Decentralized model reference adaptive control ... - Semantic Scholar
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 7, JULY 1999
[5] St. Kotsios, “Finite input/output representation of a class of Volterra polynomial systems,” Automatica, vol. 33, no. 2, pp. 257–262, 1997. [6] St. Kotsios and D. Lappas, “A description of 2-dimensional discrete polynomial dynamics,” IMA J. Math. Control and Inform., vol. 13, pp. 409–428, 1996. [7] L. Ljung, System Identification Theory for the User. Englewood Cliffs, NJ: Prentice-Hall, 1987. [8] J. W. Rugh, Nonlinear System Theory. Baltimore, MD: Johns Hopkins Univ. Press, 1981.
Decentralized Model Reference Adaptive Control Without Restriction on Subsystem Relative Degrees Changyun Wen and Yeng Chai Soh
Abstract—When the direct model reference adaptive control (MRAC) scheme with first-order local estimators is employed to design totally decentralized controllers, the stability result can only be applied to a system with all of its nominal subsystem relative degrees less than or equal to two. In this paper, this restriction is relaxed and it is achieved by employing the parameter projection together with static normalization. To implement the local controllers, no a priori knowledge of the subsystem unmodeled dynamics and no information exchange between subsystems are required. Global stability is established for the closed-loop system and small in the mean tracking error is ensured. With this analysis, the class of interactions and subsystem unmodeled dynamics can be enlarged to include those having infinite memory. Index Terms—Adaptive control, decentralized control, robustness, stability.
I. INTRODUCTION Decentralized adaptive control is an important control scheme for large scale systems, and it has continued to receive a lot of attention from control researchers over the last few decades. However, only a limited number of stability results in this area are available due to the difficulties in the analysis of ignored interactions. The first batch of results were obtained based on the direct model reference adaptive control (MRAC) approach [1]–[3]. A strong assumption for these results is that relative degrees of all the nominal subsystem models should be less than or equal to two. The stability results using the indirect pole assignment design scheme were reported later in [4]–[6] where there is no restriction on the relative degrees of the nominal subsystem. Recently, efforts on relaxing the subsystem relative degrees in the case of employing the direct adaptive control scheme have been made by using some advanced adaptive strategies. The concept of high-order tuners in [7] was applied to achieve this in [8] and [9]. In this case, a local dynamic estimator with the subsystem relative degree as its order is designed to identify the unknown parameters of each subsystem. The integrator backstepping technique of [10] was also successfully utilized to reach a similar goal in [11]–[13]. To obtain the final control for each subsystem, a number of iterative design steps should be involved to calculate Manuscript received April 24, 1996; revised March 16, 1998. Recommended by Associate Editor, M. Krstic. This work was supported in part by NTU under the Applied Research Project Grant RP 23/92. The authors are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore (e-mail: [email protected]). Publisher Item Identifier S 0018-9286(99)04544-4.
some intermediate virtual control signals. As commented in [9], the unmodeled interactions must satisfy certain structural conditions when these advanced schemes are used. However, for the conventional MRAC scheme, the problem of the relaxation of the subsystem relative degrees is still unsolved. Due to the simplicity of the conventional MRAC scheme, the solution to such a problem is of practical interest. In [14], Datta and Ioannou applied the normalization technique used in the single-loop robust adaptive controller design to achieve the required relaxation. But the proposed local normalizing signals require information from the other subsystems to bound the effects of interactions from these subsystems. Thus, only partially decentralized adaptive controllers can be designed. In this paper, the problem will be solved with totally decentralized controllers by employing the parameter projection together with a static normalization technique. Global stability is established for the closed-loop system and small in the mean tracking error is ensured. With our analysis, the class of interactions and subsystem unmodeled dynamics can be enlarged to include those having infinite memory. The remaining part of the paper is organized as follows. Section II gives the class of systems to be controlled and Section III presents the decentralized controllers. The analysis of the closed-loop system and the main result are given in Section IV. Finally, the paper is concluded in Section V. II. SYSTEM MODELS
AND
ASSUMPTIONS
In this paper, the class of interconnected systems considered consists of m single-input/single-output subsystems. The ith subsystem is modeled as yi (t)
= Hi (D)ui (t) + Hi (D) +
m j =1
m
j =1
ij (D)[uj (t) + yj (t)] ij H
ij (D)[uj (t) + yj (t)] + di (t) ij H
(1)
for i; j = 1; 1 1 1 ; m, where yi ; ui ; and di are, respectively, the output, (D ) input, and disturbance of the ith subsystem. In (1), Hi (D) = B A (D) and it is the reduced-order transfer function of subsystem i with
= Dn + ani 01 Dn 01 + 1 1 1 + a0i m m + bm 01 Dm 01 + 1 1 1 + b0 Bi (D ) = bi D i i Ai (D )
where D denotes the differentiation operator, mi < ni ; ij ; ij are ij (D) denote the subsystem interactions ij (D) and H constants, and H if i 6= j and unmodeled dynamics if i = j . Now, a reference model given below is chosen for the ith subsystem
i
ym (t)
i (D)ri (t) = Wm
(2)
i (D) = km i 1 and ri is an external reference input where Wm D (D) i is a constant and Dm i (D) is a monic Hurwitz signal. Here, km i (D) = Dn + polynomial of degree n3i = ni 0 mi , i.e., Dm i n 0 1 i i + 1 1 1 + d1 D + d0 . The control problem is to design dn 01 D totally decentralized controllers for plant (1) such that the closed-loop system is stable in the sense that all signals in the system are bounded for arbitrary bounded ri and initial conditions, and the output yi (t) i (t) of the model (2) as closely as possible. To follows the output ym solve the control problem, the following assumptions are made for the plant given in (1).
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Assumption 2.1: A1) Bi (D) is Hurwitz. A2) An upper bound for ni , the nominal relative degree n3i = ni 0 mi of subsystem i, and the sign of the high frequency gain sgn(bm i ) are known. Furthermore, the coefficients of Ai (D ) and Bi (D ) are inside a known compact convex region Ci . ij (D) and H ij (D) are stable, and Hi (D)H ij (D) and A3) H H ij (D ) are strictly proper. A4) di (t) is bounded. Remarks 2.1: 1) Note that there is no restriction on the nominal subsystem relative degrees n3i ; i = 1; 2; 1 1 1 ; m. 2) While modeling errors are assumed to satisfy A3) and A4), no a priori knowledge is required from them for the implementation of the adaptive controllers given in the later sections. Assumption A2) also implies a known lower bound for jbm i j.
Remarks 3.1: 1) As can be noted from (6)–(9), the normalization is static. Also the implementation of local adaptive controllers does not require any information exchange between subsystems and the a priori knowledge on subsystem unmodeled dynamics. 2) The results for the adaptive controller in Case 2 can be obtained by following the similar analyses as in Case 1 and [15]. Thus we just focus our attention on Case 1 without any further elaboration on Case 2.
III. DESIGN OF ROBUST DECENTRALIZED ADAPTIVE CONTROLLERS For each subsystem, we define the following filtered variables: ! _ i;1
= 3i !i;1 + qi yi ;
! _ i;2
= 3i !i;2 + qi ui
(3)
(3i ; qi ) is a controllable pair satisfying (DI 0 3i )01 qi = 1 [Dn 02 ; 1 1 1 ; 1]T (4) Fi (D ) with Fi (D) as an arbitrary Hurwitz polynomial of order ni 0 1. Both 3i and qi are chosen by users. Let (5) ! iT = !i;T1 ; !i;T2 ; yi : where
Then the control is given as
= ! iT i + ci;0 ri (6) T T T where i (t) = [i;1 (t); i;2 (t); i;3 (t)] is a (2ni 0 1)-dimensional control parameter vector and ci;0 (t) is a feedforward parameter ui
scalar. From [16], it can be shown that a desired parameter vector 3 of i and a desired parameter ci;3 0 of ci;0 exist, and they can be obtained when the nominal transfer function Hi (D) of the ith subsystem is known. When Hi (D) is unknown, an adaptive law is required to update i and ci;0 . To achieve this and to ensure the robustness of the adaptive controller in the presence of modeling errors including interactions, subsystem unmodeled dynamics, and external disturbances, we introduce parameter projection operation to the adaptive law. The adaptive law to tune i and ci;0 is divided into the following two cases. i is chosen to be Case 1: bim = 1. In this case, ci;0 = 1 if km one and
i
_i
=P
0i ei;1 i i ; 0 1 + !iT !i
0i = 0iT > 0 e i;1 = yi 0 ym + iT i 0 vi ; i T vi = Wm (D )i ! i T T T (1) T !i = ! i ; i ; i ; 1 1 1 ; i(n
(7)
where
i
= Wmi (D)I ! i (8)
)T
(9)
and P denotes the projection operation defined in [17] or [18]. Case 2: bim is unknown. In this case, ci;0 (t) is unknown and needs to be updated. The local adaptive law in this case is a modified version of that in [15] by changing the modification and the normalizing signal appropriately as in Case 1.
IV. STABILITY OF THE DECENTRALIZED ADAPTIVE CONTROL SYSTEMS We need to establish the robustness of the local adaptive controllers in the presence of ignored interactions, unmodeled dynamics, and external disturbances. Before doing this, some preliminary analysis is required. From (1)–(6), it can be shown that the ith subsystem can be expressed as yi
where
= Wmi (D) ! iT ~i + ri + mi (t)
(10)
= i 0 i3 i 3 01 mi (t) = i (t) + 1 + Wm (D )i;1 (DI 0 3i ) qi ~i
i (t)
=
m
j =1
(11) di
1ij (D)[uj (t) + yj (t)]
(12) (13)
1ij (D) = Wmi (D)ij H ij (D) 1 0 i;3 2 (DI 0 3i )01 qi + ij H ij (D) 1 + Wmi (D)(i;3 3 + i;3 1 (DI 0 3i )01 qi
:
(14) Clearly, 1ij (D) is strictly proper and stable from Assumption 2.1. From (12), we have the following result. Lemma 4.1: The modeling error mi (t) in (12) satisfies
jmi (t)j
m
ij j =1
sup k!j ( )k + d0 t
(15)
0
where ij and d0 are some nonnegative constants. Proof: Let Vi (D) be an arbitrary Hurwitz polynomial defined as Vi (D ) T vi
= Dn 02 + vi;n 03 Dn 03 + 1 1 1 + vi;0 = [1; vi;n 03 ; 1 1 1 ; vi;0 ]:
Then i (t)
= =
m j =1 m j =1
1ij Fj
Vj (uj Vj Fj
1ij Fj Vj
+ yj )
T vj (!j;1
+ !j;2 ) F
:
(16)
Thus from the stability and properness of 1ij V , the result can be established from (16). Remarks 4.1: 1) In the proof of Lemma 4.1, the effects of some exponentially decaying terms due to nonzero initial conditions have been absorbed by d0 . 2) The constant ij indicates the strength of the interactions between subsystems i and j when i 6= j , and the unmodeled dynamics of the ith subsystem are coupled to the nominal model when i = j . 3) In terms of the bounding signals, the bound for the modeling error in (15) allows the effects of the unmodeled dynamics and interactions to have infinite memory, thus it is looser than those given in existing literature, such as [1] and [2]. The class of modeling errors considered can also be enlarged to include any nonlinear unmodeled dynamics satisfying (15).
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Now, if sup0 t k!i ( )k = k!i (t)k and sup0 t k!j ( )k sup0 t k!i ( )k for all j 6= i and t > ti0 , then (15) becomes jmi (t)j k!i (t)k + d0 ; for all t ti0 (17) where is a nonnegative constant depending on ij . From (17), some useful properties of the local estimators can be obtained. In the remaining part of this section, we will use mki ; k = 1; 1 1 1 ; 5 to denote Mik (D)mi where Mik (D) is a vector containing proper stable transfer functions. Thus, mik (t); k = 1; 1 1 1 ; 5 also satisfy the same bounds as in (17) under the same conditions. Note that the constants and d0 in (17) have been made uniform for all i = 1; 2; 1 1 1 ; m and k = 1; 1 1 1 ; 5. Also in this section, all cj ; j = 1; 2; 1 1 1 stand for generic constants. We now derive an equation to describe the closed-loop system of the ith loop. It can be shown that the plant in the ith loop has the following state representation:
x_ i = Ai xi + bi ui + mi1 (t); yi = hiT xi + mi2 (t) (18) T where (Ai ; bi ; hi ) is a minimal state representation of Hi (D). Note ij (D). that Dmi2 also satisfies (17) from the strict properness of H Now the closed-loop system of the ith loop can be described as x_ ci = Aci xci + bic ! iT ~i + bic ri + mi1 (t) (19) yi = hic T xci + mi2 (t) where Aci is a stable matrix satisfying (hic )T (DI 0 Aci )01 bic = Wmi (D) and xci = [xTi ; !i;T1 ; !i;T2 ]T . (n 01) (1) Let iT = [iT ; i ; 1 1 1 ; i ]. Then i(k) = Dk Wmi (D) 3 I ! i ; k = 0; 1; 1 1 1 ; ni 0 1 can have the following realization: _i = A i + B ! i = A i + B ;1 !i;1 + B ;2 !i;2 + B ;3 hiT xi + B ;3 mi2 (t)
Lemma 4.2: The estimator (7)–(9), when applied to the plant given in (1), has the following properties. 1) j! i (t)T ~i (t)j c4 ; for t 0: (27) 1=2 2)
1 + !iT (t)!i (t) Suppose M0 is a positive constant s.t. d0 =M0 . If k!i (t)k > M0 ; sup0 t k!i ( )k = k!i (t)k and sup0 t k!j ( )k sup0 t k!i ( )k 8j 6= i and for all t > t0 with some t0 0, then t j! i ( )T ~i ( )j d c5 =0 + ( 1 + 2 )(t 0 t0 ); t 1 + !iT ( )!i ( ) 1=2 for t t0 (28) where
1 = (c6 =0 + ); 2 = (c6 =0 + ) + c7 0 and > 0; 0 2 (0; 1]. Proof: 1)
! i (t)T ~i (t) 1 + !iT (t)!i (t)
A = B
Bic
A h ;B b
0 A :
c c i i T ;3 i ;1 ;2 c c suitably augmented from i . Clearly, i is now establish the relationship between i
;B
(23)
is A a stable matrix. We ! and xci . It can be shown, by taking the modeling error mi into account and following similar steps in the proof of [16, Th. C.1], that
kxi (t)k c k!i (t)k + mi (t) : 4
1
Then
x (t) = x (t); ! (t); ! (t); c2 k!i (t)k + mi4(t) : c i
Also
(24)
T i
T i;1
T i;2
2
2
0
3
1
5
Before establishing the stability of the system, we now explore some properties of the estimator (7)–(9).
(30)
Vi = 1 ~iT 0i01 ~i : 2
(31)
Using (7) and (30) gives 2 ~T 2 V_ i 0 1 i iT + 1 (mi T) : 2 1 + !i !i 2 1 + !i !i
(32)
From the assumption of the lemma and (17), we have
Then for
Then
mi (t) + ; for t t0 : 1 + !iT !i 1=2 t t0 , (32) becomes ~iT i 2 + 1 ( + )2 : V_ i 0 1 2 1 + !iT !i 2 iT ( )~i ( ) 2 d 1 + !iT !i
t t
(33)
(34)
c + ( + ) (t 0 t ): 2
8
0
(35)
k k
From (9), we can note that (1+! ! ) ; k = 0; 1; 1 1 1 ; n3i are bounded. From (3), (6), (18), and (24), we can have
!_ i 1 + ! (t)!i (t) Now (n +1)
1
j~i (t)j
Then consider the following positive definite function:
T i
k!i (t)k = !i;T (t); !i;T (t); hic T xci (t) + mi (t); iT ; !iT 0 di I; 1 1 1 ; dni 0 I i T T c xci (t) + mi (t) : (26)
1=2
ei;1 = iT ~i + mi (t):
T T i
(25)
k!i (t)k 1 + !iT (t)!i (t) c :
2) From (8) and (10), we have
(k)
where
1=2
1
(20)
i = Ck i (21) where Ck = [0; 0; 1 1 1 ; I; 0 1 1 1 0] with zero being (2ni 01)2(2ni 01) block matrices and I an identity matrix at the (k +1)th position. A i is a stable matrix satisfying CkT (DI 0 A )01 B = Dk Wm (D)I and B = [B ;1 ; B ;2 ; B ;3 ]. Let xci = [ xci ; iT ]T . Then we have x_ ci = Aci xci + Bic ! iT ~i + Bic r + mi3 (t) (22)
(29)
i
1=2
c ( + ) + c : 9
i D Dn 0 Dm (D) I ! i + !_ i i (D) Dm = 0d0i i(1) 0 d1i i(2) 0 1 1 1 0 dni 01 i(n
10
(36)
=
)
+ !_ i :
Therefore, similar bounds to (36) can be obtained for
i(n +1) ; (1 + !iT (t)!i (t))1=2
(1 + !iT !i )1=2 (1 + !iT !i )1=2
d dt
(37)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 7, JULY 1999
and
for 31 and satisfying
d ( ~iT i ) dt (1 + !iT !i )1=2 (k )
k = 0; 1; 1 1 1 ; n3i . Note that ! i = iT (t)~i (t) 2 1 + !iT (t)!i (t) 0
n k=0
8t 2 [tj ; tj + 1tj ]; and for some tj t
1 + ! (t)!i (t)
1=2
c + ( + ) c ; 0
sup
()
sup k!i ( )k M; t M = c12 M0 + c13 with c12
where Proof: The solution of (22) is
(39)
)+ + Bic ri ( ) + mi ( ) 3
=
() =
()
e
eA (t0 ) Bic !iT ( )~i ( )
t
d : c and such (41)
k!i (t)k c ce0 t0t c !i t
+ mi t T t ! i ( )~i ( ) +c ce0 t0 (1 + k!i ( )k ) = k!i ( )k t T ~ + (1 +!ik!( i () i)(k )) = +M + mi ( ) d + mi (t) t c M + c sup k!i ( )k + c ce0 t0 t t T ~ 2 (1 +!ik!( i () i)(k )) = k!i ( )k d + c : (42) 2
1
4
0
0
)
2 1 2
12
3
0
16
5
(
17
0
)
18
2 1 2
After some rearrangement of (42), the Bellman–Grownwall lemma can be applied. Then from Lemma 4.2, we can obtain that
k!i (t)k c M + c + c sup k!i ( )k 19
0
20
21
13
1
21
21 1
1
1
1
To establish the stability result for the general case, we explore the parameter estimator further, and this gives Lemma 4.4 as follows. Lemma 4.4: If k!i t k > M0 for all t t0 , and c13 for all t 2 ; t1 and j 0 t k!j k c12 M0 ; 1 1 1 ; m, and k ! k k!i t k; 0 t k!j k i 0 t 0 t k!i k; 8j 6 i for all t t1 with some t1 t0 , then
sup 12 sup t
() () + [0 ] = sup ( ) = ( ) sup () () = j!i ( )T ~i ( )j d c = + ( + ) t 0 t ; 1 + !iT ( )!i( ) = 5
1 2
t
0
1
2
for
0
t t0
(47)
where
1 = [c6 =0 + (c12 + c13 )](c12 + c13 ):
(48)
t
0
12
(43)
13
8t 2 [t ; t ]: 0
1
(49)
1
For t t1 , (33) becomes valid. Noting that c12 , we can have (49) for all t t0 . Then replacing (33) by (49) and by c12 c13 in the proof of (28), we can establish (47). Remark 4.3: Note that the property in the above lemma is quite similar to (28) in Lemma 4.2 except that is changed to c12 c13 . From Lemmas 4.2–4.4, we can establish our main stability result as follows. Theorem 4.1: Consider the decentralized adaptive system consisting of plant (1) and local adaptive controllers (6)–(9). Under Assumption 2.1, there exists a constant 3 such that for all 3 , we have the following. 1) The closed-loop system is globally stable in the sense that all signals remain bounded 8t and for all finite initial states, any bounded ri , and arbitrarily bounded external disturbances. yi 0 ym satisfies 2) The tracking error ei;1 t
( + )
( + )
t
2 1 2
0
(45)
20
1
2 1 2
=12
15
sup k!i ( )k:
t
0
0
21
jmi (t)j (1 + k!i (t)k ) = (c + c ) + ;
Suppose that the intermediate number M0 is also such that kri k1 ; ; 1 1 1 ; m. Then using (25), (26), (17), (27), and (41) gives
(
M0 + c20 + c21
Proof: By noting the condition of the lemma and using (17), we have
ce0t :
A t
14
19
sup k!i ( )k 1 0c c M + 1 0c c (46) t if 3 with the positive constant 3 satisfying c 3 < 1. Finally, the result is proved by letting 3 = minf3 ; 3 ; g and c = maxf 0cc ; 1g; c = 0cc .
(40)
1.
As Aci is a stable matrix, there exist positive constants that
)
(44)
1
8i = 1; 2; 1 1 1 ; m
t
xci (t) = eA (t0t ) xci (t0
(
sup k!i ( )k c 0
8t 2 [tj ; tj + 1tj ]:
()
3
0
19
12
11 0
0
M0 ; 8i
c22 ( 13 + 32 ) < 31 and 32 on 3 .
Then from (45), we get
Then (28) can be established from (35) and (39). Remark 4.2: 1 can be made arbitrary small by reducing and 2 by making M0 a sufficiently large number. M0 is used here for the purpose of analysis only. It is not a design parameter. From Lemma 4.2, (25), and (26), the stability of the system can be established under a special case. This is presented in the following lemma. Lemma 4.3: Consider the decentralized adaptive system consisting of plant (1) and local adaptive controllers (6)–(9). Suppose that k!i ti0 k M0 and for all t > ti0 , 0 t k!i k k!i t k and i. Then under 0 t k!j k k!i t k for all j 6 Assumption 2.1, there exists a positive constant 31 such that for all 31 the closed-loop system ensures that
( ) = sup
1
t
(38)
we can establish T i
3 where 3 and 3 are sufficiently small constants
with 31 depending on Note that the right side of (43) is nondecreasing. Thus it can be rewritten as
dki i(k) . Thus if
0
j! i (t)T ~i (t)j
1467
t
()= ei; ( ) d + ( + d + ) t 0 ti 2
1
1
2
0
0
0
;
for all
ti0 0 (50)
where 1 ; 2 are positive constants. Proof: 1) To show the boundedness of all the trajectories k!i k; ; ; 1 1 1 ; m, we consider a function k! t k defined as
()
12
i
=
k!(t)k = maxfk! (t)k; k! (t)k; 1 1 1 ; k!m(t)kg: (51) Clearly, the result is proved if k! (t)k is bounded. It can be noted that k! (t)k is continuous and thus, starting with = 0 1
2
0
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and k = 1; 2; 1 1 1 ; we can divide the time axis [0; 1) into the following two subsequences:
0 = [ k01 ; sk ]
+ = (sk ; k )
k
k
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[5] St. Kotsios, “Finite input/output representation of a class of Volterra polynomial systems,” Automatica, vol. 33, no. 2, pp. 257–262, 1997. [6] St. Kotsios and D. Lappas, “A description of 2-dimensional discrete polynomial dynamics,” IMA J. Math. Control and Inform., vol. 13, pp. 409–428, 1996. [7] L. Ljung, System Identification Theory for the User. Englewood Cliffs, NJ: Prentice-Hall, 1987. [8] J. W. Rugh, Nonlinear System Theory. Baltimore, MD: Johns Hopkins Univ. Press, 1981.
Decentralized Model Reference Adaptive Control Without Restriction on Subsystem Relative Degrees Changyun Wen and Yeng Chai Soh
Abstract—When the direct model reference adaptive control (MRAC) scheme with first-order local estimators is employed to design totally decentralized controllers, the stability result can only be applied to a system with all of its nominal subsystem relative degrees less than or equal to two. In this paper, this restriction is relaxed and it is achieved by employing the parameter projection together with static normalization. To implement the local controllers, no a priori knowledge of the subsystem unmodeled dynamics and no information exchange between subsystems are required. Global stability is established for the closed-loop system and small in the mean tracking error is ensured. With this analysis, the class of interactions and subsystem unmodeled dynamics can be enlarged to include those having infinite memory. Index Terms—Adaptive control, decentralized control, robustness, stability.
I. INTRODUCTION Decentralized adaptive control is an important control scheme for large scale systems, and it has continued to receive a lot of attention from control researchers over the last few decades. However, only a limited number of stability results in this area are available due to the difficulties in the analysis of ignored interactions. The first batch of results were obtained based on the direct model reference adaptive control (MRAC) approach [1]–[3]. A strong assumption for these results is that relative degrees of all the nominal subsystem models should be less than or equal to two. The stability results using the indirect pole assignment design scheme were reported later in [4]–[6] where there is no restriction on the relative degrees of the nominal subsystem. Recently, efforts on relaxing the subsystem relative degrees in the case of employing the direct adaptive control scheme have been made by using some advanced adaptive strategies. The concept of high-order tuners in [7] was applied to achieve this in [8] and [9]. In this case, a local dynamic estimator with the subsystem relative degree as its order is designed to identify the unknown parameters of each subsystem. The integrator backstepping technique of [10] was also successfully utilized to reach a similar goal in [11]–[13]. To obtain the final control for each subsystem, a number of iterative design steps should be involved to calculate Manuscript received April 24, 1996; revised March 16, 1998. Recommended by Associate Editor, M. Krstic. This work was supported in part by NTU under the Applied Research Project Grant RP 23/92. The authors are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore (e-mail: [email protected]). Publisher Item Identifier S 0018-9286(99)04544-4.
some intermediate virtual control signals. As commented in [9], the unmodeled interactions must satisfy certain structural conditions when these advanced schemes are used. However, for the conventional MRAC scheme, the problem of the relaxation of the subsystem relative degrees is still unsolved. Due to the simplicity of the conventional MRAC scheme, the solution to such a problem is of practical interest. In [14], Datta and Ioannou applied the normalization technique used in the single-loop robust adaptive controller design to achieve the required relaxation. But the proposed local normalizing signals require information from the other subsystems to bound the effects of interactions from these subsystems. Thus, only partially decentralized adaptive controllers can be designed. In this paper, the problem will be solved with totally decentralized controllers by employing the parameter projection together with a static normalization technique. Global stability is established for the closed-loop system and small in the mean tracking error is ensured. With our analysis, the class of interactions and subsystem unmodeled dynamics can be enlarged to include those having infinite memory. The remaining part of the paper is organized as follows. Section II gives the class of systems to be controlled and Section III presents the decentralized controllers. The analysis of the closed-loop system and the main result are given in Section IV. Finally, the paper is concluded in Section V. II. SYSTEM MODELS
AND
ASSUMPTIONS
In this paper, the class of interconnected systems considered consists of m single-input/single-output subsystems. The ith subsystem is modeled as yi (t)
= Hi (D)ui (t) + Hi (D) +
m j =1
m
j =1
ij (D)[uj (t) + yj (t)] ij H
ij (D)[uj (t) + yj (t)] + di (t) ij H
(1)
for i; j = 1; 1 1 1 ; m, where yi ; ui ; and di are, respectively, the output, (D ) input, and disturbance of the ith subsystem. In (1), Hi (D) = B A (D) and it is the reduced-order transfer function of subsystem i with
= Dn + ani 01 Dn 01 + 1 1 1 + a0i m m + bm 01 Dm 01 + 1 1 1 + b0 Bi (D ) = bi D i i Ai (D )
where D denotes the differentiation operator, mi < ni ; ij ; ij are ij (D) denote the subsystem interactions ij (D) and H constants, and H if i 6= j and unmodeled dynamics if i = j . Now, a reference model given below is chosen for the ith subsystem
i
ym (t)
i (D)ri (t) = Wm
(2)
i (D) = km i 1 and ri is an external reference input where Wm D (D) i is a constant and Dm i (D) is a monic Hurwitz signal. Here, km i (D) = Dn + polynomial of degree n3i = ni 0 mi , i.e., Dm i n 0 1 i i + 1 1 1 + d1 D + d0 . The control problem is to design dn 01 D totally decentralized controllers for plant (1) such that the closed-loop system is stable in the sense that all signals in the system are bounded for arbitrary bounded ri and initial conditions, and the output yi (t) i (t) of the model (2) as closely as possible. To follows the output ym solve the control problem, the following assumptions are made for the plant given in (1).
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Assumption 2.1: A1) Bi (D) is Hurwitz. A2) An upper bound for ni , the nominal relative degree n3i = ni 0 mi of subsystem i, and the sign of the high frequency gain sgn(bm i ) are known. Furthermore, the coefficients of Ai (D ) and Bi (D ) are inside a known compact convex region Ci . ij (D) and H ij (D) are stable, and Hi (D)H ij (D) and A3) H H ij (D ) are strictly proper. A4) di (t) is bounded. Remarks 2.1: 1) Note that there is no restriction on the nominal subsystem relative degrees n3i ; i = 1; 2; 1 1 1 ; m. 2) While modeling errors are assumed to satisfy A3) and A4), no a priori knowledge is required from them for the implementation of the adaptive controllers given in the later sections. Assumption A2) also implies a known lower bound for jbm i j.
Remarks 3.1: 1) As can be noted from (6)–(9), the normalization is static. Also the implementation of local adaptive controllers does not require any information exchange between subsystems and the a priori knowledge on subsystem unmodeled dynamics. 2) The results for the adaptive controller in Case 2 can be obtained by following the similar analyses as in Case 1 and [15]. Thus we just focus our attention on Case 1 without any further elaboration on Case 2.
III. DESIGN OF ROBUST DECENTRALIZED ADAPTIVE CONTROLLERS For each subsystem, we define the following filtered variables: ! _ i;1
= 3i !i;1 + qi yi ;
! _ i;2
= 3i !i;2 + qi ui
(3)
(3i ; qi ) is a controllable pair satisfying (DI 0 3i )01 qi = 1 [Dn 02 ; 1 1 1 ; 1]T (4) Fi (D ) with Fi (D) as an arbitrary Hurwitz polynomial of order ni 0 1. Both 3i and qi are chosen by users. Let (5) ! iT = !i;T1 ; !i;T2 ; yi : where
Then the control is given as
= ! iT i + ci;0 ri (6) T T T where i (t) = [i;1 (t); i;2 (t); i;3 (t)] is a (2ni 0 1)-dimensional control parameter vector and ci;0 (t) is a feedforward parameter ui
scalar. From [16], it can be shown that a desired parameter vector 3 of i and a desired parameter ci;3 0 of ci;0 exist, and they can be obtained when the nominal transfer function Hi (D) of the ith subsystem is known. When Hi (D) is unknown, an adaptive law is required to update i and ci;0 . To achieve this and to ensure the robustness of the adaptive controller in the presence of modeling errors including interactions, subsystem unmodeled dynamics, and external disturbances, we introduce parameter projection operation to the adaptive law. The adaptive law to tune i and ci;0 is divided into the following two cases. i is chosen to be Case 1: bim = 1. In this case, ci;0 = 1 if km one and
i
_i
=P
0i ei;1 i i ; 0 1 + !iT !i
0i = 0iT > 0 e i;1 = yi 0 ym + iT i 0 vi ; i T vi = Wm (D )i ! i T T T (1) T !i = ! i ; i ; i ; 1 1 1 ; i(n
(7)
where
i
= Wmi (D)I ! i (8)
)T
(9)
and P denotes the projection operation defined in [17] or [18]. Case 2: bim is unknown. In this case, ci;0 (t) is unknown and needs to be updated. The local adaptive law in this case is a modified version of that in [15] by changing the modification and the normalizing signal appropriately as in Case 1.
IV. STABILITY OF THE DECENTRALIZED ADAPTIVE CONTROL SYSTEMS We need to establish the robustness of the local adaptive controllers in the presence of ignored interactions, unmodeled dynamics, and external disturbances. Before doing this, some preliminary analysis is required. From (1)–(6), it can be shown that the ith subsystem can be expressed as yi
where
= Wmi (D) ! iT ~i + ri + mi (t)
(10)
= i 0 i3 i 3 01 mi (t) = i (t) + 1 + Wm (D )i;1 (DI 0 3i ) qi ~i
i (t)
=
m
j =1
(11) di
1ij (D)[uj (t) + yj (t)]
(12) (13)
1ij (D) = Wmi (D)ij H ij (D) 1 0 i;3 2 (DI 0 3i )01 qi + ij H ij (D) 1 + Wmi (D)(i;3 3 + i;3 1 (DI 0 3i )01 qi
:
(14) Clearly, 1ij (D) is strictly proper and stable from Assumption 2.1. From (12), we have the following result. Lemma 4.1: The modeling error mi (t) in (12) satisfies
jmi (t)j
m
ij j =1
sup k!j ( )k + d0 t
(15)
0
where ij and d0 are some nonnegative constants. Proof: Let Vi (D) be an arbitrary Hurwitz polynomial defined as Vi (D ) T vi
= Dn 02 + vi;n 03 Dn 03 + 1 1 1 + vi;0 = [1; vi;n 03 ; 1 1 1 ; vi;0 ]:
Then i (t)
= =
m j =1 m j =1
1ij Fj
Vj (uj Vj Fj
1ij Fj Vj
+ yj )
T vj (!j;1
+ !j;2 ) F
:
(16)
Thus from the stability and properness of 1ij V , the result can be established from (16). Remarks 4.1: 1) In the proof of Lemma 4.1, the effects of some exponentially decaying terms due to nonzero initial conditions have been absorbed by d0 . 2) The constant ij indicates the strength of the interactions between subsystems i and j when i 6= j , and the unmodeled dynamics of the ith subsystem are coupled to the nominal model when i = j . 3) In terms of the bounding signals, the bound for the modeling error in (15) allows the effects of the unmodeled dynamics and interactions to have infinite memory, thus it is looser than those given in existing literature, such as [1] and [2]. The class of modeling errors considered can also be enlarged to include any nonlinear unmodeled dynamics satisfying (15).
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Now, if sup0 t k!i ( )k = k!i (t)k and sup0 t k!j ( )k sup0 t k!i ( )k for all j 6= i and t > ti0 , then (15) becomes jmi (t)j k!i (t)k + d0 ; for all t ti0 (17) where is a nonnegative constant depending on ij . From (17), some useful properties of the local estimators can be obtained. In the remaining part of this section, we will use mki ; k = 1; 1 1 1 ; 5 to denote Mik (D)mi where Mik (D) is a vector containing proper stable transfer functions. Thus, mik (t); k = 1; 1 1 1 ; 5 also satisfy the same bounds as in (17) under the same conditions. Note that the constants and d0 in (17) have been made uniform for all i = 1; 2; 1 1 1 ; m and k = 1; 1 1 1 ; 5. Also in this section, all cj ; j = 1; 2; 1 1 1 stand for generic constants. We now derive an equation to describe the closed-loop system of the ith loop. It can be shown that the plant in the ith loop has the following state representation:
x_ i = Ai xi + bi ui + mi1 (t); yi = hiT xi + mi2 (t) (18) T where (Ai ; bi ; hi ) is a minimal state representation of Hi (D). Note ij (D). that Dmi2 also satisfies (17) from the strict properness of H Now the closed-loop system of the ith loop can be described as x_ ci = Aci xci + bic ! iT ~i + bic ri + mi1 (t) (19) yi = hic T xci + mi2 (t) where Aci is a stable matrix satisfying (hic )T (DI 0 Aci )01 bic = Wmi (D) and xci = [xTi ; !i;T1 ; !i;T2 ]T . (n 01) (1) Let iT = [iT ; i ; 1 1 1 ; i ]. Then i(k) = Dk Wmi (D) 3 I ! i ; k = 0; 1; 1 1 1 ; ni 0 1 can have the following realization: _i = A i + B ! i = A i + B ;1 !i;1 + B ;2 !i;2 + B ;3 hiT xi + B ;3 mi2 (t)
Lemma 4.2: The estimator (7)–(9), when applied to the plant given in (1), has the following properties. 1) j! i (t)T ~i (t)j c4 ; for t 0: (27) 1=2 2)
1 + !iT (t)!i (t) Suppose M0 is a positive constant s.t. d0 =M0 . If k!i (t)k > M0 ; sup0 t k!i ( )k = k!i (t)k and sup0 t k!j ( )k sup0 t k!i ( )k 8j 6= i and for all t > t0 with some t0 0, then t j! i ( )T ~i ( )j d c5 =0 + ( 1 + 2 )(t 0 t0 ); t 1 + !iT ( )!i ( ) 1=2 for t t0 (28) where
1 = (c6 =0 + ); 2 = (c6 =0 + ) + c7 0 and > 0; 0 2 (0; 1]. Proof: 1)
! i (t)T ~i (t) 1 + !iT (t)!i (t)
A = B
Bic
A h ;B b
0 A :
c c i i T ;3 i ;1 ;2 c c suitably augmented from i . Clearly, i is now establish the relationship between i
;B
(23)
is A a stable matrix. We ! and xci . It can be shown, by taking the modeling error mi into account and following similar steps in the proof of [16, Th. C.1], that
kxi (t)k c k!i (t)k + mi (t) : 4
1
Then
x (t) = x (t); ! (t); ! (t); c2 k!i (t)k + mi4(t) : c i
Also
(24)
T i
T i;1
T i;2
2
2
0
3
1
5
Before establishing the stability of the system, we now explore some properties of the estimator (7)–(9).
(30)
Vi = 1 ~iT 0i01 ~i : 2
(31)
Using (7) and (30) gives 2 ~T 2 V_ i 0 1 i iT + 1 (mi T) : 2 1 + !i !i 2 1 + !i !i
(32)
From the assumption of the lemma and (17), we have
Then for
Then
mi (t) + ; for t t0 : 1 + !iT !i 1=2 t t0 , (32) becomes ~iT i 2 + 1 ( + )2 : V_ i 0 1 2 1 + !iT !i 2 iT ( )~i ( ) 2 d 1 + !iT !i
t t
(33)
(34)
c + ( + ) (t 0 t ): 2
8
0
(35)
k k
From (9), we can note that (1+! ! ) ; k = 0; 1; 1 1 1 ; n3i are bounded. From (3), (6), (18), and (24), we can have
!_ i 1 + ! (t)!i (t) Now (n +1)
1
j~i (t)j
Then consider the following positive definite function:
T i
k!i (t)k = !i;T (t); !i;T (t); hic T xci (t) + mi (t); iT ; !iT 0 di I; 1 1 1 ; dni 0 I i T T c xci (t) + mi (t) : (26)
1=2
ei;1 = iT ~i + mi (t):
T T i
(25)
k!i (t)k 1 + !iT (t)!i (t) c :
2) From (8) and (10), we have
(k)
where
1=2
1
(20)
i = Ck i (21) where Ck = [0; 0; 1 1 1 ; I; 0 1 1 1 0] with zero being (2ni 01)2(2ni 01) block matrices and I an identity matrix at the (k +1)th position. A i is a stable matrix satisfying CkT (DI 0 A )01 B = Dk Wm (D)I and B = [B ;1 ; B ;2 ; B ;3 ]. Let xci = [ xci ; iT ]T . Then we have x_ ci = Aci xci + Bic ! iT ~i + Bic r + mi3 (t) (22)
(29)
i
1=2
c ( + ) + c : 9
i D Dn 0 Dm (D) I ! i + !_ i i (D) Dm = 0d0i i(1) 0 d1i i(2) 0 1 1 1 0 dni 01 i(n
10
(36)
=
)
+ !_ i :
Therefore, similar bounds to (36) can be obtained for
i(n +1) ; (1 + !iT (t)!i (t))1=2
(1 + !iT !i )1=2 (1 + !iT !i )1=2
d dt
(37)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 7, JULY 1999
and
for 31 and satisfying
d ( ~iT i ) dt (1 + !iT !i )1=2 (k )
k = 0; 1; 1 1 1 ; n3i . Note that ! i = iT (t)~i (t) 2 1 + !iT (t)!i (t) 0
n k=0
8t 2 [tj ; tj + 1tj ]; and for some tj t
1 + ! (t)!i (t)
1=2
c + ( + ) c ; 0
sup
()
sup k!i ( )k M; t M = c12 M0 + c13 with c12
where Proof: The solution of (22) is
(39)
)+ + Bic ri ( ) + mi ( ) 3
=
() =
()
e
eA (t0 ) Bic !iT ( )~i ( )
t
d : c and such (41)
k!i (t)k c ce0 t0t c !i t
+ mi t T t ! i ( )~i ( ) +c ce0 t0 (1 + k!i ( )k ) = k!i ( )k t T ~ + (1 +!ik!( i () i)(k )) = +M + mi ( ) d + mi (t) t c M + c sup k!i ( )k + c ce0 t0 t t T ~ 2 (1 +!ik!( i () i)(k )) = k!i ( )k d + c : (42) 2
1
4
0
0
)
2 1 2
12
3
0
16
5
(
17
0
)
18
2 1 2
After some rearrangement of (42), the Bellman–Grownwall lemma can be applied. Then from Lemma 4.2, we can obtain that
k!i (t)k c M + c + c sup k!i ( )k 19
0
20
21
13
1
21
21 1
1
1
1
To establish the stability result for the general case, we explore the parameter estimator further, and this gives Lemma 4.4 as follows. Lemma 4.4: If k!i t k > M0 for all t t0 , and c13 for all t 2 ; t1 and j 0 t k!j k c12 M0 ; 1 1 1 ; m, and k ! k k!i t k; 0 t k!j k i 0 t 0 t k!i k; 8j 6 i for all t t1 with some t1 t0 , then
sup 12 sup t
() () + [0 ] = sup ( ) = ( ) sup () () = j!i ( )T ~i ( )j d c = + ( + ) t 0 t ; 1 + !iT ( )!i( ) = 5
1 2
t
0
1
2
for
0
t t0
(47)
where
1 = [c6 =0 + (c12 + c13 )](c12 + c13 ):
(48)
t
0
12
(43)
13
8t 2 [t ; t ]: 0
1
(49)
1
For t t1 , (33) becomes valid. Noting that c12 , we can have (49) for all t t0 . Then replacing (33) by (49) and by c12 c13 in the proof of (28), we can establish (47). Remark 4.3: Note that the property in the above lemma is quite similar to (28) in Lemma 4.2 except that is changed to c12 c13 . From Lemmas 4.2–4.4, we can establish our main stability result as follows. Theorem 4.1: Consider the decentralized adaptive system consisting of plant (1) and local adaptive controllers (6)–(9). Under Assumption 2.1, there exists a constant 3 such that for all 3 , we have the following. 1) The closed-loop system is globally stable in the sense that all signals remain bounded 8t and for all finite initial states, any bounded ri , and arbitrarily bounded external disturbances. yi 0 ym satisfies 2) The tracking error ei;1 t
( + )
( + )
t
2 1 2
0
(45)
20
1
2 1 2
=12
15
sup k!i ( )k:
t
0
0
21
jmi (t)j (1 + k!i (t)k ) = (c + c ) + ;
Suppose that the intermediate number M0 is also such that kri k1 ; ; 1 1 1 ; m. Then using (25), (26), (17), (27), and (41) gives
(
M0 + c20 + c21
Proof: By noting the condition of the lemma and using (17), we have
ce0t :
A t
14
19
sup k!i ( )k 1 0c c M + 1 0c c (46) t if 3 with the positive constant 3 satisfying c 3 < 1. Finally, the result is proved by letting 3 = minf3 ; 3 ; g and c = maxf 0cc ; 1g; c = 0cc .
(40)
1.
As Aci is a stable matrix, there exist positive constants that
)
(44)
1
8i = 1; 2; 1 1 1 ; m
t
xci (t) = eA (t0t ) xci (t0
(
sup k!i ( )k c 0
8t 2 [tj ; tj + 1tj ]:
()
3
0
19
12
11 0
0
M0 ; 8i
c22 ( 13 + 32 ) < 31 and 32 on 3 .
Then from (45), we get
Then (28) can be established from (35) and (39). Remark 4.2: 1 can be made arbitrary small by reducing and 2 by making M0 a sufficiently large number. M0 is used here for the purpose of analysis only. It is not a design parameter. From Lemma 4.2, (25), and (26), the stability of the system can be established under a special case. This is presented in the following lemma. Lemma 4.3: Consider the decentralized adaptive system consisting of plant (1) and local adaptive controllers (6)–(9). Suppose that k!i ti0 k M0 and for all t > ti0 , 0 t k!i k k!i t k and i. Then under 0 t k!j k k!i t k for all j 6 Assumption 2.1, there exists a positive constant 31 such that for all 31 the closed-loop system ensures that
( ) = sup
1
t
(38)
we can establish T i
3 where 3 and 3 are sufficiently small constants
with 31 depending on Note that the right side of (43) is nondecreasing. Thus it can be rewritten as
dki i(k) . Thus if
0
j! i (t)T ~i (t)j
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t
()= ei; ( ) d + ( + d + ) t 0 ti 2
1
1
2
0
0
0
;
for all
ti0 0 (50)
where 1 ; 2 are positive constants. Proof: 1) To show the boundedness of all the trajectories k!i k; ; ; 1 1 1 ; m, we consider a function k! t k defined as
()
12
i
=
k!(t)k = maxfk! (t)k; k! (t)k; 1 1 1 ; k!m(t)kg: (51) Clearly, the result is proved if k! (t)k is bounded. It can be noted that k! (t)k is continuous and thus, starting with = 0 1
2
0
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and k = 1; 2; 1 1 1 ; we can divide the time axis [0; 1) into the following two subsequences:
0 = [ k01 ; sk ]
+ = (sk ; k )
k
k
