Small gain theorems for large scale systems and construction of ISS Lyapunov functions

Sergey N. Dashkovskiy; Björn S. Rüffer; Fabian R. Wirth

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Small gain theorems for large scale systems and construction of ISS Lyapunov functions

Sergey N. Dashkovskiy, Björn S. Rüffer, Fabian R. Wirth

arXiv:0901.1842v2Jan 13, 2009math.OC

TL;DR

Large nonlinear interconnections are difficult to analyze because subsystem dependencies must be handled collectively. The paper encodes these dependencies with a gain matrix and monotone operator, then scales subsystem ISS Lyapunov functions to construct one for the full network under a small gain condition. The resulting framework covers general ISS formulations, including summation, maximization, and separation of external gains.

Problem
Stability analysis is difficult for large interconnections with many subsystems, arbitrary topologies, and nonlinear dynamics.
Method
A gain matrix induces a monotone operator, and a suitable path σ satisfying Γµ ◦ σ < σ scales the individual subsystem Lyapunov functions.
Results
The paper gives sufficient small gain conditions and an explicit formula for an overall ISS Lyapunov function for interconnected subsystems.
Takeaways & Limitations
The general construction unifies summation, maximization, and other formulations of ISS through monotone aggregation functions.
Takeaways & Limitations
The explicit Lyapunov construction requires the scaling function σ to be known; numerical construction is noted but not studied here.

Abstract

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We consider interconnections of n nonlinear subsystems in the input-to-state stability (ISS) framework. For each subsystem an ISS Lyapunov function is given that treats the other subsystems as independent inputs. A gain matrix is used to encode the mutual dependencies of the systems in the network. Under a small gain assumption on the monotone operator induced by the gain matrix, a locally Lipschitz continuous ISS Lyapunov function is obtained constructively for the entire network by appropriately scaling the individual Lyapunov functions for the subsystems. The results are obtained in a general formulation of ISS, the cases of summation, maximization and separation with respect to external gains are obtained as corollaries.

1. Introduction.

The paper extends small gain theory to large-scale nonlinear interconnections and constructs an overall ISS Lyapunov function by scaling subsystem Lyapunov functions. Its framework uses monotone operators to cover arbitrary subsystem counts and multiple ISS formulations.

The paper presents sufficient conditions for an ISS Lyapunov function and an explicit overall formula, while noting that numerical construction of the scaling path is left to future work.The individual subsystem functions may be easier to find because their dimensions are lower than that of the interconnection.
Small gain conditions provide sufficient conditions for ISS of feedback interconnections with respect to external inputs.
The two-subsystem construction chooses σ2 between γ21 and γ12^-1, then defines an overall function by scaling and maximizing the subsystem Lyapunov functions.The construction is V(x) = max{V1(x1), σ2^-1(V2(x2))}.
The gain relationships can be represented by a matrix that induces a monotone operator, reframing the scaling problem as finding a path σ satisfying Γ ◦ σ < σ.
The paper generalizes the construction to arbitrarily many subsystems and monotone aggregation functions, covering summation, maximization, and other ISS formulations.

2. Preliminaries.

The preliminaries define the network, ISS Lyapunov functions, and monotone aggregation framework used to analyze interconnected nonlinear systems. Subsystem gains are combined through aggregation functions and encoded in a nonlinear gain operator.

2.1. Notation and conventions.: The network consists of finitely many subsystems whose other states and common external input are treated as unrestricted inputs to each subsystem.The subsystem dynamics are assumed to have unique forward-complete solutions for bounded inputs.
2.1. Notation and conventions.: A directed graph represents the network, with edge (i, j) corresponding to an input from subsystem j to subsystem i.
2.2. Input-to-state stability.: ISS requires trajectory bounds combining a class KL decay term with a class K gain of the external input.
2.3. ISS Lyapunov functions.: An ISS Lyapunov function is continuous, proper, positive definite, and locally Lipschitz away from the origin, with a decay condition under sufficiently large Lyapunov value.
2.4. Monotone aggregation.: For interconnected systems, each subsystem Lyapunov function uses internal gains γij, an external gain γiu, and a monotone aggregation function to combine their effects.
2.4. Monotone aggregation.: The gain matrix, augmented with external gains, induces a nonlinear map whose components aggregate internal and external influences on each subsystem.Monotone aggregation functions are continuous, strictly increasing, and unbounded; subadditivity is not required for the paper’s constructions.

3. Examples for monotone aggregation.

The examples show how subsystem influences can be aggregated by maxima, sums, or other monotone functions, with explicit ISS gains and Lyapunov constructions for linear and neural-network systems.

Maximum aggregation expresses a subsystem’s combined input effect through γu(r) = r^4/2 and γw(r) = r^2/2.
The choice of aggregation function and gains is not unique, and different choices can induce the same gain operator.Some choices, including µ = max, can make checking the small gain condition computationally more convenient.
3.1. Linear systems.: For linear subsystems, quadratic Lyapunov functions Vi(xi) = x_i^T P_i x_i are constructed from positive-definite solutions of matrix equations.The resulting subsystem functions provide ISS gains for the interconnection analysis.
3.1. Linear systems.: The linear-system gain operator is nonlinear and has a useful homogeneity property induced by the choice of quadratic Lyapunov functions.
3.2. Neural networks.: For Cohen-Grossberg neural networks, the interconnection matrix describes neuron coupling while external inputs may be arbitrary measurable L∞ functions.The analysis assumes the interconnection matrix is given and focuses on stability rather than its training procedure.

4. Monotone Operators and generalized small gain conditions.

The paper formulates generalized small gain conditions through monotone gain operators and uses them to construct an ISS Lyapunov function for large interconnections.

The gain matrix and monotone aggregation function induce an operator Γ_µ that represents mutual subsystem influences.The interconnection can also be represented as a weighted directed graph whose vertices are subsystems and whose edge weights are nonlinear gain functions.
The operator’s adjacency matrix records whether each gain function is identically zero or nonzero, thereby representing the interconnection graph.
Graph properties such as primitivity, irreducibility, reducibility, and strong connectedness are characterized through the adjacency matrix.
A diagonal operator D is defined by D(s) = (s_1 + α_1(s_1), ..., s_n + α_n(s_n))^T for α_i ∈ K∞.
The condition T ≱ id requires that for every nonzero s, at least one component of T(s) is strictly less than the corresponding component of s.
Small gain conditions are defined for Γ_µ, with strong variants formulated using the diagonal operator D.Identical comparison functions α_i are sufficient for the strong small gain conditions through D = diag(id + α).
The paper’s two main results are a topological existence result for a jointly unbounded path and a Lyapunov-function construction based on that path.The path exists when Γ_µ satisfies the small gain condition and is then used to construct an ISS Lyapunov function for the interconnection.

5. Lyapunov functions.

Under small gain conditions on the monotone operator induced by subsystem gains, an Ω-path enables a constructive ISS Lyapunov function for the interconnected system. The resulting function is locally Lipschitz, generally nonsmooth, and supports ISS and 0-GAS conclusions in several external-gain formulations.

Main theorem: Theorem 5.3 constructs an overall ISS Lyapunov function from subsystem ISS Lyapunov functions, an Ω-path, and a suitable function ϕ.The construction assumes the gain operator and Ω-path satisfy the theorem’s stated small gain inequality.
Main theorem: The constructed function is V(x) = max_i σ_i^-1(V_i(x_i)), combining scaled subsystem Lyapunov functions through maximization.The scaling functions σ_i come from the Ω-path.
Main theorem: At differentiability points, V above the input-dependent threshold implies ∇V(x)f(x,u) ≤ −α(∥x∥), establishing the required decrease condition.The threshold is max_i{ϕ^-1(γ_iu(∥u∥))}, and α is positive definite.
Regularity: The overall function is not smooth even when subsystem functions are smooth, so the framework uses locally Lipschitz Lyapunov functions differentiable almost everywhere.The proof consequently works with generalized gradients and almost-everywhere differentiability.
Special cases: For strongly connected unforced networks, Γµ ≱ id yields 0-GAS, with a nonsmooth Lyapunov function supplied by the general theorem.In the absence of external inputs, ISS coincides with 0-GAS.
Special cases: The construction specializes to additive, maximization, and separated external-gain formulations, each yielding ISS under corresponding small gain assumptions.The additive and maximization cases use V from (5.4); the separated case uses an Ω-path for D ◦ Γµ.

6. The reducible case and scaling.

For reducible networks, the gain matrix can be decomposed into strongly connected diagonal blocks and treated iteratively. ISS Lyapunov functions for the blocks can then be combined into an overall function, with summation and maximization corollaries.

Block decomposition: A reducible gain matrix can be transformed into upper block triangular form by permuting the network vertices.The diagonal blocks correspond to strongly connected components, while upper-block terms describe inter-block dependencies.
Block scaling: An overall ISS Lyapunov function is obtained by constructing functions for irreducible blocks while treating states with higher indices as inputs.The construction is performed through iterative application of the block-composition result.
Block scaling: Proposition 6.2 combines block-level ISS Lyapunov functions for a reduced gain matrix into an ISS Lyapunov function for the complete interconnection.Each block function handles inputs from later blocks and the external input.
Block conditions: The reducible-network small gain requirement reduces to small gain conditions on the diagonal blocks.The paper states that Γµ ≱ id, or its D-scaled analogue, is equivalent to the corresponding diagonal-block conditions.
Corollaries: For additive gains, the reducible-network result gives ISS when the scaled gain operator satisfies the strong small gain condition.The conclusion follows by applying the additive-gain irreducible result to each diagonal block.
Corollaries: For maximization gains, the same blockwise argument yields ISS when each diagonal block satisfies the corresponding maximization small gain condition.The maximization formulation takes the componentwise maximum of subsystem and external-input gains.

7. Applications of the general small gain theorem.

The section applies the general small gain theorem to linear systems and Cohen–Grossberg neural networks, yielding ISS and explicit overall Lyapunov functions under gain conditions.

Applications to homogeneous gain operators: For homogeneous gain operators, a strong small gain condition yields ISS and a nonlinear eigenvector that defines an explicit scaling path.The construction uses a positive eigenvector whose eigenvalue is less than one.
Applications to homogeneous gain operators: For multiplicatively transformed linear gain operators, Γµ ≱ id holds exactly when the associated matrix G has spectral radius less than one.The equivalence follows from the diagonal transformation Γµ = D^-1(GD(·)).
Linear interconnected systems: If the spectral radius of G is less than 1, a linear-system interconnection is ISS and admits a nonsmooth Lyapunov function formed from scaled quadratic subsystem functions.For irreducible G, the scaling is obtained from a positive vector; reducible G requires blockwise construction followed by scaling techniques.
Cohen–Grossberg neural networks: For Cohen–Grossberg neural networks, the strong small gain condition on Γµ implies ISS from the external inputs to the network state.The result extends prior equilibrium and exponential-stability analysis to arbitrary external inputs.

8. Path construction.

The section constructs Ω-paths for monotone gain operators under small gain conditions, handling bounded and unbounded gains through connectedness, invariant sets, and transformed operators.

Basic path construction: For a point s with Γµ(s) < s, iterates Γµ^k(s) decrease to zero, providing the basis for an Ω-path construction.Convex combinations between s and Γµ(s), followed by induction over iterates, connect points in the decay set to the origin.
Basic path construction: Under the small gain condition, every point in Ω can be connected to the origin; with strict monotonicity, a strictly increasing Ω-path exists.The relevant decay set is pathwise connected, and the stronger conclusion applies when Γµ preserves strict order.
Bounded gain operators: For bounded Γµ, the small gain condition guarantees an Ω-path by combining an initial point in the decay set with the path-to-zero construction.Scaling the initial point supplies the unbounded direction through Ω.
Unbounded gain operators: For unbounded gain operators, the simple construction fails, so the proof uses a transformed operator Γµ ◦ D and an unbounded backward-invariant set Ψ∞.The additional diagonal map D creates room to convert a nondecreasing sequence into a strictly increasing path.
Extensions: The section also records existence results for reducible operators and operators with mixed bounded and unbounded class K entries.These cases are covered by the stated theorems and propositions on Ω-path existence.
Special case: Maximization: For maximization operators, contraction of all subordinated cycles implies existence of an Ω-path, with path construction avoiding the diagonal map D.The cycle condition is equivalent to Γµ ≱ id in this setting.

9. Remarks for the case of three subsystems.

For three subsystems, the paper gives a simpler constructive procedure: solve sequential scalar equations, choose the third scaling between two bounds, and obtain an Ω-path.

Constructing the first two components: For each fixed s1, equation (9.2) has exactly one solution s2 because its two sides vary strictly in opposite directions.The small gain condition ensures the relevant roots are ordered so that the solution exists.
Constructing the first two components: Defining σ1(r) = r and σ2(r) as that unique solution yields a continuous, strictly increasing dependence of σ2 on r.This follows from continuity and monotonicity of the gain functions.
Constructing the third component: The third component is chosen between h(r) and g(r), whose nonempty interval is guaranteed by the small gain condition.The construction uses σ3(r) = 1/2(g*(r) + h(r)) with h(r) < σ3(r) < g*(r).
Constructing the third component: The resulting vector σ satisfies Γµ(σ(r)) < σ(r) for all r > 0, providing the desired path for the three-subsystem case.The analogous case with a zero gain can be treated similarly.
Scope: The paper’s overall method applies to arbitrary finite interconnections, while the three-subsystem case illustrates the construction with simpler considerations.The general construction encodes network topology and subsystem interactions in Γµ.