THEORY-773: In a singularly perturbed system ẋ = f(x, z, t), εż = g(x, z, ε), if g is contracting in z at rate λ_z for every x and ε, the reduced system ẋ = f(x, γ(x), t) on the slow manifold z = γ(x) is contracting at rate λ_x, f is α-Lipschitz in z, |∂γ/∂x · f| ≤ d and g is K-Lipschitz in ε, then for all t ≥ 0 the fast state is within χ_z e^{−λ_z t/ε}|z(0) − γ(x(0))| + εχ_z(d + K)/λ_z of the manifold and the slow state within χ_x|x(0) − x_γ(0)|e^{−λ_x t} + ε[C₁(e^{−λ_x t} − e^{−λ_z t/ε}) + C₂(1 − e^{−λ_x t})] of the reduced trajectory; partial contraction of f in x alone is not enough

Source

Del Vecchio and Slotine (2013), “A Contraction Theory Approach to Singularly Perturbed Systems”, IEEE Transactions on Automatic Control 58(3):752–757, LIT-1295, read in full in NOTE-1020 from the authors’ revised manuscript. The statement is its Lemma 1, Lemma 2 and Theorem 3 (Lemma 1, Lemma 2 and Theorem 2 of the CDC-ECC 2011 version), with the correction below.

What was actually shown

Setting. ẋ = f(x, z, t), εż = g(x, z, ε), ε > 0. Contraction is certified by a matrix measure µ in a norm |Θ·| with Θ a constant invertible matrix (µ_Θ(A) = µ(ΘAΘ⁻¹)), and χ is an upper bound on the condition number |Θ||Θ⁻¹|.

Hypotheses.

• g(x, ·, ε) is contracting in z at rate λ_z, uniformly in x, t and ε: µ_{Θ_z}(∂g/∂z) ≤ −λ_z. Then g(x, z, 0) = 0 has a unique root z = γ(x) for every x (Theorem 2), and ∂γ/∂x = −(∂g/∂z)⁻¹ ∂g/∂x.
• |∂γ/∂x (x) · f(x, z, t)| ≤ d. The proof uses this only along the trajectory, so d may be taken as the supremum over it.
• |g(x, z, 0) − g(x, z, ε)| ≤ Kε.
• The reduced system ẋ = f(x, γ(x), t) is contracting at rate λ_x, in the metric Θ_x. The paper assumes instead that f is partially contracting in x (ẏ = f(y, z(t), t) contracting for every path z(t)).
• f is α-Lipschitz in z, and γ is α_γ-Lipschitz.

Fast bound (Lemma 2). |z(t) − γ(x(t))| ≤ χ_z e^{−λ_z t/ε}|z(0) − γ(x(0))| + εχ_z(d + K)/λ_z. z(t) solves the contracting system εẏ = g(x(t), y, ε); γ(x(t)) solves it with the disturbance ε∂γ/∂x f + g(x, γ, 0) − g(x, γ, ε), of size at most ε(d + K). Lemma 1 at rate λ_z/ε gives the bound.

Slow bound (Theorem 3). With C₁ = χ_x α χ_z |z(0) − γ(x(0))| / (λ_z − ελ_x) and C₂ = χ_x α χ_z (d + K)/(λ_z λ_x), |x(t) − x_γ(t)| ≤ χ_x|x(0) − x_γ(0)|e^{−λ_x t} + ε[C₁(e^{−λ_x t} − e^{−λ_z t/ε}) + C₂(1 − e^{−λ_x t})], and |z(t) − γ(x_γ(t))| is bounded by the fast bound plus α_γ times this. x solves the reduced system disturbed by f(x, z, t) − f(x, γ(x), t), of size at most α|z − γ(x)|; Lemma 1, in its differential form, integrated against the fast bound, gives the slow one. Every step is re-derived in NOTE-1020.

What could have come out otherwise. The paper’s own hypothesis, partial contraction of f in x, is not enough. In ẋ = −x + 2 tanh z, εż = −z + 2 tanh x, every hypothesis the paper states holds (λ_x = λ_z = 1, χ = 1, K = 0, α = α_γ = 2, d ≈ 4.9), but the reduced system ẋ = −x + 2 tanh(2 tanh x) is bistable. From x(0) = x_γ(0) = 0.01, z(0) = −1, the full system goes to x ≈ −1.915 and the reduced one to x ≈ +1.915: an error of 3.83 at t = 20, against a bound of 0.98 at ε = 0.1 and 0.098 at ε = 0.01 (NOTE-1020). The CDC version’s introduction states the right hypothesis, “the reduced system and the fast system are each partially contracting”; the theorems in both versions state the wrong one. In the paper’s example f does not depend on z (α = 0), so the example is unaffected.

Checks (the record’s). On ẋ = −2x + y, ẏ = d(x − y) (QUESTION-222’s memory example with c = d, ε = 1/d, γ(x) = x, reduced rate λ_x = 1), the slow error stayed below the bound at d = 5, 20, 100 and 500, at 41 to 74% of it, and sup|x − x_γ|/ε settled near 3: first order, as stated.

What this does not say

• Not for a state- or time-dependent metric. Θ is constant here. The extension to Θ(x, t), which Bousquet and Slotine’s hypotheses need, is re-derived in NOTE-1020 and is not this paper’s.
• Not a small-gain result. f’s dependence on z enters only through α, with no feedback condition, because d bounds the manifold’s drift globally. When the disturbance on the fast state grows with the slow state, as in Bousquet and Slotine, a small-gain condition ε < ε_c is needed instead (THEORY-771).
• Not for linear systems as stated. A bound on |∂γ/∂x · f| over all x and z fails when f is unbounded and γ is linear. The proof needs it only along the trajectory, which is how the paper’s example and the record’s check use it.
• Not stochastic. With noise on the fast variable z − γ(x) does not shrink to O(ε), and nothing here bounds an information measure.
• Not sharp. In the record’s check the slow bound was loose by a factor of 1.4 to 2.4, and the paper reports a factor of about 40 in its example.

Connections

• QUESTION-222. Here the coarse variable is the slow state x and the eliminated variable z is fast and contracting. The Markovian reduction, with no memory coordinate at all, then predicts x to O(ε) for all time, with explicit constants. In A5’s example, with c = κd, the memory kernel bκd e^{−dt} tends to bκδ(t) as d grows, and this theorem bounds the error of the memoryless model by O(1/d). It answers QUESTION-222 in part, for a fast, contracting eliminated variable: the finite augmentation is the empty one, and the error is controlled. It says nothing when the eliminated variables are slow or noisy, which is where the question is hard.
• THEORY-771. Bousquet and Slotine apply the same robustness lemma to each line of the system and close the loop by small gain. This paper supplies the lemma’s published proof for constant metrics.
• THEORY-661. A uniform-in-time first-order rate for the quasi-steady-state reduction, in a different frame from Simon and Ando (nonlinear, continuous time, contraction metrics, a given fast–slow split).