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104UBICCV2no4 104

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104UBICCV2no4 104 Powered By Docstoc
					                A NOVEL APPROACH TO ADAPTIVE CONTROL
                        OF NETWORKED SYSTEMS

                                          A. H. Tahoun1, Fang Hua-Jing1
                              1
                                  School of Information Technology and Engineering,
                                   Huazhong University of Science and Technology
                                                Wuhan 430074, China
                                                alitahoun@yahoo.com


                                                   ABSTRACT
               The insertion of communication network in the feedback adaptive control loops
               makes the analysis and design of networked control systems more and more
               complex. This paper addresses the stability problem of linear time-invariant
               adaptive networked control systems. Our approach is novel in that the knowledge
               of the exact values of all system parameters is not required. The case of state
               feedback is treated in which an upper bound on the norm of matrix A is required to
               be known. The priori knowledge of upper bound on norm A is not required in
               constructing the controller but it is required only to determine an upper bound on
               the transmission period h that guarantees the stability of the overall adaptive
               networked control system under an ideal transmission process, i.e. no transmission
               delay or packet dropout Rigorous mathematical proofs are established relies
               heavily on Lyapunov's stability criterion. Simulation results are given to illustrate
               the efficacy of our design approach.

               Keywords: Networked control systems, Transmission period, Adaptive control,
               Lyapunov's stability.


1   INTRODUCTION                                            delay (sensor-to-controller delay and controller-to-
                                                            actuator delay) that occurs while exchanging data
     Networked control systems (NCSs) are feedback          among devices connected to the shared medium.
control systems with network communication                  This delay, either constant (up to jitter) or time
channels used for the communications between                varying, can degrade the performance of control
spatially distributed system components like sensors,       systems designed without considering the delay and
actuators and controllers. In recent years, the             can even destabilize the system. Next, the network
discipline of networked control systems has become          can be viewed as a web of unreliable transmission
a highly active research field. The use of networks as      paths. Some packets not only suffer transmission
media to interconnect the different components in an        delay but, even worse, can be lost during
industrial control system is rapidly increasing. For        transmission [3].
example in large scale plants and in geographically              The main challenge to be addressed when
distributed systems, where the number and/or                considering a networked control system is the
location of different subsystems to control make the        stability of the overall NCSs. In this paper, we treat
use of single wires to interconnect the control system      the stability analysis of networked control adaptive
prohibitively expensive [1]. The primary advantages         systems, when the network is inserted only between
of an NCS are reduced system wiring, ease of system         sensors and the controller. Under an ideal
diagnosis                                                   transmission process, i.e. no transmission delay or
     and maintenance, and increase system agility           packet dropout, we have derived a sufficient
[2]. The insertion of the data network in the feedback      condition on the transmission period that guarantees
control loop makes the analysis and design of an            the NCS will be stable. This case is treated in [4] and
NCS more and more complex, especially for                   [5], with completely known systems.
adaptive systems in which systems parameters not                 This paper is organized as follows; the problem
completely known. Conventional control theories             is formulated in Section 2. The main result is given
with many ideal assumptions, such as synchronized           in Section 3. Section 4 presents an example and
control and non-delayed sensing and actuation, must         simulation results, finally we present our conclusions
be reevaluated before they can be applied to NCSs.          in Section 5.
Specifically; the following issues need to be
addressed. The first issue is the network induced


                                                                                                                 1
2    FORMULATION OF THE PROBLEM                                    error, Eq. (3) can be rewritten as

    Consider an NCS shown in Fig. 1, in which
                                                                         x (t ) = A x (t ) + bφ T (t ) x (t k ) − bk * e(t )
                                                                                                                      T
                                                                         &                                                           (4)
sensor is clock-driven and both controller and
actuator are event driven.
                                                                   3     MAIN RESULT
    Actuator                   Plant               Sensor               The main result of this paper will be treated in
                                                                   the following theorem.

                                                   Network         Theorem 1: Let an NCS with linear time-invariant
                                                                   plant (1), an adaptive stabilizer with control input (2)
                                                                   is globally stable if the adaptive control law takes the
                          Controller                               form [6]

           Figure 1 The block diagram of NCS                                 φ& ( t ) = −α x ( t k ) x T ( t k ) Pb                  (5)

     In Fig. 1, a class of linear time-invariant plants            and        the       transmission    period                 satisfies
is described as                                                    h < min { , h1 , h2 , h3 } ,
                                                                            1                     where, α is                  an n×n
                                                                   symmetric positive-definite adaptation gain matrix,
    x (t ) = Ax (t ) + bu (t ) t ∈ [tk, tk+1) , k = 0,1, 2, ...
    &                                                              and
                                                             (1)
                                                                                                1 ⎛ Aupp ⎞
where x(t)∈Rn is a state vector, u(tk)∈R is a control                                   h1 =       ln⎜1 +    ⎟
input vector, (A, b) is controllable, A is a constant                                          Aupp ⎜⎝    ζ1 ⎟
                                                                                                             ⎠
matrix with unknown elements, b is a known
                                                                                         1 ⎛ βλmin (Q) Aupp ⎞
constant vector. We assume that the control is
updated at the instant tk and kept constant until next                           h2 =       ln⎜1 +          ⎟
control update is received at time tk+1. Let h be the                                   Aupp ⎜⎝    ζ2       ⎟
                                                                                                            ⎠
transmission      period        between        successive
transmissions, that is, h = tk+1 – tk. For this paper, we
                                                                                ⎛ ⎛                                  ⎞      ⎞
assume that the transmission process is ideal, there                            ⎜ ⎜ (1 − β ) − (1 − β ) 2 − (1 − β ) ⎟ Aupp ⎟
                                                                                     ⎜                               ⎟
are no delays, no data losses (packet losses) during                      1 ⎜ ⎝           4         4                ⎠      ⎟
the transmission. In future work, we will relax these               h3 =      ln⎜1 +                                        ⎟
assumptions.                                                             Aupp ⎜                     ζ1                      ⎟
     Our objective is to design an adaptive stabilizer                          ⎜                                           ⎟
                                                                                ⎝                                           ⎠
for the networked system and to find an upper bound
                                                                                                  1 2
                                                                   where ζ 1 = Aupp + bk (t k ) + b p x(t k ) α
                                                                                                                    2
on the time transmission period (sampling period) h                                      T
such that the NCS is still stable.                                                                2
     The control input is of the form                                                     (
                                                                   and ζ 2 = 4ζ 1 P 1 + λmin (Q) + A + Aupp + bkT (t ) .              )
                u ( t ) = k T (t ) x (t k )                  (2)
                                                                         To prove the stability of the NCS, firstly, we
                                                                   will find an upper bound on the transmission error
where k(t) is an n-dimensional control parameter                   e(t), a lower and an upper bound on the state x(t), and
vector, T denotes transpose. From Eqs. (1) and (2),                finally, we will use these bounds in Lyapunov
we get                                                             function to prove Theorem 1.

           x (t ) = Ax (t ) + bk T (t ) x (t k )
           &                                                       Lemma 1: (Transmission Error Upper Bound) The
                                                                   transmission error e(t) is bounded between two
                               T                                   successive transmissions by
           = A x (t ) − bk * x (t ) + bk T (t ) x (t k )     (3)
                                                                                       e (t ) ≤ γ x (t k )                           (6)
                           T
where A = A + bk * is Hurwitz matrix satisfying
that A T P + P A = − Q , P and Q are symmetric and                 where
                                              *
positive–definite matrices, and k is the true value of                                   1 2
                                                                      Aupp + bkT (tk ) + b p x(tk ) α
                                                                                                   2

k(t). Define φ (t ) = k (t ) − k * as the control parameter        γ=                    2              A (t −t )
                                                                                                      (e upp k −1) ,
error vector, and e(t) = x(t) – x(tk) as the transmission                               Aupp



                                                                                                                                      2
Aupp is an upper bound on A such that; A ≤ Aupp .                                       Using Eq. (6), it can be concluded that

Proof: From the definition of e(t), it can be found                                            (1 − γ ) x (t k ) ≤ x (t ) ≤ (1 + γ ) x (t k )
that
                                                                                             Now we turn our attention to proof of Theorem
          e (t ) = x (t ) = Ax (t ) + bk T (t ) x (t k )
          &        &                                                                    1. Consider a positive-definite Lyapunov function
                                                                                        V(t) of the form
                          = Ae (t ) + Ax (t k ) + bk T (t ) x (t k )
                                                                                             V (t ) = x T (t ) Px(t ) + φ T (t )α −1φ (t )              (8)
Taking the integral on both sides, and taking into
account that e(tk) = 0, we have                                                         Differentiating V(t) with respect to t, we have

                                                                                         &                                         &
                                                                                        V (t ) = xT (t ) Px(t ) + xT (t ) Px(t ) + φ T (t )α −1φ (t )
                               (                                              )
                           t
                                                                                                 &                         &
e(t ) = e(t k ) + ∫ Ae( s ) + Ax (t k ) + bk T (t ) x (t k ) ds                                                                                       (9)
                          tk
                                                                                                                 &
                                                                                                  + φ T (t )α −1φ (t )
       = [ Ax (t k ) + bk T (t k ) x (t k )](t − t k )
                                                                                                                     &
                                                                                        Substituting for x (t ) and φ (t ) from Eqs. (4) and (5),
                                                                                                         &
           1
          − bb T p x (t k ) x T (t k )αx (t k )(t −t k ) 2                              there results
           2
                t
                                                                                         &
                                                                                        V (t ) = xT (t ) A Px(t ) + x T (t k )φ (t )bT Px(t )
          + ∫ Ae( s )ds
                tk                                                                                − eT (t )k *bT Px(t ) + x T (t ) PA x(t )

If we choose t-tk < 1, Therefore,                                                                 + xT (t ) Pbφ T (t ) x(t k )                     (10)
                                                                                                  − x (t ) Pbk e(t )
                                                                                                        T         *T

         e(t ) ≤ [ A x(t k ) + bk T (t k ) x(t k )                                                − bT Px(t k ) x T (t k )φ (t )
               +
                     1
                       b
                               2
                                   p x(t k ) α ](t − t k )
                                               3
                                                                                                  − φ T (t ) x(t k ) x T (t k ) Pb
                     2
                     t
                                                                                        Rearranging Eq. (10), yields
               + ∫ A e( s ) ds
                     tk                                                                  &
                                                                                        V (t ) = − x T (t )Qx (t ) + 2 x T (t ) Pb φ T (t ) x (t k )
If we know an upper bound of A that is; A ≤ Aupp ,
                                                                                                   − 2 x T (t ) Pbk *T e (t )                          (11)
and applying Bellman-Gronwall Lemma [2], yields
                                                                                                   − 2 x T (t k ) Pb φ T (t ) x (t k )
           t
                                                   1
 e (t ) ≤ ∫ [ Aupp + bk T (t k ) +                                                α ]
                                                         2                2
                                                     b       p x (t k )                  &
                                                                                        V (t ) becomes bounded from above as
           tk
                                                   2
                          ⎛t          ⎞                                                  &
                                                                                        V (t ) ≤ − λ min (Q ) x (t )
                                                                                                                          2
           × x (t k ) exp ⎜ ∫ Aupp dw ⎟ ds
                          ⎜           ⎟
                          ⎝s          ⎠
                                                                                                    + 2 P b φ T ( t ) e (t ) x ( t k )             (12)
Then                                                                                                + 2 P bk *T x (t ) e (t )
                                   e (t ) ≤ γ x (t k )
                                                                                        From (6) and (7), where we choose h < 1, then
Lemma 2: The state of the NCS, x(t), between
successive transmissions is bounded by                                                   &
                                                                                        V ( t ) ≤ − λ min ( Q ) x ( t )
                                                                                                                          2



       (1 − γ ) x (t k ) ≤ x (t ) ≤ (1 + γ ) x (t k )                         (7)                         2γ
                                                                                                    +            P b φ T ( t ) x ( t ) x ( t k ) (13)
                                                                                                        (1 − γ )
Proof: As e(t) = x(t) – x(tk), then                                                                 + 2γ P bk *T              x (t ) x (t k )

       x(t k ) − e(t ) ≤ x(t ) ≤ e(t ) + x(t k )                                        Using (7), and rearranging,




                                                                                                                                                         3
                                                                           satisfies h < min { , h1 , h2 , h3 } defined in Eqs. (15),
  &
 V (t ) ≤
                1
             (1 − γ )
                                   (
                      x ( t ) − (1 − γ ) 2 λ min ( Q )
                                                                                              1
                                                                           (17), and (19), respectively. Therefore, x(t), φ(t), and
              + 2 γ P bk ( t ) − bk
                                  T          *T                   (14)     V(t) are bounded for all t ≥ t0 and the over all system
                                                                           is globally stable.
              + 2 γ (1 − γ ) P bk *T         ) x (t   k   )
                                                                           4     SIMULATION RESULTS

By choosing γ < 1 to guarantee that (1-γ) > 0, we can                           Now, we demonstrate the applicability of our
conclude that h < h1, where                                                approach through the following example. Consider
                                                                           the plant parameters
                       1 ⎛ Aupp ⎞
               h1 =       ln⎜1 +    ⎟                             (15)
                      Aupp ⎜⎝    ζ1 ⎟
                                    ⎠                                                         ⎡0 2 ⎤            ⎡1⎤
                                                                                            A=⎢    ⎥,         B=⎢ ⎥
                                                                                              ⎣1 0 ⎦            ⎣1⎦
                                &
Using, bk *T ≤ A + Aupp , then V (t ) becomes

                                                                           Assume the desired plant parameters
 &
V (t ) ≤
               1
            (1 − γ )
                              (
                     x (t ) − (1 − γ ) 2 λmin (Q )
                                                                                                    ⎡− 1 0 ⎤
                       {
             + 2γ P bk (t ) + A + Aupp
                               T
                                                          }       (16)                            A=⎢      ⎥
                                                                                                    ⎣ 0 − 2⎦
             + 2γ (1 − γ ) P           { A + A }) x(t )
                                              upp             k
                                                                           Let
Again, by choosing
                                                                                              ⎡1 0⎤       ⎡2 0⎤
                              βλ min (Q )                                                   P=⎢   ⎥ , Q = ⎢0 4⎥
γ<
      (4 P )(1 + λ                                                 )                          ⎣0 1⎦       ⎣   ⎦
                                                                       ,
                       min   (Q ) + A + Aupp + bk T (t )
and                                                                        Assume A is unknown but only Aupp is known (take
                                                                           Aupp = 3).
0 < β < 1 , we have h < h2, where
                                                                                Figure 2 shows the simulation results for the
                                                                           networked control system with x(0) = [1 1]T , α = I
              1 ⎛ βλmin (Q) Aupp ⎞                                         (identity matrix), β = 0.9, k(0) = [0 0]T, it is found
      h2 =       ln⎜1 +          ⎟                                (17)
             Aupp ⎜⎝    ζ2       ⎟
                                 ⎠                                         that h1 < 0.1729s, h2 < 0.0125s, and h3 < 0.0149s.
                                                                           Before starting simulation we know that, h <
Substituting for γ in (16), we get                                         0.0125s, but with simulation proceeds, h can be
                                                                           found on-line as shown in Fig. 3 ( we take h =
                                                                           0.002s). Figure 4 shows the simulation results for the
 &     λ (Q)        ⎛                β ⎞                                   networked control system with x(0) = [1 1]T , α = I, β
 V(t) ≤ min    x(t) ⎜ − (1−γ )2 + β − γ ⎟ x(tk ) (18)
        (1−γ )      ⎝                2 ⎠                                   = 0.9, k(0) = [1 1]T, it is found that h1 < 0.1279s, h2 <
                                                                           0.0069s, and h3 < 0.0104s. Before starting simulation
Finally, by choosing                                                       we know that, h < 0.0069s (we take h = 0.001s), also
                                                                           with simulation proceeds, h can be found on-line as
       β       β                                                           shown in Fig. 5. From Eqs. (15), (17), (19) and Figs.
γ < (1− ) − (1− )2 − (1− β ) , we have h < h3,                             (3), (6), it can be concluded that h3 is the minimum
        4              4                                                   transmission period.
where

            ⎛ ⎛ β                         ⎞ ⎞
            ⎜ ⎜(1− ) − (1− β )2 − (1− β) ⎟Aupp ⎟
       1 ⎜ ⎝    ⎜ 4        4              ⎟ ⎟
 h3 =     ln⎜1+                           ⎠
                                                ⎟ (19)
      Aupp ⎜              ζ1                    ⎟
            ⎜                                   ⎟
            ⎝                                   ⎠
and, we can conclude that V         & (t ) < 0 , if h




                                                                                                                                   4
                                1.4                                                                                           0.18

                                                                                                                              0.16
                                1.2                                                                                                                                           h1
                                 1
                                                                                                                              0.14                                            h2




                                                                                               Transmission Period h1,h2,h3
                                                                                                                              0.12
                                                                                                                                                                              h3
                                0.8
      NCS States x(t)




                                                                                                                               0.1
                                0.6
                                                                                                                              0.08
                                0.4
                                                                                                                              0.06

                                0.2
                                                                                                                              0.04

                                 0                                                                                            0.02

                               -0.2                                                                                             0
                                      0     1       2    3       4        5   6   7    8                                             0     1    2    3       4        5   6   7    8
                                                             Time (Sec)                                                                                  Time (Sec)


                                                Figure 2 NCS states x(t)                                                                 Figure 5 Transmission period h



                               0.18

                               0.16
                                                                                  h1
                               0.14                                               h2       1                                    CONCLUSIONS
Transmission Period h1,h2,h3




                               0.12
                                                                                  h3
                                                                                                The paper addresses the stability analysis of
                                0.1
                                                                                           linear time-invariant adaptive networked control
                               0.08                                                        systems. The case of state feedback is in which only
                               0.06
                                                                                           an upper bound on the norm of matrix A is required.
                                                                                           As shown in theorem 1, the priori knowledge of
                               0.04
                                                                                           upper bound on norm A is not required in
                               0.02                                                        constructing the controller but it is required only to
                                 0
                                                                                           determine an upper bound on the transmission period
                                      0     1       2    3       4
                                                             Time (Sec)
                                                                          5   6   7    8   h that guarantees the stability of the overall adaptive
                                                                                           networked control system under an ideal
                                          Figure 3 Transmission period h                   transmission process, i.e. no transmission delay or
                                                                                           packet dropout. In future work we will try to relax
                                                                                           these assumptions. Rigorous mathematical proofs are
                                                                                           established relies heavily on Lyapunov's stability
                                2.5                                                        criterion. Simulation results are given to illustrate the
                                                                                           efficacy of our design approach. It is verified that, if
                                 2                                                         the sampling period of the network is less than the
                                                                                           upper bound on h, the control parameters of the
                                1.5                                                        adaptive controller are bounded and that the NCS
      NCS States x(t)




                                                                                           states converge to zero as time tends to infinity value
                                 1                                                         as time evolves.

                                0.5
                                                                                                                                          ACKNOWLEDGEMENTS
                                 0
                                                                                               This work is supported by National Natural
                               -0.5                                                        Science Foundation of China, Grant #60574088 and
                                      0     1       2    3       4        5   6   7    8
                                                             Time (Sec)
                                                                                           #60274014.

                                                Figure 4 NCS states x(t)
                                                                                                                                                REFERENCES

                                                                                           [1] L. A Montestruque, and P. Antsaklis: Stability of
                                                                                               model-based networked control systems with
                                                                                               time-varying transmission times, IEEE Trans,



                                                                                                                                                                                   5
      Automat. Contr., vol. 49, no. 9, pp. 1562-1572
      (2004).
[2]    W. Zhang: Stability analysis of networked
      control systems, PhD Thesis, Case Westem
      Reserve University (2001).
[3]   W. Zhang, M. S. Branicky, and S. M. Phillips:
      Stability of networked control systems, IEEE
      Control System Magzine, vol. 21, pp. 84-99
      (2001).
[4]   H. Ye: Research on networked control systems,
      PhD Thesis, University of Maryland (2000).
[5]   G.C. Walsh, H. Ye, and L. Bushnell: Stability
      analysis of networked control systems, in Proc.
      Amer. Control Conf., San Diego, CA, pp. 2876
      2880 (1999).
[6]   G. Tao: Adaptive Control Design and Analysis.
      John Wiley & Sons, Inc., New Jersy, (2003).




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Description: UBICC, the Ubiquitous Computing and Communication Journal [ISSN 1992-8424], is an international scientific and educational organization dedicated to advancing the arts, sciences, and applications of information technology. With a world-wide membership, UBICC is a leading resource for computing professionals and students working in the various fields of Information Technology, and for interpreting the impact of information technology on society.
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About UBICC, the Ubiquitous Computing and Communication Journal [ISSN 1992-8424], is an international scientific and educational organization dedicated to advancing the arts, sciences, and applications of information technology. With a world-wide membership, UBICC is a leading resource for computing professionals and students working in the various fields of Information Technology, and for interpreting the impact of information technology on society.