Cascaded Systems

Definition

A cascade is a pair of dynamical subsystems in which a driving subsystem ${\Sigma }_2$
perturbs a driven subsystem ${\Sigma }_1$ through an interconnection term, with no feedback from
${\Sigma }_1$ back into ${\Sigma }_2$. The cascaded-systems method certifies global/uniform asymptotic
stability of the whole by proving each subsystem stable in isolation and then verifying a
growth/integrability condition on the interconnection — avoiding the search for a single
monolithic Lyapunov function for the coupled system (Panteley–Loría 1998, 2001). For the
free-flying space manipulator the closed loop is exactly this shape: the decoupled CoM
translation loop drives the coupled attitude+EE loop through the CoM-coupling Coriolis term
${\mathbf{C}}_c{\mathbf{v}}_c$, so cascade theory is the rigorous route to a global,
time-varying (tracking) stability claim that the single-loop energy certificate
(coordinated_control_lyapunov_stability)
does not by itself cover.

Key Equations

Symbols per notation.md. The two Panteley sources are pure systems theory — no
spacecraft or manipulator model — so the free-flying vs free-floating distinction does not arise
in the sources; it enters only when we map our closed loop onto the cascade form.

Cascade canonical form — driven ${\Sigma }_1$, driving ${\Sigma }_2$, interconnection $g(t,\mathbf{x})$:

$${\Sigma }_1:{\dot{\mathbf{x}}}_1=f_1(t,{\mathbf{x}}_1)+g(t,\mathbf{x}){\mathbf{x}}_2,{\Sigma }_2:{\dot{\mathbf{x}}}_2=f_2(t,{\mathbf{x}}_2),\mathbf{x}=\mathrm{col}[{\mathbf{x}}_1,{\mathbf{x}}_2]\text{(Panteley 2001 eqs 1–2)}$$

Reduction principle — UGAS of each part plus global uniform boundedness of the cascade gives
UGAS of the whole; every theorem below reduces to a boundedness (no finite-escape) argument:

$$\left({\Sigma }_2\text{UGAS}\right)\wedge ({\dot{\mathbf{x}}}_1=f_1(t,{\mathbf{x}}_1)\text{UGAS})\wedge \left(\text{solutions globally uniformly bounded}\right)\Rightarrow \text{cascade UGAS}\text{(Panteley 2001 Lemma 2)}$$

Lyapunov derivative along the cascade — the cross term is everything; with $V(t,{\mathbf{x}}_1)$,
${\dot{V}}_{(4)}\le -W({\mathbf{x}}_1)$ the certificate of the unforced driven part:

$${\dot{V}}_{(1)}(t,\mathbf{x})\le -W({\mathbf{x}}_1)+\Vert [L_{g}V](t,\mathbf{x})\Vert \Vert {\mathbf{x}}_2\Vert \text{(Panteley 2001, proof of Thms 3–5)}$$

Growth-rate taxonomy (Panteley 2001) — classify the cascade by how the interconnection
$L_{g}V$ compares to the decay rate $W$ as $\Vert {\mathbf{x}}_1\Vert \to \infty$:

• Case 1 ($f_1$ dominates $g$): little-$o$ condition $\Vert [L_{g}V]\Vert =o(W({\mathbf{x}}_1))$;
  then the cascade is UGAS and ${\Sigma }_1$ is ISS w.r.t. ${\mathbf{x}}_2$ — see
  input_to_state_stability. (Panteley 2001 Thm 3)
• Case 2 ($f_1$ majorizes $g$): the non-escape integral condition
  ${\int }_a^{\infty }\frac{ds}{{\alpha }_6(s)}=\infty$ rules out finite escape; UGAS holds
  even when ${\Sigma }_1$ is not ISS. (Panteley 2001 Thm 4, eqs 22–23)
• Case 3 ($g$ outgrows $f_1$): mere decay of ${\mathbf{x}}_2$ is insufficient — the
  driving signal must be absolutely integrable. (Panteley 2001 Thm 5)

Integrability condition on the driving signal (the load-bearing hypothesis of the
strong-perturbation case, shared by both papers):

$${\int }_{t_0}^{\infty }\Vert {\mathbf{x}}_2(t;t_0,{\mathbf{x}}_2(t_0))\Vert dt\le \varphi (\Vert {\mathbf{x}}_2(t_0)\Vert ),\varphi \in \mathcal{K}\text{(Panteley 1998 Asm A3 eq 10 / Panteley 2001 Asm A6 eq 25)}$$

Interconnection at-most-linear-in-${\mathbf{x}}_1$ growth (the structural restriction in the
1998 GUAS theorems):

$$\Vert g(t,\mathbf{x})\Vert \le {\theta }_1(\Vert {\mathbf{x}}_2\Vert )+{\theta }_2(\Vert {\mathbf{x}}_2\Vert )\Vert {\mathbf{x}}_1\Vert \text{(Panteley 1998 Asm A2 eq 9)}$$

Our closed loop as a cascade. The circumcentroidal coordinated controller splits into the
driving inner loop (CoM error, autonomous damped second-order, hence UGES) and the driven outer
loop (coupled attitude+EE), interconnected by the CoM-coupling Coriolis term:

$$\underset{{\Sigma }_2\text{(driving: CoM inner loop)}}{\underbrace{m{\ddot{\tilde{\mathbf{x}}}}_c+{\mathbf{D}}_c{\dot{\tilde{\mathbf{x}}}}_c+{\mathbf{K}}_c{\tilde{\mathbf{x}}}_c=0}}\text{(current\_sota eq 4.9)}$$ $$\underset{{\Sigma }_1\text{(driven: attitude+EE outer loop)}}{\underbrace{\breve{\mathbf{M}}\dot{\breve{\mathbf{v}}}+\breve{\mathbf{C}}\breve{\mathbf{v}}+\breve{\mathbf{D}}\breve{\mathbf{v}}+{\mathbf{J}}_{\tilde{x}}^{\top }\breve{\mathbf{K}}\tilde{\mathbf{x}}=-({\mathbf{C}}_c+\breve{\mathbf{D}}{\breve{\mathbf{G}}}_{v_c}){\dot{\tilde{\mathbf{x}}}}_c}}\text{(current\_sota eq 4.10)}$$

The right-hand side $-({\mathbf{C}}_c+\breve{\mathbf{D}}{\breve{\mathbf{G}}}_{v_c}){\dot{\tilde{\mathbf{x}}}}_c$
plays the role of $g(t,\mathbf{x}){\mathbf{x}}_2$ with the driving signal ${\mathbf{x}}_2\sim {\dot{\tilde{\mathbf{x}}}}_c$.
Because ${\Sigma }_2$ is a stable linear damped oscillator, ${\dot{\tilde{\mathbf{x}}}}_c$ decays
exponentially and is therefore absolutely integrable, satisfying the integrability hypothesis above;
the energy function $V$ of ${\Sigma }_1$
(coordinated_control_lyapunov_stability)
supplies the driven-subsystem certificate. This is precisely the gap the single-loop proof leaves
open: at the inner-loop equilibrium the CoM forcing vanishes, but the measured CoM error settles
near $10^{-3}\mathrm{m}$ (not zero), so the residual ${\mathbf{C}}_c{\mathbf{v}}_c$ is a
persistent perturbation that only a cascade argument certifies away.

Source Support

• panteley1998global — Lyapunov sufficient conditions under which
  a GUS/GUAS driven NLTV system stays GUS/GUAS when perturbed by a GUAS driving system; the
  integrability hypothesis (A3) and the at-most-linear interconnection growth (A2). Theorem 2 is the
  workhorse; GES of the driver recovers the earlier ECC’97 result as a corollary.
• panteley2001growth — the growth-rate taxonomy (little-$o$ /
  majorization / dominant interconnection) giving three complementary UGAS theorems plus the
  reduction principle (Lemma 2) and the ISS corollary (Case 1); Example 4 shows GAS+GAS+forward-completeness
  alone does not imply GAS.

Related Topics

• coordinated_control — our controller is this cascade: CoM inner loop
  (eq 4.9) driving the coupled attitude+EE outer loop (eq 4.10).
• lyapunov_stability — supplies the per-subsystem certificates the cascade
  theorems compose.
• input_to_state_stability — the Case-1 corollary: little-$o$
  interconnection makes the driven loop ISS w.r.t. the driving signal.
• circumcentroidal_motion — the $\oplus$ coordinate split that
  produces the inner/outer decoupling, hence the cascade structure.
• trajectory_tracking — the time-varying closed loop whose global UGAS
  these theorems certify (a reference enters ${\Sigma }_1$ explicitly through time).

Result page certifying the single (outer) loop:
coordinated_control_lyapunov_stability.

Open Questions

• Which Panteley case applies to our loop? The interconnection
  $-({\mathbf{C}}_c+\breve{\mathbf{D}}{\breve{\mathbf{G}}}_{v_c}){\dot{\tilde{\mathbf{x}}}}_c$
  is bounded and ${\dot{\tilde{\mathbf{x}}}}_c$ decays exponentially — does this place us in Case 1
  (little-$o$, hence ISS of the attitude+EE loop) or only Case 3 (integrability)? Resolving this
  fixes whether the outer loop is ISS in the disturbance.
• The single-loop certificate is only asymptotic on $\Omega$, not exponential. Does the cascade
  composition upgrade or merely preserve the rate when the driver is UGES?
• Both papers give sufficient, not necessary conditions; the residual CoM coupling at the real
  (non-zero) inner-loop equilibrium leaves a persistent forcing — is the integrability hypothesis
  genuinely met, or only the milder $L_2$-type relaxation (Panteley 1998 eq 11)?
• How does the margin degrade as ${\sigma }_G\to {\sigma }_{\mathrm{crit}}$, where the joint-rate inverse is
  regularized and the closed loop the certificate describes is perturbed?