A Construction Method of Switching Lyapunov

A Construction Method of Switching Lyapunov Function for Nonlinear Systems Hiroshi Ohtake and Kazuo Tanaka Department of Mechanical Engineering and Intelligent Systems The University of Electro-Communications 1-5-1 Chofugaoka, Chofu, Tokyo 182-8585 Japan Email [email protected], [email protected]

Hua O. Wang Department of Electrical and Computer Engineering Duke University Durham, NC 27708-0291 USA Email [email protected]

Abstract— This paper presents a new stability condition based on a switching Lyapunov function for a class of nonlinear systems. In our previous paper [11], we propsed a switching fuzzy model based on the sector nonlinearity concept. The switching fuzzy model can exactly represent a class of nonlinear systems globally or semi-globally. This paper describes a construction method of a switching Lyapunov function for the switching fuzzy model. The switching Lyapunov function is constructed by mirroring structure of the switching fuzzy model. An example illustrates the utility of this approach.

y

a1x

w2 (x) w1 (x)+w2 (x) w1(x) h 2 (x)= w1 (x)+w2 (x)

y=f(x)

h 1 (x)=

w1(x) w2 (x)

a2 x x

I. Introduction A lot of research on fuzzy model-based control has been reported. In the fuzzy model-based control, a fuzzy model of a system is constructed exactly or approximately. The complexity of a system makes the number of rules of a fuzzy model exponentially increase. The curse of the number of rules makes controller design difficult. To loosen the curse, we proposed a switching fuzzy model based on the so-called sector nonlinearity concept [11]. The switching fuzzy model has two key features. One is to switch local Takagi-Sugeno (T-S) fuzzy models represented in each region. The other is to decrease the number of rules which fire simultaneously in comparison with an ordinary fuzzy model. On the other hand, the stability of fuzzy control systems is mainly analyzed via quadratic Lyapunov functions. However, the conservatism of quadratic stability conditions has been often discussed as a typical problem. This paper presents a new stability condition for switching fuzzy models by introducing a switching Lyapunov function. The switching Lyapunov function switches local Lyapunov functions constructed in each region as well as the switching fuzzy model. The condition is less conservative than quadratic stability conditions. An example illustrates the utility of this approach. II.

Switching fuzzy model based on the sector nonlinearity concept

In [7], we proposed a switching fuzzy model which consists of local T-S fuzzy models. Furthermore, we extended it by introducing a fuzzy model construction based on the sector nonlinearity concept [11]. This section recalls the sector nonlinearity concept. 0-7803-7280-8/02/$10.00 ©2002 IEEE

Fig. 1. Sector nonlinearity concept.

A. Sector nonlinearity concept in single variable system case Consider a nonlinear continuous function with single variable. y = f (x), (1) where f is known. Figure 1 illustrates the sector nonlinearity concept. The model construction based on the sector nonlinearity guarantees to exactly represent f (x) with the sector (straight lines: a1 x and a2 x) globally or semiglobally. From a1 , a2 and f (x), membership functions h1 (x) and h2 (x) are constructed as h1 (x) =

f (x) − a2 x a1 x − f (x) , h2 (x) = , (a1 − a2 )x (a1 − a2 )x

where h1 (x) + h2 (x) = 1 and h1 (x), h2 (x) ≥ 0 for all x. From a1 , a2 and these membership functions, the output y is reconstructed by y

= h1 (x)a1 x + h2 (x)a2 x =

2
i=1

hi (x)ai x.

(2)



where

sector 1 40 sector 2

30 20 10

y

Original nonlinear function

0 -10

-30 10 -5

0

0 5

x1

10 -10

In our approach, we define a quadrant as a region. The total number of regions is Q = 2n . A switching fuzzy model based on the sector nonlinearity concept is represented as Region q : Rule 1 : IF (x1 , · · · , xn ) is hq1 THEN yq1 = a1q1 x1 + · · · + anq1 xn , Rule 2 : IF (x1 , · · · , xn ) is hq2 (5) THEN yq2 = a1q2 x1 + · · · + anq2 xn ,

-20

-40 -10

x2

where q = 1, 2, · · · , Q and Q = 2n . The local T-S fuzzy model in each region has two rules. The membership functions are calculated as

Fig. 2. Limitation of global sector.

As shown in Figure 1, to exactly represent the function (1) with (2), the constructed sector is required to exactly cover the function. If it is difficult to find a global sector, a semi-global sector is employed [10]. B. Sector nonlinearity concept in multi-variable system case [11] Consider a nonlinear continuous function with n variables. (3) y = f (x) = f (x1 , x2 , · · · , xn ),

(4)

hq1 (x) =

f (x) − yq2 (x) , yq1 (x) − yq2 (x)

(6)

hq2 (x) =

yq1 (x) − f (x) , yq1 (x) − yq2 (x)

(7)

where hq1 (x) ≥ 0, hq2 (x) ≥ 0 and hq1 (x) + hq2 (x) = 1. The fuzzy reasoning process is defined as y

=

Q
2


vq (x)hqi (x)

q=1 i=1

where f is known. A sector for the function (3) is constructed by n dimensional linear functions y = a1 x1 + a2 x2 + · · · + an xn .

sj = 1 (xj ≥ 0), sj = 0 (xj ≤ 0).

× {a1qi x1 + · · · + anqi xn } , 

where vq (x) =

In the single variable system case, it is possible to exactly represent the nonlinear function (1) with the sector (a1 x and a2 x) globally or semi-globally. However, in the multivariable system case, it is generally impossible to exactly represent the nonlinear function (3) with a sector (two linear hyperplanes represented by (4)). For example, Figure 2 shows the case where it is impossible to exactly cover the function via the sector (planes) even if any coefficients of a1 and a2 are selected. To overcome the problem, we defined each quadrant on inputs space as an (local) area. By finding a local sector for each (local) area, the nonlinear function (3) can be globally or semi-globally covered.

1, x ∈ Rq , 0, x ∈ / Rq .

(8)

(9)

Remark 1: The ordinary Takagi-Sugeno fuzzy model can exactly represent the nonlinear function (3) globally or semi-globally. However, in the ordinary Takagi-Sugeno fuzzy model, many rules have to be fired simultaneously if the nonlinear function (3) is complicated. On the other hand, the switching fuzzy model (5) has only two rules in each region. The switching fuzzy model can decrease the number of rules (which fire simultaneously) in comparison with the ordinary fuzzy model. This means that the number of LMIs for analysis and design of fuzzy control systems can be reduced.

Definition 1: Consider a quadrant III. Construction of multi-dimensional sectors x1 ≥ 0, x2 ≤ 0, x3 ≤ 0, x4 ≥ 0, · · · , xn ≥ 0. We define the quadrant as R(s1 , s2 , s3 , s4 , · · · , sn ), s1 = 1, s4 , s5 , · · · , sn = 1, s2 , s3 = 0 or R(1, 0, 0, 1, · · · , 1), 0-7803-7280-8/02/$10.00 ©2002 IEEE

We proposed a method to construct multi-dimensional sectors by partially differentiating the nonlinear function (3) [11]. However, the method does not guarantee the optimal sector construction which tightly covers the nonlinear function as much as possible. A tighter sector brings us a shaper solution of analysis and design for a system. This section proposes a method to construct a multi-dimensional sector which has the minimum distance between the sector and the function (3).

solving the following condition, where |x1 | ≤ d1 , |x2 | ≤ d2 , · · ·, |xn | ≤ dn .

A. Minimum distance sector Consider qth region Rq (s1q , s2q , · · · , snq )

minimize a ´111 , · · · , a ´n11 a ´112 , · · · , a ´n12 a ´121 , · · · , a ´n21 a ´122 , · · · , a ´n22

skq = 1 or 0, k = 1, 2, · · · , n Suppose that |x1 | ≤ d1 , |x2 | ≤ d2 , · · ·, |xn | ≤ dn . We can find sector coefficients by numerically solving the following condition: |Yq1 (aq1 ) − Yq2 (aq2 )| minimize a1q1 , · · · , anq1 a1q2 , · · · , anq2 subject to yq1 (aq1 , x) − f (x) ≥ 0, f (x) − yq2 (aq2 , x) ≥ 0,

×

d1 (s1q −1)

dn (snq −1)

d2 (s2q −1)



×

x dx1 dx2 · · · dxn ,

yqi (aqi , x) = a1ql(1,q,i) x1 + a2ql(2,q,i) x2 + · · · + anql(n,q,i) xn , 2

d1 s1q

d1 (s1q −1)

x dx1 dx2 · · · dxn

yqi (aqi , x) = a ´1l(1,q,i)ζ(1,q) x1 + a ´2l(2,q,i)ζ(2,q) x2 +···a ´nl(n,q,i)ζ(n,q) xn , 2

l(j, q, i) = 2(2−sjq −i) = 1 or 2, ζ(j, q) = 2 − sjq = 1 or 2. The sector in qth region is represented as

l(j, q, i) = 2(2−sjq −i) = 1 or 2. The sector in qth region is represented as yq1 (x) = a1ql(1,q,1) x1 + a2ql(2,q,1) x2 + · · · + anql(n,q,1) xn , yq2 (x) = a1ql(1,q,2) x1 + a2ql(2,q,2) x2 + · · · + anql(n,q,2) xn .

x ∈ Rq , x ∈ Rq ,

aqi = [´ a1l(1,q,i)ζ(1,q) a ´2l(2,q,i)ζ(2,q) · · · a ´nl(n,q,i)ζ(n,q) ], Yqi (aqi ) = aqi D q  dn snq  d2 s2q Dq = ···

d2 (s2q −1) d1 s1q

(13)

where

x ∈ Rq , x ∈ Rq ,

aqi = [a1ql(1,q,i) a2ql(2,q,i) · · · anql(n,q,i) ], Yqi (aqi ) = aqi D q ,  dn snq  d2 s2q Dq = ··· 

|Yq1 (aq1 ) − Yq2 (aq2 )|

q=1

subject to yq1 (aq1 , x) − f (x) ≥ 0, f (x) − yq2 (aq2 , x) ≥ 0, ∀q

(10)

where

dn (snq −1)

Q


(11) (12)

B. Compatibility condition of sectors on region boundary Since each region corresponds to each quadrant, each region is partitioned by each axis. If outputs of neighbor local T-S fuzzy models on the region boundary are different, the fuzzy model is discontinuous although the nonlinear function (3) is continuous. To avoid the problem, we consider a compatibility condition of sectors on region boundary. To satisfy a compatibility condition, some of coefficients on region boundary between Rq1 and Rq2 should be equal each other.  ´j11 , ajq1 1 = ajq2 1 = a s = sjq2 = 1, ´j21 , jq1 ajq1 2 = ajq2 2 = a  ajq1 1 = ajq2 1 = a ´j12 , s = sjq = 0, 2 ajq1 2 = ajq2 2 = a ´j22 , jq1 where j = 1, 2, · · · , n. By combining (10) and the compatibility condition, we can find a sector by numerically 0-7803-7280-8/02/$10.00 ©2002 IEEE

´1l(1,q,1)ζ(1,q) x1 + a ´2l(2,q,1)ζ(2,q) x2 yq1 (x) = a +···+ a ´nl(n,q,1)ζ(n,q) xn , yq2 (x) = a ´1l(1,q,2)ζ(1,q) x1 + a ´2l(2,q,2)ζ(2,q) x2 +···+ a ´nl(n,q,2)ζ(n,q) xn .

(14) (15)

Remark 2: The conditions (10) and (13) are a nonlinear optimization problem. It is difficult to analytically solve the problem. For instance, MATLAB optimization toolbox can be employed to solve the problem. IV. Dynamic state equation In the previous section, we described the construction method of the switching fuzzy model for the n-input 1output (static) function. This section considers the following dynamic state equation which has n nonlinear or linear functions which consist of n state variables.   f1 (x(t))  f2 (x(t))    ˙ x(t) = (16) , ..   . fn (x(t)) where x(t) = [x1 (t) x2 (t) · · · xn (t)]T , fj (x(t)) is a scalar nonlinear or linear function, where j = 1, 2, · · ·, n. Suppose that m is the number of nonlinear functions in (16). Hence,

the number of linear functions in (16) is n − m. We apply the sector nonlinearity concept to each fj (x(t)).   f1 (x(t))  f2 (x(t))    ˙ = x(t)  ..   . fn (x(t))



hqi1 (x(t))aqi1 x(t)    i=1   ρ2  
  Q hqi2 (x(t))aqi2 x(t) 
  , vq (x(t))  =  i=1    .. q=1   . ρ  n 
  h (x(t))a x(t) qin

Theorem 1: [7] The switching fuzzy system (17) is asymptotically stable if there exists a positive definite matrix P satisfying the following LMI conditions. ATqi P + P Aqi < 0

qin

i=1

=

ρ1 Q



ρ2


···

q=1 i1 =1 i2 =1

ρn


Q
r


vq (x(t))

(17)

q=1 i=1

where r = 2m ,  1, ρj = 2,

fj is linear, fj is nonlinear,

aqi1 1 = [a1l(1,q,i1 )i1 1 a2l(2,q,i1 )i1 1 · · · anl(n,q,i1 )i1 1 ], aqi2 2 = [a1l(1,q,i2 )i2 2 a2l(2,q,i2 )i2 2 · · · anl(n,q,i2 )i2 2 ], .. . aqin n = [a1l(1,q,in )in n a2l(2,q,in )in n · · · anl(n,q,in )in n ],

where Rq (q = 1, 2, · · · , Q) denotes each quadrant. An important problem is the continuity of (19). It will be addressed in the proof of Theorem 2. Theorem 2: The switching fuzzy system (17) is asymptotically stable if there exist matrices P q (q = 1, 2, · · · , Q) satisfying (20), (21) and (22). P q > 0, ATqi P q

ˆ qi (x(t)) := h

i1 =1 i2 =1

i=1

···

ρn
in =1

hqi1 1 (x(t))

There are r rules in each region. Remark 3: Suppose that fj (x(t)) has σj nonlinear terms, where σj = 0 if fj (x(t)) is a linear function. Then, the maximum

n number of rules that simultaneously fire is σ generally 2 j=1 j in an ordinary T-S fuzzy model that represents the nonlinear dynamics (16). On the other hand, the maximum number of rules that simultaneously fire is 2m in the switching

fuzzy model (17). Note that it is indeσ

∀i, q,

(21)

skq = 1 or 0, k = 1, 2, · · · , n,

×hqi2 2 (x(t)) · · · hqin n (x(t)).

n

+ P q Aqi < 0

Rq (s1q , s2q , · · · , snq )

2

ρ2 ρ1



(20)

where P q belongs to qth region

l(j, q, ik ) = 2(2−sjq −ik ) , r


(18)

This section presents a stability condition based on switching Lyapunov function. The switching Lyapunov function is defined as  T   xT (t)P 1 x(t), x(t) ∈ R1 ,   x (t)P 2 x(t), x(t) ∈ R2 , (19) V (x(t)) = ..  .    T x (t)P Q x(t), x(t) ∈ RQ ,

in =1

ˆ qi (x(t))Aqi x(t), vq (x(t))h

∀i, q

VI. Stability analysis based on switching Lyapunov function

×hqi1 1 (x(t))hqi2 2 (x(t)) × · · ·   aqi1 1  aqi 2   2  ×hqin n (x(t))   x(t) ..   . aqin n =

V. Stability condition based on common Lyapunov function This section recalls a stability condition (based on common Lyapunov function) for the switching fuzzy system (17).



ρ1


p = {j|fj (x(t)) is nonlinear}. Thus, the maximum number of fired rules of the switching fuzzy model is less than or equal to that of the ordinary fuzzy model.

pendent of σj . 2 j=1 j = 2m if and only if σp = 1, where 0-7803-7280-8/02/$10.00 ©2002 IEEE

and P q is defined as    Pq = 

P11ζ(1,q) P12ψ(1,2,q) .. .

P12ψ(1,2,q) P22ζ(2,q) .. .

· · · P1nψ(1,n,q) · · · P2nψ(2,n,q) .. .. . .

P1nψ(1,n,q)

P2nψ(2,n,q)

···

   , 

Pnnζ(n,q) (22)

ζ(k, q) = 2 − skq = 1 or 2, ψ(k, w, q) = −skq − 2swq + 4 = 1, 2, 3 or 4, due to guarantee of the continuity of (19). (proof)

We focus on the derivative only in qth region. The switching fuzzy system (17) in qth region is represented as r
ˆ qi Aqi x(t), ˙ x(t) = x(t) ∈ Rq . (24) h

0.25

x1

0.2 0.15 x1 (t), x2 (t)

Consider (19) as a candidate of Lyapunov function. The time derivative of (19) is  T ˙ x˙ (t)P 1 x(t) + xT (t)P 1 x(t),     x(t) ∈ R1 ,    T T  ˙ ˙ x x(t), (t)P x(t) + x (t)P  2 2  x(t) ∈ R2 , V˙ (x(t))= (23)  .  .   .    x˙ T (t)P x(t) + xT (t)P x(t),  Q Q˙   x(t) ∈ RQ .

0.1 0.05 0 -0.05 -0.1 -0.15 -0.2

x

2

-0.25

0

i=1

1

2

3

4

5 6 time [sec.]

7

8

9

10

Fig. 3. State transient of the system (27)

By substituting (24) to (23) in qth region, we obtain ˙ V˙ (x(t)) = x˙ T (t)P q x(t) + xT (t)P q x(t) r  
ˆ = hqi xT (t) ATqi P q + P q Aqi x(t).

. ˙ THEN x(t) = A12 x(t), (25)

Region2 R(0, 1) : Rule1 : IF (x1 (t), x2 (t)) is h21 ˙ THEN x(t) = A21 x(t), Rule2 : IF (x1 (t), x2 (t)) is h22 ˙ THEN x(t) = A22 x(t), Region3 R(1, 0) : Rule1 : IF (x1 (t), x2 (t)) is h31 ˙ THEN x(t) = A31 x(t), Rule2 : IF (x1 (t), x2 (t)) is h32 ˙ THEN x(t) = A32 x(t), Region4 R(0, 0) : Rule1 : IF (x1 (t), x2 (t)) is h41 ˙ THEN x(t) = A41 x(t), Rule2 : IF (x1 (t), x2 (t)) is h42 ˙ THEN x(t) = A42 x(t),

i=1

Next, we consider the continuity of the candidate (19) on region boundary. The continuity of (19) is guaranteed if xT (t)P q1 x(t) = xT (t)P q2 x(t),

(26)

where x(t) ∈ Rq1 ∩ Rq2 . Note that (26) is guaranteed if P q satisfies (22). The candidate (19) becomes a Lyapunov function if (20), (21) and (22) hold. (Q. E. D.) Remark 4: Theorem 2 is reduced to Theorem 1 when P q = P for all q. Therefore, Theorem 2 is less conservative than Theorem 1. VII. Application example Consider the following nonlinear system:   x˙ (t) ˙ = 1 x(t) , x˙ 2 (t)   x2 (t) =−x31 (t) − x32 (t) + 5x21 (t)x2 (t) + 5x1 (t)x22 (t) , (27) −3x1 (t)x2 (t) − x1 (t) − x2 (t)

where

 A11 = A41 =  A12 = A42 =

−d ≤ x1 ≤ d, −d ≤ x2 ≤ d. Figure 3 shows the state transient of the system (27), where initial state is x(0) = [0.23 − 0.23]T . When d = 0.23, the quadratic Lyapunov function based on the ordinary TS fuzzy model does not exist. On the other hand, when d=0.23, the switching fuzzy model (17) are constructed as Region1 R(1, 1) : Rule1 : IF (x1 (t), x2 (t)) is h11 ˙ THEN x(t) = A11 x(t), Rule2 : IF (x1 (t), x2 (t)) is h12 0-7803-7280-8/02/$10.00 ©2002 IEEE

 A21 = A31 =  A22 = A32 =  h11 =h41 =

0 1 −1.19 −1.18 0 1 −1.19 −0.45 0 1 −0.43 −1.18

 ,  ,  ,  ,

x˙ 2 (t)+1.19x1 (t)+1.18x2 (t) , (x1 (t), x2 (t)) 0.76x1 (t)+0.73x2 (t)

1, 

h12 =h42 =

0 1 −0.43 −0.45

= (0, 0),

otherwise,

−0.43x1 (t)−0.45x2 (t)−x˙ 2 (t) , (x1 (t), x2 (t)) 0.76x1 (t)+0.73x2 (t)

0,

otherwise,

= (0, 0),

20 18 35

16 30

14 25

12

V(x(t))

V(x(t)) 20

10

15

8

10

6

5

4

-0.2 0

2

-0.2

0 0

1

2

3

4

5

6

7

8

9

10

time [sec.]

0 -0.1

0

x2

0.1

0.2

0.2

x1

Fig. 5. Switching Lyapunov function.

Fig. 4. Time trajectory of the switching Lyapunov function.

 h21 =h31 =

x˙ 2 (t)+0.43x1 (t)+1.18x2 (t) −0.76x1 (t)+0.73x2 (t) , (x1 (t), x2 (t))

1, 

h22 =h32 =

= (0, 0),

otherwise,

−1.19x1 (t)−0.45x2 (t)−x˙ 2 (t) , (x1 (t), x2 (t)) 0.76x1 (t)+0.73x2 (t)

0,

= (0, 0),

otherwise,

where x˙ 2 (t)

= −x31 (t) − x32 (t) + 5x21 (t)x2 (t) +5x1 (t)x22 (t) − 3x1 (t)x2 (t) − x1 (t) − x2 (t).

By solving the LMI conditions (20), (21) and (22), the following positive definite matrices are obtained.   201.3 73.3 P1 = P4 = > 0, 73.3 247.0   201.3 47.3 > 0. P2 = P3 = 47.3 247.0 Figures 4 and 5 show the time trajectory and the the switching Lyapunov function. We can guarantee the stability of the nonlinear system (27) under −0.23 ≤ x1 ≤ 0.23 and −0.23 ≤ x2 ≤ 0.23. VIII. Conclusions We have derived a new stability condition based on switching Lyapunov function for open-loop systems represented by a switching fuzzy model. An application example has illustrated the utility of this approach. A future work is to extend this approach for closed-loop systems. References [1] [2]

[3]

K. Tanaka, and M. Sugeno, “Stability Analysis and Design of Fuzzy Control System,” FUZZY SETS AND SYSTEM, vol.45, no.2, pp.135-156, 1992. H. O. Wang, K. Tanaka and M. Griffin, “An Approach to Fuzzy Control of Nonlinear Systems: Stability and Design Issues,” IEEE Transactions on Fuzzy Systems, vol.4, no.1, pp.14-23, 1996. K. Tanaka, T. Ikeda and H. O. Wang, “Fuzzy Regulators and Fuzzy Observers: Relaxed Stability Conditions and LMI based

0-7803-7280-8/02/$10.00 ©2002 IEEE

Designs,” IEEE Transactions on Fuzzy Systems, vol.6, no.2, pp.250-265, 1998. [4] M. Johansson and A. Rantzer, “Computation of Piecewise Quadratic Lyapunov function for Hybrid systems,” IEEE Transactions on Automatic Control, Vol.43, No.4, pp.555-559, 1998. [5] A. Rantzer and M. Johansson, “Piecewise Linear Quadratic Optimal Control,” IEEE Transactions on Automatic Control, Vol.45, No.4, pp.629-637, 2000. [6] M. Johansson, A. Rantzer and K. - E. Arzen, “Piecewise Quadratic Stability of Fuzzy Systems,” IEEE Transactions on Fuzzy Systems, Vol.7, No.6, pp.713-722, 1998. [7] K. Tanaka, M. Iwasaki and H. O. Wang, “Switching Control of an R/C Hovercraft: Stabilization and Smooth Switching,” IEEE Transactions on Systems, Man, and Cybernetics, Part C, 2001 (in print). [8] T. Takagi and M. Sugeno, “Fuzzy Identification of Systems and Its Applications to Modeling and Control,” IEEE Transactions on Systems, Man and Cybernetics, Vol.15, pp.116-132, 1985. [9] S. Kawamoto et al., “An Approach to Stability Analysis of Second Order Fuzzy Systems,” Proceeding of First IEEE International Conference on Fuzzy Systems, Vol.1, pp.1427-1434, 1992. [10] K. Tanaka and H. O. Wang, Fuzzy Control Systems Design and Analysis, JOHN WILEY & SONS, INC., 2000. [11] H. Ohtake, K. Tanaka and H. O. Wang, “Fuzzy Modeling via Sector Nonlinearity Concept,” Proceeding of Joint 9th IFSA World Congress and 20th NAFIPS International Conference, pp.127-132, Vancouver, 2001. [12] K. Tanaka, T. Hori and H. O. Wang, “A Fuzzy Lyapunov Approach to Fuzzy Control System Design,” Proceeding of 2001 American Control Conference, pp.4790-4795, Arlington, 2001.