VIEWS: 8 PAGES: 6 CATEGORY: Communications & Networking POSTED ON: 1/3/2011
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. 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