261]. The sampling period T is assumed to be a design parameter which can be arbitrarily assigned. Since we are dealing with a family of approximate discrete-time models FTa , parameterized by T , in order to achieve a certain objective we need in general to obtain a family of controllers, parameterized by T . We consider a family of dynamic feedback controllers
( ( ) ( ) ( ))
( + 1) = ( ) +
where
x~ 2
z (k + 1) =GT (x(k); z (k)) u(k) =uT (x(k); z (k))
2
(6)
~ := (xT zT )T ,
n . To shorten notation, we introduce x n , where n nx nz and x~ FTi x k ; uT x k ; z k ; i T x k ; GT x k ; z k
z
:= + ( ( ) ( ( ) ( )) 1) F (~( ) 1) := ( ( ) ( ))
:
(7)
The superscript i may be either e or a, where e stands for exact model, a for approximate model. We omit the superscript if we refer to a general model. The second argument of FTi x; 1 (third argument of FTi ) is either a vector w k or a piece of function wT k . Similar to [10], we define the following. Definition II.1: [Lyapunov semiglobally practically input-to-state stable (Lyapunov-SP-ISS)] The family of systems x k FT x k ; wT k is Lyapunov-SP-ISS if there exist functions1 1 ; 2 ; 3 2 K and 2 K, and for any strictly positive-real numbers there exist strictly positive-real numbers 1; 2 ; 1 ; 2 T 3 and L, such that for all T 2 ; T 3 there exists a function VT n ! 0 such that for all x 2 n with jxj 1 and all w 2 L1 with kwk1 2 the following holds:
(~ )
()
(~( )
[ ]) 1 (1 1
:
[]
~( + 1) =
~ )
(0 ) ~
1
~
1
(jx~j) VT (~x) (jx~j) (8) [VT (FT (~x; wT )) 0 VT (~x)] 0 (jx~j)+~ (kwT k )+ (9) 1 T and, moreover, for all x ; x ; z with (xT z T )T ; (xT z T )T 2 [ ; 1 ] and all T 2 (0; T 3 ), we have jVT (x ; z) 0 VT (x ; z)j 1
2
3
1
2
2
1
L jx1 0 x2 j. The function VT
1
1
2
1
2
is called an ISS-Lyapunov function for
the family FT . Remark II.1: In the case when the family of parameterized closed-loop discrete-time nonlinear systems is an ordinary difference equation x k FT x k ; w k , the condition (9) is replaced
~( + 1) =
(~( ) ( ))
1A function : is of class- if it is continuous, zero at zero and strictly increasing. It is of classif it is of class- and is unbounded. A continuous function : is of classif ( ) is of class- for each 0 and ( ) is decreasing to zero for each 0.
The following definition is a semiglobal-practical version of the ISS property used in [13] and [15], and we use it in the case when we consider measurable disturbances w . Definition II.2: [Semiglobal practical ISS ((SP-ISS)] The family of systems x k FT x k ; wT k is SP-ISS if there exist 2 KL and 2 K1 such that for any strictly positive-real num3 3 bers x~ ; w ; there exists T > such that for all T 2 ; T , jx j x~ and w 1 with kwk1 w , the solutions of the system satisfy jx k j jx j ; kT
kwk1 , 8 k 2 . The following semiglobal practical “ISS-like property” was used in [9], and we use it when the disturbances are continuously differentiable. Definition II.3: [Semiglobal practical derivative ISS (SP-DISS)] The family of systems x k FT x k ; wT k is SP-DISS if there exist 2 KL and 2 K1 such that for any strictly positive-real 3 such that for all numbers x~ ; w ; w_ ; there exists T > 3 T 2 ; T , jx j x~ and all continuously differentiable w 1 such that kwk1 w , kw k1 w_ , the solutions of the family FT satisfy jx k j jx j ; kT
kwk1 , 8 k 2 . Note that a similar property to SP-ISS, called input-to-state practical stability (ISpS) was defined in [5] and [14] when considering nonparameterized systems. Definition II.4: uT is said to be locally uniformly bounded if for any x~ > there exist strictly positive numbers T 3 and u such that for all T 2 ; T 3 and all jxj x~ we have juT x j u . In order to prove our main results, we need to guarantee that the mismatch between FTe and FTa is small in some sense. We define two consistency properties, which will be used to limit the mismatch. Similar definitions can be found in numerical analysis literature [16, Def. 3.4.2], and recently in the context of sampled-data systems (see [11, Def. 1] and [10, Def. 2]). In the sequel, we use the notation x x k , u u k , w w k , and wT wT k . Definition II.5: (One-step weak consistency) The family FTa is said to be one-step weakly consistent with FTe if given any strictly positive real numbers x ; u ; w ; w_ , there exist a function 2 K1 and T 3 > such that, for all T 2 ; T 3 , all x 2 n ; u 2 m with jxj x , juj u and functions w 1 that are continuously differentiable and satisfy kwT k1 w and kwf k1 w_ , we have jFTe 0 FTa j T T . Definition II.6: (One-step strong consistency) The family FTa is said to be one-step strongly consistent with FTe if given any strictly positive real numbers x ; u ; w , there exist a function 2 K1 and T 3 > such that, for all T 2 ; T 3 , all x 2 n ; u 2 m ; w 2 L1 with jxj x , juj u , kwT k1 w , we have jFTe 0 FTa j T T . Sufficient checkable conditions for one-step weak and strong consistency are given next (similar results for systems without disturbances are [10, Lemma 1] and [11, Lemma 1]). Lemma II.1: FTa is one-step weakly consistent with FTe if the following conditions hold: 1) FTa is one-step weakly consistent with FTEuler x; u; w x T f x; u; w , and 2) given any strictly positive-real numbers x ; u ; w ; w_ , there exist 1 2 K1 , 2 2 K1 , T 3 > , such that, for all T 2 ; T 3 , all x1 ; x2 2 n with fjx1j; jx2 jg x , all u 2 m with juj u and all w1 ; w2 2 p with fjw1 j ; jw2 jg w , the following holds: jf x1; u; w1 0 f ax2 ; u; w2 j 1 jx1 0 x2 j 2 jw1 0 we 2 j . Lemma II.2: FT is one-step strongly consistent with FT if the following conditions hold: 1) FTa is one-step strongly consistent
~( + 1) = (1 1 ) ~(0) 1 () ~( ) ( ~(0)
(~( )
)+ (
[ ])
0 1
(0 )
)+
~( + 1) = (~( ) [ ]) 0 (1 1 1 ) (0 ) ~(0) 1 1 1 _ ~( ) ( ~(0) ) + ( )+
1
0 (0 )
= () = () 0 1
0
1
1 (~) 1
~ 1
=
()
= ()
[]
(1 1 1 1 ) (0 ) 1 () 1 _ ( )
1
(1 1 1 ) (0 ) 1 1
) := + ( ) (1 1 1 1 ) 0 (0 ) max 1 1 max 1 ( ) ( ) ( )+ (
( )
(
)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 7, JULY 2002
~
( +1)
with FTEuler x; u; wT x f x; u; w s ds, and 2) kT given any strictly positive-real numbers x; u; w , there exist 1 2 K , T 3 > , such that, for all T 2 ; T 3 and for all fjpx1 j ; jx2 jg x , all u 2 m with x1 ; x2 2 n with juj u and all w 2 with jwj w , the following holds: jf x1 ; u; w 0 f x2 ; u; w j 1 jx1 0 x2 j . The proofs of Lemmas II.1 and II.2 are similar to the proofs of [10, Lemma 1] and [11, Lemma 1]. Proof of Lemma II.1: Let x; u; w; w _ be given. Using the numbers Rx ; u ; w ; w_ , where Rx , let the second x condition of the lemma generate T13 > , 1 2 K1 and 2 2 K1 . Since f is locally Lipschitz, it is locally bounded and there exists a number M > such that for all jxj Rx , juj u , jwj1 w we have jf x; u; w j M . Let T 3 fT13 ; =M g. It follows that for each jxj x , kwT k1 w and all t 2 kT; k T;T2 ; T 3 , the solution x t of
(
) := 0 max ) (
1 1 )
(
(
+
(
(0 )
( ( )) (1 1 1 ) (0 ) 1 1 ) T
(1 1 1 1 ) = 1 +1 0 1 1 := min 1 1 [ ( + 1) ]
1 1 1)
0 ) 1
(
k
() x_ (t) = f (x(t); u; w(t)) x0 = x(k) = x (11) satisfies jx(t)j R and jx(t) 0 xj M (t 0 kT ) MT and since w(1) is continuously differentiable by definition, we have jw(t) 0 w(k)j 1 _ (t 0 kT ) 1 _ T , for all t 2 [kT; (k + 1)T ] and T 2 (0; T 3 ). It then follows from condition 2) of the lemma that, for all jxj 1 , juj 1 , kw k1 1 , kw_ [k]k1 1 _ , and all T 2 (0; T 3 )
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Theorem III.2: Suppose that i) the family of approximate discrete-time models FTa x; wT is Lyapunov-SP-ISS (where (9) holds), ii) FTa is one-step strongly consistent with FTe , and iii) uT is uniformly locally bounded. Then, the family of exact discrete-time models FTe x; wT is SP-ISS. The following lemmas are needed to complete proofs of both theorems. We prove only Lemma III.1 for the case of ordinary difference equations (i.e., when (10) holds) and then comment on the changes in the proof for the case of functional difference equations (i.e., when (9) holds) and the proof of Lemma III.2. Lemma III.1: If all conditions in Theorem III.1 are satisfied, there exist 2 K1 such that for any strictly positive numbers Cx~ ; Cw ; Cw_ ; , there exists T 3 > such that for all T 2 ; T 3 , we have
(~
(~
)
)
^
(
)
0
(0 )
jx~j C~ ; kwk1 C ; kw_ k1 C _ maxfV (F (~x; w )); V (~x)g ^(kwk1 ) + =) V (F (~x; wT )) 0 V (~x) 0 14 3 (jx~j): x
T
w
e T
T
w
T
T
e T
T
T
(14)
x
w
w
x
( +1) k
T
kT
u
T
w
f
w
[f (x( ); u; w( )) 0 f (x; u; w)]d
( +1)
k
kT
0
T
(
(0 )
)
jx~j C~ ; kwk1 C maxfV (F (~x; w )); V (~x)g ^(kwk1 ) + =) V (F (~x; wT )) 0 V (~x) 0 14 3 (jx~j): x
e T
T
w
T
T
( +1)
T
^
T
1 (jx( ) 0 xj)d + 2 (jw( ) 0 wj)d T 1 (MT ) + T 2 (1 _ T ) T ~(T ) (12) where ~(s) := 1 (Ms) + 2 (1 _ s) is a K1 function since 1 and 2 are K1 . Since k
Lemma III.2: If all conditions in Theorem III.2 are satisfied, there exist 2 K1 such that for any strictly positive numbers Cx~ ; Cw ; , there exists T 3 > such that for all T 2 ; T 3 , we have
e T
T
T
(15)
kT
w
w
F
e T
= x + T f (x; u; w) F
( +1)
+
k
T
kT
[f (x( ); u; w( )) 0 f (x; u; w)]d
(13)
Proof of Lemma III.1: First, we prove the following fact. Fact 1: Suppose that for any strictly positive numbers 1 ; 2 ; 1 there exists Tw3 > such that for all T 2 ; Tw3 , jxj 1 and jwj 2 we have that (10) holds. Then, for any strictly positive numbers 1 ; 2 ; 3 ; 1 there exists Ts3 > such that for all T 2 ; Ts3 , jxj 1 and continuously differentiable disturbances with kwk1 2 and kwk1 3 we have that
0
~( )
~
= ~+ ~
s
T
T
T
;
w
s
III. MAIN RESULTS In this section, we state and prove our main results (Theorems III.1 and III.2). The results specify conditions on the approximate model, the controller and the plant, which guarantee that the family of controllers GT ; uT that input-to-state stabilize FTa would also input-to-state stabilize FTe for sufficiently small T . We emphasize that our results are given for general approximate discrete-time models FTa (not only for the Euler approximation). We remark that under certain mild conditions on the plant and the controller, our results can be extended to include inter-sample behavior, to conclude SP-ISS results for the closed-loop sampled-data systems (see the results in [12]). Finally, an example is presented to illustrate our approach. Theorem III.1: Suppose that i) the family of approximate discrete-time models FTa x; 1 is Lyapunov-SP-ISS (where either (9) or (10) holds), ii) FTa is one-step weakly consistent with FTe , and iii) uT is uniformly locally bounded. Then, the family of exact discrete-time models FTe x; wT is SP-DISS.
(
)
(~ )
(~
)
(1 1 ) ) ~ 1 (0 )
1 (1 1 1 ) 0 ~ 1 _ 1 1 V (F (~x; w)) 0 V (~x) 0 (jx~j) + ~(kw k ) + : (16) 3 1 1 T Proof of Fact 1: Let (11 ; 12 ; 13 ; 1 ) be given. Let be such that sup 2[0 1 ] ~(s + ) 0 ~(s) (1 =2). Let 11 := 11 , 12 := 12 , 1 := (1 =2) and let the numbers 11 ; 12 ; 1 generate T 3 > 0 from the condition of Fact 1. Let T 3 := min T 3 ; ( 13 ) . Consider arbitrary T 2 (0; T 3 ), jx ~j 11 and any continuously differentiable disturbance with kwk1 12 and kw_ k1 13 . From the mean value theorem and our choice of T 3 , it follows that for all t 2 [kT; (k +1)T ], k 2 we have that jwj = jw(k)j jw(t) 0 w(kT )j + jw(t)j kw_ k1 (t 0 kT ) + kw k1 13 T + kw k1 13 T 3 + kw k1 + kw k1 . Finally, using our definitions of ; 1 we can write 1 [V (F (~x; w)) 0 V (~x)] 0 3 (jx~j) + ~(jwj) + 1 T = 0 3 (jx~j) + ~(kw k1 ) + ~(jwj) 0 ~(kw k1 ) + 21 0 3 (jx~j) + ~(kw k1 ) + ~ + kw k1 0 ~(kw k1 ) + 21 (17) 0 3 (jx~j) + ~(kw k1 ) + 21 + 21 T
the result follows from (12) and the first condition of the lemma, which implies the existence of 1 2 K1 , such that FTa 0 FTEuler T 1 T . Finally, by letting 1 we prove that FTa is one-step weakly consistent with FTe . The proof of Lemma II.2 is omitted since it follows closely the proof of Lemma II.1.
(0
w
s
s
T
f
T
s
T
T
T
T
T
T
T
T
T
T
T
which completes the proof of the fact. Now we continue the proof of Lemma III.1.
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Suppose that all conditions in Theorem III.1 (where (10) holds) are satisfied. Using Fact 1 it follows that all conditions in Theorem III.1 2 03 1 s . (where (16) holds) are also satisfied. Let s Let arbitrary strictly positive numbers Cx~ ; Cw ; Cw_ ; be given. Using Cx~ ; Cw ; Cw_ ; , we define = 201 = ; 1 = 1 = ; = 3 02 1 = 1 ; 0 1 2 2 = 1 ; and 01 1 2 Cx~ Cw 1 . Let the numbers 1 ; 2 ; ; generate the numbers T13 > and L > from condition i) of Theorem III.1 (where (16) holds). Let and T23 > generate u > from condition iii) of Theorem III.1. Let the quadruple ; u ; Cw ; Cw_ generate T33 > and from condition ii) of Theorem III.1. Let strictly positive numbers T43 ; T53 ; T63 ; T73 be such that L T43 = 3 2 ; T53 T53 ; T63 Cw = 1 = ; and 3 T7 = 3 Cx~ Cw 1 L T73 = . Finally, we take T 3 fT13 ; T23 ; T33 ; T43 ; T53 ; T63 ; T73 ; g. In the calculations that follow, we consider arbitrary T 2 ; T 3 , jxj Cx~ , kwk1 Cw and 3kwk1 Cw_ . From (8), (9), the definition of , and the fact that T , we have
^( ) := (4~( )) ( ) ( := (1 2) ( 2) ) := min (1 4) ( 4) (1 4) ((1 2) ( )) := ((1 2) ( )) 1 := ( ( )+ ~( )+ )+ 0 ( 1 ) 0 0 0 1 1 (1 1 ) 0 ( ) (1 4) ( ) ( ) ~( ) (1 2) ((1 4) ) ((1 4) ( ) + ~( ) + + ( )) ( 2) = min 1 (0 ) ~ _ 1 1 jFTa (~x; w)j 01 1 (VT (FTa (~x; w))) 01 1 (VT (~x) + T ~(kwk1 ) + T 1 ) 01 1 ( 2 (Cx~ ) + ~(Cw ) + 1 ) < 1:
choice of 1 and 2 , and using (18)–(23), we deduce that VT
Cw = implies
^( ) + ( 2)
^(Cw ) + VT (FTe ) 0 VT (~x) + VT (~x) 2 + VT (FTa ) 0 VT (FTa ) VT (FTa ) 0 VT (~x) + jVT (FTe ) 0 VT (FTa )j + VT (~x) T ~(Cw ) + T 1 + LT (T ) + VT (~x) + VT (~x):
2
VT (FTe ) ^(Cw ) +
2 =) VT (~x) ^(Cw ):
VT (FTe (~x; wT )) 0 VT (~x) VT (FTa (~x; w)) 0 VT (~x) + jVT (FTe (~x; wT )) 0 VT (FTa (~x; w))j 0T 3 (jx~j) + T ~(kwk1 ) + T 1 + LT (T ) 0 T4 3 (jx~j) 0 34T 3 (jx~j) + T ~(Cw ) + T ( ) + T ( )
(18)
4 3 2 4 3 2 0 T4 3 (jx~j) 0 T4 3 02 1 (VT (~x)) + T ~(Cw )
(19)
)) ^( ) + (=2). From (8), the defi-
Suppose that VT FTe x; wT Cw nition of and the choice of T 3 , we have
jFTe (~x; wT )j 02 1 2 = 2 >
jFTa (~x; w)j 0 jFTe (~x; wT ) 0 FTa (~x; w)j + jFTe (~x; wT )j (21) 0 T (T ) + 02 1 2 0 + 2 = : From our choice of T 3 (16), it follows that:
0 T 3 (jx~j) + T 3 (2 )
2
0 T 0 3 (jx~j):
(20)
and then using the condition ii) of Theorem III.1 and our choice of T53 , we have
(22)
which implies
jx~j 02 1 12 1 () = 2 :
(23)
Note that 2 . From the conditions i) and ii) of Theorem III.1 and from the choice of T 3 (in particular, the choice of T43 and T73 ), the
0
2
4
(26)
Suppose now that VT (FTe (~ x; wT )) ^(Cw ) + (=2) and VT (~x)
^(Cw ) + . From our choice of T 3 (in particular the choice of T73 ), it follows that:
VT (FTe (~x; wT )) 0 VT (~x) ^(Cw ) +
2 0 VT (~x) + 2 0 2
1, T63 , 1 , and , and using the inequality
1 1 () 1 1 () + 1 1 () 0 1 1 0 1 1 2 2 2 4 4 4 4 1 () 0 T ~(Cw ) 0 T 1 1 (jFTa (~x; w)j) 0 T ~(Cw ) 0 T 1 VT (FTa (~x; w)) 0 T ~(kwk1 ) 0 T 1 VT (~x) 2 (jx~j)
(25)
Again using the conditions i) and ii) of Theorem III.1 and from the choice of T 3 (in particular the choice of T43 ), the choice of 1 and 2 , and using (18)–(25), we can write
1
( (~
(24)
Hence, we can conclude that
Using the condition ii) of Theorem III.1, inequality (18) and our choice of and T 3 (in particular the choice of T53 ), we can write
jFTe (~x; wT )j jFTa (~x; w)j + jFTe (~x; wT ) 0 FTa (~x; w)j 01 1 ( 2 (Cx~ ) + ~(Cw ) + 1 ) + T (T ) 01 1 ( 2 (Cx~ ) + ~(Cw ) + 1 ) + = 1:
(FTe )
0
2
0 T 0 (jx~j)
4
(27)
3
which shows that (14) is valid, and this completes the proof of Lemma III.1. The proof of Lemma III.1 for the case of functional difference equations and the proof of Lemma III.3 follow the same steps, except that we do not need to use Fact 1 since (9) holds. Also, in the case of functional difference equations of Lemma III.1 we use one-step weak consistency and in the case of Lemma III.3 we use one-step strong consistency. The next lemma is needed in proofs of Theorems III.1 and III.2, and it was proved as a part of the proof of [11, Th. 2]. Lemma III.3: Let W L1 and let 1 ; 2 ; 3 2 K1 . Let strictly positive-real numbers d; D be such that 1 D > d and let T 3 > be such that for any T 2 ; T 3 there exists a function VT n ! 0 such that for all T 2 ; T 3 and all x 2 n we have 1 jxj VT x 2 jxj and, moreover, for all x 2 n
:
0 (~)
(
(~)
(~)
)
(0 ) (0 )
( ) ~
~
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 7, JULY 2002
maxfV (F 3(~x; w )); V (~x)g d and jx~j D, all w 2 W 2 (0; T ), the following holds: V (F (~x; w )) 0 V (~x) 0 1 (jx~j): T 4 3 Then, there exist a function 2 KL such that for all T 2 (0; T 3 ), jx~(0)j 02 1 1 (D) and w 2 W and all k 2 the solutions of the family of discrete-time models x ~(k +1) = F (~x(k); w [k]) exist and satisfy jx ~(k)j (jx~(0)j ; kT ) + 01 1 (d). Proof of Theorem III.1: Let arbitrary strictly positive-real numbers (1~ ; 1 ; 1 _ ; ) be given and let all conditions in Theorem III.1 hold. Let ^ 2 K come from Lemma III.1. We define (C~ ; C ; C _ ; 0)1 as C := 1 , 0C1 _ := 1 _ , > 0 is such that sup 2[0 1 ] [ 101(^ (s) + ) 0 1 0 ^1(s)] , and the number C~ := maxf 1 (^ (1 ) + ) + 1; 1 2 (1~ )g. Using Lemma III.1, let (C~ ; C ; C _ ; ) generate T 3 > 0, such that (14) holds. Introduce D := C~ and d := ^(kwk1 ) + , and from the choice of (C~ ; C ; C _ ; ) we have that 1 (D) > d. Let W be a set of continuously differentiable functions defined as follows: W := fw 2 L1 j kwk1 C ; kw_ k1 C _ g. With these definitions of (D; d) and W , together with (8), we have that all conditions of Lemma III.3 hold. Hence, we can conclude that for all T 2 (0; T 3 ), x ~(0) 2 , jx~(0)j 1~ and w 2 L1 with kwk1 1 , kw_ k1 1 _ and all k 0 we have that the solutions of F (~x; w ) exist and satisfy jx~(k)j (jx~(0)j ; kT ) + 01 1 (d) (jx~(0)j ; kT ) + 01 1 (^ (kwk1 ) + ) (jx~(0)j ; kT ) + 01 1 ^(kwk1 ) + = (jx~(0)j ; kT ) + (kwk1 ) + (28) 0 1 where (s) := 1 ^(s). This completes the proof of Theorem III.1. with and all T
T
T
T
T
T
x
s
w
w
TABLE I ROAS IN DISTURBANCE FREE CASE
T
T
T
T
x
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T
TABLE II PERFORMANCE WITH A DISTURBANCE
w
w
w
w
w
w
;
x
w
x
x
w
w
x
x
w
w
w
w
n
x
w
e T
w
T
The proof of Theorem III.2 is omitted since it follows closely the proof of Theorem III.1. We illustrate below our results via an example. Example III.1: Consider the scalar continuous-time plant xt x3 t ut w t and its approximate discrete-time (k+1)T w s ds model x k x k T x3 k u k kT a FT x k ; u k ; wT k , which can be obtained from numerical integration schemes described in [3]. The following three controllers:
_( ) = ( ) + ( ) + ( ) ( + 1) = ( ) + ( ( ) + ( )) + ( ( ) ( ) [ ])
( ) =:
u1 (x) = 0 x3 0 x u2 (x) = 0 x3 0 x 0 T x p (29) u3 (x) = 0 1 1 + 2T x2 0 1 0 4T x 2T can be shown to yield, respectively, the following three dissipation inequalities with V (x) = (1=2)x2 : 1 [V (F (x; u1 (x); w )) 0 V (x)] T 2 + T kw k2 + T x2 0 12 x2 + 21 kw k1 (30) 1 1 [V (F (x; u2 (x); w )) 0 V (x)] 0 1 x2 T 2 2 1 T + 2 kw k12 + T + 2 kw k12 2 3 + T+T (31) + T x2 T T T
a T
T
T
T
a T
T
1 4 =1 2 + ( )
_
()
() (1 2) + (1 2)
T
T
T
T
2 2 1 [V (F (x; u3 (x); w )) 0 V (x)] T 2 + T kw k2 : 0 12 x2 + 21 kw k1 (32) 1 From our choice of V (x) and (30)–(32) it follows that the approxia T
~
same as FTEuler in the first condition of Lemma II.2, it follows that: FTa is one-step strongly consistent with FTe . Finally, all of the controllers in (29) are locally uniformly bounded (for uT1 and uT2 this is 3 obvious expanp and for uT this can be seen2by using the Taylor series 0 T 0 T O T ). Therefore, for FTa , V x and sion any controller in (29), we have that all conditions of Theorem III.2 hold. Hence, we can conclude using Theorem III.2 that each of controllers (29) semiglobally practically input-to-state stabilizes the exact discrete-time plant model. We applied the controllers (29) via a sampler and zero order hold to the continuous-time plant model and compared the performance of the three controllers via simulations in SIMULINK.2 Note that the controller uT1 x may be obtained using a continuous-time design (obtain = w2 for the continuous-time closed-loop) and V 0 = x2 controller discretization. In Table I we estimated regions of attraction (ROA) of the closed-loop sampled-data system with controllers (29) for different sampling periods. The controller uT1 gives the largest ROA for all tested sampling periods. In Table II we summarize simulations for different sampling periods and fixed initial states with a sinusoidal disturbance of frequency 1 rad/s. The values of amplitude of the sinusoidal disturbance recorded in Table II are the largest values for which solutions of the sampled-data closed-loop system stay bounded. It is obvious that the controller uT3 is the most robust with respect to the test disturbance for all tested sampling periods. Similar observations were obtained for other initial states and disturbances that are not presented in Table II. From Tables I and II, we see that the performance of all controllers (29) becomes very similar for small sampling periods which can be expected since the dissipation inequalities in (30)–(32) differ only in terms of order T , which can be made arbitrarily small on compact sets by reducing T . Difference in performance of controllers (29) is more pronounced for larger sampling periods (see Tables I and II).
T
T
T
T
mate discrete-time model with any of the controllers (29) is Lyapunov SP-ISS. Moreover, since the approximate discrete-time model is the
REFERENCES [1] D. Dochain and G. Bastin, “Adaptive identification and control algorithms for nonlinear bacterial growth systems,” Automatica, vol. 20, pp. 621–634, 1984. [2] G. C. Goodwin, B. McInnis, and R. S. Long, “Adaptive control algorithm for waste water treatment and pH neutralization,” Optimiz. Control Appl. Meth., vol. 3, pp. 443–459, 1982. [3] L. Grüne and P. E. Kloeden, “Higher order numerical schemes for affinely controlled nonlinear systems,” Numer. Math., vol. 89, pp. 669–690, 2001. 2We used the following parameters in simulations: variable step size; ode-45; relative tolerance 10 , absolute tolerance 10 ; max step size auto; and initial step size auto.
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[4] R. Ferretti, “Higher order approximations of linear control systems via Runge Kutta schemes,” Computing, vol. 58, pp. 351–364, 1997. [5] Z. P. Jiang, A. R. Teel, and L. Praly, “Small gain theorem for ISS systems and applications,” Math. Control, Signals, Syst., vol. 7, pp. 95–120, 1994. [6] M. Krstic´, I. Kanellakopoulos, and P. V. Kokotovic´, Nonlinear and Adaptive Control Design. New York: Wiley, 1995. [7] I. M. Y. Mareels, H. B. Penfold, and R. J. Evans, “Controlling nonlinear time varying systems via Euler approximations,” Automatica, vol. 28, pp. 681–696, 1992. [8] D. Neˇsic´ and A. R. Teel, “Changing supply functions in input-to-state stable systems: The discrete-time case,” IEEE Trans. Automat. Contr., vol. 46, pp. 960–962, June 2001. , “Input-to-state stability for nonlinear time-varying systems via av[9] eraging,” Math. Control, Signals, Syst., vol. 14, pp. 257–280, 2001. , “Set stabilization of sampled-data nonlinear differential inclusions [10] via their approximate discrete-time models,” in Proc. 39th Conf. Decision Control, Sydney, Australia, 2000, pp. 2112–2117. [11] D. Neˇsic´, A. R. Teel, and P. Kokotovic´, “Sufficient conditions for stabilization of sampled-data nonlinear systems via discrete-time approximations,” Syst. Control Lett., vol. 38, pp. 259–270, 1999. stability [12] D. Neˇsic´, A. R. Teel, and E. Sontag, “Formulas relating estimates of discrete-time and sampled-data nonlinear systems,” Syst. Control Lett., vol. 38, pp. 49–60, 1999. [13] E. D. Sontag, “The ISS philosophy as a unifying framework for stability-like behavior,” in Nonlinear Control in the Year 2000, A. Isidori, F. Lamnabhi-Lagarrigue, and W. Respondek, Eds. Berlin, Germany: Springer-Verlag, 2000, pp. 443–468. [14] E. D. Sontag and Y. Wang, “On characterizations of the input-to-state stability with respect to compact sets,” in Proc. IFAC Non-Linear Control Systems Design Symp. (NOLCOS ’95), Tahoe City, CA, June 1995, pp. 226–231. [15] E. D. Sontag, “Smooth stabilization implies coprime factorization,” IEEE Trans. Automat. Contr., vol. 34, pp. 435–443, Apr. 1989. [16] A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis. Cambridge, U.K.: Cambridge Univ. Press, 1996.
Sequential Versus Concurrent Languages of Labeled Conflict-Free Petri Nets Hsu-Chun Yen
Abstract—For structurally deterministic labeled conflict-free Petri nets (PNs), we show that two PNs have identical sequential languages if and only if their concurrent languages are identical as well, and whether a given labeled conflict-free PN is structurally deterministic or not can be checked in polynomial time. We also investigate a number of language-related problems in supervisory control theory for this class of PNs. As it turns out, the properties of controllability, observability, and normality in the sequential framework coincide with that in the concurrent framework. Index Terms—Concurrency, equivalence, Petri net (PN), supervisory control.
I. INTRODUCTION Based upon the framework of Ramadge and Wonham [1], the investigation of Petri net (PN) languages has emerged as a research area of increasing importance in supervisory control theory (see, e.g., [2]–[6]). Once modeled by a PN, one way to characterize the behavior of a system is to view the set of all executable transition sequences of the Manuscript received March 23, 2000; revised May 9, 2001. Recommended by Associate Editor R. S. Sreenivas. The author is with the Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan 106, Republic of China (e-mail: [email protected]). Publisher Item Identifier 10.1109/TAC.2002.800664.
PN, which, according to the underlying transition firing semantics, can be classified as either a sequential language or a concurrent language (cf. [7]). The former associates a system’s event to a single PN’s transition, whereas in the latter, a system’s event is captured by a set of concurrently fireable transitions. The contribution of this note is twofold. First, our work generalizes the equivalence theorem of structurally deterministic labeled marked graphs (reported in [8]) in the sense that for a wider class, namely, the class of structurally deterministic labeled conflict-free PNs, the equivalence theorem is shown to hold as well. (The equivalence theorem says that two PNs have identical concurrent languages if and only if their sequential languages are also identical). A polynomial time algorithm is also derived in this note to decide whether a labeled conflict-free PN is structurally deterministic or not. Our work also supplements the existing results concerning deterministic PN languages [9], in which the equivalence of two arbitrary, deterministically labeled, sequential PN languages has been shown to be decidable. Such a result, together with the equivalence theorem derived in this note, implies the decidability of equivalence for any two conflict-free, deterministically labeled, concurrent PN languages. The second contribution of our work lies in the applications to various problems in the theory of supervisory control in the framework of concurrent PN languages. As pointed out in [3], most research on supervisory control of labeled PNs was based upon the so-called no concurrency (NC) assumption. (See [4] for more). Here, we study three of the most important issues, namely, controllability, observability, and normality, in supervisory control of PNs with respect to both the sequential and the concurrent firing semantics. For structurally deterministic conflict-free controlled PNs (CPNs), it turns out that a PN being controllable (observable or normal) in the sequential framework coincides with that in the concurrent framework. As concurrent PN languages are less understood than their sequential counterparts, our result offers an alternative in reasoning about concurrent languages regarding controllability, observability and normality for the aforementioned class of PNs. Hopefully, our work will provide more insights into the theory of supervisory control for labeled PNs with the NC assumption lifted.
II. PRELIMINARIES A PN is a three-tuple (P , T , '), where P is a finite set of places, T is a finite set of transitions, and ' is a flow function ' P 2T [ T 2 P ! f ; g. We assume that the reader is familiar with the basic 0 to denote that definitions of PNs. (See [10] for more). We write 0! 0 is reachable from through the firing of transition sequence 2 to denote that can be fired from ). T 3 . (We also write 0! t represents the number of occurrences of transition t in . , the 0 . displacement of , is defined as 0 0, provided that 0! T r ftjt 2 T; t > g, denoting the set of transitions used in . p ftj' p; t ; t 2 T g (resp., p ftj' t; p ; t 2 T g) is the set of output (resp., input) transitions of place p. 0 0 is defined inductively as follows. Suppose 0 t1 ; ; tn . Let 0 be . If ti is in i01 , let i be i01 with the leftmost occurrence of ti deleted; otherwise, let i i01 . Finally, let 0 0 n . For example, if t1 t2 t3 t4 t5 and 0 t4 t3 t1 , then 00 t2 t5 . Given a computation 0 , a sequence 0 is said to be a rearrangement of if 8 t 2 T , 0! 0 t t and 00! . Conflict-free PNs (cf-PN) [10]: A PN P P; T; ' is said to be conflict-free iff for every place p, either jp j , or 8 t 2 p , t and p are on a self-loop (i.e., t 2 p \ p ). In words, a PN is conflict-free if every place which is an input of more than one transition is also
(
)
:(
01
( # () 1( )
)
( )= # () 0 = ( ) 1
= = # ()=# ()
0018-9286/02$17.00 © 2002 IEEE
(
)
1( ) =
= ( ) 1 _ = ... _ = _ = )
=( 1
)
=
[10] J. C. Geromel, P. L. D. Peres, and J. Bernoussou, “On a convex parameter space method for linear control design of uncertain systems,” SIAM J. Control Optimiz., vol. 29, no. 2, pp. 381–402, 1991. [11] F. Blanchini, “The gain scheduling and the robust state feedback stabilization problems,” IEEE Trans. Automat. Contr., vol. 45, pp. 2061–2070, Nov. 2000. [12] M. M. Seron, G. A. De Doná, and G. C. Goodwin, “Global analytical model predictive control under input constraints,” in Proc. 39th IEEE Conf. Decision Control, Sydney, Australia, Dec. 2000. [13] T. Parisini and R. Zoppoli, “A receding-horizon regulator for nonlinear systems and a neural approximation,” Automatica, vol. 31, pp. 1443–1451, 1995. [14] B. D. O. Anderson and J. B. Moore, “Detectability and stabilizability of time-varying discrete-time linear systems,” SIAM J. Optimiz., vol. 5, pp. 20–32, 1981. [15] K. Takaba, “Robust servomechanism with preview action for polytopic uncertain systems,” Int. J. Robust Nonlinear Control, vol. 10, pp. 101–111, 2000. [16] H. O. Wang, K. Tanaka, and F. Griffin, “An approach to fuzzy control of nonlinear systems: Stability and design issues,” IEEE Trans. Fuzzy Syst., vol. 4, pp. 14–23, Feb. 1996.
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for the design of input-to-state stabilizing (ISS) controllers based on approximate discrete-time plant models (for more details on ISS see [6], [15], [13], and [14]). In particular, we provide sufficient conditions on the continuous-time plant model, the controller and the approximate discrete-time model, which guarantee that if the controller input-to-state stabilizes the approximate discrete-time plant model it would also input-to-state stabilize the exact discrete-time plant model. Our results apply to dynamic controllers and our approach benefits from the results on numerical integration schemes in [16], [3], and [4]. Related results were investigated in [8], on changes of supply rates for ISS discrete-time systems. II. PRELIMINARIES
A Note on Input-to-State Stabilization for Nonlinear Sampled-Data Systems
Sets of real and natural numbers (including 0) are denoted, respectively, as and . For a given function w: 0 ! n , we use the following notation: wT [k] := w(t); t 2 [kT ; (k + 1)T ] where k 2 and T > 0, and wT [k] is zero elsewhere (in other words wT [k] is a piece of function w(t) in the k th sampling interval [kT ; (k + 1)T ]); and w(k) is the value of the function w(1) at t = kT , k 2 . We denote the norms kwT [k]k1 = sup 2[kT;(k+1)T ] jw( )j and kwk1 := sup 0 jw ( )j and in the case when w (1) is a measurable function (in the Lebesgue sense) we use the essential supremum in the definitions. If kwk1 < 1, then we write w 2 L1 . Consider a continuous-time nonlinear plant with disturbances
Dragan Neˇsic´ and Dina S. Laila
x_ (t) = f (x(t); u(t); w (t))
Abstract—We provide a framework for the design of stabilizing controllers via approximate discrete-time models for sampled-data nonlinear systems with disturbances. In particular, we present sufficient conditions under which a discrete-time controller that input-to-state stabilizes an approximate discrete-time model of a nonlinear plant with disturbances would also input-to-state stabilize (in an appropriate sense) the exact discrete-time plant model. Index Terms—Input-to-state stability, nonlinear, sampled-data.
I. INTRODUCTION A stumbling block in controller design for nonlinear sampled-data control systems is the absence of a good model for the design. Indeed, even if the continuous-time plant model is known, we can not in general compute the exact discrete-time model of the plant since this requires an explicit analytic solution of a nonlinear differential equation. This has motivated research on controller design via approximate discrete-time models for sampled-data nonlinear systems [1], [2], [7]. A drawback of these early results was their limited applicability: they investigate a particular class of plant models, a particular approximate discrete-time plant model (usually Euler) and a particular controller. A more general framework for stabilization of disturbance-free sampled-data nonlinear systems via their approximate discrete-time models that is applicable to general plant models, controllers and approximate discrete-time models was first presented in [10], [11]. In this note, we generalize results in [11] by i) considering sampled-data nonlinear systems with disturbances, and ii) providing a framework Manuscript received April 26, 2001; revised November 19, 2001 and January 8, 2002. Recommended by Associate Editor Z. Lin. This work was supported by the Australian Research Council under the Large Grants Scheme. The authors are with the Department of Electrical and Electronic Engineering, The University of Melbourne, Parkville 3010, Victoria, Australia (e-mail: [email protected]; [email protected]). Publisher Item Identifier 10.1109/TAC.2002.800663.
(1)
where x 2 n , u 2 m and w 2 p are respectively the state, control input and exogenous disturbance. It is assumed that f is locally Lipschitz and f (0; 0; 0) = 0. We will consider two cases: w(1) are measurable functions (in the Lebesgue sense), and w(1) are continuously differentiable functions. We will always make precise which case we consider. The control is taken to be a piecewise constant signal u(t) = u(kT ) =: u(k); 8 t 2 [kT ; (k +1)T ), k 2 , where T > 0 is the sampling period. Also, we assume that some combination (output) or all of the states (x(k) := x(kT )) are available at sampling instant kT , k 2 . The exact discrete-time model for the plant (1), which describes the plant behavior at sampling instants kT , is obtained by integrating the initial value problem x_ (t) = f (x(t); u(k); w(t))
(2)
with given wT [k], u(k) and x0 = x(k), over the sampling interval [kT ; (k + 1)T ]. If we denote by x(t) the solution of the initial value problem (2) at time t with given x0 = x(k), u(k) and wT [k], then the exact discrete-time model of (1) can be written as x(k + 1) =x(k) +
(k+1)T
f (x( ); u(k); w ( ))d kT e =:FT (x(k ); u(k ); wT [k ]):
(3)
We refer to (3) as a functional difference equation since it depends on
wT [k]. We emphasize that FTe is not known in most cases. Indeed, in order to compute FTe we have to solve the initial value problem (2) analytically and this is usually impossible since f in (1) is nonlinear.
Hence, we will use an approximate discrete-time model of the plant to design a controller. Different approximate discrete-time models can be obtained using different methods. For example, we may first assume that the disturbances w(1) are constant during sampling intervals, w(t) = w(k); 8 t 2 [kT ; (k + 1)T ] and then use a classical Runge–Kutta numerical integration scheme (such as Euler) for the
0018-9286/02$17.00 © 2002 IEEE
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initial value problem (2). In this case, the approximate discrete-time model can be written as xk FTa x k ; u k ; w k : (4)
( + 1) = ( ( ) ( ) ( ))
We refer to the approximate model (4) as an ordinary difference equation since FTa does not depend on wT k but on w k . For instance, the Euler approximate model is x k x k Tf x k ;u k ;w k . Recently, numerical integration schemes for systems with measurable disturbances were considered in [3] and [4]. Using these numerical integration techniques, we can obtain an approximate discrete-time model
by: for all T with jwj
2 (0; T 3), all x~ 2
1
2
n with
we have
jx~j 1
1
and all w
1 [VT (FT (~x; w)) 0 VT (~x)] 0 (jx~j) + ~(jwj) +
T
3
1
2
p
(10)
[] () and the function VT is called an ISS-Lyapunov function for the family ( +1) = ( )+ ( ( ) ( ) ( )) FT .
x(k + 1) = FTa (x(k); u(k); wT [k])
(5)
which is in general a functional difference equation. For instance, the simplest such approximate discrete-time model, which is analogous to Euler model, has the following form (k+1)T f x k ; u k ; w s ds (see [3]). Since xk xk kT we will consider semiglobal ISS (see Definition II.2), we will think of FTe and FTa as being defined globally for all small T , even though the initial value problem (2) may exhibit finite escape times [11, p. 261]. The sampling period T is assumed to be a design parameter which can be arbitrarily assigned. Since we are dealing with a family of approximate discrete-time models FTa , parameterized by T , in order to achieve a certain objective we need in general to obtain a family of controllers, parameterized by T . We consider a family of dynamic feedback controllers
( ( ) ( ) ( ))
( + 1) = ( ) +
where
x~ 2
z (k + 1) =GT (x(k); z (k)) u(k) =uT (x(k); z (k))
2
(6)
~ := (xT zT )T ,
n . To shorten notation, we introduce x n , where n nx nz and x~ FTi x k ; uT x k ; z k ; i T x k ; GT x k ; z k
z
:= + ( ( ) ( ( ) ( )) 1) F (~( ) 1) := ( ( ) ( ))
:
(7)
The superscript i may be either e or a, where e stands for exact model, a for approximate model. We omit the superscript if we refer to a general model. The second argument of FTi x; 1 (third argument of FTi ) is either a vector w k or a piece of function wT k . Similar to [10], we define the following. Definition II.1: [Lyapunov semiglobally practically input-to-state stable (Lyapunov-SP-ISS)] The family of systems x k FT x k ; wT k is Lyapunov-SP-ISS if there exist functions1 1 ; 2 ; 3 2 K and 2 K, and for any strictly positive-real numbers there exist strictly positive-real numbers 1; 2 ; 1 ; 2 T 3 and L, such that for all T 2 ; T 3 there exists a function VT n ! 0 such that for all x 2 n with jxj 1 and all w 2 L1 with kwk1 2 the following holds:
(~ )
()
(~( )
[ ]) 1 (1 1
:
[]
~( + 1) =
~ )
(0 ) ~
1
~
1
(jx~j) VT (~x) (jx~j) (8) [VT (FT (~x; wT )) 0 VT (~x)] 0 (jx~j)+~ (kwT k )+ (9) 1 T and, moreover, for all x ; x ; z with (xT z T )T ; (xT z T )T 2 [ ; 1 ] and all T 2 (0; T 3 ), we have jVT (x ; z) 0 VT (x ; z)j 1
2
3
1
2
2
1
L jx1 0 x2 j. The function VT
1
1
2
1
2
is called an ISS-Lyapunov function for
the family FT . Remark II.1: In the case when the family of parameterized closed-loop discrete-time nonlinear systems is an ordinary difference equation x k FT x k ; w k , the condition (9) is replaced
~( + 1) =
(~( ) ( ))
1A function : is of class- if it is continuous, zero at zero and strictly increasing. It is of classif it is of class- and is unbounded. A continuous function : is of classif ( ) is of class- for each 0 and ( ) is decreasing to zero for each 0.
The following definition is a semiglobal-practical version of the ISS property used in [13] and [15], and we use it in the case when we consider measurable disturbances w . Definition II.2: [Semiglobal practical ISS ((SP-ISS)] The family of systems x k FT x k ; wT k is SP-ISS if there exist 2 KL and 2 K1 such that for any strictly positive-real num3 3 bers x~ ; w ; there exists T > such that for all T 2 ; T , jx j x~ and w 1 with kwk1 w , the solutions of the system satisfy jx k j jx j ; kT
kwk1 , 8 k 2 . The following semiglobal practical “ISS-like property” was used in [9], and we use it when the disturbances are continuously differentiable. Definition II.3: [Semiglobal practical derivative ISS (SP-DISS)] The family of systems x k FT x k ; wT k is SP-DISS if there exist 2 KL and 2 K1 such that for any strictly positive-real 3 such that for all numbers x~ ; w ; w_ ; there exists T > 3 T 2 ; T , jx j x~ and all continuously differentiable w 1 such that kwk1 w , kw k1 w_ , the solutions of the family FT satisfy jx k j jx j ; kT
kwk1 , 8 k 2 . Note that a similar property to SP-ISS, called input-to-state practical stability (ISpS) was defined in [5] and [14] when considering nonparameterized systems. Definition II.4: uT is said to be locally uniformly bounded if for any x~ > there exist strictly positive numbers T 3 and u such that for all T 2 ; T 3 and all jxj x~ we have juT x j u . In order to prove our main results, we need to guarantee that the mismatch between FTe and FTa is small in some sense. We define two consistency properties, which will be used to limit the mismatch. Similar definitions can be found in numerical analysis literature [16, Def. 3.4.2], and recently in the context of sampled-data systems (see [11, Def. 1] and [10, Def. 2]). In the sequel, we use the notation x x k , u u k , w w k , and wT wT k . Definition II.5: (One-step weak consistency) The family FTa is said to be one-step weakly consistent with FTe if given any strictly positive real numbers x ; u ; w ; w_ , there exist a function 2 K1 and T 3 > such that, for all T 2 ; T 3 , all x 2 n ; u 2 m with jxj x , juj u and functions w 1 that are continuously differentiable and satisfy kwT k1 w and kwf k1 w_ , we have jFTe 0 FTa j T T . Definition II.6: (One-step strong consistency) The family FTa is said to be one-step strongly consistent with FTe if given any strictly positive real numbers x ; u ; w , there exist a function 2 K1 and T 3 > such that, for all T 2 ; T 3 , all x 2 n ; u 2 m ; w 2 L1 with jxj x , juj u , kwT k1 w , we have jFTe 0 FTa j T T . Sufficient checkable conditions for one-step weak and strong consistency are given next (similar results for systems without disturbances are [10, Lemma 1] and [11, Lemma 1]). Lemma II.1: FTa is one-step weakly consistent with FTe if the following conditions hold: 1) FTa is one-step weakly consistent with FTEuler x; u; w x T f x; u; w , and 2) given any strictly positive-real numbers x ; u ; w ; w_ , there exist 1 2 K1 , 2 2 K1 , T 3 > , such that, for all T 2 ; T 3 , all x1 ; x2 2 n with fjx1j; jx2 jg x , all u 2 m with juj u and all w1 ; w2 2 p with fjw1 j ; jw2 jg w , the following holds: jf x1; u; w1 0 f ax2 ; u; w2 j 1 jx1 0 x2 j 2 jw1 0 we 2 j . Lemma II.2: FT is one-step strongly consistent with FT if the following conditions hold: 1) FTa is one-step strongly consistent
~( + 1) = (1 1 ) ~(0) 1 () ~( ) ( ~(0)
(~( )
)+ (
[ ])
0 1
(0 )
)+
~( + 1) = (~( ) [ ]) 0 (1 1 1 ) (0 ) ~(0) 1 1 1 _ ~( ) ( ~(0) ) + ( )+
1
0 (0 )
= () = () 0 1
0
1
1 (~) 1
~ 1
=
()
= ()
[]
(1 1 1 1 ) (0 ) 1 () 1 _ ( )
1
(1 1 1 ) (0 ) 1 1
) := + ( ) (1 1 1 1 ) 0 (0 ) max 1 1 max 1 ( ) ( ) ( )+ (
( )
(
)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 7, JULY 2002
~
( +1)
with FTEuler x; u; wT x f x; u; w s ds, and 2) kT given any strictly positive-real numbers x; u; w , there exist 1 2 K , T 3 > , such that, for all T 2 ; T 3 and for all fjpx1 j ; jx2 jg x , all u 2 m with x1 ; x2 2 n with juj u and all w 2 with jwj w , the following holds: jf x1 ; u; w 0 f x2 ; u; w j 1 jx1 0 x2 j . The proofs of Lemmas II.1 and II.2 are similar to the proofs of [10, Lemma 1] and [11, Lemma 1]. Proof of Lemma II.1: Let x; u; w; w _ be given. Using the numbers Rx ; u ; w ; w_ , where Rx , let the second x condition of the lemma generate T13 > , 1 2 K1 and 2 2 K1 . Since f is locally Lipschitz, it is locally bounded and there exists a number M > such that for all jxj Rx , juj u , jwj1 w we have jf x; u; w j M . Let T 3 fT13 ; =M g. It follows that for each jxj x , kwT k1 w and all t 2 kT; k T;T2 ; T 3 , the solution x t of
(
) := 0 max ) (
1 1 )
(
(
+
(
(0 )
( ( )) (1 1 1 ) (0 ) 1 1 ) T
(1 1 1 1 ) = 1 +1 0 1 1 := min 1 1 [ ( + 1) ]
1 1 1)
0 ) 1
(
k
() x_ (t) = f (x(t); u; w(t)) x0 = x(k) = x (11) satisfies jx(t)j R and jx(t) 0 xj M (t 0 kT ) MT and since w(1) is continuously differentiable by definition, we have jw(t) 0 w(k)j 1 _ (t 0 kT ) 1 _ T , for all t 2 [kT; (k + 1)T ] and T 2 (0; T 3 ). It then follows from condition 2) of the lemma that, for all jxj 1 , juj 1 , kw k1 1 , kw_ [k]k1 1 _ , and all T 2 (0; T 3 )
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Theorem III.2: Suppose that i) the family of approximate discrete-time models FTa x; wT is Lyapunov-SP-ISS (where (9) holds), ii) FTa is one-step strongly consistent with FTe , and iii) uT is uniformly locally bounded. Then, the family of exact discrete-time models FTe x; wT is SP-ISS. The following lemmas are needed to complete proofs of both theorems. We prove only Lemma III.1 for the case of ordinary difference equations (i.e., when (10) holds) and then comment on the changes in the proof for the case of functional difference equations (i.e., when (9) holds) and the proof of Lemma III.2. Lemma III.1: If all conditions in Theorem III.1 are satisfied, there exist 2 K1 such that for any strictly positive numbers Cx~ ; Cw ; Cw_ ; , there exists T 3 > such that for all T 2 ; T 3 , we have
(~
(~
)
)
^
(
)
0
(0 )
jx~j C~ ; kwk1 C ; kw_ k1 C _ maxfV (F (~x; w )); V (~x)g ^(kwk1 ) + =) V (F (~x; wT )) 0 V (~x) 0 14 3 (jx~j): x
T
w
e T
T
w
T
T
e T
T
T
(14)
x
w
w
x
( +1) k
T
kT
u
T
w
f
w
[f (x( ); u; w( )) 0 f (x; u; w)]d
( +1)
k
kT
0
T
(
(0 )
)
jx~j C~ ; kwk1 C maxfV (F (~x; w )); V (~x)g ^(kwk1 ) + =) V (F (~x; wT )) 0 V (~x) 0 14 3 (jx~j): x
e T
T
w
T
T
( +1)
T
^
T
1 (jx( ) 0 xj)d + 2 (jw( ) 0 wj)d T 1 (MT ) + T 2 (1 _ T ) T ~(T ) (12) where ~(s) := 1 (Ms) + 2 (1 _ s) is a K1 function since 1 and 2 are K1 . Since k
Lemma III.2: If all conditions in Theorem III.2 are satisfied, there exist 2 K1 such that for any strictly positive numbers Cx~ ; Cw ; , there exists T 3 > such that for all T 2 ; T 3 , we have
e T
T
T
(15)
kT
w
w
F
e T
= x + T f (x; u; w) F
( +1)
+
k
T
kT
[f (x( ); u; w( )) 0 f (x; u; w)]d
(13)
Proof of Lemma III.1: First, we prove the following fact. Fact 1: Suppose that for any strictly positive numbers 1 ; 2 ; 1 there exists Tw3 > such that for all T 2 ; Tw3 , jxj 1 and jwj 2 we have that (10) holds. Then, for any strictly positive numbers 1 ; 2 ; 3 ; 1 there exists Ts3 > such that for all T 2 ; Ts3 , jxj 1 and continuously differentiable disturbances with kwk1 2 and kwk1 3 we have that
0
~( )
~
= ~+ ~
s
T
T
T
;
w
s
III. MAIN RESULTS In this section, we state and prove our main results (Theorems III.1 and III.2). The results specify conditions on the approximate model, the controller and the plant, which guarantee that the family of controllers GT ; uT that input-to-state stabilize FTa would also input-to-state stabilize FTe for sufficiently small T . We emphasize that our results are given for general approximate discrete-time models FTa (not only for the Euler approximation). We remark that under certain mild conditions on the plant and the controller, our results can be extended to include inter-sample behavior, to conclude SP-ISS results for the closed-loop sampled-data systems (see the results in [12]). Finally, an example is presented to illustrate our approach. Theorem III.1: Suppose that i) the family of approximate discrete-time models FTa x; 1 is Lyapunov-SP-ISS (where either (9) or (10) holds), ii) FTa is one-step weakly consistent with FTe , and iii) uT is uniformly locally bounded. Then, the family of exact discrete-time models FTe x; wT is SP-DISS.
(
)
(~ )
(~
)
(1 1 ) ) ~ 1 (0 )
1 (1 1 1 ) 0 ~ 1 _ 1 1 V (F (~x; w)) 0 V (~x) 0 (jx~j) + ~(kw k ) + : (16) 3 1 1 T Proof of Fact 1: Let (11 ; 12 ; 13 ; 1 ) be given. Let be such that sup 2[0 1 ] ~(s + ) 0 ~(s) (1 =2). Let 11 := 11 , 12 := 12 , 1 := (1 =2) and let the numbers 11 ; 12 ; 1 generate T 3 > 0 from the condition of Fact 1. Let T 3 := min T 3 ; ( 13 ) . Consider arbitrary T 2 (0; T 3 ), jx ~j 11 and any continuously differentiable disturbance with kwk1 12 and kw_ k1 13 . From the mean value theorem and our choice of T 3 , it follows that for all t 2 [kT; (k +1)T ], k 2 we have that jwj = jw(k)j jw(t) 0 w(kT )j + jw(t)j kw_ k1 (t 0 kT ) + kw k1 13 T + kw k1 13 T 3 + kw k1 + kw k1 . Finally, using our definitions of ; 1 we can write 1 [V (F (~x; w)) 0 V (~x)] 0 3 (jx~j) + ~(jwj) + 1 T = 0 3 (jx~j) + ~(kw k1 ) + ~(jwj) 0 ~(kw k1 ) + 21 0 3 (jx~j) + ~(kw k1 ) + ~ + kw k1 0 ~(kw k1 ) + 21 (17) 0 3 (jx~j) + ~(kw k1 ) + 21 + 21 T
the result follows from (12) and the first condition of the lemma, which implies the existence of 1 2 K1 , such that FTa 0 FTEuler T 1 T . Finally, by letting 1 we prove that FTa is one-step weakly consistent with FTe . The proof of Lemma II.2 is omitted since it follows closely the proof of Lemma II.1.
(0
w
s
s
T
f
T
s
T
T
T
T
T
T
T
T
T
T
T
which completes the proof of the fact. Now we continue the proof of Lemma III.1.
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 7, JULY 2002
Suppose that all conditions in Theorem III.1 (where (10) holds) are satisfied. Using Fact 1 it follows that all conditions in Theorem III.1 2 03 1 s . (where (16) holds) are also satisfied. Let s Let arbitrary strictly positive numbers Cx~ ; Cw ; Cw_ ; be given. Using Cx~ ; Cw ; Cw_ ; , we define = 201 = ; 1 = 1 = ; = 3 02 1 = 1 ; 0 1 2 2 = 1 ; and 01 1 2 Cx~ Cw 1 . Let the numbers 1 ; 2 ; ; generate the numbers T13 > and L > from condition i) of Theorem III.1 (where (16) holds). Let and T23 > generate u > from condition iii) of Theorem III.1. Let the quadruple ; u ; Cw ; Cw_ generate T33 > and from condition ii) of Theorem III.1. Let strictly positive numbers T43 ; T53 ; T63 ; T73 be such that L T43 = 3 2 ; T53 T53 ; T63 Cw = 1 = ; and 3 T7 = 3 Cx~ Cw 1 L T73 = . Finally, we take T 3 fT13 ; T23 ; T33 ; T43 ; T53 ; T63 ; T73 ; g. In the calculations that follow, we consider arbitrary T 2 ; T 3 , jxj Cx~ , kwk1 Cw and 3kwk1 Cw_ . From (8), (9), the definition of , and the fact that T , we have
^( ) := (4~( )) ( ) ( := (1 2) ( 2) ) := min (1 4) ( 4) (1 4) ((1 2) ( )) := ((1 2) ( )) 1 := ( ( )+ ~( )+ )+ 0 ( 1 ) 0 0 0 1 1 (1 1 ) 0 ( ) (1 4) ( ) ( ) ~( ) (1 2) ((1 4) ) ((1 4) ( ) + ~( ) + + ( )) ( 2) = min 1 (0 ) ~ _ 1 1 jFTa (~x; w)j 01 1 (VT (FTa (~x; w))) 01 1 (VT (~x) + T ~(kwk1 ) + T 1 ) 01 1 ( 2 (Cx~ ) + ~(Cw ) + 1 ) < 1:
choice of 1 and 2 , and using (18)–(23), we deduce that VT
Cw = implies
^( ) + ( 2)
^(Cw ) + VT (FTe ) 0 VT (~x) + VT (~x) 2 + VT (FTa ) 0 VT (FTa ) VT (FTa ) 0 VT (~x) + jVT (FTe ) 0 VT (FTa )j + VT (~x) T ~(Cw ) + T 1 + LT (T ) + VT (~x) + VT (~x):
2
VT (FTe ) ^(Cw ) +
2 =) VT (~x) ^(Cw ):
VT (FTe (~x; wT )) 0 VT (~x) VT (FTa (~x; w)) 0 VT (~x) + jVT (FTe (~x; wT )) 0 VT (FTa (~x; w))j 0T 3 (jx~j) + T ~(kwk1 ) + T 1 + LT (T ) 0 T4 3 (jx~j) 0 34T 3 (jx~j) + T ~(Cw ) + T ( ) + T ( )
(18)
4 3 2 4 3 2 0 T4 3 (jx~j) 0 T4 3 02 1 (VT (~x)) + T ~(Cw )
(19)
)) ^( ) + (=2). From (8), the defi-
Suppose that VT FTe x; wT Cw nition of and the choice of T 3 , we have
jFTe (~x; wT )j 02 1 2 = 2 >
jFTa (~x; w)j 0 jFTe (~x; wT ) 0 FTa (~x; w)j + jFTe (~x; wT )j (21) 0 T (T ) + 02 1 2 0 + 2 = : From our choice of T 3 (16), it follows that:
0 T 3 (jx~j) + T 3 (2 )
2
0 T 0 3 (jx~j):
(20)
and then using the condition ii) of Theorem III.1 and our choice of T53 , we have
(22)
which implies
jx~j 02 1 12 1 () = 2 :
(23)
Note that 2 . From the conditions i) and ii) of Theorem III.1 and from the choice of T 3 (in particular, the choice of T43 and T73 ), the
0
2
4
(26)
Suppose now that VT (FTe (~ x; wT )) ^(Cw ) + (=2) and VT (~x)
^(Cw ) + . From our choice of T 3 (in particular the choice of T73 ), it follows that:
VT (FTe (~x; wT )) 0 VT (~x) ^(Cw ) +
2 0 VT (~x) + 2 0 2
1, T63 , 1 , and , and using the inequality
1 1 () 1 1 () + 1 1 () 0 1 1 0 1 1 2 2 2 4 4 4 4 1 () 0 T ~(Cw ) 0 T 1 1 (jFTa (~x; w)j) 0 T ~(Cw ) 0 T 1 VT (FTa (~x; w)) 0 T ~(kwk1 ) 0 T 1 VT (~x) 2 (jx~j)
(25)
Again using the conditions i) and ii) of Theorem III.1 and from the choice of T 3 (in particular the choice of T43 ), the choice of 1 and 2 , and using (18)–(25), we can write
1
( (~
(24)
Hence, we can conclude that
Using the condition ii) of Theorem III.1, inequality (18) and our choice of and T 3 (in particular the choice of T53 ), we can write
jFTe (~x; wT )j jFTa (~x; w)j + jFTe (~x; wT ) 0 FTa (~x; w)j 01 1 ( 2 (Cx~ ) + ~(Cw ) + 1 ) + T (T ) 01 1 ( 2 (Cx~ ) + ~(Cw ) + 1 ) + = 1:
(FTe )
0
2
0 T 0 (jx~j)
4
(27)
3
which shows that (14) is valid, and this completes the proof of Lemma III.1. The proof of Lemma III.1 for the case of functional difference equations and the proof of Lemma III.3 follow the same steps, except that we do not need to use Fact 1 since (9) holds. Also, in the case of functional difference equations of Lemma III.1 we use one-step weak consistency and in the case of Lemma III.3 we use one-step strong consistency. The next lemma is needed in proofs of Theorems III.1 and III.2, and it was proved as a part of the proof of [11, Th. 2]. Lemma III.3: Let W L1 and let 1 ; 2 ; 3 2 K1 . Let strictly positive-real numbers d; D be such that 1 D > d and let T 3 > be such that for any T 2 ; T 3 there exists a function VT n ! 0 such that for all T 2 ; T 3 and all x 2 n we have 1 jxj VT x 2 jxj and, moreover, for all x 2 n
:
0 (~)
(
(~)
(~)
)
(0 ) (0 )
( ) ~
~
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 7, JULY 2002
maxfV (F 3(~x; w )); V (~x)g d and jx~j D, all w 2 W 2 (0; T ), the following holds: V (F (~x; w )) 0 V (~x) 0 1 (jx~j): T 4 3 Then, there exist a function 2 KL such that for all T 2 (0; T 3 ), jx~(0)j 02 1 1 (D) and w 2 W and all k 2 the solutions of the family of discrete-time models x ~(k +1) = F (~x(k); w [k]) exist and satisfy jx ~(k)j (jx~(0)j ; kT ) + 01 1 (d). Proof of Theorem III.1: Let arbitrary strictly positive-real numbers (1~ ; 1 ; 1 _ ; ) be given and let all conditions in Theorem III.1 hold. Let ^ 2 K come from Lemma III.1. We define (C~ ; C ; C _ ; 0)1 as C := 1 , 0C1 _ := 1 _ , > 0 is such that sup 2[0 1 ] [ 101(^ (s) + ) 0 1 0 ^1(s)] , and the number C~ := maxf 1 (^ (1 ) + ) + 1; 1 2 (1~ )g. Using Lemma III.1, let (C~ ; C ; C _ ; ) generate T 3 > 0, such that (14) holds. Introduce D := C~ and d := ^(kwk1 ) + , and from the choice of (C~ ; C ; C _ ; ) we have that 1 (D) > d. Let W be a set of continuously differentiable functions defined as follows: W := fw 2 L1 j kwk1 C ; kw_ k1 C _ g. With these definitions of (D; d) and W , together with (8), we have that all conditions of Lemma III.3 hold. Hence, we can conclude that for all T 2 (0; T 3 ), x ~(0) 2 , jx~(0)j 1~ and w 2 L1 with kwk1 1 , kw_ k1 1 _ and all k 0 we have that the solutions of F (~x; w ) exist and satisfy jx~(k)j (jx~(0)j ; kT ) + 01 1 (d) (jx~(0)j ; kT ) + 01 1 (^ (kwk1 ) + ) (jx~(0)j ; kT ) + 01 1 ^(kwk1 ) + = (jx~(0)j ; kT ) + (kwk1 ) + (28) 0 1 where (s) := 1 ^(s). This completes the proof of Theorem III.1. with and all T
T
T
T
T
T
x
s
w
w
TABLE I ROAS IN DISTURBANCE FREE CASE
T
T
T
T
x
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T
TABLE II PERFORMANCE WITH A DISTURBANCE
w
w
w
w
w
w
;
x
w
x
x
w
w
x
x
w
w
w
w
n
x
w
e T
w
T
The proof of Theorem III.2 is omitted since it follows closely the proof of Theorem III.1. We illustrate below our results via an example. Example III.1: Consider the scalar continuous-time plant xt x3 t ut w t and its approximate discrete-time (k+1)T w s ds model x k x k T x3 k u k kT a FT x k ; u k ; wT k , which can be obtained from numerical integration schemes described in [3]. The following three controllers:
_( ) = ( ) + ( ) + ( ) ( + 1) = ( ) + ( ( ) + ( )) + ( ( ) ( ) [ ])
( ) =:
u1 (x) = 0 x3 0 x u2 (x) = 0 x3 0 x 0 T x p (29) u3 (x) = 0 1 1 + 2T x2 0 1 0 4T x 2T can be shown to yield, respectively, the following three dissipation inequalities with V (x) = (1=2)x2 : 1 [V (F (x; u1 (x); w )) 0 V (x)] T 2 + T kw k2 + T x2 0 12 x2 + 21 kw k1 (30) 1 1 [V (F (x; u2 (x); w )) 0 V (x)] 0 1 x2 T 2 2 1 T + 2 kw k12 + T + 2 kw k12 2 3 + T+T (31) + T x2 T T T
a T
T
T
T
a T
T
1 4 =1 2 + ( )
_
()
() (1 2) + (1 2)
T
T
T
T
2 2 1 [V (F (x; u3 (x); w )) 0 V (x)] T 2 + T kw k2 : 0 12 x2 + 21 kw k1 (32) 1 From our choice of V (x) and (30)–(32) it follows that the approxia T
~
same as FTEuler in the first condition of Lemma II.2, it follows that: FTa is one-step strongly consistent with FTe . Finally, all of the controllers in (29) are locally uniformly bounded (for uT1 and uT2 this is 3 obvious expanp and for uT this can be seen2by using the Taylor series 0 T 0 T O T ). Therefore, for FTa , V x and sion any controller in (29), we have that all conditions of Theorem III.2 hold. Hence, we can conclude using Theorem III.2 that each of controllers (29) semiglobally practically input-to-state stabilizes the exact discrete-time plant model. We applied the controllers (29) via a sampler and zero order hold to the continuous-time plant model and compared the performance of the three controllers via simulations in SIMULINK.2 Note that the controller uT1 x may be obtained using a continuous-time design (obtain = w2 for the continuous-time closed-loop) and V 0 = x2 controller discretization. In Table I we estimated regions of attraction (ROA) of the closed-loop sampled-data system with controllers (29) for different sampling periods. The controller uT1 gives the largest ROA for all tested sampling periods. In Table II we summarize simulations for different sampling periods and fixed initial states with a sinusoidal disturbance of frequency 1 rad/s. The values of amplitude of the sinusoidal disturbance recorded in Table II are the largest values for which solutions of the sampled-data closed-loop system stay bounded. It is obvious that the controller uT3 is the most robust with respect to the test disturbance for all tested sampling periods. Similar observations were obtained for other initial states and disturbances that are not presented in Table II. From Tables I and II, we see that the performance of all controllers (29) becomes very similar for small sampling periods which can be expected since the dissipation inequalities in (30)–(32) differ only in terms of order T , which can be made arbitrarily small on compact sets by reducing T . Difference in performance of controllers (29) is more pronounced for larger sampling periods (see Tables I and II).
T
T
T
T
mate discrete-time model with any of the controllers (29) is Lyapunov SP-ISS. Moreover, since the approximate discrete-time model is the
REFERENCES [1] D. Dochain and G. Bastin, “Adaptive identification and control algorithms for nonlinear bacterial growth systems,” Automatica, vol. 20, pp. 621–634, 1984. [2] G. C. Goodwin, B. McInnis, and R. S. Long, “Adaptive control algorithm for waste water treatment and pH neutralization,” Optimiz. Control Appl. Meth., vol. 3, pp. 443–459, 1982. [3] L. Grüne and P. E. Kloeden, “Higher order numerical schemes for affinely controlled nonlinear systems,” Numer. Math., vol. 89, pp. 669–690, 2001. 2We used the following parameters in simulations: variable step size; ode-45; relative tolerance 10 , absolute tolerance 10 ; max step size auto; and initial step size auto.
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[4] R. Ferretti, “Higher order approximations of linear control systems via Runge Kutta schemes,” Computing, vol. 58, pp. 351–364, 1997. [5] Z. P. Jiang, A. R. Teel, and L. Praly, “Small gain theorem for ISS systems and applications,” Math. Control, Signals, Syst., vol. 7, pp. 95–120, 1994. [6] M. Krstic´, I. Kanellakopoulos, and P. V. Kokotovic´, Nonlinear and Adaptive Control Design. New York: Wiley, 1995. [7] I. M. Y. Mareels, H. B. Penfold, and R. J. Evans, “Controlling nonlinear time varying systems via Euler approximations,” Automatica, vol. 28, pp. 681–696, 1992. [8] D. Neˇsic´ and A. R. Teel, “Changing supply functions in input-to-state stable systems: The discrete-time case,” IEEE Trans. Automat. Contr., vol. 46, pp. 960–962, June 2001. , “Input-to-state stability for nonlinear time-varying systems via av[9] eraging,” Math. Control, Signals, Syst., vol. 14, pp. 257–280, 2001. , “Set stabilization of sampled-data nonlinear differential inclusions [10] via their approximate discrete-time models,” in Proc. 39th Conf. Decision Control, Sydney, Australia, 2000, pp. 2112–2117. [11] D. Neˇsic´, A. R. Teel, and P. Kokotovic´, “Sufficient conditions for stabilization of sampled-data nonlinear systems via discrete-time approximations,” Syst. Control Lett., vol. 38, pp. 259–270, 1999. stability [12] D. Neˇsic´, A. R. Teel, and E. Sontag, “Formulas relating estimates of discrete-time and sampled-data nonlinear systems,” Syst. Control Lett., vol. 38, pp. 49–60, 1999. [13] E. D. Sontag, “The ISS philosophy as a unifying framework for stability-like behavior,” in Nonlinear Control in the Year 2000, A. Isidori, F. Lamnabhi-Lagarrigue, and W. Respondek, Eds. Berlin, Germany: Springer-Verlag, 2000, pp. 443–468. [14] E. D. Sontag and Y. Wang, “On characterizations of the input-to-state stability with respect to compact sets,” in Proc. IFAC Non-Linear Control Systems Design Symp. (NOLCOS ’95), Tahoe City, CA, June 1995, pp. 226–231. [15] E. D. Sontag, “Smooth stabilization implies coprime factorization,” IEEE Trans. Automat. Contr., vol. 34, pp. 435–443, Apr. 1989. [16] A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis. Cambridge, U.K.: Cambridge Univ. Press, 1996.
Sequential Versus Concurrent Languages of Labeled Conflict-Free Petri Nets Hsu-Chun Yen
Abstract—For structurally deterministic labeled conflict-free Petri nets (PNs), we show that two PNs have identical sequential languages if and only if their concurrent languages are identical as well, and whether a given labeled conflict-free PN is structurally deterministic or not can be checked in polynomial time. We also investigate a number of language-related problems in supervisory control theory for this class of PNs. As it turns out, the properties of controllability, observability, and normality in the sequential framework coincide with that in the concurrent framework. Index Terms—Concurrency, equivalence, Petri net (PN), supervisory control.
I. INTRODUCTION Based upon the framework of Ramadge and Wonham [1], the investigation of Petri net (PN) languages has emerged as a research area of increasing importance in supervisory control theory (see, e.g., [2]–[6]). Once modeled by a PN, one way to characterize the behavior of a system is to view the set of all executable transition sequences of the Manuscript received March 23, 2000; revised May 9, 2001. Recommended by Associate Editor R. S. Sreenivas. The author is with the Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan 106, Republic of China (e-mail: [email protected]). Publisher Item Identifier 10.1109/TAC.2002.800664.
PN, which, according to the underlying transition firing semantics, can be classified as either a sequential language or a concurrent language (cf. [7]). The former associates a system’s event to a single PN’s transition, whereas in the latter, a system’s event is captured by a set of concurrently fireable transitions. The contribution of this note is twofold. First, our work generalizes the equivalence theorem of structurally deterministic labeled marked graphs (reported in [8]) in the sense that for a wider class, namely, the class of structurally deterministic labeled conflict-free PNs, the equivalence theorem is shown to hold as well. (The equivalence theorem says that two PNs have identical concurrent languages if and only if their sequential languages are also identical). A polynomial time algorithm is also derived in this note to decide whether a labeled conflict-free PN is structurally deterministic or not. Our work also supplements the existing results concerning deterministic PN languages [9], in which the equivalence of two arbitrary, deterministically labeled, sequential PN languages has been shown to be decidable. Such a result, together with the equivalence theorem derived in this note, implies the decidability of equivalence for any two conflict-free, deterministically labeled, concurrent PN languages. The second contribution of our work lies in the applications to various problems in the theory of supervisory control in the framework of concurrent PN languages. As pointed out in [3], most research on supervisory control of labeled PNs was based upon the so-called no concurrency (NC) assumption. (See [4] for more). Here, we study three of the most important issues, namely, controllability, observability, and normality, in supervisory control of PNs with respect to both the sequential and the concurrent firing semantics. For structurally deterministic conflict-free controlled PNs (CPNs), it turns out that a PN being controllable (observable or normal) in the sequential framework coincides with that in the concurrent framework. As concurrent PN languages are less understood than their sequential counterparts, our result offers an alternative in reasoning about concurrent languages regarding controllability, observability and normality for the aforementioned class of PNs. Hopefully, our work will provide more insights into the theory of supervisory control for labeled PNs with the NC assumption lifted.
II. PRELIMINARIES A PN is a three-tuple (P , T , '), where P is a finite set of places, T is a finite set of transitions, and ' is a flow function ' P 2T [ T 2 P ! f ; g. We assume that the reader is familiar with the basic 0 to denote that definitions of PNs. (See [10] for more). We write 0! 0 is reachable from through the firing of transition sequence 2 to denote that can be fired from ). T 3 . (We also write 0! t represents the number of occurrences of transition t in . , the 0 . displacement of , is defined as 0 0, provided that 0! T r ftjt 2 T; t > g, denoting the set of transitions used in . p ftj' p; t ; t 2 T g (resp., p ftj' t; p ; t 2 T g) is the set of output (resp., input) transitions of place p. 0 0 is defined inductively as follows. Suppose 0 t1 ; ; tn . Let 0 be . If ti is in i01 , let i be i01 with the leftmost occurrence of ti deleted; otherwise, let i i01 . Finally, let 0 0 n . For example, if t1 t2 t3 t4 t5 and 0 t4 t3 t1 , then 00 t2 t5 . Given a computation 0 , a sequence 0 is said to be a rearrangement of if 8 t 2 T , 0! 0 t t and 00! . Conflict-free PNs (cf-PN) [10]: A PN P P; T; ' is said to be conflict-free iff for every place p, either jp j , or 8 t 2 p , t and p are on a self-loop (i.e., t 2 p \ p ). In words, a PN is conflict-free if every place which is an input of more than one transition is also
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