Decentralized Parametric Stabilization of Quantized Interconnected Systems with Application to Coupled Inverted Pendulums

13th IFAC Symposium on Large Scale Complex Systems: Theory and Applications July 7-10, 2013. Shanghai, China

Decentralized Parametric Stabilization of Quantized Interconnected Systems with Application to Coupled Inverted Pendulums Ning Chen*, Guisheng Zhai**, Yuqian Guo*, Weihua Gui*, Xiaoyu Shen*

* School of Information Science and Engineering, Central South University, Changsha 410083, China ( e-mail:ningchen@ csu.edu.cn). ** Department of Mathematical Sciences, Shibaura Institute of Technology, Saitama 337-8570, Japan (e-mail: [email protected]) Abstract: This paper studies the analysis of parametric stability and decentralized state feedback control of a kind of quantized interconnected systems. The output of each controller is quantized logarithmically before it is input to the subsystem, and the quantized density would affect the stability of the systems. First, a decentralized state feedback controller is designed for interconnected systems without quantization and the corresponding stable region is obtained. Second, for a given controller, the lower bound of the quantization density is evaluated from parameters of local controllers. Finally, the proposed method is applied to coupled inverted pendulums systems which can be viewed as quantized interconnected systems. The simulation results show that by using the proposed quantized controllers, the interconnected inverted pendulum systems are parametrically stabilized.

disappears in some cases like heavily stressed electric power systems (Ikeda and Siljak,1980;Ikeda et al., 1991). Hence, parametric stability becomes the first concern for moving equilibrium. The concept of parametric stability was first introduced in competitive equilibrium systems by Siljak where the existence and the stability of a moving equilibrium are simultaneously captured. Lotka-Volterra models of multispecies communities claim that system parameter uncertainty often induces equilibrium location uncertainty (Kwatny et al., 1986). For uncertain parameters cause equilibrium state shift or even destroy equilibrium, parametric stability jointly considers feasibility and stability of equilibrium when parameters belong to a bounded uncertainty set.

1. INTRODUCTION The effect of quantization has become a big concern in the design of computer-based control systems and digital communication systems (Delchamps, 1990; Brockett and Liberzon, 2000). In classical feedback control theory, various signals or data in the control loop have been assumed to be passed directly without data loss. But, in practice, signals must first be converted into a discrete form before being transferred to controllers and the systems, such as A/D or D/A transform in computer control. Quantization is a common source of errors which may cause the system performance to deteriorate. Earlier work on quantized feedback control can be traced back to as early as 1956, when Kalman investigated the effect of quantization in a sampled data control system (Kalman,1956). In addition, the classical control theory, usually assuming that data transmission required by the system can be performed with infinite precision, may not work well in the presence of signal quantization or capacity-limited feedback. Therefore, there is a demand to develop tools for the analysis and design of quantized feedback systems.

As far as parametric stability is concerned, many scholars have investigated the problem of parametric stability (Ohta and Siljak,1994; Wada et al.,1998; Wada et al., 2000; Sundarapandian, 2006; Zecevic and Siljak, 2003; Zecevic and Siljak, 2009; Chen et al., 2007; Chen et al., 2012). Wada, et al. provided state-space and frequency-domain criteria for parametric absolute stability of Lurie systems with uncertain constant parameters and constant reference inputs (Wada et al.,1998; Wada et al., 2000). Zecevic and Siljak proposed an optimal method for the stabilization of nonlinear systems with parametric uncertainty, and selected the reference input to enlarge the size of the stability region in the parameter space (Zecevic and Siljak, 2003; Zecevic and Siljak, 2009). Chen et al. formulated the parametric absolute stability of the interconnected systems and proposed the existing sufficient condition of polytopic interconnected Lurie systems (Chen et al., 2007; Chen et al., 2012).

Large-scale systems are viewed as a dynamical system composed of several subsystems with connection (Siljak, 1978; Jamshidi, 1983; Bakule, 2008). The interconnection among the subsystems is generally uncertain and nonlinear. Uncertainties always exist and are one of reasons of instability of systems. In nonlinear systems, control schemes developed from traditional analysis methods separately treat the existence and stability of equilibrium, with the implicit assumption that equilibrium remains fixed for the entire range of parameter values. In most practical applications, however, it is not realistic to make such assumption. It is often the case that variations of system parameters result in a moving equilibrium, whose stability properties can vary substantially. It is reported that the equilibrium even 978-3-902823-39-7/2013 © IFAC

Since signal quantization always exists in computer based control systems, many researchers have begun to study the analysis and design problems for control systems involving various quantization methods (Ishii and Francis, 2002). There 43

10.3182/20130708-3-CN-2036.00021

IFAC LSS 2013 July 7-10, 2013. Shanghai, China

are two types of quatizer, one is dynamic, the other is static. It is proved in Elia and Mitter (2001) that for a quadratically stabilizable system, a static quantizer needs to be logarithmic (i.e., the quantization levels are linear in logarithmic scale). By using a sector bounded approach, Fu and Xie investigated the linear system with general types of quantizations involving state quantization, measured output quantization and input quantization, so that the quantized feedback design problem was converted into a robust control problem with sector bound uncertainties (Fu and Xie, 2005). They also proposed a simple dynamic scaling method for the quantizer based on a known logarithmic quantization scheme and a corresponding quantized dynamic output feedback controller was designed (Fu and Xie, 2009). Gao and Chen presented a new approach to quantized feedback control systems which, both single- and multiple-input cases considered, provided for stability and H∞ performance analysis as well as controller synthesis for discrete-time state-feedback control systems with logarithmic quantizers (Gao and Chen, 2008). Liu et al studied the feedback control problem of nonlinear systems in strict-feedback form with state quantizers by a setvalued map based recursive control design approach (Liu et al., 2012). These papers did not consider parametric stability.

nonlinear function in x , pi ∈ R li are uncertain parameter vector and it can be limited to a small neighborhood according to parameter estimation. qi (ui ) (i = 1, 2, , N ) are logarithmic quantizer of ith subsystem. Quantizer qi (ui ) (i = 1, 2, the form [20]

qi = {±qi (l ) : qi (l ) = ρil qi (0), l = ±1, ±2, }

(2) ∪{±qi (0)} ∪ {0}, 0 < ρi < 1, qi (0) > 0 The associated quantizers qi (i) are defined as follows: ⎧ 1 1 ⎪ ⎪qi (l ), if qi (l ) < ui ≤ qi (l ), ui > 0 ⎪ ⎪ 1 + δi 1− δi ⎪ ⎪ if ui = 0 qi (ui ) = ⎪ ⎨0, ⎪ ⎪ ⎪− qi (−ui ),if ui < 0 ⎪ ⎪ ⎪ ⎪ ⎩ (3) where 1− ρi δi = (4) 1 + ρi

The coupled pendulums compose of a spring is one of the types of interconnected system. There are two equilibria in coupled pendulums, one is a stable point vertically downwards, the other is unstable vertically upwards. Gavel and Siljak studied stability of coupled inverted pendulums system with unknown uncertainties by decentralized adaptive control (Gavel and Siljak,1989). Gong et al considered the decentralized control for coupled inverted pendulums when uncertainties exist both in interconnection among the subsystems and subsystem itself (Gong et al., 1996). Song et al studied the control of two inverted pendulums coupled with a spring using a progressive fuzzy fusion control method (Song, 2003). A controller was designed based on the idea of fuzzy logic and classical control methods, such that the second pendulum can reject the disturbance from the first pendulum significantly even without precise mathematical model of the system. These papers did not consider the quantization in the system.

ρi are quantization density. A logarithmic quantizer is shown as Fig. 1.

ui (0) 1 + δi

ui = ri + K i ( xi − xir ),

ui

i = 1, 2,

,N ,

(5)

r i

where ri and x denote user-defined reference values. K i ∈ R mi ×ni are constant matrices.

The closed-loop system after quantization can be described as xi = Aii xi + Bi qi (ri + K i ( xi − xir )) + hi ( x, pi ), i = 1, 2, , N (6) According to sector bound method (Fu and Xie, 2005)，the quantization error of the ith subsystem is defined as: ei (ui ) = qi (ui ) − ui = Δ i (ui )ui

(1)

(7)

where Δ i (ui ) ∈ [−δ i , δ i ] are the relative quantization error of logarithmic quantizer qi (ui ) . In the range of quantization, we can get

appropriate dimensional matrices. hi ( x, pi ) : R n × R li → R ni T

ui (0) 1 − δi

We consider decentralized controllers as follows:

where xi ∈ R ni , ui ∈ R mi are the states and input of the ith subsystem respectively, Aii , Bi (i = 1, 2, , N ) are of ( x = ⎡⎢⎣ x1T , x2T ,

ui (0)

Fig. 1 Logarithmic quantizer

Consider the interconnected system with quantization composed of N subsystems as ,N

qi (l ) = (1 − δ i )ui

ui (0)

2. PROCEDURE FOR PAPER SUBMISSION

i = 1, 2,

qi (l ) = (1 + δ i )ui qi (l ) = ui

qi (l )

This paper studies decentralized parametric stabilization for a kind of quantized interconnected systems. A logarithmic quatizer is designed to make the system parametric stable. The proposed method is applied to coupled inverted pendulums systems by the jumps of the coupled spring.

xi = Aii xi + Bi qi (ui ) + hi ( x, pi )

, N ) is called logarithmic if it has

N

qi (ui ) = ( I + Δ i )ui , i = 1, 2,L , N .

, xNT ⎤⎥⎦ , n = ∑ ni ) are a piecewise-continuous i =1

44

(8)

IFAC LSS 2013 July 7-10, 2013. Shanghai, China

( )

(ii) Equilibria xie ( p ) are stable for any pi ∈ Ωi pi* .

Thus, the system (6) can be rewritten as: x&i = Aii xi + Bi ( I + Δ i )(ri + K i ( xi − x )) + hi ( x, pi ), r i

Then, the closed-loop system composed of system (1) and decentralized controllers (5) is said to be parametric stable. We call controller (5) as decentralized parametric controller of system (1).

(9)

i = 1, 2, L, N where Δ i ≤ δ i .

With the above preparation, we formulate our control problem as follows:

We call system (9) as a nominal system when Δ i =0 are satisfied. As parameter pi shift, the equilibrium of the nominal system changes which make the equilibrium of the closed-loop system need not be confined to the origin. Thus, the equilibrium is function of parameter pi , denoted as

Decentralized parametric stabilization problem of quantized interconnected system: Design a decentralized parametric controller (5) of system (1), under Assumption 1, 2 and 3, which makes the closed-loop quantized interconnected system (10) parametric stable by adjusting feasible range of quantization density (5) of logarithmic quantizer, depending on the state of each subsystem.

xie ( pi ) .

To develop an appropriate mathematical framework for this type of problem, let us introduce a new state vector zi = xi − xie ( pi ) . System (9) can be rewritten as z&i = ( Ai + Bi ( I + Δ i ) K i ) zi + gi ( x e ( pi ) , pi , zi ) ， i = 1, 2,

3 Design of decentralized parametric controller

,N

(10)

3.1 Parametric stability and controller design without quantization

where g i ( x e ( pi ) , pi , zi ) ≡ hi ( x, pi ) − hi ( x e ( pi ) , pi ) .

We first consider parametric stability and controller design without quantization. In this case, system (1) becomes xi = Aii xi + Bi ui + hi ( x, pi ), i = 1, 2, , N (13) The whole system can be written as x = Ax + Bu + h( x, p) (14)

We introduce three key assumptions. Assumption 1: The variation of parameter pi , i = 1, 2, , N is limited to ball Ωi centered around some known nominal value pi*

{

Ωi = pi ∈ R

li

}

pi − p ≤ ηi . ∗ i

where u = ⎡⎣⎢u1T , u2T ,

(11)

Assumption 2: The closed-loop equilibria x ( pi ) are continuous functions of pi for all Ωi . Its nominal values

where r = ⎡⎢⎣ r , r ,

( p ) , pi ) can be bounded

≤ z T H iT ( x e ( p ) , pi ) H i ( x e ( p ) , pi ) z

(12)

hold for some matrices H i ( x e ( p ) , pi ) whose elements are continuous functions of

x e ( p ) and pi . We define

H i ( x e ( p ) , pi ) as a block matrix as

H i = [ H i1 , H i 2 ,L, H iN ] ， i = 1, 2,

, N.

T

, hNT ⎤⎥⎦ .

T 2

T T N

r

r T 1

(15)

r T T N

r T 2

z = x − x e ( p ) , and system (16) becomes z = ( A + BK ) z + g ( x e ( p ) , p, z )

We now introduce the concept of decentralized parametric stabilization for quantized interconnected system (1).

where z = ⎡⎣⎢ z1T , z2T ,

T T N

(18) T T

, z ⎤⎦⎥ ， g = ⎡⎣⎢ g1T , g 2 T , , g N ⎤⎦⎥ , and g ( x e ( p ) , p , z ) ≡ h ( x, p ) − h ( x e ( p ) , p ) (19)

Definition 1. For quantized interconnected system (1), if decentralized controllers (5) are designed in neighborhoods

Ωi ( pi* ) under Assumption 1, 2 and 3, so that the following

conditions hold:

, ANN } ,

, r ⎤⎥⎦ , x = ⎡⎢⎣ ( x ) , ( x ) , , ( x ) ⎤⎥⎦ , K = diag{K1 , K 2 , , K N } . Then, we obtain the closed-loop system as x = Ax + B[r + K ( x − x r )] + h( x, p ) (16) The equilibrium T x e ( p) = ⎡⎣⎢ ( x1e ( p1 ))T , ( x2e ( p2 ))T , , ( xNe ( pN ))T ⎤⎦⎥ will satisfy the following equation Ax e ( p ) + B[r + K ( x e ( p) − x r )] + h( x e ( p), p ) = 0 (17) The equilibrium of the system (13) is not at origin when parameter p shifts. We introduce a new state vector T 1

in such a way that inequalities

gi T ( x e ( p ) , pi ) gi ( x e ( p ) , pi )

, A = diag{ A11 , A22 ,

Also, a decentralized controller can be written as u = r + K ( x − xr )

xie ( p ) will be denoted in the following by xi* . Assumption 3: The functions gi ( x

T

B = diag{B1 , B2 ,L BN } , h = ⎡⎢⎣ h1T , h2T ,

e i

e

, u NT ⎤⎦⎥

Based on Assumption 3, (12) can be rewritten as ⎛ N ⎞ g T g ≤ z T ⎜ ∑ H iT ( x e ( pi ) , pi ) H i ( x e ( pi ) , pi ) ⎟ z . (20) ⎝ i =1 ⎠ Using Assumption 1-3, it can be shown that there exist constants μi > 0, i = 1, 2, , N such that，

( )

(i) Equilibria xie ( p ) ∈ R n exist for any pi ∈ Ωi pi* ;

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IFAC LSS 2013 July 7-10, 2013. Shanghai, China

⎛ N ⎞ z T ⎜⎜∑ H iT ( x e ( pi ) , pi ) H i ( x e ( pi ) , pi )⎟⎟⎟ z ⎜⎝ i =1 ⎠⎟ ⎛ N ⎞ ≤ z T ⎜⎜∑ μi2 H iT ( x∗ , pi∗ ) H i ( x∗ , pi∗ )⎟⎟⎟ z ⎜⎝ i =1 ⎠⎟ ∗

where P = diag{P1 , P2 ,

We calculate the derivative of V ( x ) as

(21)

V ( x − xie ( p )) = z T Pz + z T Pz = z T ( A + BK )T Pz + g T Pz

∗ i

for any pi ∈ Ωi , provided that matrices H i ( x , p ) are non-

+ z T P ( A + BK ) z + z T Pg

singular. For simplicity, H i ( x ( pi ) , pi ) in this context are e

N

≤ z T (( A + BK )T P +P ( A + BK ) + ∑ μi2 H i∗T H i∗ ) z

∗ i

denoted as H .

i =1

+ z T Pg + g T Pz − g T g ⎡ ( A + BK )T P +P ( A + BK ) ⎤ ⎢ ⎥ T ⎡ z ⎤ ⎢ N 2 ∗T ∗ P ⎥⎥ ⎡ z ⎤ ⎢ ⎢ ⎥ = ⎢ ⎥ ⎢+∑ μi H i H i ⎥ ⎢g⎥ ⎢g⎥ ⎣ ⎦ ⎢ i =1 ⎥⎣ ⎦ ⎢ −I ⎥⎦⎥ P ⎢⎣ If the LMI (27) holds, N ⎡ ⎤ T 2 ∗T ∗ ⎢ ( A + BK ) P +P( A + BK ) + ∑ μi H i H i P ⎥ < 0 i =1 ⎢ ⎥ − I ⎥⎦ P ⎢⎣

Next, we give the equilibrium analysis of the system with uncertain parameter pi . We first obtain the equilibria pi of nominal system when uncertain parameter pi change. From (17) it follows that xi* depend on xir and K i . On the other hand, the construction of matrices H i ( x∗ , pi∗ ) (therefore, the computation of K i as well) require prior knowledge of xi* . To avoid this circularity, the reference inputs xir are proposed to be determined in the following way: Step 1 Let r = 0 . Fix pi* , and solve

then, V ( x − xie ( p )) < 0 is satisfied.

We derive the condition of parametric stabilization and give the region of parametric stabilization.

γ i (i = 1, 2， L，N ) ，such that the LMI holds

3.2 Decentralized quantizer design

YH1∗T L YH N∗ T ⎤ ⎥ 0 0 ⎥ −I L 0 −γ 1 I L 0 ⎥<0 ⎥ M M O M ⎥ 0 0 L −γ N I ⎥⎦ I

Since (27) is a strict matrix inequality in the sense of negative definiteness, we can always find a positive definite matrix Q = diag{Q1 , Q2 , , QN } such that N ⎡ ⎤ ⎢ ( A + BK )T P +P ( A + BK ) + ∑ μi2 H i∗T H i∗ + Q P ⎥ ⎢ ⎥<0. i =1 ⎢ ⎥ ⎢ P −I ⎦⎥ ⎣ (29) Pre-multiplying and post-multiplying diag{P−1 , I } to (29)， we can get (30) by Schur complement.

(24) then, system (13) is parametric stable. The controllers can be obtained by K i = LiYi −1 , where Y = diag{Y1 , Y2 , , YN } ，

μi = 1/ γ i (i = 1, 2,L , N )

, LN } .

are

admissible nonlinear bound. Proof ： We choose Lyapunov function of system (18) as follows: V ( x − x e ( p )) = ( x − x e ( p )) P ( x − x e ( p )) = z T Pz T

(27)

When the quantizer is added to interconnected system (1), the quantization density of quantizer will affect the stability of the system. In the following, we will study the condition of parametric stability of quantized interconnected systems.

Theorem 1 For system (13)，if there exist positive definite matrices Yi , any matrices Li and positive scalars

⎡ AY + YAT + BL + LT B T ⎢ I ⎢ ⎢ H1∗Y ⎢ M ⎢ ⎢ H N∗ Y ⎣

(26)

Pre-multiplying and post-multiplying diag{P−1 , I } to (27), we get ⎡ ⎤ ⎛ N 2 ∗T ∗ ⎞ T T T ⎢ AY + YA + BL + L B + Y ⎜ ∑ μi H i H i ⎟ Y I ⎥ < 0 ⎝ i =1 ⎠ ⎢ ⎥ ⎢⎣ − I ⎥⎦ I (28) where Y = P−1 , L = KY . By Schur complement, (28) is equivalent to (24). This completes the proof.

Ax + h( x, p∗ ) = 0 (22) Under relative mild assumptions on h , it can be done using Newton’s method [20]. Step 2 Set x r = x* , then, for any choice of Kσ , x* will be the equilibria of the closed-loop system x = Ax + BK ( x − x∗ ) + h( x, p∗ ) (23)

L = diag{L1 , L2 ,

, PN } is a positive definite matrix.

(25)

46

IFAC LSS 2013 July 7-10, 2013. Shanghai, China

⎡ AY + YAT + BL + LT B T + Qˆ I YH1∗T ⎢ 0 −I I ⎢ ∗ ⎢ 0 −γ 1 I H1 Y ⎢ M M M ⎢ ∗ ⎢ 0 0 H Y N ⎣

L YH N∗ T ⎤ ⎥ 0 ⎥ L 0 ⎥<0 L ⎥ O M ⎥ L −γ N I ⎥⎦

N

V& < −∑ ( ziT Qi zi − 2 ziT Pi Bi Δ i K i zi ) i =1

N

≤ ∑ (2 ziT i =1 N

=∑ z

(30)

i =1

≤ ∑ ziT

2

i =1

(2 Pi Bi K i δ i − λmin (Qi ))

(37)

4 Applications to Coupled Inverted Pendulums

(32)

Let us consider two inverted pendulums that are coupled by a spring and subject to two distinct inputs as shown in Fig.1.

where Δ = diag{Δ1 , Δ 2 ,L , Δ N } . We choose the Lyanpunov function of system (32) as V ( x − x e ( p )) = ( x − x e ( p )) P ( x − x e ( p )) = z T Pz . T

u1

(33)

We calculate the derivative of V ( x ) as

(2 Pi Bi Δ i K i − λmin (Qi ))

From (4), we note that δ i is inversely proportional to ρi . That is to say, the quantization density of logarithmic quantizer should have an infimum, which makes the closedloop system (10) parametric stable.

where Pi = Yi −1 ， K i = LiYi−1 , then, the closed-loop quantized interconnected system (10) is parametric stable. Proof：System (10) can be rewritten as z = ( A + B( I + Δ) K ) z + g ( x e ( p ) , p, z )

(36)

(37) is equivalent to (31). This completes the proof.

(31)

2 Pi Bi K i

2

Pi Bi Δ i K i − λmin (Qi ) zi )

If (37) is satisfied, then V < 0 . 2 Pi Bi K i δ i < λmin (Qi )

Theorem 2 Assume that for each local quantizer, the range of relative quantization error δ i (i = 1, 2,L , N ) such that

δi <

T 2 i

N

where Qˆ = YQY . We obtain the main result in this paper.

λmin (Qi )

2

m

u2

θ1

V& ( x − x e ( p ) )

m

θ2

k

= z& T Pz + z T Pz&

l

a1

= z T ( A + B ( I + Δ ) K )T Pz + g T Pz

a2

+ z P ( A + B ( I + Δ ) K ) z + z Pg T

T

Fig. 1 Coupled inverted pendulums In Fig.1, m is mass of pendulum rod； u1 , u2 are input of two pendulums, respectively; θ1 and θ 2 are the angle of two

≤ z T ( A + B ( I + Δ ) K )T Pz + g T Pz + z T P ( A + B ( I + Δ ) K ) z + z T Pg N

+ z T (∑ μi2 H i∗T H i∗ ) z − g T g

pendulums, respectively; k is the spring constant. The equations of motion are ml 2θ&&1 = mg lθ1 − ka12 (θ1 − θ 2 ) + u1， y1 = θ1 , ml 2θ&& = mg lθ − ka 2 (θ − θ ) + u ，y = θ .

i =1

⎡ ( A + BK )T P +P( A + BK ) ⎤ ⎢ N ⎥ z P z ⎡ ⎤ ⎥⎡ ⎤ = ⎢ ⎥ ⎢ + ∑ μi2 H i∗T H i∗ + 2 PBΔK ⎢ ⎥ ⎢⎣ g ⎥⎦ g ⎣ ⎦ i =1 ⎢ ⎥ −I ⎦ P ⎣ T

N

< −∑ ( ziT Qi zi − 2 ziT Pi Bi Δ i K i zi )

2

(34)

λmin (Qi ) zi

≤ z Qi zi ≤ λmax (Qi ) zi

2

.

2

1

2

2

2

(38)

u = (u1 , u2 ) T , y = ( y1 , y2 ) T , (38) can be rewritten as

⎡ 0 ⎢ 2 ⎢ g − ka1 ⎢ l ml 2 x& = ⎢ ⎢ 0 ⎢ ka22 ⎢ ⎣ ml 2 ⎡1 0 0 y=⎢ ⎣0 0 1

where | Δ i |≤ δ i , i = 1, 2,L , N . Assume that λmax (Qi ) and λmin (Qi ) denote the maximum and minimum eigenvalue of Qi respectively. For Qi > 0 , then for any zi , we can get T i

2

Choosing the state vector as x = (θ1 ,θ&1 ,θ 2 ,θ&2 )T and denoting

i =1

2

2

(35)

From (34),

1

0 2 1 2

ka ml 0

0 0 0

g ka22 − l ml 2

0⎤ ⎡ 0 ⎥ ⎢ 1 ⎢ 0⎥ ⎥ ⎢ ml 2 x+⎢ ⎥ 1⎥ 0 ⎢ ⎥ ⎢ 0⎥ ⎢⎣ 0 ⎦

0 ⎤ ⎥ 0 ⎥ ⎥ u, 0 ⎥ ⎥ 1 ⎥ ml 2 ⎥⎦

0⎤ x. 0 ⎥⎦

System (39) can be divided into two subsystems form as

47

(39)

IFAC LSS 2013 July 7-10, 2013. Shanghai, China

⎧ ⎡ 0 1⎤ ⎡0⎤ ⎡0 x1 + ⎢ ⎥ u1 + p1 ⎢ ⎪ x&1 = ⎢ ⎥ ⎣α 0 ⎦ ⎣β ⎦ ⎣ −γ ⎨ ⎪y = 1 0 x ]1 ⎩ 1 [ ⎧ ⎡ 0 1⎤ ⎡0⎤ ⎡0 ⎪ x&2 = ⎢ ⎥ x2 + ⎢ β ⎥ u2 + p2 ⎢ −γ α 0 ⎣ ⎦ ⎣ ⎦ ⎣ ⎨ ⎪y = 1 0 x ] 2 ⎩ 2 [

(43) Using Assumption 1-3 ， it can be shown that there exist constants μi > 0, i = 1, 2 such that，

0⎤ ⎡ 0 0⎤ x1 + p1 ⎢ ⎥ ⎥ x2 0⎦ ⎣γ 0 ⎦ 0⎤ ⎡ 0 0⎤ x2 + p2 ⎢ ⎥ x1 0 ⎥⎦ ⎣γ 0⎦

⎛ 2 ⎞ z T ⎜ ∑ H iT ( x e ( p ) , pi ) H i ( x e ( p ) , pi ) ⎟ z ⎝ i =1 ⎠

(40)

where x1 = (θ1 ,θ&1 )T ， x2 = (θ 2 ,θ&2 )T are the state vectors of

H11 ( x1e ( p1 ) , p1 ) = H 21 ( x1e ( p1 ) , p1 ) =

two subsystems respectively; y1 ， y2 the outputs of two subsystems respectively; the system parameters are α = g / l ， β = 1/ ml 2 ， γ = a 2 k / ml 2 . We assume that the spring can slide up and down the rods of the pendulums in sudden jumps of unpredictable size and direction between the support and a height a . This means that a1 (t ) and a2 (t ) are assumed as a piecewise continuous function in both time and state, such that 0 ≤ a1 (t ) ≤ a , 0 ≤ a2 (t ) ≤ a . p1 (t ) =

(

⎤ ⎥ (45a) 1/ 2 ⎥ 2 ⎡0.2 + xe ( p ) ⎤ ⎥ 11 1 ⎢⎣ ⎥⎦ ⎦ 0

H 22 ( x2e ( p2 ) , p2 ) = H12 ( x2e ( p2 ) , p2 )

(

)

⎡ ⎡0.1 xe p + 1 ⎤1/ 2 21 ( 2 ) ⎢⎣ ⎦ =⎢ ⎢ 0 ⎣

respectively，such that p1 (t ) ∈ [0,1] , p2 (t ) ∈ [0,1] .

∗

⎤ ⎥ (45b) 1/ 2 ⎥ 2 ⎡0.2 + xe ( p ) ⎤ ⎥ 21 2 ⎢⎣ ⎥⎦ ⎦ 0

∗

Putting x11 = x21 = 0 into (45)-(46), we get

When we control the coupled inverted pendulums by computer, we need to use a quantizer to the control input of the system. Let g / l = 1 ， 1/ ml 2 = 1 , and a 2 k / ml 2 = 1 . System (40) becomes ⎡0 1 ⎤ ⎡0⎤ ⎡ 0 0⎤ ⎡0 0⎤ x&1 = ⎢ x1 + ⎢ ⎥ q1 (u1 ) + p1 ⎢ x1 + p1 ⎢ ⎥ ⎥ ⎥ x2 , ⎣1 0 ⎦ ⎣1 ⎦ ⎣ −1 0 ⎦ ⎣1 0⎦

H11 ( x1∗ , p1∗ ) = H12 ( x2∗ , p2∗ ) = H 21 ( x1∗ , p1∗ ) = H 22 ( x2∗ , p2∗ )

. 0 ⎤ ⎡ 0.32 =⎢ ⎥ 0.45⎦ ⎣ 0 Based on Theorem 1, by solving (24), we get μ1 = 1.1 ，

μ2 = 1.24 , K1 = K 2 = [−2.145 − 2.0731] , K1 = K 2 = 2.98 .

⎡0 1 ⎤ ⎡0⎤ ⎡0 0⎤ ⎡ 0 0⎤ x&2 = ⎢ x2 + ⎢ ⎥ q2 (u2 ) + p2 ⎢ x1 + p2 ⎢ ⎥ ⎥ ⎥ x2 , ⎣1 0 ⎦ ⎣1 ⎦ ⎣1 0 ⎦ ⎣ −1 0 ⎦ (41) where q1 (u1 ) and q2 (u2 ) are the logarithmic quantizer. The nonlinear terms of interconnected system are shown as ⎡ 0 0 0 0⎤ h1 ( x, p1 ) = h11 ( x1 , p1 ) + h12 ( x2 , p1 ) = ⎢ ⎥ p1 x ⎣ −1 0 1 0 ⎦

The nonlinear terms need to satisfy the following inequalities by (43)-(44).

(

)

2 H111 ( x1e ( p1 ), p1 ) = 0.1 x11e ( p1 ) + 1 ≤ 0.1μ12 2 H112 ( x1e ( p1 ), p1 ) = 0.2 + x11e ( p1 ) ≤ 0.2 μ12 2

(

)

2 e H121 ( x2e ( p2 ), p2 ) = 0.1 x21 ( p2 ) + 1 ≤ 0.1μ12 2 e H122 ( x2e ( p2 ), p2 ) = 0.2 + x21 ( p2 ) ≤ 0.2μ12 2

⎡0 0 0 0⎤ h2 ( x, p2 ) = h21 ( x1 , p2 ) + h22 ( x2 , p2 ) = ⎢ ⎥ p2 x ⎣1 0 −1 0 ⎦

(

)

2 H 211 ( x1e ( p1 ), p1 ) = 0.1 x11e ( p1 ) + 1 ≤ 0.1μ 22

(42) ⎡ 0 0⎤ ⎡ 0 0⎤ where h11 ( x1 , p1 ) = p1 ⎢ ⎥ x1 , h22 ( x2 , p2 ) = p2 ⎢ −1 0 ⎥ x2 ⎣ ⎦ ⎣ −1 0 ⎦ are the nonlinear term of each subsystem, while ⎡0 0⎤ ⎡0 0⎤ h12 ( x2 , p1 ) = p1 ⎢ x2 and h21 ( x1 , p2 ) = p2 ⎢ ⎥ ⎥ x1 are ⎣1 0 ⎦ ⎣1 0 ⎦ nonlinear terms between two subsystems. First, we give parametric stability of the system (41) without quantization and decentralized controller design. The equilibrium of the system (41) without quantization are

(46)

2 H 212 ( x1e ( p1 ), p1 ) = 0.2 + x11e ( p1 ) ≤ 0.2 μ 22 2

(

)

e 2 H 221 ( x2e ( p2 ), p2 ) = 0.1 x21 ( p2 ) + 1 ≤ 0.1μ22 e 2 H 222 ( x2e ( p2 ), p2 ) = 0.2 + x21 ( p2 ) ≤ 0.2μ22 2

We can obtain x11e ( p1 ) ≤ 0.2 , x21e ( p2 ) ≤ 0.2 , based on (46). The corresponding range of parameters p1 and p2 are p1 ≤ 0.5832 p2 ≤ 0.5832 . Putting u1 = K1 ( x1 − x1∗ ) and u2 = K 2 ( x2 − x2∗ ) into system (41), then,

x1∗ = [ 0 0 ] , x2∗ = [ 0 0 ] . It is obvious that the system is

the stability range of closed-loop system is p1 ∈ [0 0.5832] ， p2 ∈ [0 0.5832] .

T

not stable at this point. According to Assumption 1, function g ( x e ( p ) , p, z ) is satisfied

Secondly, based on Theorem δ = diag{δ1 , δ 2 } = diag{0.3, 0.3} .

⎛ 2 ⎞ g g ≤ z ⎜ ∑ H iT ( x e ( p ) , pi ) H i ( x e ( p ) , pi ) ⎟ z. ⎝ i =1 ⎠ T

)

⎡ ⎡0.1 x e p + 1 ⎤1/ 2 11 ( 1 ) ⎢⎣ ⎦ ⎢ ⎢ 0 ⎣

( a1 (t ) / a ) 2 , p2 (t ) = ( a2 (t ) / a ) 2 are uncertain parameters

T

(44)

⎛ 2 ⎞ ≤ z ⎜ ∑ μi2 H iT ( x∗ , pi∗ ) H i ( x∗ , pi∗ ) ⎟ z 1 i = ⎝ ⎠ for any pi ∈ Ω i , where T

T

48

2,

we

obtain

IFAC LSS 2013 July 7-10, 2013. Shanghai, China

Suppose that p10 = p20 = 0.5 , θ10 = −0.2rad , θ 20 = 0.1rad , θ&10 = θ&20 = 0 . From the Fig.2-4, we can see that the decentralized quantizer makes the system stable when p10 = p20 = 2.5 . In this case, the spring is in the stretching state. When t = 15s , parameters p1 and p2 are changed to 0.58, the decentralized quantizer still make the system stable. While in this case, the spring is in the compressing state. When t = 30 s , parameters p1 and p2 are changed to 0.42, the decentralized quantizer still make the system stable. The rods of pendulums are vertical upwards and the spring is in the compressing state. Although the state of system is changing as the position change of the spring, the designed controller can parametrically stabilize the system within the range of parameters.

is still parametric stable under structural perturbations. Finally, the proposed method is applied to coupled inverted pendulum systems. The simulation results show that the designed controllers can parametric stabilize the coupled inverted pendulum systems effectively. 0.15 x21 x22 0.1

x2

0.05

0

-0.05

-0.1 p1 p2

0.6

0

5

10

15

20

25 t(s)

30

35

40

45

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Fig.4 Trajectory of state vector x2

0.5

Acknowledgment

0.3

This work was supported in part by National Natural Science Foundation of China (61074001, 61074002), and in part by the Fundamental Research Funds for the Central Universities (2010QZZD016).

p

0.4

0.2

0.1

0

0

5

10

15

20

25 t(s)

30

35

40

45

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REFERENCES

Fig.2 Functions p1 and p2

Bakule L.(2008). Decentralized control: An overview, Annual Reviews in Control, 32: 87-98. Brockett R. W., Liberzon D.(2000). Quantized feedback stabilization of linear systems, IEEE Transactions on Automatic Control, 45: 1279–1289. Chen N., Gui W.H., Liu B.(2007). Parametric absolute stability of interconnected lurie control systems, Acta Automatica Sinica, 33(6), 1283-1289(in Chinese). Chen N., Liu Y.T, Liu B, and Gui W.H.(2012). Parametric absolute stabilization of Lurie time-delay systems with polytopic uncertainty based on state feedback, Asian Journal of Control, Accepted. Delchamps D. F.(1990). Stabilizing a linear system with quantized state feedback, IEEE Transactions on Automatic Control, 35: 916–924. Elia N. and Mitter K.(2001). Stabilization of linear systems with limited information. IEEE Transactions on Automatic Control, 46( 9), 1384–1400. Fu M., Xie L.(2005). The sector bound approach to quantized feedback control, IEEE Transactions on Automatic Control, 50, 1698-1710. Fu M. and Xie L.(2009). Finite-level quantized feedback control for linear systems, IEEE Transactions on Automatic Control, 54(5), 1165–1170. Gao H., Chen T. (2008). A new approach to quantized feedback control systems, Automatica, 44, 534-542.

x11 x12

0.25 0.2 0.15 0.1

x1

0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25 0

5

10

15

20

25 t(s)

30

35

40

45

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Fig.3 Trajectory of state vector x1 5 Conclusion We have studied parametric stability and decentralized control of a kind of quantized interconnected systems. First, based on the solution of LMI, the decentralized state feedback controller is designed to make the closed-loop system without quantizer parametric stable. Then, the logarithm quantizer is used to quantizing the control input of each subsystem and the range of quantized density is regulated by local information, so that the closed-loop system 49

IFAC LSS 2013 July 7-10, 2013. Shanghai, China

Gavel D. T. and Siljak D. D.(1989). Decentralized Adaptive Control: Structural Conditions for Stability[J]. IEEE Transactions on Automatic Control, 34(4): 413-426. Gong Z. M., Wen C. Y. and Mital D. P.(1996). Decentralized robust controller design for a class of interconnected uncertain systems: with unknown bound of uncertainty[J]. IEEE Transactions on Automatic Control, 1996, 41(6): 850-854. Ikeda M., Ohta Y., and Siljak D. D.(1991). Parametric stability, New trends in system theory, Boston, Birkhauser. Ikeda M., Siljak D. D.(1980). Lotka-Volterra equtions: Decomposition, stability and structure, J. math. Biol., 9, 65-83. Ishii H. and Francis B.(2002). Limited Data Rate in Control Systems with Networks, Berlin: Springer. Jamshidi M.(1983). Large-scale systems modeling and control, North-Holland. Kalman R. E.(1956). Nonlinear aspects of sampled-data control systems, In: Proceedings of the Symposium on Nonlinear Circuit Theory. Brooklyn, NY. Kwatny H. G., Pasrija A. K., Bahar L. Y.(1986). Static bifurcations in electric power networks: Loss of steadystate stability and voltage collapse, IEEE Trans. Circuits Systems, 33(10), 981-991. Liu Tengfei, Jiang Zhongping, Hill J. David. (2012) A sector bound approach to feedback control of nonlinear systems with state quantization. Automatica, 46(1): 145-152. Ohta Y., and Siljak D. D.(1994). Parametric quadratic stabilizability of uncertain nonlinear systems, System Control Letters, 22: 437-444. Siljak D. D. (1978). Large-scale dynamic systems: stability and structure, North-Holland, Amsterdam. Song Z, Sukthankar P, Chen Y. Q., et al.(2003). Progressive fuzzy fusion control of two coupled inverted penduli. Proceedings 2003 IEEE International Symposium on Computational Intelligence in Robotics and Automation, 3: 1457-1462. Sundarapandian V.(2006). New results on the parametric stability of nonlinear systems, Mathematical and Computer Modelling, 43(1), 9-15. Wada T., Ikeda M., Ohta Y., and Siljak D. D.(1998). Parametric absolute stability of Lur’e systems, IEEE Transactions Automatic Control, 43(11): 1649-1653. Wada T., Ikeda M., Ohta Y., and Siljak D. D.(2000). Parametric absolute stability of multivariable Lur’e systems, Automatica, 36, 1365-1372. Zecevic A. I., Siljak D. D.(2003). Stabilization of nonlinear systems with moving equilibria, IEEE Transactions Automatic Control, 48(6): 1036-1040. Zecevic A. I., Siljak D. D.(2009). Control of Complex Systems, Structural Constraints and Uncertainty, Springer.

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