Time Delay Systems Part 8 pdf

the unknown scalar function τ j (t) denotes any nonnegative, continuous and bounded time-varying delay satisfying ˙ τ j (t) ≤ ¯ τ j < 1, (4) where ¯ τ j are known constants. For each decoupled local system, we make the following assumptions. Assumption 1: The triple (A i , b i , c i ) are completely controllable and observable. Assumption 2: For every 1 ≤ i ≤ N, the polynomial b i,m i s m i + ···+ b i,1 s + b i,0 is Hurwitz. The sign of b i,m i and the relative degree ρ i (= n i −m i ) are known. Assumption 3: The nonlinear interaction terms satisfy |f ij (t, y j )|≤ ¯ γ ij ¯ f j (t, y j )y j ,(5) where ¯ γij are constants denoting the strength of interactions, and ¯ f j (y j ), j = 1, 2, . . . , N are known positive functions and differentiable at least ρ i times. Assumption 4: The unknown functions h ij (y j (t)) satisfy the following properties |h ij (y j (t))|≤ ¯ ι ij ¯ h j (y j (t))y j ,(6) where ¯ h j are known positive functions and differentiable at least ρ i times, and ¯ ι k ij are positive constants. Remark 1. The effects of the nonlinear interactions f ij and time-delay functions h ij from other subsystems to a local subsystem are bounded by functions of the output of this subsystem. With these conditions, it is possible for the designed local controller to stabilize the interconnected systems with arbitrary strong subsystem interactions and time-delays. The control objective is to design a decentralized adaptive stabilizer for a large scale system (1) with unknown time-varying delay satisfying Assumptions 1-4 such that the closed-loop system is stable. 3. Design of adaptive controllers 3.1 Local state est imation filters In this section, decentralized filters using only local input and output will be designed to estimate the unmeasured states of each local system. For the ith subsystem, we design the filters as ˙v i,ι = A i,0 v i,ι + e n i ,(n i −ι) u i , ι = 0, ,m i (7) ˙ ξ i,0 = A i,0 ξ i,0 + k i y i ,(8) ˙ Ξ i = A i,0 Ξ i + Φ i (y i ),(9) where v i,ι ∈ n i , ξ i,0 ∈ n i , Ξ i ∈ n i ×r i ,thevectork i =[k i,1 , ,k i,n i ] T ∈ n i is chosen such that the matrix A i,0 = A i −k i (e n i ,1 ) T is Hurwitz, and e i,k denotes the kth coordinate vector in  i .ThereexistsaP i such that P i A i,0 +(A i,0 ) T P i = −3I, P i = P T i > 0. With these designed filters, our state estimate is ˆx i (t)=ξ i,0 + Ξ i θ i + m i ∑ k=0 b i,k v i,k , (10) 129 Decentralized Adaptive Stabilization for Large-Scale Systems with Unknown Time-Delay and the state estimation error  i = x i − ˆx i satisfies ˙ i = A i,0  i + N ∑ j=1 f ij (t, y j )+ N ∑ j=1 h ij (y j (t −τ j (t))). (11) Let V  i =  T i P i  i .Itcanbeshownthat ˙ V  i ≤− T i  i + 2N  P i  2 N ∑ j=1  f ij (t, y j )  2 +2N  P i  2 N ∑ j=1  h ij (y j (t −τ j (t)))  2 . (12) Now system (1) is expressed as ˙ y i = b i,m i v i,(m i ,2) + ξ i,(0,2) + ¯ δ T i Θ i +  i,2 + N ∑ j=1 f ij,1 (t, y j ) + N ∑ j=1 h ij,1  y j (t −τ j (t))  , (13) ˙ v i,(m i ,q) = v i,(m i ,q+1) −k i,q v i,(m i ,1) , q = 2, ,ρ i −1 (14) ˙ v i,(m i ,ρ i ) = v i,(m i ,ρ i +1) −k i,ρ i v i,(m i ,1) + u i , (15) where ¯ δ i =[0, v i,(m i −1,2) , ,v i,(0,2) , Ξ i,2 + Φ i,1 ] T , Θ i =[b i,m i , ,b i,0 , θ T i ] T , (16) and v i,(m i ,2) ,  i,2 , ξ i,(0,2) , Ξ i,2 denote the second entries of v i,m i ,  i , ξ i,0 , Ξ i respectively, f ij,1 (t, y j ) and h ij,1 (y j (t − τ j (t))) are respectively the first elements of vectors f ij (t, y j ) and h ij (y j (t − τ j (t))). Remark 2. It is worthy to point out that the inputs to the designed filters (7)-(9) are only the local input u i and output y i and thus t otally decentralized. Remark 3. Even though the estimated state is given in (10), it is still unknown and thus not employed in our controller design. Instead, the outputs v i,ι , ξ i,0 and Ξ i from filters (7)-(9) are used to design controllers, while the state estimation error (11) will be considered in system analysis. 3.2 Adaptive decentralized controller design In this section, we develop an adaptive backstepping design scheme for decentralized output tracking. There is no a priori information required from system parameter Θ i and thus they can be allowed totally uncertain. As usual in backstepping approach in Krstic et al. (1995), the following change of coordinates is made. z i,1 = y i , (17) z i,q = v i,(m i ,q) −α i,q−1 , q = 2, 3, . . . , ρ i , (18) where α i,q−1 is the virtual control at the q-th step of the ith loop and will be determined in later discussion, ˆ p i is the estimate of p i = 1/b i,m i . To illustrate the controller design procedures, we now give a brief description on the first step. 130 Time-Delay Systems • Step 1: Starting with the equations for the tracking error z i,1 obtained from (13), (17) and (18), we get ˙ z i,1 = b i,m i v i,(m i ,2) + ξ i,(0,2) + ¯ δ T i Θ i +  i,2 + N ∑ j=1 f ij,1 (t, y j ) + N ∑ j=1 h ij,1 (t, y j (t −τ j (t))) = b i,m i α i,1 + b i,m i z i,2 + ξ i,(0,2) + ¯ δ T i Θ i +  i,2 + N ∑ j=1 f ij,1 (t, y j ) + N ∑ j=1 h ij,1 (t, y j (t −τ j (t))). (19) The virtual control law α i,1 is designed as α i,1 = ˆ p i ¯ α i,1 , (20) ¯ α i,1 = −  c i,1 + l i,1  z i,1 −l ∗ i z i,1  ¯ f i (y i )  2 −λ ∗ i z i,1  ¯ h i (y i )  2 −ξ i,(0,2) − ¯ δ T i ˆ Θ i , (21) where c i,1 , l i,1 , l ∗ i and λ ∗ i are positive design parameters, ˆ Θ i and ˆ p i are the estimates of Θ i and p i , respectively. Using ˜ p i = p i − ˆ p i , we obtain b i,m i α i,1 = b i,m i ˆ p i ¯ α i,1 = ¯ α i,1 −b i,m i ˜ p i ¯ α i,1 , (22) ¯ δ T i ˜ Θ i + b i,m i z i,2 = ¯ δ T i ˜ Θ i + ˜ b i,m i z i,2 + ˆ b i,m i z i,2 = ¯ δ T i ˜ Θ i +(v i,(m i ,2) −α i,1 )(e (r i +m i +1),1 ) T ˜ Θ i + ˆ b i,m i z i,2 =(δ i − ˆ p i ¯ α i,1 e (r i +m i +1),1 ) T ˜ Θ i + ˆ b i,m i z i,2 , (23) where δ i =[v i,(m i ,2) , v i,(m i −1,2) , ,v i,(0,2) , ξ i,2 + Φ i,1 ] T . (24) From (20)-(23), (19) can be written as ˙ z i,1 = −c i,1 z i,1 −l i,1 z i,1 −l ∗ i z i,1  ¯ f i (y i )  2 −λ ∗ i z i,1  ¯ h i (y i )  2 + i,2 +(δ i − ˆ p i ¯ α i,1 e r i +m i +1,1 ) T ˜ Θ i −b i,m i ¯ α i,1 ˜ p i + ˆ b i,m i z i,2 + N ∑ j=1 f ij,1 (t, y j )+ N ∑ j=1 h ij,1 (t, y j (t −τ j (t))), (25) where ˜ Θ i = Θ i − ˆ Θ i ,ande (r i +m i +1),1 ∈ r i +m i +1 . We now consider the Lyapunov function V 1 i = 1 2 (z i,1 ) 2 + 1 2 ˜ Θ T i Γ −1 i ˜ Θ i + | b i,m i | 2γ  i ( ˜ p i ) 2 + 1 2 ¯ l i,1 V  i , (26) 131 Decentralized Adaptive Stabilization for Large-Scale Systems with Unknown Time-Delay where Γ i is a positive definite design matrix and γ  i is a positive design parameter. Examining the derivative of V 1 i gives ˙ V 1 i = z i,1 ˙ z i,1 − ˜ Θ T i Γ −1 i ˙ ˆ Θ i − | b i,m i | γ  i ˜ p i ˙ ˆ p i + 1 2 ¯ l i,1 ˙ V  i ≤−c i,1 (z i,1 ) 2 −l i,1 (z i,1 ) 2 −l ∗ i (z i,1 ) 2  ¯ f i (z i,1 )  2 −λ ∗ i (z i,1 ) 2  ¯ h i (y i )  2 − 1 2 ¯ l i,1  T i  i + ˆ b i,m i z i,1 z i,2 −|b i,m i | ˜ p i 1 γ  i [γ  i sgn(b i,m i ) ¯ α i,1 z i,1 + ˙ ˆ p i ] + ˜ Θ T i Γ −1 i [Γ i (δ i − ˆ p i ¯ α i,1 e (r i +m i +1),1 )z i,1 − ˙ ˆ Θ i ] +( N ∑ j=1 f ij,1 (t, y j )+ N ∑ j=1 h ij,1 (t, y j (t −τ j (t))) +  i,2 )z i,1 + 1 ¯ l i,1 N  P i  2   N ∑ j=1 h ij (t, y j (t −τ j (t)))  2 + N ∑ j=1  f ij (t, y j )  2  . (27) Then we choose ˙ ˆ p i = −γ  i sgn(b i,m i ) ¯ α i,1 z i,1 , (28) τ i,1 =  δ i − ˆ p i ¯ α i,1 e (r i +m i +1),1  z i,1 . (29) Let l i,1 = 3 ¯ l i,1 and using Young’s inequality we have − ¯ l i,1 (z i,1 ) 2 + N ∑ j=1 f ij,1 (t, y j )z i,1 ≤ N 4 ¯ l i,1 N ∑ j=1  f ij,1 (t, y j )  2 , (30) − ¯ l i,1 (z i,1 ) 2 + N ∑ j=1 h ij,1 (t, y j (t −τ j (t)))z i,1 ≤ N 4 ¯ l i,1  N ∑ j=1 h ij,1 (t, y j (t −τ j (t)))  2 , (31) − ¯ l i,1 (z i,1 ) 2 +  i,2 z i,1 − 1 4 ¯ l i,1  T i  i ≤− ¯ l i,1 (z i,1 ) 2 +  i,2 z i,1 − 1 4 ¯ l i,1 ( i,2 ) 2 = − ¯ l i,1 (z i,1 − 1 2 ¯ l i,1  i,2 ) 2 ≤ 0. (32) Substituting (28)-(32) into (27) gives ˙ V 1 i ≤−c i,1 (z i,1 ) 2 − 1 4 ¯ l i,1  T i  i −l ∗ i (z i,1 ) 2  ¯ f i (y i )  2 −λ ∗ i (z i,1 ) 2  ¯ h i (y i )  2 + ˆ b i,m i z i,1 z i,2 + ˜ Θ T i (τ i,1 −Γ −1 i ˙ ˆ Θ i )+ N ¯ l i,1  P i  2 N ∑ j=1  f ij (t, y j )  2 + N 4 ¯ l i,1 N ∑ j=1  f ij,1 (t, y j )  2 + N ¯ l i,1  P i  2 N ∑ j=1  h ij (t, y j (t −τ j (t)))  2 + N 4 ¯ l i,1  N ∑ j=1 h ij,1 (t, y j (t −τ j (t)))  2 . (33) 132 Time-Delay Systems • Step q ( q = 2, ,ρ i ,i= 1, ,N): Choose virtual control laws α i,2 = − ˆ b i,m i z i,1 −  c i,2 + l i,2  ∂α i,1 ∂y i  2  z i,2 + ¯ B i,2 + ∂α i,1 ∂ ˆ Θ i Γ i τ i,2 , (34) α i,q = −z i,q−1 −  c i,q + l i,q  ∂α i,q−1 ∂y i  2  ]z i,q + ¯ B i,q + ∂α i,q−1 ∂ ˆ Θ i Γ i τ i,q −  q −1 ∑ k=2 z i,k ∂α i,k−1 ∂ ˆ Θ i  Γ i ∂α i,q−1 ∂y i δ i , (35) τ i,q = τ i,q−1 − ∂α i,q−1 ∂y i δ i z i,q , (36) where c q i , l i,q , q = 3, ,ρ i are positive design parameters, and ¯ B i,q , q = 2, ,ρ i denotes some known terms and its detailed structure can be found in Krstic et al. (1995). Then the local control and parameter update laws are finally given by u i = α i,ρ i −v i,(m i ,ρ i +1) , (37) ˙ ˆ Θ i = Γ i τ i,ρ i . (38) Remark 4. The crucial terms l ∗ i z i,1  ¯ f i (y i )  2 in (21) an d λ ∗ i z i,1  ¯ h i (y i )  2 are proposed in the controller design to compensate for the effects of interactions from other subsystems or the un-modelled part of its own subsystem, and for the effects of time-delay functions, respectively. The detailed analysis will be given in Section 4. Remark 5. When going through the details of the design procedures, we note that in the equations concerning ˙ z i,q , q = 1, 2, . . . , ρ i , just functions ∑ N j =1 f ij,1 (t, y j ) from the interactions and ∑ N j =1 h ij,1 (t, y j (t −τ j (t))) appear, and they are always together with  i,2 . This is because only ˙ y i from theplantmodel(1)wasusedinthecalculationof ˙ α i,q for steps q = 2, ,ρ i . 4. Stability analysis In this section, the stability of the overall closed-loop system consisting of the interconnected plants and decentralized controllers will be established. Now we define a Lyapunov function of decentralized adaptive control system as V i = ρ i ∑ q =1  1 2 (z i,q ) 2 + 1 2 ¯ l i,q  T i P i  i  + 1 2 ˜ Θ T i Γ −1 i ˜ Θ i + | b i,m i | 2γ  i ˜ p 2 i . (39) From (12), (20), (33), (35)-(38), and (49), the derivative of V i in (39) satisfies ˙ V i ≤− ρ i ∑ q =1 c i,q (z i,q ) 2 −l ∗ i (z i,1 ) 2 ( ¯ f i (y i )) 2 −λ ∗ i (z i,1 ) 2  ¯ h i ( y i )  2 + ρ i ∑ q =1 1 ¯ l i,q N  P i  2 ⎛ ⎝ N ∑ j=1  h ij (t, y j (t − τ j ))  2 + N ∑ j=1  f ij (t, y j )  2 ⎞ ⎠ 133 Decentralized Adaptive Stabilization for Large-Scale Systems with Unknown Time-Delay + 1 4 ¯ l i,1 ⎛ ⎝ N N ∑ j=1  f ij,1 (t, y j )  2 +N N ∑ j=1  h ij,1 (t, y j (t −τ j ))  2 ⎞ ⎠ − 1 4 ¯ l i,1  T i  i + ρ i ∑ q =2  −l i,q  ∂α i,q−1 ∂y i  2 (z i,q ) 2 − 1 2 ¯ l i,q  T i  i + ∂α i,q−1 ∂y i ⎛ ⎝ N ∑ j=1 f ij,1 (t, y j )+ N ∑ j=1 h ij,1  t, y j (t − τ j )  +  i,2 ⎞ ⎠ z i,q ⎤ ⎦ . (40) Using Young’s inequality and let l i,q = 3 ¯ l i,q ,wehave − ¯ l i,q  ∂α i,q−1 ∂y i  2 (z i,q ) 2 + ∂α i,q−1 ∂y i N ∑ j=1 f ij,1 (t, y j )z i,q ≤ N 4 ¯ l i,q N ∑ j=1  f ij,1 (t, y j )  2 , (41) − ¯ l i,q  ∂α i,q−1 ∂y i  2 (z i,q ) 2 + ∂α i,q−1 ∂y i  i,2 z i,q − 1 4 ¯ l i,q  T i  i ≤ 0, (42) − ¯ l i,q  ∂α i,q−1 ∂y i  2 (z i,q ) 2 + ∂α i,q−1 ∂y i N ∑ j=1 h ij,1 (t, y j (t − τ j ))z i,q ≤ N 4 ¯ l i,q N ∑ j=1  h ij,1 (t, y j (t −τ j ))  2 . (43) Then from (40), ˙ V i ≤− ρ i ∑ q =1 c i,q (z i,q ) 2 − ρ i ∑ q =1 1 4 ¯ l i,q  T i  i −l ∗ i (z i,1 ) 2  ¯ f i (y i )  2 −λ ∗ i (z i,1 ) 2  ¯ h i (y i (t)  2 + ρ i ∑ q =1 N 4 ¯ l i,q ⎛ ⎝ 4  P i  2 N ∑ j=1  f ij (t, y j )  2 + N ∑ j=1  f ij,1 (t, y j )  2 ⎞ ⎠ + ρ i ∑ q =1 N 4 ¯ l i,q ⎛ ⎝ 4  P i  2 N ∑ j=1  h ij (t, y j (t − τ j ))  2 + N ∑ j=1  h ij,1 (t, y j (t −τ j ))  2 ⎞ ⎠ . (44) From Assumptions 3 and 4, we can show that ρ i ∑ q =1 N 4 ¯ l i,q ⎛ ⎝ 4  P i  2 N ∑ j=1  f ij (t, y j )  2 + N ∑ j=1  f ij,1 (t, y j )  2 ⎞ ⎠ ≤ N ∑ j=1 γ ij  ¯ f j (y j )  2 (y j ) 2 , (45) ρ i ∑ q =1 N 4 ¯ l i,q ⎛ ⎝ 4  P i  2 N ∑ j=1  h ij (t, y j (t −τ j ))  2 + N ∑ j=1  h ij,1 (t, y j (t − τ j ))  2 ⎞ ⎠ ≤ N ∑ j=1 ι ij  ¯ h j (y j )(t −τ j )  2 (y j (t −τ j )) 2 , (46) 134 Time-Delay Systems where γ ij = O( ¯ γ 2 ij ) indicates the coupling strength from the jth subsystem to the ith subsystem depending on ¯ l i,q ,  P i  and O( ¯ γ 2 ij ) denotes that γ ij and O( ¯ γ 2 ij ) are in the same order mathematically, and ι ij = O( ¯ ι 2 ij ). Then the derivative of V i is given as ˙ V i ≤− ρ i ∑ q =1 c i,q (z i,q ) 2 − ρ i ∑ q =1 1 4 ¯ l i,q  T i  i −l ∗ i (z i,1 ) 2  ¯ f i (y i )  2 −λ ∗ i (z i,1 ) 2  ¯ h i (y i (t)  2 + N ∑ j=1 γ ij  ¯ f j (y j )y j  2 + N ∑ j=1 ι ij  ¯ h j (y j )(t −τ j )y j (t − τ j )  2 . (47) To tackle the unknown time-delay problem, we introduce the following Lyapunov-Krasovskii function W i = N ∑ j=1 ι ij 1 − ¯ τ j  t t −τ j (t)  ¯ h 1 j  y j (s)  y j (s)  2 ds. (48) ThetimederivativeofW i is given by ˙ W i ≤ N ∑ j=1  ι ij 1 − ¯ τ j  ¯ h j  y j (t)  y j (t)  2 −ι ij  ¯ h j  y j (t −τ j (t))  y j (t −τ j (t))  2  . (49) Now define a new control Lyapunov function for each local subsystem V ρ i = V i + W i = ρ i ∑ q =1  1 2 (z i,q ) 2 + 1 2 ¯ l i,q  T i P i  i  + 1 2 ˜ Θ T i Γ −1 i ˜ Θ i + | b i,m i | 2γ  i ˜ p 2 i + N ∑ j=1 ι ij 1 − ¯ τ j  t t −τ j (t)  ¯ h 1 j  y j (s)  y j (s)  2 . (50) Therefore, the derivative of V ρ i ˙ V ρ i ≤− ρ i ∑ q =1 c i,q (z i,q ) 2 − ρ i ∑ q =1 1 4 ¯ l i,q  T i  i −l ∗ i  ¯ f i (y i )z i,1  2 −λ ∗ i  ¯ h i (y i (t)z i,1  2 + N ∑ j=1 γ ij  ¯ f j (y j )y j  2 + N ∑ j=1 ι ij 1 − ¯ τ j  ¯ h j (y j )y j  2 . (51) Clearly there exists a constant γ ∗ ij such that for each γ ij satisfying γ ij ≤ γ ∗ ij ,and l ∗ i ≥ N ∑ j=1 γ ji if l ∗ i ≥ N ∑ j=1 γ ∗ ji . (52) Constant γ ∗ ij standsforaupperboundofγ ij . Simialy, there exists a constant ι ∗ ij such that for each ι ij satisfying ι ij ≤ ι ∗ ij ,and λ ∗ i ≥ N ∑ j=1 ι ji 1 1 − ¯ τ i if λ ∗ i ≥ N ∑ j=1 ι ∗ ji 1 1 − ¯ τ i . (53) 135 Decentralized Adaptive Stabilization for Large-Scale Systems with Unknown Time-Delay Now we define a Lyapunov function of overall system V = N ∑ i=1 V ρ i . (54) Now taking the summation of the last four terms in (51) and using (52) and (53), we get N ∑ i=1 ⎡ ⎣ −l ∗ i  ¯ f i (y i )z i,1  2 −λ ∗ i  ¯ h i (y i (t)z i,1  2 + N ∑ j=1 γ ij  ¯ f j (y j )y j  2 + N ∑ j=1 ι ij 1 − ¯ τ j  ¯ h j (y j )y j  2 ⎤ ⎦ = N ∑ i=1 ⎡ ⎣ − ⎛ ⎝ (l ∗ i − N ∑ j=1 γ ji ⎞ ⎠  ¯ f i (y i )y i  2 − ⎛ ⎝ λ ∗ i − N ∑ j=1 ι ji 1 − ¯ τ i ⎞ ⎠  ¯ h i (y i )y i  2 ⎤ ⎦ ≤ 0. (55) Therefore, ˙ V ≤− N ∑ i=1 ρ i ∑ q =1 c i,q (z i,q ) 2 − N ∑ i=1 ρ i ∑ q =1 1 4 ¯ l i,q  T i  i ≤ 0. (56) This shows that V is uniformly bounded. Thus z i,1 , ,z i,ρ i , ˆ p i , ˆ Θ i ,  i are bounded. Since z i,1 is bounded, y i is also bounded. Because of the boundedness of y i ,variablesv i,j , ξ i,0 and Ξ i are bounded as A i,0 is Hurwitz. Following similar analysis to Wen & Zhou (2007), we can show that all the states associated with the zero dynamics of the ith subsystem are bounded under Assumption 2. In conclusion, boundedness of all signals is ensured as formally stated in the following theorem. Theorem 1. Consider the closed-loop adaptive system consisting of the plant (1) under Assumptions 1-4, the controller (37), the estimator (28) and (38), and the filters (7)-(9). There exist a constant γ ∗ ij such that for each constant γ ij satisfying γ ij ≤ γ ∗ ij and ι ij satisfying ι ij ≤ ι ∗ ij i, j = 1, ,N, all the signals in the system are globally uniformly bounded. We now derive a bound for the vector z i (t) where z i (t)=[z i,1 , z i,2 , ,z i,ρ i ] T . Firstly, the following definitions are made. c 0 i = min 1≤q≤ρ i c i,q (57)  z i  2 =   ∞ 0  z i (t)  2 dt. (58) From (56), the derivative of V can be given as ˙ V ≤−c 0 i  z i  2 . (59) Since V is nonincreasing, we obtain  z i  2 2 =  ∞ 0  z i (t)  2 dt ≤ 1 c 0 i  V (0) −V ( ∞)  ≤ 1 c 0 i V(0). (60) Similarly, the output y i is bounded by  y i  2 2 =  ∞ 0 (y i (t)) 2 dt ≤ 1 c i,1 V(0). (61) 136 Time-Delay Systems Theorem 2. The L 2 norm of the state z i is bounded by  z i (t)  2 ≤ 1  c 0 i  V(0), (62)  y i  2 2 ≤ 1 √ c i,1  V(0). (63) Remark 6. Regarding the output bound in (63), the following conclusions can be drawn by noting that ˜ Θ i (0), ˜ p i (0),  i (0) and y i (0) are independent of c i,1 , Γ i , γ  i . • The t ransient output performance in the sense of truncated norm given in (62) depends on the initial estimation errors ˜ Θ i (0), ˜ p i (0) and  i (0). The closer the initial estimates to the true valu es, the better the transient output performance. • This bound can also be systematically reduced down to a lower bound by increasing Γ i , γ  i , c i,1 . 5. Simulation example We consider the following interconnected system with two subsystems. ˙x 1 =  01 00  x 1 +  2y 1 y 2 1 0 y 1  θ 1 +  0 b 1  u 1 + f 1 + h 1 , y 1 = x 1,1 (64) ˙x 2 =  01 00  x 2 +  00 y 2 1 + y 2  θ 2 +  0 b 2  u 2 + f 2 + h 2 , y 2 = x 2,1 , (65) where θ 1 =[1, 1] T , θ 2 =[0.5, 1] T , b 1 = b 2 = 1, the nonlinear interaction terms f 1 =[0, y 2 2 + sin(y 1 )] T ,f 2 =[0.2y 2 1 + y 2 ,0] T , the external disturbance h 1 = 0, h 2 =[y 1 (t − τ 1 ), y 2 (t − τ 2 (t)] T . The parameters and the interactions are not needed to be known. The objective is to make the outputs y 1 and y 2 converge to zero. The design parameters are chosen as c 1,1 = c 1,2 = 2, c 2,1 = c 2,2 = 3, l 1,1 = l 1,2 = 1, l 2,1 = l 2,2 = 2, l ∗ 1 = l ∗ 2 = 5, λ ∗ 1 = λ ∗ 2 = 5, γ  1 = 2, γ  2 = 2, Γ 1 = 0.5I 3 , Γ 2 = I 3 , l i,p = l i,Θ = 1, p 1,0 = p 2,0 = 1, Θ 1,0 =[1, 1, 1] T , Θ 2,0 =[0.6, 1, 1] T . The initials are set as y 1 (0)=0.5, y 2 (0)= 1, ˆ Θ 1 (0)=[0.5, 0.8, 0.8] T , ˆ Θ 2 (0)=[0.6, 0.8, 0.8] T . The block diagram in Figure 1 shows the proposed control structure for each subsystem. The input signals to the designed ith local adaptive controller are y i , ξ i,0 , Ξ i , v i,0 . Figures 2-3 show the system outputs y 1 and y 2 .Figures 4-5 show the system inputs u 1 and u 2 (t) . All the simulation results verify that our proposed scheme is effective to cope with nonlinear interactions and time-delay. 6. Conclusion In this chapter, a new scheme is proposed to design totally decentralized adaptive output stabilizer for a class of unknown nonlinear interconnected system in the presence of time-delays. Unknown time-varying delays are compensated by using appropriate Lyapunov-Krasovskii functionals. It is shown that the designed decentralized adaptive controllers can ensure the stability of the overall interconnected systems. An explicit bound in terms of L 2 norms of the output is also derived as a function of design parameters. This implies that the transient the output performance can be adjusted by choosing suitable design parameters. 137 Decentralized Adaptive Stabilization for Large-Scale Systems with Unknown Time-Delay Subsystem i Backstepping controller ,0i v ,0ii ,1 ,1 ii ˆ i ˆ i i u ,2i i u Step 1 Step 2 i y ˆ i p ˆ i p Parameter update laws () j y ji Interactions Filter i v Filter Filter 0, i i Time-delay Interactions )(t y j Fig. 1. Control block diagram. 138 Time-Delay Systems [...]... delay- independent stabilization and control of time- delay systems Delay- dependent stabilization and H∞ control of time- delay systems have been studied in De Souza & Li (1999); Fridman (19 98) ; Fridman & Shaked (2003); He et al (19 98) ; Lee et al (2004); Mahmoud (2000); Wang (2004) In Mahmoud (2000), the author discussed stabilization conditions and analyzed passivity of continuous and discrete time- delay. .. controllers for nonlinear time- delay systems, IEE Proceedings on Control Theory and Applications 146: 91–97 Zhou, J (20 08) Decentralized adaptive control for large-scale time- delay systems with dead-zone input, Automatica 44: 1790–1799 142 Time- Delay Systems Zhou, J & Wen, C (2008a) Adaptive Backstepping Control of Uncertain Systems: Nonsmooth Nonlinearities, Interactions or Time- Variations, Lecture Notes... Springer-Verlag Zhou, J & Wen, C (2008b) Decentralized backstepping adaptive output tracking of interconnected nonlinear systems, IEEE Transactions on Automatic Control 53: 23 78 2 384 Zhou, J., Wen, C & Wang, W (2009) Adaptive backstepping control of uncertain systems with unknown input time- delay, Automatica 45: 1415–1422 8 Resilient Adaptive Control of Uncertain Time- Delay Systems Hazem N Nounou and Mohamed... on Automatic Control 48: 4524–4529 Hua, C C., Guan, X P & Shi, P (2005) Robust backstepping control for a class of time delayed systems, IEEE Transactions on Automatic Control 50: 89 4 89 9 Hua, C., Guan, X & Shi, P (2007) Robust output feedback tracking control for time- delay nonlinear systems using neural network, IEEE Transactions on Neural Networks 18: 495–505 Ioannou, P (1 986 ) Decentralized adaptive... Stabilization for Large-Scale Systems with Unknown Time- Delay 141 7.References Chou, C H & Cheng, C C (2003) A decentralized model reference adaptive variable structure controller for large-scale time- varying delay systems, IEEE Transactions on Automatic Control 48: 1213–1217 Ge, S S., Hang, F & Lee, T H (2003) Adaptive neural network control of nonlinear systems with unknown time delays, IEEE Transactions... Texas A&M University at Qatar, Doha Qatar 1 Introduction Time delay systems are widely encountered in many real applications, such as chemical processes and communication networks Hence, the problem of controlling time- delay systems has been investigated by many researchers in the past few decades It has been found that controlling time- delay systems can be a challenging task, especially in the presence... problems, the bounds of the uncertainties are unknown For such a class of systems, the author in Wu (1999) has developed a continuous time state 144 Time- Delay Systems feedback adaptive controller to guarantee uniform ultimate boundedness for systems with partially known uncertainties For a class of systems with multiple uncertain state delays that are assumed to satisfy the matching condition, an adaptive... large-scale systems including delayed state perturbations in the interconnections, IEEE Transactions on Automatic Control 47: 1745–1751 Wu, S., Deng, F & Xiang, S (2006) Backstepping controller design for large-scale stochastic systems with time delays, Proceedings of the World Congress on Intelligent Control and Automation (WCICA), pp 1 181 –1 185 Wu, W (1999) Robust linearising controllers for nonlinear time- delay. .. conditions and analyzed passivity of continuous and discrete time- delay systems with time- varying delay and norm-bounded parameter uncertainties The results in Mahmoud (2000) have been extended in Nounou (2006) to consider designing delay- dependent adaptive controllers for a class of uncertain time- delay systems with time- varying delays in the presence of nonlinear perturbation In Nounou (2006), the... Stabilization for Large-Scale Systems with Unknown Time- Delay 1.5 1 y 1 0.5 0 −0.5 −1 0 5 10 15 t(sec) 20 25 30 0 5 10 15 t(sec) 20 25 30 Fig 2 Output y1 0.5 0.4 0.3 0.2 y 2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 Fig 3 Output y2 139 140 Time- Delay Systems 5 4 3 u 1 2 1 0 −1 −2 −3 0 5 10 15 t(sec) 20 25 30 5 10 15 t(sec) 20 25 30 Fig 4 Input u1 1 0 .8 0.6 0.4 u2 0.2 0 −0.2 −0.4 −0.6 −0 .8 −1 Fig 5 Input u2 0 Decentralized . class of time delayed systems, IEEE Transactions on Automatic Control 50: 89 4 89 9. Hua, C., Guan, X. & Shi, P. (2007). Robust output feedback tracking control for time- delay nonlinear systems. above, the authors investigated delay- independent stabilization and control of time- delay systems. Delay- dependent stabilization and H ∞ control of time- delay systems have been studied in De. uncertain systems with unknown input time- delay, Automatica 45: 1415–1422. 142 Time- Delay Systems Hazem N. Nounou and Mohamed N. Nounou Texas A&M University at Qatar, Doha Qatar 1. Introduction Time