Lyapunov Stability

Definition

Lyapunov stability concerns the qualitative behaviour of an equilibrium of a dynamical
system $\dot{\mathbf{x}}=\mathbf{f}(t,\mathbf{x})$ without solving the equations of
motion. The direct (second) method certifies stability through a scalar energy-like
function $V(t,\mathbf{x})$ that is positive definite at the equilibrium and whose
time-derivative $\dot{V}$ along trajectories is non-positive: $\dot{V}\le 0$ gives stability,
$\dot{V}<0$ (or LaSalle’s invariance argument when $\dot{V}$ is only negative semidefinite)
gives asymptotic stability. For non-autonomous (time-varying) systems — the generic case for a
tracking controller, where the reference enters through $t$ — the stronger uniform notions
(GUS / GUAS / UGAS) are required, and the certificate $V$ must be sandwiched between class-${\mathcal{K}}_{\infty }$
functions. The free-flying vs free-floating distinction is immaterial to the method itself; it
becomes a tool for whichever closed loop is posed.

Key Equations

Direct-method certificate (the energy template used in this project). For the closed-loop
circumcentroidal tracking error of the free-flying manipulator, the certificate is the total
mechanical energy of the reduced ($9\times 9$) attitude+EE subsystem in error coordinates:

V=\tfrac12\,\breve{\boldsymbol{v}}^{T}\breve{\boldsymbol{M}}\,\breve{\boldsymbol{v}} +\tfrac12\,\tilde{\boldsymbol{x}}^{T}\breve{\boldsymbol{K}}\,\tilde{\boldsymbol{x}}, \tag{current_sota eq 4.15 / Giordano eq 37}

with $\breve{\mathbf{v}}=[{\mathbf{\omega }}_b;{\mathbf{\nu }}_e^{\oplus }]\in {\mathbb{R}}^9$ the
stacked attitude+EE velocity, $\tilde{\mathbf{x}}=[{\tilde{\mathbf{x}}}_b;{\tilde{\mathbf{x}}}_e]\in {\mathbb{R}}^9$
the stacked outer-loop error, and $\breve{\mathbf{M}},\breve{\mathbf{K}}\succ 0$ the reduced
inertia / stiffness (notation.md). $V>0$ and radially unbounded on the
singularity-free region $\Omega =\{\mathbf{z}:{\sigma }_{\min }({\mathbf{J}}_{{\nu }_e}^{\oplus }(\mathbf{q}))>0\}$.

Negative-semidefinite derivative. Along the closed loop, the Coriolis quadratic cancels by
passivity (skew-symmetry of $\dot{\breve{\mathbf{M}}}-2\breve{\mathbf{C}}$) and the
stiffness cross-terms cancel via the error-rate map $\dot{\tilde{\mathbf{x}}}={\mathbf{J}}_{\tilde{x}}\breve{\mathbf{v}}$,
leaving only the damping dissipation:

\dot{V}=-\,\breve{\boldsymbol{v}}^{T}\breve{\boldsymbol{D}}\,\breve{\boldsymbol{v}}\le 0, \qquad \breve{\boldsymbol{D}}\succ0. \tag{current_sota eq 4.16 / Giordano eq 38}

Because $\dot{V}$ is only negative semidefinite (vanishing on $\breve{\mathbf{v}}=0$,
not on $\tilde{\mathbf{x}}$), asymptotic stability follows by LaSalle’s invariance
principle: the largest invariant set in $\{\dot{V}=0\}$ is the origin, since
${\mathbf{J}}_{\tilde{x}}^T\breve{\mathbf{K}}\tilde{\mathbf{x}}=0$ with
$\breve{\mathbf{K}}\succ 0$ and ${\mathbf{J}}_{\tilde{x}}$ full rank on $\Omega$ forces
$\tilde{\mathbf{x}}=0$. The full statement, assumptions, and proof sketch live in
coordinated_control_lyapunov_stability.

Class-function sandwich (general template). The uniform versions of the method require, for
some ${\alpha }_1,{\alpha }_2,{\alpha }_3\in {\mathcal{K}}_{\infty }$ (or $\mathcal{K}$),

$${\alpha }_1(\Vert \mathbf{x}\Vert )\le V(t,\mathbf{x})\le {\alpha }_2(\Vert \mathbf{x}\Vert ),{\dot{V}}_{(\cdot )}(t,\mathbf{x})\le -{\alpha }_3(\Vert \mathbf{x}\Vert )(\text{or }\le -W(\mathbf{x})\le 0),$$

(Panteley & Loría A1). The lower/upper sandwich is what upgrades plain stability to uniform
stability for the time-varying case.

Cascade (interconnection) tools. When the closed loop decomposes into a UGAS driven loop
perturbed by the output of a UGAS driving loop,

$${\Sigma }_1:{\dot{\mathbf{x}}}_1={\mathbf{f}}_1(t,{\mathbf{x}}_1)+\mathbf{g}(t,\mathbf{x}){\mathbf{x}}_2,{\Sigma }_2:{\dot{\mathbf{x}}}_2={\mathbf{f}}_2(t,{\mathbf{x}}_2),$$

the cascade is UGAS provided the solutions stay globally uniformly bounded (Panteley & Loría 2001,
Lemma 2). Boundedness is secured either by an absolute-integrability condition on the perturbing
signal,

$${\int }_{t_0}^{\infty }\Vert {\mathbf{x}}_2(t)\Vert dt\le \varphi (\Vert {\mathbf{x}}_2(t_0)\Vert ),\varphi \in \mathcal{K},$$

(Panteley & Loría 1998 A3, eq. 10; 2001 A6, eq. 25) — used when the interconnection grows at least
as fast as the drift — or, when the drift dominates, by a little-o growth condition
$\Vert [L_{\mathbf{g}}V](t,\mathbf{x})\Vert =o(W({\mathbf{x}}_1))$ that additionally yields
ISS of ${\Sigma }_1$ w.r.t. ${\mathbf{x}}_2$ (Panteley & Loría 2001 Thm 3). These supply the rigorous
route to global, uniform asymptotic stability for the project’s tracking loops, which the single
energy $V$ above certifies only locally about a settled inner loop; see
cascaded_systems.

Source Support

• panteley1998global — Lyapunov conditions under which a UGAS
  time-varying cascade preserves GUS/GUAS via an absolute-integrability bound on the perturbing
  signal; the standard tool for global (not merely local) UAS of time-varying tracking loops.
• panteley2001growth — three growth-rate theorems (drift
  dominates / majorizes / is dominated by the interconnection) that replace the search for a single
  strict Lyapunov function; includes the ISS corollary and the reduction-to-boundedness lemma.
• giordano2019coordinated — supplies the concrete energy
  certificate $V$, $\dot{V}$ (eqs 36–38) for the circumcentroidal coordinated control law on a
  free-flying (fully-actuated base) manipulator.

Related Topics

• coordinated_control_lyapunov_stability — the
  named result: this method applied to our closed loop, with assumptions, LaSalle step, and the three
  Giordano corrections the certificate depends on.
• cascaded_systems — the interconnection machinery (Panteley–Loría)
  that upgrades the single-$V$ argument to global, uniform asymptotic stability of the time-varying loop.
• input_to_state_stability — ISS is the natural strengthening
  obtained from the little-o growth condition; the bridge to bounded disturbances in the risk layer.
• task_space_error_dynamics — supplies the error-rate map
  $\dot{\tilde{\mathbf{x}}}={\mathbf{J}}_{\tilde{x}}\breve{\mathbf{v}}$ that makes the
  stiffness cross-terms cancel in $\dot{V}$.
• trajectory_tracking — the time-varying setting that demands the
  uniform notions and motivates the cascade decomposition rather than a monolithic $V$.

Open Questions

• The single energy $V$ certifies the regulator form about a settled inner CoM loop; the rigorous
  global/uniform claim for the full time-varying tracking loop (with persistent CoM coupling
  ${\mathbf{C}}_c{\mathbf{v}}_c$) needs the cascade route — which growth-rate case (Panteley–Loría
  Thm 3/4/5) actually applies to our interconnection is not yet pinned down.
• Stability is asserted only on $\Omega$; near the singular boundary the regularized joint-rate
  inverse perturbs the loop the certificate describes — a Lyapunov characterization that survives the
  Tikhonov / damped-${\mathbf{J}}^{\oplus }$ switching is open.
• The method as used here gives asymptotic, not exponential, stability; whether a strict (exponential)
  certificate exists under the SPD gains is unresolved.