Input To State Stability

Definition

Input-to-state stability (ISS), introduced by Sontag (1989), is a property of a non-linear
system $\dot{\mathbf{x}}=\mathbf{f}(t,\mathbf{x},\mathbf{u})$ with state
$\mathbf{x}$ and exogenous input $\mathbf{u}$: the state stays bounded by a decaying
function of the initial condition plus a class-$\mathcal{K}$ function of the input magnitude, and
in particular $\mathbf{u}\to 0\Rightarrow \mathbf{x}\to 0$. It is a regime-agnostic
systems-theory notion (it does not assume any space/manipulator dynamics). panteley2001growth
does not restate Sontag’s definition; it uses ISS as one sufficient mechanism to guarantee
bounded solutions of a cascade — when the driven subsystem ${\Sigma }_1$ is
ISS with respect to the interconnection input ${\mathbf{x}}_2$ and the driving subsystem
${\Sigma }_2$ is (U)GAS, the composite cascade is (U)GAS. ISS thus answers the question the authors
raise for cascades — building on Sontag’s CIBS result — “how to guarantee boundedness of the
solutions?”. The paper notes ISS was originally
posed for autonomous systems; for the time-varying case (relevant to trajectory tracking) it
points to Kanellakopoulos, Krstić & Kokotović (1995).

Key Equations

Symbols per notation.md.

Generic systems-theory symbols here are reproduced source-faithfully from Panteley/Loría and are
not canonicalized in notation.md. NOTE: the input is written $\mathbf{u}$ below, which
collides with the helix parameter $u\in [0,1]$ in notation.md — distinct meaning, disambiguated
locally to this page; likewise $\gamma ,\beta$ here are comparison functions, unrelated to the
guidance scalars of the same glyph.

Standard ISS bound (the textbook Sontag characterization — stated here for orientation; it is
not written in panteley2001growth, which uses ISS without
restating its definition): the system is ISS w.r.t. $\mathbf{u}$ if there exist
$\beta \in \mathcal{K}\mathcal{L}$ and $\gamma \in \mathcal{K}$ such that, for all initial states and all bounded
inputs,

$$\Vert \mathbf{x}(t)\Vert \le \beta (\Vert \mathbf{x}(t_0)\Vert ,t-t_0)+\gamma (\underset{t_0\le s\le t}{\sup }\Vert \mathbf{u}(s)\Vert ).$$

The condition panteley2001growth actually verifies (its Theorem 3,
Case 1) is an ISS-Lyapunov / growth-rate test: with $V(t,x_1)$ the Lyapunov function of the
unforced driven subsystem ${\dot{x}}_1=f_1(t,x_1,0)$ and $W(x_1)$ its decay margin, the
interconnection $g(t,x)$ must be dominated by the drift,

$$\Vert [L_{g}V](t,x)\Vert =o(W(x_1))\text{as}\Vert x_1\Vert \to \infty ,$$

i.e. $\Vert [L_{g}V](t,x)\Vert \le \lambda (\Vert x_1\Vert )W(x_1)$ with $\lambda (s)\to 0$. If additionally
$W(x_1)\ge {\alpha }_3(\Vert x_1\Vert )$ for some ${\alpha }_3\in \mathcal{K}$, the driven subsystem is ISS with
respect to the input ${\mathbf{x}}_2$ (the drift “grows faster than” the coupling). Here
$L_{g}V=\partial V/\partial x\cdot g$ is the Lie derivative and "$o(\cdot )$" is Panteley’s small-o
ordering (Definition 1: $\Vert \phi \Vert \le \lambda (\Vert x\Vert )\Vert \varrho \Vert$ with $\lambda \to 0$) — distinct from
the weaker majorizes relation (Definition 2, $\mathrm{limsup}<\infty$) that governs Case 2.

Source Support

• panteley2001growth — invokes ISS as a boundedness mechanism for cascaded non-linear (time-varying) systems; its Theorem 3 gives the drift-dominates-coupling ($L_{g}V=o(W)$) growth condition under which the driven subsystem is ISS w.r.t. the interconnection input, and contrasts ISS with the alternative majorization / integral-growth route (Case 2) when the system is not ISS.

Related Topics

• cascaded_systems — the setting in which the source uses ISS: ISS of the driven subsystem (plus GAS of the driver) certifies bounded, asymptotically stable cascade solutions.
• lyapunov_stability — ISS is verified through an ISS-Lyapunov function; the $L_{g}V=o(W)$ test is a Lyapunov-derivative condition, so ISS is the input-driven generalization of Lyapunov asymptotic stability.
• trajectory_tracking — the practical motivation: tracking-error dynamics form a cascade of a stable nominal loop perturbed by a vanishing signal, where ISS (time-varying form) guarantees the error stays bounded as the perturbation decays.
• barrier_lyapunov_function — a related Lyapunov-based certificate, but for constraint satisfaction (error confined to a set) rather than input-bounded-state; both quantify robustness through a Lyapunov argument.
• prescribed_performance_control — enforces transient/steady-state error envelopes; ISS-type bounds underwrite the boundedness such schemes assume.

Open Questions

• The source treats ISS for generic (autonomous and, by extension, time-varying) cascades. Does the free-flying closed loop actually decompose into an ISS driven subsystem driven by a vanishing input, and which physical signal plays the role of ${\mathbf{x}}_2$ (e.g. the dynamic_coupling wrench from the arm onto the actuated base)?
• The paper warns the unqualified ISS notion is “originally proposed… in the context of autonomous systems”; for our time-varying tracking problem, is the time-varying ISS of Kanellakopoulos–Krstić–Kokotović (1995) the correct property to certify, rather than autonomous ISS?
• When a coupled subsystem is not ISS w.r.t. its input, the source falls back to the Case-2 majorization (integral growth) condition — which of the two routes is satisfiable for the circumcentroidal attitude+EE block of our dynamics?