Abstract
In analysis and design of nonlinear dynamical systems, (nonlinear) scaling of Lyapunov functions has been a central idea. This paper proposes a set of tools to make use of such scalings and illustrates their benefits in constructing Lyapunov functions for interconnected nonlinear systems. First, the essence of some scaling techniques used extensively in the literature is reformulated in view of preservation of dissipation inequalities of integral input-to-state stability (iISS) and input-to-state stability (ISS). The iISS small-gain theorem is revisited from this viewpoint. Preservation of ISS dissipation inequalities is shown to not always be necessary, while preserving iISS which is weaker than ISS is convenient. By establishing relationships between the Legendre–Fenchel transform and the reformulated scaling techniques, this paper proposes a way to construct less complicated Lyapunov functions for interconnected systems.
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Notes
The decay rate \({x^2}/{(1+x^2)}\) in (2) is not of class \(\mathcal {K}_\infty \) (i.e., the decay rate does not approach infinity as x approaches infinity), although the formal definition in [37, Definition 2.2] employs \(\mathcal {K}_\infty \). However, as indicated in [37] and verified easily, a positive definite decay rate, e.g., (2), can imply ISS.
in the sense of [37, Definition 2.2], i.e., an implication-form characterization
The use of \(\mu ^\prime (s)=b\) is sufficient for verifying stability of interconnections of components admitting linear gains [5].
The condition \(V(x)\ge \alpha ^{\ominus }\circ (\mathbf {Id}+\omega )\circ \sigma (|w|)\) gives a hypothesizing clause of the implication-form characterization of ISS from (9).
It is assumed in [23] that \(\mu ^\prime \) is strictly increasing. However, as demonstrated in this section, \(\mu ^\prime \) can be taken to be non-decreasing.
It is not necessary to use \(\tau =1.1\). There exists \(\tau \) satisfying (79) if and only if \(\varphi >21.419...\)
References
Angeli D, Astolfi A (2007) A tight small gain theorem for not necessarily ISS systems. Syst Control Lett 56:87-91
Angeli D, Sontag ED, Wang Y (2000) A characterization of integral input-to-state stability. IEEE Trans Autom Control 45:1082-1097
Chaillet A, Angeli D, Ito H (2014) Combining iISS and ISS with respect to small inputs: the strong iISS property. IEEE Trans Autom Control 59:2518-2524
Freeman RA, Kokotović PV (1996) Robust nonlinear control design: State-space and Lyapunov techniques. Birkhäuser, Boston, Massachusetts
Hill DJ, Moylan PJ (1977) Stability results for nonlinear feedback systems. Automatica 13:377-382
Isidori A (1999) Nonlinear control systems II. Springer, London
Ito H (2002) A constructive proof of ISS small-gain theorem using generalized scaling. In: Proceedings of the 41th IEEE Conf. Decision Control, pp 2286-2291
Ito H (2006) State-dependent scaling problems and stability of interconnected iISS and ISS systems. IEEE Trans Autom Control 51:1626-1643
Ito H (2008) A degree of flexibility in Lyapunov inequalities for establishing input-to-state stability of interconnected systems. Automatica 44:2340-2346
Ito H (2010) A Lyapunov approach to cascade interconnection of integral input-to-state stable systems. IEEE Trans Autom Control 55:702-708
Ito H (2012) Necessary conditions for global asymptotic stability of networks of iISS systems. Math Control Signals Syst 24:55-74
Ito H (2013) Utility of iISS in composing Lyapunov functions. In: Proceedings of the 9th IFAC Sympo. Nonlinear Control Systems,Toulouse, France, pp 723-730
Ito H, Dashkovskiy S, Wirth F (2012) Capability and limitation of max- and sum-type construct ion of Lyapunov functions for networks of iISS systems. Automatica 48:1197-1204
Ito H, Jiang ZP (2009) Necessary and sufficient small gain conditions for integral input-to-state stable systems: A Lyapunov perspective. IEEE Trans Autom Control 54:2389-2404
Ito H, Jiang ZP, Dashkovskiy S, Rüffer B (2013) Robust stability of networks of iISS systems: construction of sum-type Lyapunov functions. IEEE Trans Autom Control 58:1192-1207
Ito H, Kellett CM (2015) Preservation and interconnection of iISSand ISS dissipation inequalities by scaling. In: Proceedings of the1st IFAC Conference on Modelling, Identification and Control of Nonlinear Systems, Saint Petersburg, Russia, pp 776-781
Ito H, Nishimura Y (2014) Stochastic robustness of interconnected nonlinear systems in an iISS Framework. In: Proceedings of the 2014 American Control Conf., Portland, USA, pp 5210-5216
Ito H, Nishimura Y (2014) Stability criteria for cascaded nonlinear stochastic systems admitting not necessarily unbounded decay rate. In: Proceedings of the 19th IFAC World Congress, pp 8616-8622
Ito H, Rüffer BS, Rantzer A (2014) Max- and sum-separable Lyapunov functions for monotone systems and their level sets. In: Proceedings of the 53rd IEEE Conf. Decision Control, Los Angeles,USA, pp 2371-2377
Jiang ZP, Mareels I, Wang Y (1996) A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems. Automatica 32:1211-1215
Jiang ZP, Teel AR, Praly L (1994) Small-gain theorem for ISS systems and applications. Math Control Signals Syst 7:95-120
Karafyllis I, Jiang ZP (2011) Stability and stabilization of nonlinear systems. Springer, London
Kellett CM, Wirth FR (2016) Nonlinear scalings of (i)ISS-Lyapunov functions. IEEE Trans Autom Control 61:1087-1092
Kellett CM (2014) A compendium of comparison function results. Math Control Signals Syst 26(3):339-374
Krstić M, Kanellakopoulos I, Kokotović PV (1995) Nonlinear and adaptive control design. Wiley, New York
Krstić M, Li Z (1998) Inverse optimal design of input-to-state stabilizing nonlinear controllers. IEEE Trans Autom Control 43:336-350
Liu T, Jiang JP (2014) Nonlinear control of dynamic networks. CRC Press, Boca Raton
Mazenc F, Praly L (1996) Adding integrations, saturated controls and stabilization for feedforward systems. IEEE Trans Autom Control 41:1559-1578
Mironchenko A, Ito H (2014) Integral input-to-state stability ofbilinear infinite-dimensional systems. In: Proceedings of the 53rdIEEE Conf. Decision Control, Los Angeles, USA, pp 3155-3160
Praly L, Carnevale D, Astolfi A (2010) Dynamic vs static scaling: anexistence result. In: Proceedings of the 8th IFAC Symp. NonlinearControl Systems, Bologna, Italy, pp 1075-1080
Praly L, Jiang ZP (1993) Stabilization by output feedback for systems with ISS inverse dynamics. Syst Control Lett 21:19-34
Rüffer BS, Kellett CM, Weller SR (2010) Connection between cooperative positive systems and integral input-to-state stability of large-scale systems. Automatica 46:1019-1027
Sepulchre R, Janković M, Kokotović PV (1997) Constructive nonlinear control. Springer, New York
Sontag ED (1989) Smooth stabilization implies coprime factorization. IEEE Trans Autom Control 34:435-443
Sontag ED (1998) Comments on integral variants of ISS. Syst Control Lett 34:93-100
Sontag ED, Teel AR (1995) Changing supply functions in input/state stable systems. IEEE Trans Autom Control 40:1476-1478
Sontag ED, Wang Y (1995) On characterizations of input-to-state stability property. Syst Control Lett 24:351-359
Teel A (1996) A nonlinear small gain theorem for the analysis of control systems with saturation. IEEE Trans Autom Control 41:1256-1270
Willems JC (1972) Dissipative dynamical systems. Arch Ration Mech Anal 45:321-393
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The work is supported in part by JSPS KAKENHI Grant Number 26420422. C. M. Kellett is supported by the Australian Research Council under Future Fellowship FT110100746.
Appendix
Appendix
1.1 Appendix 1: Proof of Theorem 1
Property (17) implies the existence of \(w_L\in (0,\infty )\) and a sequence \(\{s_i\}\) of real numbers such that \(\lim _{i\rightarrow \infty }s_i=\infty \) and \(\lim _{i\rightarrow \infty }\alpha (s_i)<\sigma (w_L)\). By virtue of (8), if \(\liminf _{s\rightarrow \infty }\mu ^\prime (s)=\infty \) holds, then
On the other hand, the assumptions \(\hat{\alpha }\in \mathcal {P}\) and \(\hat{\sigma }\in \mathcal {K}\) imply
The contradiction between (99) and (100) arising from (18) indicates that (19) must hold. If \(\lim _{s\rightarrow \infty }\alpha (s)\) exists, property \(\lim _{i\rightarrow \infty }\alpha (s_i)<\sigma (w_L)\) holds for any sequence \(\{s_i\}\) of real numbers such that \(\lim _{i\rightarrow \infty }s_i=\infty \). Hence, the claim (20) follows from (100).
1.2 Appendix 2: Proof of Theorem 2
The decomposition (11) yields
Obviously, in the case of \(\lambda (s)\equiv 0\), inequality (15) holds with \(\hat{\alpha }=b\alpha \circ \mu ^{-1}\in \mathcal {P}\) and \(\hat{\sigma }=b\sigma \in \mathcal {K}\) which are identical with the pair (23a), (23b). The assertions about \(\hat{\alpha }\in \mathcal {K}\) and \(\hat{\alpha }\in \mathcal {K}_\infty \) are straightforward. Note that \(\omega \) is irrelevant in this case. Hence, the rest of the proof assumes \(\lambda (s)\not \equiv 0\). Property (12) implies \(\lambda (s)>0\) for all \(s\in (0,\infty )\). Since \(\mu ^\prime \) is non-decreasing, so is \(\lambda \).
First, suppose that \(\liminf _{l\rightarrow \infty }\alpha (l)>0\). This clearly guarantees the existence of \(\tilde{\alpha }\in \mathcal {K}\) satisfying (25). It is also straightforward that there exists a continuous function \(\omega : \mathbb {R}_+\rightarrow \mathbb {R}_+\) satisfying (26). Following the idea in [36], we evaluate \(\lambda (V(x))\left[ -\alpha (V(x))+\sigma (|w|)\right] \) in (101) in the two cases \(\tilde{\alpha }(V(x))\ge (\mathbf {Id}+\omega )\circ \sigma (|w|)\) and \(\tilde{\alpha }(V(x))\le (\mathbf {Id}+\omega )\circ \sigma (|w|)\) separately. Due to the non-decreasing property of \(\lambda \) and \(\mathbf {Id}+\omega \in \mathcal {K}_\infty \), the combination of the evaluation in the two cases yields (15) with (23). Notice that (22) implying \(\lim _{s\rightarrow \infty }\lambda (s)< \infty \) for \(\lim _{s\rightarrow \infty }\tilde{\alpha }(s)< \infty \) ensures \(\lambda \circ \tilde{\alpha }^{\ominus }\circ (\mathbf {Id}+\omega )\circ \sigma (s)\) is well-defined for all \(s\in \mathbb {R}_+\). The non-decreasing property of \(\lambda \) and \(\mathbf {Id}+\omega \in \mathcal {K}_\infty \) yields \(\hat{\sigma }\in \mathcal {K}\). It is verified that
Due to \(\mathbf {Id}+\omega \in \mathcal {K}_\infty \), we have
which gives
From (26) it follows that \(\omega \circ (\mathbf {Id}+\omega )^{-1}\circ \tilde{\alpha }(s)>0\) holds for all \(s\in (0,\infty )\), and \(\omega \circ (\mathbf {Id}+\omega )^{-1}\circ \tilde{\alpha }\in \mathcal {P}\). Thus, we have \(\hat{\alpha }\in \mathcal {P}\). Finally, Eq. (103) also implies \(\hat{\alpha }\in \mathcal {K}\) (resp. \(\hat{\alpha }\in \mathcal {K}_\infty \)) if \(\omega , \tilde{\alpha }, \alpha \in \mathcal {K}\) (resp. \(\omega , \tilde{\alpha }, \alpha \in \mathcal {K}_\infty \)).
Next, suppose that \(\liminf _{l\rightarrow \infty }\alpha (l)=0\). Then property (24) implies that \(\tilde{\alpha }^\ominus (s)=\infty \) for all \(s\in \overline{\mathbb {R}}_+\) by virtue of the definition of \(\ominus \). Since \(L:=\lim _{l\rightarrow \infty }\lambda (l)< \infty \) is ensured by (22), the formula (23b) gives \(\hat{\sigma }=(b+L)\sigma \in \mathcal {K}\) which is independent of \(\omega \). The choice (24) also implies \(\hat{\alpha }\in \mathcal {P}\) for (23a) for each given \(\omega \). On the other hand,
holds. From (24) we also obtain
for all \(x\in \mathbb {R}^N\) and \(w\in \mathbb {R}^M\) Applying these inequalities to (101), we arrive at not only (23), but also (23) with (27) with \({\varOmega }\rightarrow \infty \).
Finally, suppose that \(\lim _{s\rightarrow \infty }\mu ^\prime (s)< \infty \). Defining \(L:=\lim _{l\rightarrow \infty }\mu ^\prime (l)< \infty \) again yields (104). Independently, (105) follows from (25). Thus, (101) is bounded from above by \(-\hat{\alpha }(W(x))+\hat{\sigma }(|w|)\) defined by
These functions are identical with taking \({\varOmega }\rightarrow \infty \) in (23) with (27) for each \(s\in \mathbb {R}_+\).
1.3 Appendix 3: Proof of Theorem 3
In the case of \(\lambda (s)\equiv 0\), the claim holds true obviously from \(\alpha , \sigma \in \mathcal {K}\) and (10) since (34) gives \(\hat{\alpha }=b\alpha \circ \mu ^{-1}\in \mathcal {K}\) and \(\hat{\sigma }=b\sigma \in \mathcal {K}\). Therefore, the rest considers the case of \(\lambda (s)\not \equiv 0\). First, suppose that (30)–(32) are satisfied with a continuous function \(\omega : \mathbb {R}_+\rightarrow \mathbb {R}_+\). Following the proof of Theorem 2 with \(\tilde{\alpha }=\alpha \), we obtain \(\hat{\sigma }\in \mathcal {K}\) in (34b). Note that property (28) is guaranteed by (32). The function \(\hat{\alpha }\in \mathcal {K}\) which is obtained as in (23a) with \(\tilde{\alpha }=\alpha \) and satisfies (15) is only of class \(\mathcal {P}\). Hence, write (23a) as \(\eta + b\alpha \circ \mu ^{-1}\) by defining \(\eta \) as in (35). Rewrite \(\eta \in \mathcal {P}\) as
Applying (30) and (32) to (107) with the help of \(\mu \in \mathcal {K}_\infty \), one arrives at
Since this inequality implies \(\liminf _{s\rightarrow \infty }\eta (s)>0\) and we have \(\alpha \circ \mu ^{-1}\in \mathcal {K}\) in addition, there always exists a continuous function \(k: \mathbb {R}_+\rightarrow \mathbb {R}_+\) such that (36) and (37) are fulfilled. Defining \(\hat{\alpha }\) as in (34a) with (36) and (37) ensures \(\hat{\alpha }\in \mathcal {K}\) and \(\hat{\alpha }(s)\le \eta (s)+ b\alpha \circ \mu ^{-1}(s)\) for all \(s\in \mathbb {R}_+\). Thus, the preservation of the iISS dissipation inequality (9) under the scaling \(\mu \) is established by \(\hat{\alpha }, \hat{\sigma }\in \mathcal {K}\) given in (34). Furthermore, by virtue of \(\limsup _{l\rightarrow \infty } k(l)=1\) and \(\hat{\alpha }\in \mathcal {K}\), property (33) follows from (10), (30) and (107). This proves the preservation of the ISS dissipation inequality.
Finally, replace the pair of (31) and (32) by (27) with \({\varOmega }\rightarrow \infty \) in the case of \(\lim _{s\rightarrow \infty }\mu ^\prime (s)< \infty \). Define \(L:=\lim _{l\rightarrow \infty }\lambda (l)< \infty \). Then (34) becomes
and clearly satisfies \(\hat{\sigma }\), \(\hat{\sigma }\in \mathcal {K}\). Using (36) and (37) to modify (23) of Theorem 2 verifies that the functions in (108) achieve the preservation the iISS dissipation inequality (9) under the scaling \(\mu \). For (108), by virtue of \(\limsup _{l\rightarrow \infty } k(l)=1\) and \(\hat{\alpha }\in \mathcal {K}\), property (33) is implied by (10). This establishes the preservation the ISS dissipation inequality.
1.4 Appendix 4: Proof of Proposition 2
The existence of \(\tau , \varphi \ge 0\) fulfilling (45) is straightforward from \(c>1\). Property (44) with \(\tau <c\) implies (39). Due to the property \(\alpha _{i}\circ \alpha ^{\ominus }(s)\le s\) for all \(s\in \mathbb {R}_+\), property (40) follows if
The non-decreasing property of \(\beta \) and (39) guarantee
Thus, if
is met, property (109) is satisfied. In the case where (45) holds for \(\varphi =0\), we can easily verify (110). Therefore, we next assume that (45) holds for some \(\varphi >0\). Property (110) is satisfied if we have
This property is achieved if
since we have (44). Property (111) is secured by (45).
1.5 Appendix 5: Proof of Proposition 4
In the case of \(\mu ^\prime (s)\equiv b\), the implications (52) and (22) do not require anything, which proves the claim. Suppose that \(\mu ^\prime (s)\not \equiv b\). Since \(\mu \in \mathcal {K}_\infty \), property (49) is equivalent to
Clearly, this property implies
Recalling \(\mu ^\prime (s)\not \equiv b\), properties \(\lim _{s\rightarrow \infty }\lambda (s)>0\) and \(\kappa \in \mathcal {K}_\infty \) imply \(\lim _{s\rightarrow \infty }\kappa \circ \lambda (s)>0\). Since \(\kappa ^\prime \) is of \(\mathcal {K}_\infty \), property (113) requires \(\liminf _{s\rightarrow \infty }\alpha (s)>0\). Hence, property (52) must hold. Next, suppose \(\lim _{s\rightarrow \infty }\mu ^\prime (s)=\infty \) which means \(\lim _{s\rightarrow \infty }\lambda (s)=\infty \). Property \(\kappa ^\prime \in \mathcal {K}_\infty \) in (113) again implies \(\lim _{s\rightarrow \infty }\alpha (s)=\infty \). Therefore, (22) must hold.
1.6 Appendix 6: Proof of Theorem 4
First, we assume that (64) holds. Let \(L:=\lim _{l\rightarrow \infty }\lambda (l)\le \infty \). Suppose that \(\alpha \circ \lambda ^\ominus \) is piecewise differentiable on the interval [0, L). Let \(\kappa ^\prime \) be any class \(\mathcal {K}_\infty \) function satisfying
where the second inequality is evaluated at all differentiable points. Note that \((\alpha \circ \lambda ^\ominus )^\prime (s)\ge 0\) holds for almost all \(s\in [0,L)\) due to \(\alpha \), \(\lambda \in \mathcal {K}\). Therefore, in the case of \(\lim _{s\rightarrow \infty }\alpha (s)< \infty \), the existence of a function \(\kappa ^\prime \in \mathcal {K}_\infty \) satisfying (114) follows from assumption (22) since \(\lim _{s\rightarrow \infty }\lambda (s)< \infty \) and \(\alpha , \lambda \in \mathcal {K}\). In the case of \(\lim _{s\rightarrow \infty }\alpha (s)= \infty \), the existence is guaranteed by \(\alpha \circ \lambda ^{\ominus }\in \mathcal {K}_\infty \) under the assumption of (64).
Let \(\kappa \) denote the antiderivative of \(\kappa ^\prime \) satisfying \(\kappa (0)=0\). Then \(\kappa \in \mathcal {K}_\infty \) follows from \(\kappa ^\prime \in \mathcal {K}_\infty \). Define the map \(\overline{\mathbb {R}}_+\rightarrow \overline{\mathbb {R}}_+\) as
which satisfies
for almost all \(s\in [0,L)\). Thus, from (114) we obtain
This property together with (115) yields
Hence, we have
and \(\hat{\alpha }_{D,\tau }\in \mathcal {K}\) by virtue of \(\mu \in \mathcal {K}_\infty \) and \(\tau >1\). Therefore, we arrive at (61) with \(\hat{\alpha }_L\in \mathcal {K}\).
Next, applying \(\lambda \in \mathcal {K}\) to both sides of the first inequality in (114) from the right, one obtains
Applying the non-decreasing function \(\alpha ^\ominus (\tau s)\) defined for \(s\in [0,\lim _{l\rightarrow \infty }\alpha (l)/\tau )\) again yields
Applying \((\kappa ^\prime )^{-1}\in \mathcal {K}_\infty \) to the above from the left, one obtains
Here, recalling \(\ell \kappa (s)=s(\kappa ^\prime )^{-1}(s)-\kappa \circ (\kappa ^\prime )^{-1}(s)\) and \(\kappa , \kappa ^\prime \in \mathcal {K}_\infty \), we have \(\ell \kappa (s)\le s(\kappa ^\prime )^{-1}(s)\) for all \(s\in \mathbb {R}_+\). Thus,
Hence, we have
Since we have the implication
by virtue of (22), we arrive at
Comparing this with (42) yields
which implies (62) with \(R=\infty \) and (63).
If \(\alpha \circ \lambda ^\ominus : [0,L)\rightarrow \mathbb {R}_+\) is not piecewise differentiable, the above arguments hold true by replacing (114) with
Note that (117) is guaranteed again since \(\kappa ^\prime \in \mathcal {K}_\infty \) is chosen, due to (22).
Finally, suppose that (64) does not hold, i.e., assume that \(\lim _{s\rightarrow \infty }\lambda (s)< \infty \) and \(\lim _{s\rightarrow \infty }\alpha (s)=\infty \) are satisfied. Let \(q>0\) be arbitrary. Consider \(p\in (0,\infty )\) which has yet to be determined. Let \(\tilde{\lambda }\in \mathcal {K}_\infty \) be defined as
Obviously, \(\tilde{L}:=\lim _{l\rightarrow \infty }\tilde{\lambda }(l)=\infty \), \(\tilde{L}>L:=\lim _{l\rightarrow \infty }\lambda (l)\) and in addition,
Assume that \(\alpha \circ \tilde{\lambda }^{-1}: \mathbb {R}_+\rightarrow \mathbb {R}_+\) is piecewise differentiable. Let \(\kappa ^\prime \) be any class \(\mathcal {K}_\infty \) function satisfying
where the second inequality is evaluated at all differentiable points. Due to \(\alpha , \tilde{\lambda }\in \mathcal {K}_\infty \), we have \((\alpha \circ \tilde{\lambda }^{-1})^\prime (s)\ge 0\) for almost all \(s\in \mathbb {R}_+\). Thus, the existence of a function \(\kappa ^\prime \in \mathcal {K}_\infty \) satisfying (126) is guaranteed by virtue of \(\alpha \circ \tilde{\lambda }^{-1}\in \mathcal {K}_\infty \). Let \(\kappa \) denote the antiderivative of \(\kappa ^\prime \) satisfying \(\kappa (0)=0\). Then \(\kappa \in \mathcal {K}_\infty \) follows from \(\kappa ^\prime \in \mathcal {K}_\infty \). Define \(\overline{\kappa }\in \mathcal {K}_\infty \) by
Since \(\overline{\kappa }^\prime (s)= \frac{1}{\tau }\alpha \circ \tilde{\lambda }^{-1}(s)+ \frac{1}{\tau } s\,[(\alpha \circ \tilde{\lambda }^{-1})^\prime (s)]\) holds for almost all \(s\in \mathbb {R}_+\), from (126) we obtain
This property, (127) and (124) give
Hence, by virtue of \(\mu \in \mathcal {K}_\infty \) and \(\tau >1\), we arrive at (61) with \(\hat{\alpha }_L\in \mathcal {K}\). On the other hand, taking inverse of both sides of the first inequality in (126) yields
From \(\ell \kappa (s)\le s(\kappa ^\prime )^{-1}(s)\) for all \(s\in \mathbb {R}_+\) it follows that
Due to (125), we have
Since \(\alpha \) is of class \(\mathcal {K}_\infty \), for any given \(R\in (0,\infty )\), there exists \(p\in (0,\infty )\) such that \(R=\sigma ^\ominus \circ \tau ^{-1}\alpha (p)\) holds. Therefore,
Using \(L=\lim _{l\rightarrow \infty }\lambda (l)\) and (42), we obtain (62) and (63). If \(\alpha \circ \tilde{\lambda }^\ominus : \mathbb {R}_+\rightarrow \mathbb {R}_+\) is not piecewise differentiable, the above arguments hold true by replacing (126) with
This completes the proof.
1.7 Appendix 7: Proof of Proposition 6
Apply Theorem 2 and Remark 3 to each subsystem \({\varSigma }_i\) with \(\omega _i(s)=(\tau -1)s\) for \(\tau >1\), \(\tilde{\alpha }_i=\alpha _i\) and \(\lambda _i=\mu _i^\prime \). It can be verified that (52), (25) and (26) are satisfied. Property (77) also guarantees (28). The formulas in (29) with \(\tilde{\alpha }_i=\alpha _i\in \mathcal {K}\) yield
Property \(\alpha _{i}\circ \alpha ^{\ominus }(s)\le s\) for all \(s\in \mathbb {R}_+\) and property (77) with \(1<\tau \le c\) in (79) imply
Thus, we have
where \(W_i=\mu _i(V_i)\) and
The existence of \(\epsilon >0\) such that
is guaranteed by (79). From (134), (135), (73) and (74) it follows that W is a Lyapunov function proving GAS of \(x=0\) of (72).
1.8 Appendix 8: Proof of Proposition 7
Property (80) together with (78) implies
In the case of \(\lim _{s\rightarrow \infty }\mu _i^\prime (s)<\infty \), \(i=1,2\), iISS and ISS of system (72) are easily verified by incorporating
into the proof of Proposition 6. In the remaining case, property (136) allows one to invoke a technique proposed in [8, 14] as indicated by [12, Proposition 12].
1.9 Appendix 9: Proof of Theorem 5
Suppose that \(\tau , \varphi \ge 0\) satisfy (45). Then \(\tau <c\) implies \(({\tau }/{c})^\varphi >({\tau }/{c})^{\varphi +1}\). Hence, property (79) is met. By virtue of (77), Propositions 6 and 7 prove all the claims.
1.10 Appendix 10: Proof of Theorem 6
First, the function \(\lambda _{i,\psi }\) defined by (87) for each \(i=1,2\) is of class \(\mathcal {K}\) for all \(\psi >0\) since \(\alpha _i, \sigma _{3-i}\in \mathcal {K}\). With the help of (80), property (87) with \(\psi >0\) also ensures
for all \(\psi >0\), which corresponds to (64) as well as (22). Thus, for arbitrary given \(\psi >0\), Corollary 4 is applicable to the two pairs \((\alpha _i,\sigma _i)\), \(i=1,2\), and the formula (66) with (65) and \(k=0\), which is exactly (88), guarantees (61) and (62) with \(R=\infty \), provided that \(\kappa _{i,\psi }: [0,L_{i,\psi })\rightarrow \mathbb {R}_+\) is continuously differentiable, and that (67), (68) and (69) hold in terms of \(\kappa _{i,\psi }\) for each \(i\in \{1,2\}\). Recall that the arguments to derive (119) and (121) allow \(\kappa _{i,\psi }\) to be defined on only the interval \([0,{L_i,\psi })\) in (88) instead of the entire \(\mathbb {R}_+\). To confirm the continuous differentiability of \(\kappa _{i,\psi }\) and (67), (68) and (69), using (87) and (88) and continuous differentiability of \(\alpha _i\) and \(\sigma _{3-i}\), we first obtain
The chain rule \([\kappa _{i,\psi }\circ \lambda _{i,\psi }]^\prime (s)= \left( \kappa _{i,\psi }^\prime \circ \lambda _{i,\psi }(s)\right) \lambda _{i,\psi }^\prime (s)\) yields
where
Property \(\alpha _i, \sigma _{3-i}\in \mathcal {K}\) implies \(\alpha _i(s)\ge 0\) and \(\sigma _{3-i}(s)\ge 0\) for \(s\in \mathbb {R}_+\). Hence, for each \(\psi >0\), assumption (86) and \(\alpha _i\in \mathcal {K}\) guarantee that \(\kappa _{i,\psi }^\prime \circ \lambda _{i,\psi }(s)\) given by (142) exits and is strictly increasing for all \(s\in \mathbb {R}_+\). By definition, it also holds that
for any \(\psi >0\). Therefore, properties \(\kappa _{i,\psi }^\prime \circ \lambda _{i,\psi }(0)=0\) and \(\lambda _{i,\psi }\in \mathcal {K}\) ensure for all \(\psi >0\) that \(\kappa _{i,\psi }^\prime (s)\) exists for \(s\in [0,L_{i,\psi })\), and
Recalling that \(L_{i,\psi }< \infty \) implies \(\lim _{s\rightarrow \infty }\alpha _i(s)< \infty \) due to (138), from (142) and (143) it follows that
Therefore, it is proved that \(\kappa _{i,\psi }\) is continuously differentiable and fulfills and (67), (68) and (69) for each \(i=1,2\). We can now invoke Corollary 4. By virtue of \(\tilde{\alpha }_{i,\psi }=\lambda _{i,\psi }\alpha -\kappa _{i,\psi }\circ \lambda _{i,\psi }\) that can be verified from (118) with \(\kappa =\overline{\kappa }\), substituting (119) and (121) into (134) and (135) in the proof of Proposition 6, there exists \(\epsilon >0\) satisfying (90). Hence, (i) is proved. Finally, following the arguments used to prove Propositions 6 and 7, the proof of (ii) is completed.
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Ito, H., Kellett, C.M. iISS and ISS dissipation inequalities: preservation and interconnection by scaling. Math. Control Signals Syst. 28, 17 (2016). https://doi.org/10.1007/s00498-016-0169-2
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DOI: https://doi.org/10.1007/s00498-016-0169-2