In this section, we turn our attention to the stability analysis of interconnected continuous time ISS systems. First, for each subsystem a CPA ISS Lyapunov function in dissipative formulation is computed by our proposed approach. Then stability of interconnected systems is looked into via the small gain theorem in linear form.
In the following we investigate the stability of interconnected continuous time systems which are described by the following equations
We treat each subsystem as a dynamic system with perturbations by regarding the inputs of other states as perturbations.
We assume thatfi satisfies (H1) or (H2) and that each subsystem is locally ISS.
Define eij = 0 if the statexj does not influencexi and eij = 1 otherwise.
4.4 Stability of interconnected ISS systems and estimate of the domain of
attraction 111
Theorem 4.4.1. Given a suitable triangulation Ti of a subset Di of state space xi and a suitable triangulationTj of a subset Dj of the input perturbation value space xj. If the linear optimization problem (4.36) has a feasible solution for each subsystem, then the function Vi defined by (4.37)is a CPAISS Lyapunov function in dissipative formulation, i.e., it satisfies (4.4)and
h∇Vν,i(xi), fi(x1, x2, . . . , xs)i ≤ −kxik2+
s
X
j=1,j6=i
√njeijrijkxjk2 (4.81)
for allxi ∈ Sνi ⊂ Dii
T and all xj ∈ DjT (i6=j),Dii
T :=DiT \ B2(0, i), i >0 . Proof. This result is directly obtained from Theorem 4.2.6.
Theorem 4.4.2. If the conditions of Theorem 4.4.1 and Theorem 1.6.1 are satisfied, then the interconnected system (4.80)is locally asymptotically stable on D defined by (4.82).
Proof. LetV = (V1, . . . , Vs)>. From the proof of Theorem 1.6.1 it is known that there exists an s-vector b > 0 such that W =hb, Vi is a CPA Lyapunov function for the whole system.
Define the set
D={x∈ D1T ×. . .× DsT \
B2(0, 1)×. . .× B2(0, s)
: W(x)< min
x∈∂(D1T×...×DsT)W(x)}.
(4.82) The interconnected system (4.80) is locally asymptotically stable on D, since W is a CPA Lyapunov function for system (4.80) onD.
4.4.1 Examples
In this section we present three examples to demonstrate how to analyse the stability of interconnected systems by Theorem 4.4.2. In order to compare results of Chapter 3 and this chapter, the first example is the academic example studied in Chapter 3. The second and third examples show that the proposed approach for computing CPA ISS Lyapunov functions in dissipative formulation is very useful in stability analysis of complicated and practical systems for which it is difficult to construct a Lyapunov function.
Example 1
We consider the academic example studied in Chapter 3.
(∆1: ˙x1=−x1+x31+x1x22,
∆2: ˙x2=−x2+x32+x2x21, (4.83) where x= (x1, x2)> inD= [−0.70225,0.70225]2 ⊂R2.
The suitable triangulation of [−0.70225,0.70225] is obtained in the same way as described in Section 2.1.3 with K = 265, k= 1 and the map ρ:R7→R.
ρ(s) = 10−5|s|s. (4.84)
System (4.83) is considered as two interconnected one dimensional systems. For subsystem
∆i, we obtain a CPA ISS Lyapunov function in dissipative formulation Vi (see Figure 4.9,
112
4. Computation of ISS Lyapunov functions and stability of interconnected systems
V1 =V2) by solving the corresponding linear optimization problem (4.36). The gain parameter isri = 0, which means that the perturbation has no influence in the stability of the statexi. Furthermore, Vi satisfies
hξ,x˙ii ≤ −kxik2+rikxjk1
≤ −kxik2+rikxjk2, ∀ξ∈∂ClVi(xi), i6=j. (4.85) According to inequalities (4.85), we define
A=
−1 0 0 −1
.
0 0.5 1 1.5 2 2.5 3 3.5 4
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 xi
Figure 4.9: CPA ISS Lyapunov function in dissipative formulation Vi delivered by the algorithm for subsystem ∆i.
It is obvious that the conditions of Theorem 1.6.1 hold. Thus system (4.83) is locally asymptotically stable inD1\(−10−5,10−5)2 withD1 defined by (4.86).
Let c= (1,1)>. According to Theorem 1.6.1 we have b= (1,1)> such that c> =−b>A.
LetV(x) =hb,(V1, V2)>i. Therefore, V is a CPA Lyapunov function for system (4.87). An estimate of the domain of attraction of (−10−5,10−5)2 (see Figure 4.10) is given by
D1={x∈[−0.70225,0.70225]2 : V(x)< min
x∈∂DV(x) = 3.7166738}. (4.86)
4.4 Stability of interconnected ISS systems and estimate of the domain of
Figure 4.10: An estimate of the domain of attraction of system (4.83) is the inside of the red curve.
Remark 4.4.3. Compared with the estimate of the domain of attraction by the local version of the small gain theorem in comparison form in Section 3.4, the estimate obtained here is not bigger. In order to obtain a bigger D1, we construct suitable triangulations of a set bigger than D and then solve the linear optimization problem (4.36) again. But the gain parameters ri yielded by the algorithm for system ∆i on the bigger D1 do not satisfy the conditions of Theorem 1.6.1. However, the domain of attraction obtained here is bigger than the estimate of the domain of attraction obtained by the small gain theorem in dissipative form. Moreover, the proposed approach in this chapter can deal with more complicate cases, which is demonstrated by the following two examples.
Example 2
We study stability of the following system adapted from the practical model of [87]. Here the practical model of [87] is investigated with fixed control, i.e. (33), (34) of [87] and without perturbations.
We consider this model as two interconnected subsystems, i.e., subsystem S1 with states x1= (lB, vB)>, subsystem S2 with statesx2= (lS, vB)>.
Consider each subsystem of (4.87) on D = B2(0,0.05) ⊂ R2 with UR = D. A suitable triangulation of D is constructed as described in Section 2.1.3 with K = 5, k = 4 and the
114
4. Computation of ISS Lyapunov functions and stability of interconnected systems
mapF :R2 7→R2,
F(s) =
(0.2s10−2kskksk2∞
2 , s6= 0,
0, s= 0. (4.88)
By our proposed approach we get a CPA ISS Lyapunov function in dissipative formulation for system S1 on D \ B2(0,0.032) as shown in Figure 4.11. The gain parameter is r1 = 0 and V1 satisfies
hξ,x˙1i ≤ −kx1k2+r1kx2k1
≤ −kx1k2+√
2r1kx2k2, ∀ξ∈∂ClV1(x1). (4.89) Similarly, a CPA ISS Lyapunov function in dissipative formulation for system S2 on D \ B2(0,0.032) is obtained by the same proposed approach. It is shown in Figure 4.12. The gain parameter isr2= 3.78 and V2 satisfies
hξ,x˙2i ≤ −kx2k2+r2kx1k1
≤ −kx2k2+√
2r2kx1k2, ∀ξ∈∂ClV2(x2) (4.90)
-0.06 -0.04 -0.02
0 0.02 0.04 0.06-0.06-0.04-0.02 0 0.02 0.04 0.06 0
20 40 60
lB vB
Figure 4.11: CPA ISS Lyapunov function in dissipative formulationV1 delivered by the algo-rithm for subsystem S1.
Based on inequalities (4.89) and (4.90) we define A=
−1 0 5.3457 −1
.
Through calculation, we have that the conditions of Theorem 1.6.1 are satisfied. Therefore, the whole system is locally asymptotically stable in D0d\ B2(0,0.032)× B2(0,0.032) with D0d defined by (4.91).
Let c = (0.6453,1)>. According to Theorem 1.6.1 we have b = (6,1)> such that c> =
−b>A. Then a CPA Lyapunov function for system (4.87) is defined by V = hb,(V1, V2)>i.
4.4 Stability of interconnected ISS systems and estimate of the domain of
attraction 115
-0.06 -0.04 -0.02 20 40 60 80 0 0 0.02 0.04 0.06-0.06-0.04-0.02 0 0.02 0.04 0.06 100 120
lS
vS
Figure 4.12: CPA ISS Lyapunov function in dissipative formulationV2 delivered by the algo-rithm for subsystem S2.
Based on this Lyapunov function an estimate of the domain of attraction ofD2 ={(x1, x2)>∈ D × D : V(x)≤ max
x∈∂(B2(0,0.032)×B2(0,0.032))V(x) = 46.7922} is obtained, i.e., D0d={(x1, x2)>∈ D × D : V(x1, x2)< min
x∈∂(D×D)V(x1, x2) = 46.8102}. (4.91) Remark 4.4.4. Compared with the setD, the excluded neighbourhood of the origin is quite big. When choosing a smaller neighbourhood of the origin, we cannot get that the whole system is locally asymptotically stable. This means the origin of the overall system is not locally asymptotically stable. When we consider system (4.87) on a bigger set than D, the obtained gain parametersr1, r2 will not satisfy the conditions of Theorem 1.6.1.
Example 3
Consider the system
˙
z1 =−(2 + sinz3)z1+z3,
˙
z2 = (0.1 sinz5−1)z2+ 0.1z2e−z23,
˙
z3 =−(sinz1+ 2)z3+ 0.1z4,
˙
z4 = sin(0.1z3+z4)−z4(2 + sin(0.1z1)),
˙
z5 = (sinz2−1)z5+ 0.1z3e−z12,
(4.92)
which is adapted from [106, Example 5.3]. We divide this model into three interconnected systems, systemS1 with states x1 = (z1, z3)>, system S2 with state x2 =z4 and system S3 with statesx3= (z2, z5)>.
We consider each subsystem as a dynamic system with perturbations. We study systems S1,S3 on D=B2(0,0.072)⊂R2 and systemS2 on D1 = [−0.072,0.072]⊂R.
A suitable triangulation ofD is obtained as described in Section 2.1.3 withK = 6,k= 1 and the map F from (4.88). A suitable triangulation of D1 is attained with K = 6, k = 1 and map G1:R7→R,
G1(s) = 0.002s|s|. (4.93)
116
4. Computation of ISS Lyapunov functions and stability of interconnected systems
With the proposed method we compute a CPA ISS Lyapunov function in dissipative formulationV1for systemS1onB2(0,0.072)\B2(0,0.002), see Figure 4.13. The gain parameter isr1 = 0.0620467. V1 satisfies
hξ,x˙1i ≤ −kx1k2+r1kx2k2, ∀ξ∈∂ClV1(x1). (4.94)
-0.09-0.06-0.03
0 0.03 0.06 0.09-0.09-0.06-0.03 0 0.03 0.06 0.09 0.02 0
0.04 0.06 0.08
z1 z3
Figure 4.13: CPA ISS Lyapunov function in dissipative formulation V1 computed by the algorithm for subsystemS1,r1= 0.0620467.
A CPA ISS Lyapunov function in dissipative formulation V2 (see Figure 4.14) is com-puted by our proposed approach for systemS2 on [−0.072,0.072]\(−0.002,0.002). The gain parameter is r2= 0.114601 and V2 satisfies
hξ,x˙2i ≤ −kx2k2+r2kx1k1
≤ −kx2k2+√
2r2kx1k2, ∀ξ∈∂ClV2(x2). (4.95)
0 0.02 0.04 0.06 0.08
-0.09 -0.06 -0.03 0 0.03 0.06 0.09 z4
Figure 4.14: CPA ISS Lyapunov function in dissipative formulation V2 computed by the algorithm for subsystemS2,r2= 0.114601.
4.4 Stability of interconnected ISS systems and estimate of the domain of
attraction 117
Similarly, using our proposed approach we obtain a CPA ISS Lyapunov function in dissi-pative formulationV3 (see Figure 4.15) for systemS3 on B2(0,0.072)\ B2(0,0.002). The gain parameter is r3= 0.107921 and V3 satisfies
hξ,x˙3i ≤ −kx3k2+r3kx1k1
≤ −kx3k2+
√
2r3kx1k2, ∀ξ∈∂ClV3(x3). (4.96)
-0.09-0.06-0.03 0 0 0.03 0.06 0.09-0.09-0.06-0.03 0 0.03 0.06 0.09 0.02
0.04 0.06 0.08 0.1
z2 z5
Figure 4.15: CPA ISS Lyapunov function in dissipative formulation V3 computed by the algorithm for subsystemS3,r3= 0.107921.
Based on inequalities (4.94), (4.95) and (4.96), we define
A=
−1 0.06240467 0
0.1621 −1 0
0.15026 0 −1
.
Remark 4.4.5. Let ˜D=D × D1× D. The conditions of Theorem 1.6.1 are fulfilled. Thus system (4.92) is asymptotically stable on D0d\ B2(0,0.002)×(−0.002,0.002)× B2(0,0.002) with Dd0 defined by (4.97).
Let c = (0.6853,0.9389533,1)>, x = (x>1, x>2, x>3)>. According to Theorem 1.6.1 there exists a vectorb= (1,1,1)>such thatc>=−b>A. Then a CPA Lyapunov function is defined by V(x) = hb,(V1, V2, V3)>i for system (4.92). An estimate of the domain of attraction of B2(0,0.002)×(−0.002,0.002)× B2(0,0.002) is given by
D0d={x∈D˜ : V(x)< min
x∈∂D˜
V(x) = 0.0786}. (4.97)
Remark 4.4.6. It is difficult to construct Lyapunov functions for systems (4.87) and (4.92) without considering them as interconnected systems. Even for the subsystems, there is no known analytic ISS Lyapunov function except for the one dimensional system. By our pro-posed method, CPA ISS Lyapunov functions in dissipative formulation are computed for subsystems, and then CPA Lyapunov functions are defined for systems (4.87) and (4.92) and estimates of domain of attraction are obtained.
118
4. Computation of ISS Lyapunov functions and stability of interconnected systems
4.4.2 Conclusion
In this section, the stability of interconnected continuous time ISS systems was investigated.
We assume each subsystem is locally ISS. Then for each subsystem, a CPA ISS Lyapunov function in dissipative formulation is computed by the method described in Section 4.2. Using the small gain theorem in linear form (Theorem 1.6.1) we analysed the stability of the whole system.
For interconnected discrete time ISS systems, stability of the overall system can be studied by a similar method and parallel results can be obtained.
In the inequalities of the small gain theorem in linear form (Theorem 1.6.1),k · k2 is used.
However,kuk1 is utilized in the inequality obtained from the linear optimization problem. In order to analyse stability of the interconnected systems by the small gain theorem in linear form, a new inequality withk · k2is deduced from the obtained inequality from the algorithm, see (4.85), (4.89), (4.90), (4.95) and (4.96). In order to avoid this step,kxk1,kuk1 could be used in the linear optimization problem and the linear inequalities of the small gain theorem in linear form.