Distributed Consensus Observers Based H

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REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < temperature distribution of forest fire [23]. However, in the case of using SNs, a centralized observer may be impractical or impossible for the state estimation or control, due to high dimensionality of the target system or the limit of power supply and communication capacity of the sensor nodes. In order to overcome this difficulty, the consensus-based distributed estimation problem of SNs has gained rapidly increasing interest in the past few years (see, e.g., [24]-[31], and the references therein), whose objective is to develop a set of distributed local observers for achieving a common estimate of state at each SN node. These observers are usually called distributed consensus observers (DCOs). Compared with the centralized estimation approach, the distributed one has its own advantages such as low communication burden, fast implementation, and low cost [24]. Until now, the existing works on distributed consensus estimation have been mainly developed for ODE systems, most of which focused on proposing different mechanisms for combining the Kalman filter [25],[26] or H filter [27],[28] with a consensus filter to enforce the consensus of the estimation outcomes of all local filters. As regards SDPs, more recently, some distributed consensus estimation schemes have been proposed in [29]-[31], which enforce consensus of the spatially distributed estimators by dynamically minimizing the disagreement between them. The problem of consensus control based on distributed estimators for a network of linear parabolic PDEs has been studied to ensure that the states of all agents are synchronized in [32]. Despite these efforts, however, very little research has directly addressed the problem of H control design based on DCOs for a nonlinear SDP by using an SN with given topology, which motivates this study. In this paper, we are concerned with the problem of finite dimensional DCOs-based H control design for nonlinear dissipative PDE systems with SNs. The topology of the adopted SN is described by a direct graph, according which each node communicates with its neighboring nodes so that the DCOs of the SN perform the estimation task in a distributed way. The modal decomposition and singular perturbation techniques are initially applied to the PDE system to derive a finite dimensional ODE model, which accurately captures the dynamics of the dominant (slow) modes of the PDE system. Subsequently, based on the slow system and the SN topology, a set of finite dimensional DCOs are constructed to estimate the state of the slow system and enforce the agreement of all estimates. Then, a DCOs-based centralized control scheme is proposed for the stabilization of the PDE system, which only uses the available estimates from the specified group of SN nodes. An H control design method is developed in terms of bilinear matrix inequality (BMI) to ensure the closed-loop exponential stability of the original PDE system while satisfying a prescribed level of disturbance attenuation for the slow system. Furthermore, to make the attenuation level as small as possible, a suboptimal H controller design problem is also addressed, which can be solved by a local optimization algorithm that treats the BMI as double linear matrix inequality

2

(LMI). Finally, a simulation study on the control of one dimensional KSE system is given to show the effectiveness of the proposed design method. The main innovations and contributions can be summarized as follows. 1) This paper aims at solving the DCOs-based H control design problem for a class of nonlinear dissipative PDE systems via an SN. To the best authors’ knowledge, this problem is rarely studied. 2) A set of finite dimensional DCOs accounting for the complex communication between sensor nodes are proposed to compute the slow mode estimates for the control design. 3) Compared with the existing infinite dimensional results in [29]-[31], the developed finite dimensional control result can significantly improve the computational efficiency and reduce the communication burden of the SNs. Notations:  and   denote the set of real and nonnegative real numbers, respectively.  n and  nm are the n-dimensional Euclidean space and the set of all real n  m matrices, respectively.  ,  , and ,  n stand for the absolute value for scalars, Euclidean norm and inner product for vectors, respectively. The superscript ‘T’ is used for the transpose of a vector or a matrix. Let   be the vector space of infinite

sequences

α  1    

equipped with the norm α

T







 i 1

of

real

numbers

 i2   , which is a

natural generalization of  n . For a symmetric matrix M , M  (, , )0 means that it is positive definite (positive semi-definite, negative definite, negative semi-definite, respectively). min () ( max () ) denotes the minimum (maximum) eigenvalue of a matrix. The identity matrix of dimension n is denoted by I n (or I , if the dimension is clear from the context). The N-dimensional column vector of 1’s is denoted by 1N . The Kronecker product of matrices A   mn and B   p q is a matrix in  mp nq and denoted as A  B . diag in1{ Ai } stands for the block diagonal matrix diag{ A1 , A2 , , An } . colim1{ Ai } stands for the block column

vector of m block matrices Ai , i  1, , m . The symbol  is used as an ellipsis for terms in matrix expressions that are induced by symmetry, e.g.,  S  [ M  ]    S  [ M  M T ] X     X Y   XT Y  . II.

PRELIMINARIES AND PROBLEM STATEMENT

A. Description of dissipative PDE Systems with SNs We consider a class of SDPs described by the following highly dissipative and nonlinear PDEs:

x ( z, t )  x ( z, t ) + f ( x ( z, t ))  ku buT ( z )u(t )  kw bwT ( z ) w (t ) (1) t subject to the boundary conditions

> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < x  nz 1 x , , nz 1 )  0 on  z z and the initial condition l (t , x ,

(2)

x ( z , t )  x0 ( z )

(3)

where x ( z , t )  [ x1 ( z , t )  xl ( z , t )]T  l is the vector of state variables, t  0 is the time variable, z    [ z1 , z2 ]   is the spatial variable,  is the spatial domain of definition of the SDP and  is its boundary, u(t )   qu is the manipulated input vector of the actuators, and w (t )   qw denotes the bounded process disturbance.  is a dissipative, self-adjoint, linear spatial differential operator of the form  2  nz  =a1  a2 2    anz nz z z z in which ai , i  1, 2, , nz are known constants , nz is the highest order of spatial derivatives in the PDE and usually an even number (e.g., nz  2 for the parabolic PDE [2] and the

3

mij  0  (i, j )   . Moreover, we assume mii  0 for all i   . The set of neighbors of the i-th SN node is denoted by  i  { j   : (i, j )   } .

B. Infinite-dimensional singular perturbation formulation of the PDE system To simplify the presentation, we define the following Hilbert space:



2,  2,    :    l and 

with inner product

1

2, 





1 , 2   1 ( z )2 ( z )dz 

and norm

1 2

2, 

 1 , 1 , where 1 and 2 are two elements of 2, .

The domain of the operator  is denoted by   nz 1 , , nz 1 )  0 on  }. z z To present the theoretical results, the PDE system of (1)-(3) will be formulated as an infinite dimensional singular perturbation model of ODEs through modal decomposition technique. For the operator  , the eigenvalue problem is defined as  j ( z )   j  j ( z ) , j  1, 2, ,  where  j is the

 ( )  {   2, and l ( ,

NSE [5], nz  4 for the KSE [4]). f ( x ( z, t )) is a locally Lipschitz continuous nonlinear vector function satisfying f (0)  0 . ku and k w are known constant vectors.

j-th eigenvalue and  j ( z )   ( ) is the corresponding

bu ( z )  [bu ,1 ( z )  bu , qu ( z )] and bw ( z )  [bw,1 ( z )  bw, qw ( z )]

orthonormal eigenfunction, i.e., k ( z ),  j ( z )   (k  j ) , in

are known smooth vector functions of z , where bu ,i ( z ) denotes

which  ()

how the control action ui (t ) is distributed in the spatial domain

eigenfunctions form an orthonormal basis for domain  ( ) .

, bw,i ( z ) specifies the position of action of the exogenous

Moreover, all eigenvalues of the self-ajoint operator  are real. To facilitate the subsequent development, we give the following assumption. Assumption 1: All eigenvalues of  are ordered so that  j   j 1 , and there is a finite number n so that n 1  0 and

T

T

nz 1

x  x , , nz 1 ) z z is a nonlinear vector function which is assumed to be sufficiently smooth. x z  denotes the normal derivative on

disturbance wi (t ) in the spatial domain . l ( x ,

the boundary  , and x0 ( z ) is a smooth vector function of z. The state of the SDP (1) is observed by an SN of p nodes distributed along the spatial extent of the SDP, whose sensing models are given by yi (t )   Si ( z ) x ( z, t )dz  vi (t ) , i    {1, 2, , p} 

(4)

where yi (t )   q y ,i is the measured output of the i-th SN node equipped with q y ,i sensors, vi (t )  

q y ,i

is the bounded

measurement disturbance of the i-th SN node, Si ( z )  [ s jk ( z )]q y ,i l is the measurement distribution matrix whose j-th row ( j  1, , q y ,i ) is a known smooth vector

is

the

Kronecker

delta

function.

These

  L n 1  1 is a small positive number, where L is the largest non-zero eigenvalue. Without loss of generality, we assume x ( z, t )  x ( z , t )   , and denote f ( x ( z, t ))  f ( x ( z, t )) , ku  ku , k w  kw . Expand the solution of the system of (1) into an infinite series in terms of the basis functions  j ( z ) as follows: x ( z , t )   j 1 x j (t ) j ( z ) 

(5)

where x j (t ) ( j  1, 2, ,  ) are time-varying coefficients called the modes of the PDE system. Taking the inner product of both sides of (5) with  j ( z ) , we can immediately write down the following relation:

function of z and determined by the location and shape (point or distributed) of the j-th sensor in the i-th SN node. The topology of the SN can be represented by a direct graph   ( ,  , ) of order p with the set of nodes  , the set of edges      , and the weighted adjacency matrix   [mij ] p p . An edge of  is denoted by (i, j ) . The adjacent

Differentiating both sides of (6) with respect to time and denoting the state of the slow subsystem by xs (t )  [ x1 (t )  xn (t )]T  R n and the state of the fast

elements associated with the edges of the graph are positive, i.e.,

subsystem by x f (t )  [ xn 1 (t )  x (t )]T  R  , we have the

x j (t )  x (, t ),  j () .

(6)

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REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < computing the control inputs of the actuators. Fig. 1 shows the diagram of the DCOs-based centralized controller for the SDP with an SN. As is well known, H control is an effective control methodology to attenuate the effect of uncertain external disturbance on the desired control performance. Thus, here we consider the following H control performance index for the slow system (10) under zero-initial condition (i.e., x0 ( z )  0 ):



tf

0

tf

[ xsT (t )Qxs (t )  uT (t ) Ru(t )]dt   2  w T (t ) w (t )dt 0

where w   w T

T

(14)

v T  and v  v1  v p  , t f is the final T

time of control, Q  0 , R  DRT DR  0 are given weighting matrices, and   0 is a prescribed attenuation level. Remark 3: It should be pointed out that the performance (14) can be transformed into an H performance for the original PDE systems by making some additional assumptions in a similar way as in [34]. Therefore, the control design problem considered in this paper is to find a centralized controller of the form (13) based on the finite dimensional DCOs in (12), such that the closed-loop PDE system is exponentially stable in the absence of external disturbance w and measurement disturbances vi , i   , and the H control performance defined by (14) is achieved for some prescribed level   0 in the presence of disturbance w . In general, it is desirable to make the attenuation level as small as possible. To facilitate this study, we make the following assumption. Assumption 2: There exists a known positive constant 1 such that the nonlinear dynamics f s ( xs , 0) in the slow system (10) satisfies f s ( x s , 0)  1 xs . III. FINITE DIMENSIONAL DCOS-BASED H CONTROL DESIGN For convenience, we let Gij  mij Gij , i, j   .

(15)

5

T

Defining the vector es  esT,1  esT, p    np and using   (18), we have es  ( As  LC s  G )es  (1p  I n ) f s ( xs , 0)  (1p  Bw, s ) w  Lv

(19)

where As  I n  As , L  diag {Li } , C s  diag {C s ,i } , p i 1

p i 1

  G1 j G12  G1 p   j   G  G G    21 2j 2p    j  G  , xs ,0  1p  xs ,0 .         G p 2   G pj   G p1 j   From (10), (17) and (19), we have the following augmented closed-loop system:    (1  I ) f ( x , 0)  Bw  x s  Ax s p 1 n s s (20) where  As  Bu , s  K i x  i x s   s    n ( p 1) , A     es  0 

 , As  LC s  G   Bu , s K

0   Bw, s B    , K  [ K1  K p ] . 1p  Bw, s  L  It is observed that Gij  0 when mij  0 , and K i  0 when

i   . Thus, the matrices G and K are structured, meaning that they have sparsity constraints determined by the topology of the SN, controller and actuators. Furthermore, since the matrix L is block-diagonal, it can be viewed as a structured matrix with special sparsity constraint. In this sense, throughout this paper we will define  to be the set of all 3-tuples np  q ( K , G , L )   qu np   npnp    i y ,i satisfying the sparsity constraints. Let us choose a Lyapunov function candidate for the system (20) as

Then, (12) can be rewritten as

V ( x s )  x sT Px s

xˆ s ,i  As xˆ s ,i  Bu , s u  Li C s ,i ( xs  xˆ s ,i )   Gij ( xˆ s ,i  xˆ s , j )

where P  0   n ( p 1)n ( p 1) is a matrix to be determined. Calculating the time derivative of V along the trajectory of the system (20), yields

j

 Li vi , xˆ s ,i (0)  0 , i   .

(16)

Setting K i  0 when i   , we can write (13) as u   K i xˆ s ,i   K i xˆ s ,i . i

 V ( x s )  x sT [ PA s  ] x s  2 x sT P (1p 1  I n ) f s ( xs , 0)  2 x sT PBw

(17)

i

Denoting the state estimation error of the slow system by es ,i  xs  xˆ s ,i for the i-th local observer, and subtracting (16) from (10) give es ,i  ( As  Li C s ,i   Gij )es ,i   Gij es , j  f s ( xs , 0) j

j

 Bw, s w (t )  Li vi , es ,i (0)  xs ,0 .

(21)

(18)

2

 ς T Ω1ς   f s ( x s , 0)   2 w

2

(22)

 [ PA s  ]     x s      where ς   f s ( x s , 0)  , Ω1  (1p 1  I n )T P  I     w  0  2 I  B T P  and   0 is a scalar. It is immediate from Assumption 2 that f s ( xs , 0)  1 xs  1 H1 x s

(23)

> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < where H1   I n

0  0   nn ( p 1) . Then, from (22) and

(23) we have

V ( x s )  ς T ( Ω1  Ω2 )ς   2 w

2

(24)

where Ω2  diag{12 H1T H1 , 0, 0} . Moreover, it is easy to rewrite (17) as u(t )   K i xˆ s ,i (t )   K i Fi x s (t )  KFx s (t ) i

where

Fi  [ I n

(25)

i

0  0 In   i 1

0  0]   nn ( p 1) and   p i

F  colip1{Fi } . Thus, using (24) and (25), we can obtain that

V ( x s )  xsT Qxs  uT Ru   2 w T w  ς T Λς

(26)

where Λ  Ω1  Ω2  Ω3 and Ω3  diag{H1T QH1  ( DR KF )T DR KF , 0, 0} . Obviously, if the following inequality holds: Λ 0 then we have V ( x s )  xsT Qxs  uT Ru   2 w T w  0 .

(27) (28)

Therefore, we have the following result. Theorem 1: Consider the system (20) where matrices K , G , and L satisfy the given sparsity constraints, i.e., ( K , G , L )   . For some given attenuation level   0 , if there exist a scalar   0 and a matrix P  0 satisfying the inequality (27), then the system is exponentially stable in the absence of w , and the H control performance (14) is guaranteed in the presence of w under zero-initial condition. Proof: Assume that the inequality (27) holds for some   0 and P  0 . Integrating (28) from t  0 to t  t f yields tf

V ( x s (t f ))  V ( x s (0))   [ xsT (t )Qxs (t )  uT (t ) Ru(t )]dt 0

tf

 2  w T (t ) w (t ) dt  0 . 0

(29)

6

where  2  0.5 1 / max ( P ) . Thus, V ( x s (t ))  V ( x s (0))e 2 2 t , so that x s (t )   3 e  2t x s (0) (31) for all trajectories of x s (t ) , where  3  max ( P ) / min ( P ) . Hence, the system (20) with w (t )  0 is exponentially stable. □ Remark 4: When the system (20) is exponentially stable in the absence of w , it is clear that the estimation error dynamics in (19) is also stable, which means that the estimates of all local DCOs can converge to the actual state of the slow system exponentially. By the definition of x s (t ) in (20) and considering (31), we have that the consensus dynamics of i-th local observer and j-th one satisfy xˆ s ,i  xˆ s , j  es , j  es ,i  es , j  es ,i  x s   3 e  2t x s (0) , which implies that the convergence rate of the consensus is characterized by the parameter  2 . In this case, the exponential convergence of the consensus in estimates of DCOs is guaranteed and thus the consensus is enforced. Moreover, it is easy to see from (30) that different Li , i   may require different matrix P  0 and scalar  1  0 , which can give rise to different parameter  2 . Hence, the appropriate choice of Li , i   can yield a larger scalar  2 and then improve the convergence rate of the consensus. Let us define    2 and partition P as       (32) 0      Ppp  where Pij   nn , i, j  0   , j  i . Then, based on Theorem  P00 P 10 P     Pp 0

 P11  Pp1

1, we have the following theorem. Theorem 2: Consider the PDE system (1)-(4) with the slow system (10) and the H control performance index in (14). For some given scalar   0 , suppose there exist a scalar   0 ,

Since V ( x s (0))  0 under the zero-initial condition and V ( x s (t f ))  0 , we have (14) from (29).

matrices Pij , i, j  0   , j  i , and matrices K i , Li , Gij ,

Moreover, it is clear from (27) that [ PA s  ]  12 H1T H1    0 T  I  (1p 1  I n ) P  which implies that there exists a sufficiently small scalar  1  0 such that

 Ξ (1,1)  (2,1) Ξ  Ξ (3,1)  (4,1)  Ξ

[ PA s  ]  12 H1T H1  (1p 1  I n )T P 

     1 I .  I 

which gives

V ( x s )   1 x sT x s  2 2V ( x s )

  I 0 0

   I 0

   0    I 

(33)

where

(30)

Thus, by setting w (t )  0 , we have from (24) and (30) that

V ( x s )   1[ x sT x s  f sT ( xs , 0) f s ( xs , 0)]

j  i   satisfying LMI (32) and the BMI

Ξ (1,1)

 Ξ 00(1,1)  12 I n  Q  Ξ10(1,1)     Ξ (1,1)  p0

 Ξ (2,1)    P0Ti   Pi 0 i{1,, p}  i{0}

 (1,1) 11

Ξ

 Ξ p(1,1) 1



i{0,1}

       ,     (1,1)  Ξ pp  P1Ti 



i{2,, p}

Pi1 

> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT)
REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < If   0 , then go to Step 5. Otherwise, set k  k  1 and go to Step 3. Step 3: Using , K i , Li and Gij , j  i   obtained in the previous step, solve the following LMI optimization problem for scalar  and matrices Pij , i, j  0   , j  i :

OP 2: Minimize  subject to LMIs (32) and (36). If   0 , then go to Step 4. Otherwise, set k  k  1 and go to Step 2. Step 4: Using Pij , i, j  0   , j  i , obtained in previous step, solve the following LMI optimization problem for positive scalars  and , and matrices K i , Li , Gij , j  i   .

OP 3: Minimize  subject to LMI (33). Then set l  l  1 , l   . If l  l 1    , where   is predetermined tolerance, go to Step 6; Else go to Step 5. Step 5: Using , K i , Li and Gij , j  i   obtained previously, solve the following LMI optimization problem for scalar   0 and matrices Pij , i, j  0   , j  i .

OP4: Minimize  subject to LMIs (32) and (33). Then set l  l  1 , l   . If l  l 1    , go to Step 6; Else go to Step 4. Step 6: A suboptimal solution of (35) is obtained and the optimized level is  opt   ; STOP. It is observed that Steps 1-3 of Algorithm 1 provide an iterative LMI algorithm to find an initially feasible solution for solving the BMI optimization problem (35) via (36). Clearly, when   0 in Step 2 (or Step 3) of the algorithm, it implies that the resulting solution   0 and K i , Li and Gij ,

j  i   (or Pij , i, j  0   , j  i ) also satisfies (32) and (33). Thus, a feasible initial solution to the problem (35) is obtained. As a consequence, Steps 4-6 of Algorithm 1 can be executed to find a suboptimal solution to the problem (35) in an iterative manner. However, it should be mentioned that if   0 cannot be obtained, Steps 1-3 of the algorithm cannot find a feasible initial solution to the problem (35). IV. APPLICATION TO KSE SYSTEM In this section, we will consider the control problem of one dimensional KSE system with an SN. The KSE is a nonlinear dissipative PDE, which can describe incipient instabilities in a variety of physical and chemical processes, such as falling liquid films [4], [39], the surface evolution of the sputtering process [11], and the interfacial instabilities between two viscous fluids [4], [39]. Recently, the control of fluid flow processes has attracted a lot of attention, whose practical significance ranges from feedback control of turbulence for drag reduction, to suppression of fluid mechanical instabilities in coating processes and suppression of waves exhibited by falling liquid films [39]. Moreover, we notice that an active control scheme of the fluid flow processes has been proposed by using wireless SNs [40]. These facts motivate us to apply the

8

proposed method to the control of one dimensional KSE system with an SN to verify its effectiveness. The KSE system is described by the following nonlinear dissipative PDE: U ( z, t )  U ( z, t )  f (U ( z , t ))  buT ( z )u(t )  bwT ( z ) w (t ) (37) t subject to the periodic boundary conditions  jU ( , t ) z j   jU ( , t ) z j , j  0, 1, 2, 3 (38) and the initial condition U ( z , 0)  3sin z  2sin 2 z  sin 3 z

(39)

where U ( z , t ) denotes the state variable, z    [   ,  ] is

4 2  2 is a 4 z z dissipative, linear spatial differential operator,  is the U ( z , t ) instability parameter, and f (U ( z, t ))  U ( z , t ) is the z nonlinear function. u(t )   2 is the manipulated input vector, the spatial coordinate, t is the time,   

w (t )   2 is the process disturbance. The distribution functions bu ( z ) and bw ( z ) are respectively taken to be bu ( z )   ( z  0.2 )  ( z  0.4 )  , T

bw ( z )   ( z  0.1 )  ( z  0.2 )  . T

The KSE system is measured via an SN with four nodes, whose measurement equations are given as 

yi (t )   si ( z )U ( z, t )dz  vi (t ) , i    {1, 2, 3, 4} 

(40)

where yi (t )   and vi (t )   are the measured output and the measurement disturbance of the i-th node equipped with a single sensor. The distribution functions si ( z ) , i   are chosen as s1 ( z )   ( z  0.6 ) , s2 ( z )   ( z  0.3 ) , s3 ( z )   ( z  0.2 ) , s4 ( z )   ( z  0.5 ) . These four nodes constitute an SN whose topology is represented by a directed graph   ( ,  , ) with the set of sensors  , the set of edges  ={(1, 2), (1, 4), (2, 1), (3, 1), (4, 3)}, and the weighted adjacency matrix   [mij ]44 , where adjacency elements mij  1 , when (i, j )   ; otherwise mij  0 . Fig. 2 shows the topology of the adopted SN.

Fig. 2 Topology of sensor network

The eigenvalue problem for the spatial differential operator of the KSE system of the form  4U  2U U   4  2 , z z

> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <  jU ( )  jU ( )  , j  0, 1, 2,3} z j z j can be solved analytically and its solution is given by (41)  j   j 4  j 2 ,  j ( z )  sin( jz )  , j  1, 2, ,  .  ( )  {U  2,[  ,  and

From (41), it can be found that when the parameter   1 , there exist positive eigenvalues, i.e., the system (37) is unstable. Without loss of generality, we take   0.4 for the system (37) to show the effectiveness of the proposed control method. For this system, we consider the first two eigenvalues as the dominant ones (and thus,   1 3  0.0256 ). Then, a 2-dimensional slow system is derived as follows: x s (t )  As xs (t )  Bu , s u(t )  f s ( xs , 0)  Bw, s w (t )

(42)

12.7132  L4   ,  4.2852   6.0126 G14    2.3492  9.0659 G31    5.3902

 0.3431 7.7177  G12   ,  1.5052 8.5611 1.5679   6.7274 , G21    4.1418  3.9281 3.2268   1.3052 , G43   4.4743  0.8020

9

5.5842  , 4.3532  2.6255 . 7.9942 

Thus, we have  DCO   DCO  0.8933 . To compare with the proposed DCOs-based H controller (Controller 1), a single observer (SO) based H controller (Controller 2) is also considered, where the observer is chosen to be the first local one with neglecting the consensus gains Gij in (44). By letting i  j  1 and neglecting the consensus gains Gij in (44), we can run Algorithm 1 with     900 and

with the measurement equations

yi (t )  C s ,i xs (t )  vi (t ) , i  

(43)

where x  5.3174  xs   1  , xs (0)  xs ,0    , As  diag{0.6,  2.4} , 3.5449   x2   1 sin( z )  f s ( xs , 0)  f ( sT () xs ),  s () ,  s ( z )   1 ,   sin(2 z )   0.3316 0.5366   0.1734 0.3316  , Bw, s   Bu , s    ,  0.5366 0.3316   0.3316 0.5366  C s ,1   0.5366 0.3316 , C s ,2   0.4564 0.5366 , C s ,3   0.3316 0.5366 , C s ,4   0.5364 0 .

Based on the slow system (42) and the measurement equations (43), the local DCOs of the SN are taken as xˆ s ,i  As xˆ s ,i  Bu , s u  Li ( yi  C s ,i xˆ s ,i )   Gij ( xˆ s ,i  xˆ s , j ) , i i

xˆ s ,i (0)  0 , i  

(44)

where 1  {2, 4} ,  2  {1} ,  3  {1} , and  4  {3} . Assume that only the first node can transmit the state estimate of the slow system to controller, i.e.,   {1} . Then we can adopt the following feedback control law: u(t )  K1 xˆ s ,1 .

(45)

   0.01 to obtain the following results for Controller 2:  6.5255 0.9326   11.7498  , L1   0.1848  , 10.8377 2.8401     

SO  4.9617 , K1  

 SO  SO  2.2275 . It is clear that  DCO   SO , which implies that Controller 1 can provide better H control performance than Controller 2. Now, we apply Controllers 1 and 2 to the KSE system (37)-(39). The closed-loop state evolution profiles of the disturbance-free KSE system under these two controllers are shown in Figs. 3 and 4, respectively, from which we observe that although both controllers can regulate the PDE state at the desired steady state U ( z , t )  0 , Controller 1 gives a faster convergence speed than Controller 2. Fig. 5 shows the actual state trajectory of the slow system and its estimates of the DCOs under Controller 1. Fig. 6 presents the actual state trajectory of the slow system and its estimate of the SO under Controller 2. It is observed from Figs. 5 and 6 that Controller 1 can achieve faster state convergence of the slow system than Controller 2.

U

Let Q  diag{0.1, 0.1} and R  0 . Selecting     900 ,

   0.01 , and running Steps 1-3 of Algorithm 1, we find that   6.8600 for k  2 . Then continue the algorithm, i.e., run Steps 4-6 iteratively. When l  2 , the algorithm is terminated and a suboptimal solution of the optimization problem (35) can be obtained as follows:  10.9647 5.9002    88.8032 ,  DCO  0.7980 , K1   ,  17.8487 0.9818  15.8461  8.3935   6.7134  L1    , L2   12.5294  , L3  12.8918 , 12.9702      

-π z π

t

Fig. 3 Closed-loop state evolution profile of disturbance-free KSE system under Controller 1

> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT)
REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <  x s  As xs  Bu ,s  K i xˆ s ,i  f s ( xs , x f )  i   es  ALG es  If s ( xs , 0)  Ly f   x  A x   B ˆ f f u , f  K i x s ,i   f f ( x s , x f )  f i  y f   y Tf ,1 

where

yTf , p     i T

q y ,i

(A3)

f s ( xs , x f ) and f f ( xs , x f ) are also Lipschitz continuous. Thus, for some given positive real numbers  ,  such that  1

xf



 2

  , then there exist positive real  2

numbers  2 ,  3 ,  4 such that

2  2,

2

xs ( t ) )  0

Si ( z ) x ( z, t )dz  C s ,i xs (t )

 x (, t )

2 2,

2

 xs (t )



1 2

 0,



(A4)



Pick a4  1 and b4   2 . Since the closed-loop system (20) is exponentially stable in the absence of w , from the converse Lyapunov theorem, we have that there exists a smooth Lyapunov function Vs : 

n ( p 1)

   and a set of numbers a1 ,

a2 , a3 , a4 , a5 such that for all x s   n ( p 1) satisfying x s  a4 , the following conditions hold:

 a1 x s  Vs ( x s )  a2 x s  V ( x )     ( x , 0)]  a x  s  If Vs ( xs )  s s [ Ax s s s 3 x s    Vs ( x s )  a x 5 s  x s 2

where  

p

2

y f ,i

i 1



 xf

has been



, i   , which imply that (A7)



 i2 .

p

i 1

Consider the smooth function V :  2       given by (A8)

V ( x s , x f )  Vs ( x s )  0.5q f x Tf x f

as a Lyapunov function candidate for system (A3) where q f  0 is some given constant. Computing the time derivative

considering (A4)-(A7), give V ( x ) V ( x s , x f )  s s x s  q f xTf x f x s 

f ( x , x )  f s ( xs , 0)  Vs ( x s )   ( x , 0)]  Vs ( x s )  s s f [ Ax s  If  s s  Ly f x s x s    q f x Tf A f x f  q f x Tf [ f f ( xs , x f )  Bu , f KFx s ]

 a3 x s

2

(A5)

   x s 



3

xs   4 x f

 4 x s x f

xf





    







q  f



 13 x s

 q f (n1   4 ) x f

x s xf

n 1

xf

2 

2 

    

1 2

2



 . q f (n1   4 ) 

By considering the fact n 1   1 L and defining

ku bu ( z ) dz  BuT,s Bu ,s



(A6) Let us define the following induced norms:



 2 x f

 a3 3  max ( F T K T KF ) ,     0.54

2







1

ku bu ( z ) dz  BuT,s Bu ,s  BuT, f Bu , f . Thus, we have 1 2

2



 a5 x s  2 x f

2 where 2  max ( L T L) , 4  a5 ( 2  2 )  q f ( 3  13 ) ,

By considering the orthogonality of eigenfunctions and , it follows that ku bu ( z )   Ts ( z ) Bu ,s   Tf ( z ) Bu , f

1   ( Bu , f )  max ( BuT, f Bu , f )  max

2

q f x f   a3 x s

2

1 2





2

of V ( x s , x f ) along the trajectories of system (A3), and

 f s ( xs , x f )  f s ( xs , 0)   2 x f    f f ( xs , x f )    3 x s   4 x f





used. Thus, y f ,i   i x f

yf 

f ( x ( z, t )) is locally Lipschitz continuous, we have that





2 2 i   , where the fact x (, t ) 2  xs (t )  x f (t )

where A and I are defined in (20). Noting the condition that

xs  



sup ( x (,t )

, which can be

 f ( x , x )  f s ( xs , 0)     If  ( x , 0)   s s f  x s  Ax  s s s  Ly f          x  A x  B KFx  f ( x , x ) f f u, f s f s f  f

and

xf

0 x f  



rewritten as

 1

y f ,i

 i  sup (A2)

11

* 

L a3 q f , a3 q f  4  0.2542

we have that if   (0,  * ) , then

(A9) 0

and thus

> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < V ( x s , x f )  min ( )( x s

2

 xf

2 

) , which directly implies

that the system (A3) is exponentially stable. Obviously, this implies that the system (7) is also exponentially stable. Then, the exponential stability of the system (7) implies that the closed-loop PDE system is exponentially stable. For example, for x ( z, t )  x ( z, t )   , if x (t )   c2 e  c3t x (0)  , t  0 for all x (t )  [ xsT (t )

x Tf (t )]T satisfying x (t )



 c1 , where

c1 , c2 and c3 are positive real numbers, then considering the

fact

x (, t )

that

x (, t )

2,

 c2 e c3t x0 ()

satisfying x (, t )

2,

2,

2,

 x (t )



yields

, t  0 for all x ( z , t )  2,

 c1 . This completes the proof. □ REFERENCES

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

[14] [15] [16] [17]

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