Time Delay Systems Part 8 pdf

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 th

the unknown scalar function τ j(t) denotes any nonnegative, continuous and bounded time-varying delay satisfying

˙

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,0is Hurwitz

The sign of b i,m iand the relative degreeρ 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 leastρ itimes

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 jare known positive functions and differentiable at leastρ itimes, and ¯ι k

ijare 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 estimation 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

˙vi,ι=Ai,0vi,ι+en i,(ni −ι) u i, ι=0, , m i (7)

˙

˙

where vi,ι∈ n i,ξi,0∈ n i,Ξi ∈ n i ×r i, the vector ki = [k i,1 , , k i,n i]T ∈ n iis chosen such

that the matrix Ai,0=Ai−ki(en i,1)Tis Hurwitz, and ei,k denotes the kth coordinate vector in

i There exists a Pisuch that PiAi,0+ (Ai,0)TPi= −3I, Pi =PT i >0 With these designed filters, our state estimate is

ˆxi(t) =ξi,0+Ξiθi+∑m i

k=0

and the state estimation errori=xi−ˆxisatisfies

˙

i=Ai,0i+∑N

j=1

fij(t, y j) +∑N

j=1

hij(y j(t−τ j(t))) (11)

Let V  i =T

iPii It can be shown that

˙

V  i ≤ −T

ii+2NPi2 ∑N

j=1fij(t, y j) 2+2NPi2 ∑N

j=1hij(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 Θi+ i,2+∑N

j=1 f ij,1(t, y j) +∑N

˙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,ρi) =v i,(m i,ρi+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),Ξi,2+Φi,1]T, Θi= [b i,m i , , b i,0,θT

and v i,(m i,2), i,2,ξ i,(0,2),Ξi,2denote the second entries of vi,m i,i,ξi,0,Ξi respectively, f ij,1(t, y j)

and h ij,1(y j(t−τ j(t)))are respectively the first elements of vectors fij(t, y j) and hij(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 totally 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 ξi,0 andΞi 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 parameterΘiand 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,q=v i,(m i ,q)−α i,q−1 , q=2, 3, ,ρ i, (18) whereα i,q−1 is the virtual control at the q-th step of the ith loop and will be determined in later discussion, ˆp 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

•Step 1: Starting with the equations for the tracking error z i,1obtained 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

iΘi+ 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

iΘi+ i,2+∑N

j=1 f ij,1(t, y j) +∑N

The virtual control lawα i,1is designed as

¯

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

iΘˆi, (21)

where c i,1 , l i,1 , l∗iandλ∗i are positive design parameters, ˆΘi and ˆp iare the estimates ofΘiand

p i , respectively Using ˜p i=p i−ˆp i, we obtain

b i,m i α i,1=b i,m i ˆp i α¯i,1=α¯i,1−b i,m i ˜p i α¯i,1, (22)

¯

δT

iΘ˜i+b i,m i z i,2=δ¯T

iΘ˜i+˜b i,m i z i,2+ˆb i,m i z i,2

=δ¯T

iΘ˜i+ (v i,(m i,2)−α i,1)(e(r i +m i+1),1)TΘ˜i+ˆb i,m i z i,2

= (δi−ˆp i α¯i,1e(r i +m i+1),1)TΘ˜i+ˆb i,m i z i,2, (23) where

δi = [v i,(m i,2), v i,(m i−1,2), , v i,(0,2),ξi,2+Φi,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+ (δi−ˆp i α¯i,1er i +m i+1,1)TΘ˜i−b i,m i α¯i,1 ˜p i+ˆb i,m i z i,2

+∑N

j=1 f ij,1(t, y j) +∑N

where ˜Θi=Θi−Θˆi, and e(r i +m i+1),1∈ r i +m i+1 We now consider the Lyapunov function

V i1= 1

2(z i,1)2+1

2Θ˜T

iΓ−1i Θ˜i+|b i,m i|

2γ

i

(˜p i)2+ 1

2¯l i,1 V  i, (26)

whereΓiis a positive definite design matrix andγiis a positive design parameter Examining

the derivative of V1

i gives

˙

V i1 =z i,1 ˙z i,1−Θ˜T

iΓ−1i ˙ˆΘi−|b i,m i|

γ

i

˜p i ˙ˆp 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,1T

ii+ˆb i,m i z i,1 z i,2− |b i,m i|˜p i1

γ

i

[γi sgn(b i,m i)α¯i,1 z i,1+ ˙ˆp i] +Θ˜T

iΓ−1i [Γi(δi− ˆp i α¯i,1e(r i +m i+1),1)z i,1− ˙ˆΘi] +(∑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

+¯l1

i,1

NPi2

 ∑N

j=1 h ij(t, y j(t−τ j(t))) 2+∑N

j=1fij(t, y j) 2 (27) Then we choose

˙ˆp i = −γi sgn(b i,m i)α¯i,1 z i,1, (28)

τ i,1 = δi− ˆp i α¯i,1e(r i +m i+1),1

Let l i,1=3¯l i,1and 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,1T

ii ≤ −¯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 i1 ≤ −c i,1(z i,1)2− 1

4¯l i,1T

ii−l i∗(z i,1)2

¯f i(y i) 2−λ∗i(z i,1)2

¯h i(y i) 2+ˆb i,m i z i,1 z i,2

+Θ˜T

i(τ i,1−Γ−1i ˙ˆΘi) +¯l N

i,1 Pi2 ∑N

j=1fij(t, y j) 2+ N

4¯l i,1

N

∑

j=1 f ij,1(t, y j) 2

+¯l N

i,1 Pi2 ∑N

j=1hij(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)

•Step q (q=2, ,ρ i , i=1, , N): Choose virtual control laws

α i,2= −ˆb 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

∂ ˆΘ iΓi 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

∂ ˆΘ i Γi τ i,q

−q−1∑

k=2 z i,k

∂α i,k−1

∂ ˆΘ i Γi

∂α i,q−1

τ i,q=τ i,q−1−∂α ∂y i,q−1

where c q i , l i,q , q=3, ,ρ iare positive design parameters, and ¯B i,q , q=2, ,ρ idenotes 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,ρi+1), (37)

Remark 4. The crucial terms l i∗z i,1

¯f i(y i) 2

in (21) and 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 the plant model (1) was used in the calculation of ˙ 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,qT

iPii+1

2Θ˜T

iΓ−1i Θ˜i+|b i,m i|

2γ

i

˜p2i (39)

From (12), (20), (33), (35)-(38), and (49), the derivative of V iin (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 NPi2

⎛

⎝∑N

j=1hij(t, y j(t−τ j)) 2+∑N

j=1fij(t, y j) 2

⎞

⎠

+ 1

4¯l i,1

⎛

⎝N∑N

j=1 f ij,1(t, y j) 2+N

N

∑

j=1h ij,1(t, y j(t−τ j)) 2

⎞

⎠

− 1

4¯l i,1T

ii+∑ρ i

q=2



−l i,q  ∂α i,q−1

∂y i

2 (z i,q)2− 1

2¯l i,qT

ii

+∂α ∂y i,q−1

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

⎤

Using Young’s inequality and let l i,q=3¯l i,q, we have

−¯l i,q  ∂α i,q−1

∂y i

2

(z i,q)2+∂α ∂y i,q−1

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 ∂y i,q−1

i  i,2 z i,q− 1

4¯l i,qT

−¯l i,q  ∂α i,q−1

∂y i

2

(z i,q)2+ ∂α ∂y i,q−1

i

N

∑

j=1 h ij,1(t, y j(t−τ j))z i,q

≤ N

4¯l i,q

N

∑

j=1h 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,qT

ii−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

⎛

⎝4Pi2 ∑N

j=1fij(t, y j) 2+∑N

j=1 f ij,1(t, y j) 2

⎞

⎠

+∑ρ i

q=1

N 4¯l i,q

⎛

⎝4Pi2 ∑N

j=1hij(t, y j(t−τ j)) 2+∑N

j=1h 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

⎛

⎝4Pi2 ∑N

j=1fij(t, y j) 2+∑N

j=1f 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

⎛

⎝4Pi2 ∑N

j=1hij(t, y j(t−τ j)) 2+∑N

j=1h 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)

whereγ ij=O(γ¯2

ij)indicates the coupling strength from the jth subsystem to the ith subsystem depending on ¯l i,q, Pi  and O(γ¯2

ij) denotes that γ ij and O(γ¯2

ij) are in the same order mathematically, andι ij=O(¯ι2

ij)

Then the derivative of V iis given as

˙

V i ≤ −∑ρ i

q=1 c i,q(z i,q)2−∑ρ i

q=1

1

4¯l i,qT

ii−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) ¯h1

j

y j(s) y j(s)2ds. (48)

The time derivative of W iis 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 i ρ =V i+W i

= ∑ρ i

q=1 2(z i,q)2+ 1

2¯l i,qT

iPii+1

2Θ˜T

iΓ−1i Θ˜i+|b i,m i|

2γ

i

˜p2i

+∑N

j=1

ι ij

1−τ¯j

t

t−τ j (t) ¯h1

j

y j(s) y j(s)2 (50)

Therefore, the derivative of V i ρ

˙

V i ρ≤ −∑ρ i

q=1 c i,q(z i,q)2−∑ρ i

q=1

1

4¯l i,qT

ii−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 j2

+∑N

j=1

ι ij

1−τ¯j ¯h j(y j)y j2

Clearly there exists a constantγ∗

ijsuch that for eachγ ijsatisfyingγ ij≤γ∗

ij, and

l∗i ≥ ∑N

j=1 γ ji i f l i∗≥∑N

Constantγ∗

ijstands for a upper bound ofγ ij

Simialy, there exists a constantι∗

ijsuch that for eachι ijsatisfyingι ij≤ι∗

ij, and

λ∗i ≥∑N

j=1 ι ji 1

1−τ¯i i f λ∗i ≥∑N

j=1 ι∗ji 1

Now we define a Lyapunov function of overall system

V= ∑N

i=1 V

ρ

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,qT

This shows that V is uniformly bounded Thus z i,1 , , z i,ρ i , ˆp i, ˆΘi,i are bounded Since z i,1is

bounded, y i is also bounded Because of the boundedness of y i, variables vi,j,ξi,0andΞiare

bounded as Ai,0is 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

z i2=

∞

From (56), the derivative of V can be given as

˙

Since V is nonincreasing, we obtain

z i2 =∞

0 z i(t) 2dt≤ 1

c0

i

V(0) −V(∞)≤ 1

c0

i

Similarly, the output y iis bounded by

y i2 =∞

0 (y i(t))2dt≤ 1

Theorem 2. The L2norm of the state z i is bounded by

z i(t) 2 ≤ 1

c0

i



y i2 ≤ √1c

i,1



Remark 6. Regarding the output bound in (63), the following conclusions can be drawn by noting that ˜Θi(0), ˜p i(0),i(0)and y i(0)are independent of c i,1,Γi,γ i.

•The transient output performance in the sense of truncated norm given in (62) depends on the initial estimation errors ˜Θi(0), ˜p i(0)andi(0) The closer the initial estimates to the true values, the better the transient output performance.

•This bound can also be systematically reduced down to a lower bound by increasingΓi,γi , c i,1

5 Simulation example

We consider the following interconnected system with two subsystems

˙x1 =



0 1

0 0



x1+



2y1y21

0 y1



θ1+

 0

b1



u1+f1+h1, y1=x1,1 (64)

˙x2 =



0 1

0 0



x2+



y21+y2



θ2+

 0

b2



u2+f2+h2, y2=x2,1, (65)

whereθ1 = [1, 1]T,θ2= [0.5, 1]T , b1=b2=1, the nonlinear interaction terms f1= [0, y2+ sin(y1)]T,f2 = [0.2y2+y2, 0]T, the external disturbance h1 = 0 , h2 = [y1(t−τ1), y2(t−

τ2(t)]T The parameters and the interactions are not needed to be known The objective is to

make the outputs y1and y2converge to zero

The design parameters are chosen as c1,1 = c1,2 = 2, c2,1 = c2,2 = 3, l1,1 = l1,2 = 1, l2,1 =

l2,2 = 2, l1∗ = l2∗ = 5,λ∗1 = λ∗2 = 5,γ1 = 2,γ2 = 2,Γ1 = 0.5I3,Γ2 = I3, l i,p = l i,Θ = 1,

p1,0=p2,0=1,Θ1,0= [1, 1, 1]T,Θ2,0= [0.6, 1, 1]T The initials are set as y1(0) =0.5, y2(0) =

1, ˆΘ1(0) = [0.5, 0.8, 0.8]T, ˆΘ2(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,ξi,0,Ξi, vi,0 Figures 2-3 show the system outputs y1and y2 Figures

4-5 show the system inputs u1and u2(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 L2norms 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

Subsystem i

Backstepping controller

,0

i

i

i

i

u

Step 1

Step 2

i

y

p

ˆi

p

Parameter update laws

j

Filter

i

v

Filter

Filter 0 ,

i i

Time-delay Interactions

)

(t

yj

Fig 1 Control block diagram

1, ˆΘ1(0) = [0.5, 0 .8, 0 .8] T, ˆΘ2(0) = [0.6, 0 .8, 0 .8] T The block diagram in Figure shows... 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