Constructive Nonlinear Control
1 · Why “constructive”?
Linear control is a mature engineering discipline: given a model $\dot x = Ax + Bu$, you can systematically produce a stabilizer — pole placement, LQR, $H_\infty$ — and the procedure tells you how robust the result is (gain and phase margins). Nonlinear control, by contrast, has long been split into two halves that don't cooperate:
The 1997 book closes the gap with a toolkit attitude: instead of one universal method, it builds a family of design procedures — recipes that take a system of a known structure and output two things at once: the feedback law and the Lyapunov function that certifies it. That pairing is the meaning of “constructive”: the certificate is manufactured by the same steps that produce the controller, so robustness is a by-product of the design, not an afterthought.
relative degree, zero dynamics,
triangular forms
feedback law + Lyapunov function
together
→ margins & robustness
The book's architecture follows the tools (chapter numbers in parentheses): passivity as the basic certificate (Ch. 2), stability margins through inverse optimality (Ch. 3), cascade interconnections and the ISS glue that holds them (Ch. 4), Lyapunov functions assembled from pieces — including a “cross-term” (Ch. 5), and the two recursive engines, backstepping and forwarding, plus their combination (Ch. 6). Two structural classes drive everything: systems that are lower-triangular in the state (“strict feedback”) and systems that are upper-triangular (“strict feedforward”).
2 · The language: relative degree, zero dynamics, and what feedback can't change
Before any design, the book fixes a vocabulary. Everything is phrased for systems affine in the input with a chosen output:
where $x$ is the state, $u$ the input, $y$ the output (a signal we choose to measure/define — the choice of $h$ is part of the design, a point that recurs throughout). The Lie derivative notation $L_f h(x) = \frac{\partial h}{\partial x}f(x)$ means “rate of change of $h$ along the vector field $f$”; $L_gL_f h$ means differentiate $h$ along $f$, then along $g$.
Relative degree — the number of integrators between $u$ and $y$
The system has relative degree $r$ at $x=0$ if differentiating $y$ repeatedly makes $u$ appear for the first time at the $r$-th derivative, i.e. $L_gL_f^{\,k}h \equiv 0$ for $k \lt r-1$ and the matrix $L_gL_f^{\,r-1}h(0)$ is nonsingular. For a linear system this is exactly the number of integrators between input and output: the excess of poles over zeros. A double integrator $\ddot y = u$ has relative degree 2: the input needs two differentiations of the output to show up.
Normal form and zero dynamics — what hides behind the output
When the relative degree is well defined, a coordinate change $(z,\xi) = (T(x),\,\text{outputs})$ brings the system to its normal form: a chain of $r$ integrators from $u$ to $y$, plus internal states $z$. The zero dynamics are the dynamics of $z$ when the output is held at zero ($y \equiv 0$) — the motion the system executes while looking perfectly calm from the outside. The system is minimum phase if those hidden dynamics are asymptotically stable, and weakly minimum phase if they are merely Lyapunov-stable.
(feedback lives here)
constraint $y\equiv 0$, untouched by $u$
3 · Passivity: energy as a Lyapunov certificate (Ch. 2)
The book's common thread starts with a physical observation the authors make with disarming candor: think of baking a potato in a microwave — as long as it doesn't burn, the potato's energy can increase only by what the oven supplies. Systems with that property are called dissipative; the special case where the supplied “power” is the product $u^{\!\top}y$ is called passivity.
$S$ is the storage function (think: energy stored in the system); $\int w\,dt$ is the energy supplied from outside. The available storage $S_a(x) = \sup_{u,T}\big[-\int_0^T w\,dt\big]$ — the most energy you could ever extract from state $x$ — is itself the smallest storage function; any other $S \ge S_a$.
For the affine systems used in designs, passivity has a differential characterization that is the nonlinear cousin of the Kalman–Yakubovich–Popov lemma:
$L_fS \le 0$ says the unforced system never generates energy; $L_gS = h^{\!\top}$ says the input enters exactly along the gradient of the storage — the input port is “collocated” with the energy gradient. The second identity is what makes $y$ a natural output: it is the direction in state space in which $u$ does work on the system.
cross level sets inward
Interconnections: passivity composes
Why engineers should care about a definition: passive blocks compose. The parallel connection of two passive systems is passive (storages add), and the negative-feedback interconnection of two passive systems is passive (the loop's supply telescopes: $u_1^{\!\top}y_1 + u_2^{\!\top}y_2$ with $u_1 = -y_2,\, u_2 = y_1$ cancels). The book also grades passivity: a system can have an excess or shortage of passivity, written OFP($-\alpha$) / IFP($\rho$) — a shortage of $\alpha$ in the output channel is exactly a disk margin in disguise (Section 5).
Stability from passivity
With the port closed by $u = -y$, the storage is nonincreasing; stability follows if $S$ is positive definite, and asymptotic stability needs one extra property: ZSD (zero-state detectability — the only solution that keeps $y \equiv 0$ while $u \equiv 0$ is $x \equiv 0$). This is Theorem 2.28, proved by LaSalle's invariance principle: $\dot S \le -y^{\!\top}y$ confines trajectories to the set $\{y = 0\}$, and ZSD says nothing can loiter there except the origin.
$\dot x_1 = x_2,\quad \dot x_2 = -\sin x_1 - x_2 + u,\quad y = x_2$
Motivation for this choice of output: the velocity is the variable along which the motor does work, so we expect an energy balance. Take the physical energy as storage $S = \tfrac12 x_2^2 + 1 - \cos x_1 \ge 0$. Differentiate along the dynamics (chain rule, then the $\sin$ terms cancel — that cancellation is the KYP condition $L_gS = h$):
So the pendulum is passive with excess $-y^2$ (it is OFP: strictly dissipative even at $u=0$). Closing the loop with $u = -ky$, $k>0$: $\dot S = -(1+k)y^2 \le 0$. By LaSalle, trajectories converge to the largest invariant set inside $\{y=0\}$; there $y \equiv 0$ forces $x_2 \equiv 0$ and $\dot x_2 = -\sin x_1 = 0$, so the invariant set is the two equilibria $x_1 = 0$ (hanging) and $x_1 = \pm\pi$ (upright), and the energy level $S < 2$ excludes the upright one — within that basin, $x=0$ is GAS. No cancellation of the nonlinearity $\sin x_1$ was needed or performed — the damping injection $u=-ky$ is robust to everything the model got wrong about gravity.
4 · Feedback passivation — and the two doors it can't open (Ch. 2.4)
A plant is rarely passive as given. Feedback passivation asks: can a feedback transformation $u = \alpha(x) + \beta(x)v$, with $\beta$ invertible, make the system passive from the new input $v$ to the same output $y$? If yes, the recipe above finishes the job with $v = -\kappa y$. The book's central structural result — Theorem 2.47 — says exactly when this is possible:
Both conditions are feedback-invariant (Section 2), so they are structural: they tell you not “this design fails” but “no design of this type can work.” The book illustrates both doors:
$\ddot y = u$ (double integrator), any output with two integrators in front
$\dot y$ does not contain $u$, so no static feedback makes $\tfrac12\xi^2$-style passivation work: the storage would need the term $\tfrac12\dot y^2$ whose derivative involves $\ddot y = u$ — the wrong sign, unfixable without reshaping the storage, which feedback can't do. Key to open it: backstepping (§8).
$\dot x_1 = x_2,\; \dot x_2 = u,\; y = x_1 - x_2$
Relative degree one ($\dot y = x_2 - u$ contains $u$). But holding $y \equiv 0$ forces $x_1 = x_2$ and $u = x_1$, giving $\dot x_1 = x_1$: the zero dynamics escape. Any passive storage restricted to $y \equiv 0$ would be a Lyapunov function for $\dot x_1 = x_1$ — impossible. Key: forwarding (§9).
$\dot x_1 = x_1^2 x_2, \qquad \dot x_2 = u, \qquad y = x_2$
Step 0 — check the doors. $\dot y = \dot x_2 = u$: relative degree one ✓. Zero dynamics: setting $y \equiv 0$ gives $\dot x_1 = x_1^2 \cdot 0 = 0$ — stable but not attractive: weakly minimum phase ✓. The doors are open; passivation should work.
Step 1 — read off the normal form. Here $(z,\xi) = (x_1, x_2)$: $\dot z = 0 + \underbrace{z^2\xi}_{p(z,\xi)\xi}$, $\dot\xi = u$, $y = \xi$.
Step 2 — apply the passivating feedback. The zero-dynamics Lyapunov function is any $W = \tfrac12 z^2$ ($\dot W = 0 \le 0$ along $\dot z = 0$). The formula gives $u = -\big(L_pW\big)^{\!\top} + v = -z^3 + v$ (since $L_pW = z \cdot z^2 = z^3$), i.e. the transformed system
Step 3 — close the loop. $v = -y$: $\dot S = -y^2$; and $y \equiv v \equiv 0$ forces $x_2 = 0$, then $\dot x_2 = -x_1^3 = 0 \Rightarrow x_1 = 0$ (ZSD) — global asymptotic stability, certified by $S$. Note what was not done: the term $x_1^2x_2$ was never cancelled; the design only shaped the input channel and let the storage's cross-terms cancel.
5 · Stability margins and inverse optimality (Ch. 3)
A stabilizer tuned to the exact model is a museum piece. Linear control quantifies robustness with gain and phase margins; Chapter 3 builds the nonlinear heirs — sector and disk margins — and shows they are passivity in disguise: a loop has a disk margin precisely when a passivity shortage of the linear block fits inside the nonlinearity's sector. The punchline of the chapter is an economics result:
Inverse optimality: optimality without solving the HJB equation
Solving the Hamilton–Jacobi–Bellman PDE for a nonlinear plant is generally hopeless. Freeman–Kokotović's inversion flips the task: design first, certify second. The book's Definition (3.5.2): a stabilizer is inverse optimal if it has the gradient form
with $V$ positive (semi)definite and $\dot V \le 0$ already at the detuned gain $u = -\tfrac12 k(x)$ (condition 3.5.3: $L_fV - \tfrac12 L_gV\,k \le 0$). Then the trick: define
holds identically (substitute $k = \tfrac12R^{-1}(L_gV)^{\!\top}$: $l + L_fV = \tfrac14(L_gV)R^{-1}(L_gV)^{\!\top}$, exactly the HJB remainder). So the feedback is genuinely optimal for the cost with penalty $l(x)$ — a function that was never chosen, only certified. The practical moral: every gradient-form design with a Lyapunov function is automatically optimal for some meaningful cost, hence carries the disk/sector margin — the recursive designs of Sections 8–9 are built to land in exactly this form, $u = -L_gV$.
$\dot x = x^2 + u$ — stabilize the origin globally
The cancellation design. Feedback linearization “sees” the destabilizer $x^2$ and erases it: $u_{\rm l} = -x^2 - x$, giving the deceptively clean $\dot x = -x$. Now perturb the loop gain, the cheapest imaginable modeling error — the implemented gain is $(1+\varepsilon)$ times the design:
For any $\varepsilon \neq 0$ this has a finite-escape solution: for $\varepsilon > 0$ and $x < -\tfrac{1+\varepsilon}{\varepsilon}$, $\dot x < 0$ with $- \varepsilon x^2$ winning — $x \to -\infty$ in finite time (integrate $\dot x \approx -\varepsilon x^2$: blow-up at $t = 1/\varepsilon|x|$, quadratically divergent). The margin is zero.
The dominating (inverse optimal) design. $u^{*}(x) = -x^2 - x\sqrt{x^2+1}$, which dominates rather than cancels: $\dot x = -x\sqrt{x^2+1}$, and under loop-gain error $\dot x = -\varepsilon x^2 - (1+\varepsilon)x\sqrt{x^2+1}$ stays globally stable for every $\varepsilon > -\tfrac12$ (for $x<0$ both terms push $x$ up toward the origin). It also respects the nonlinearity: for $x \to -\infty$, $u^* \to \tfrac12$ — the plant's own $x^2$ already stabilizes on that side, so the controller spends almost nothing there. Same nominal closed-loop behavior, categorically different robustness. Try it:
Closed loop $\dot x = x^2 + (1+\varepsilon)u(x)$ simulated (RK4) from $x(0) \in \{\pm 0.5, \pm 2, \pm 7, \pm 20\}$; a curve leaves the panel when $|x|>60$ (finite escape). At $\varepsilon = 0$ both designs look identical; any $\varepsilon \neq 0$ separates them. Presentation only — no claims are checked by this code.
(stability lost)
nonlinearity works alone
Control Lyapunov functions: the target the recursions shoot for
The book's designs all manufacture functions of one special kind. $V$ is a control Lyapunov function (CLF) if it is positive definite, radially unbounded, and its derivative can be made negative by some feedback — for input-affine systems, the Artstein–Sontag condition: $L_gV(x)=0 \Rightarrow L_fV(x)<0$ for $x \neq 0$. When a CLF exists, Sontag's formula gives a universal stabilizer:
— smooth away from the origin, and of inverse-optimal form. The recursive procedures of Sections 8–10 are best read as CLF factories: each step extends a CLF of a subsystem into a CLF of the augmented system, so the certificate and the gradient-form feedback are produced in lockstep.
6 · Cascades: the workhorse interconnection (Ch. 4)
Large systems decompose. The book's canonical decomposition is the cascade: a driving subsystem $\xi$, directly actuated, feeding a driven subsystem $z$ through an interconnection $\psi$:
driving: actuated,
to be stabilized first
interconnection:
the disturbance $\xi$ inflicts on $z$
driven: no direct input
$\dot z = -z^3 + z^2\,\xi$
With $W = \tfrac12z^2$: $\dot W = -z^4 + \xi z^3$. The interconnection term $z^2\xi$ grows superlinearly in $z$ — the classical growth-restriction route is blocked. Young's inequality $\xi z^3 \le \tfrac34 z^4 + \tfrac14\xi^4$ gives $\dot W \le -\tfrac14 z^4 + \tfrac14\xi^4$: whenever $|z| \ge 2|\xi|$, $\dot W \le -\tfrac{15}{64}z^4 < 0$. That is exactly the ISS-Lyapunov condition, so any converging driver — for instance the driving subsystem $\dot\xi = \xi^2 u$ stabilized by $u = -\xi$, which gives $\dot\xi = -\xi^3$ — drags $z \to 0$: global stabilization certified despite the superlinear growth.
Peaking: when “make $\xi$ faster” backfires
The tempting shortcut — if $\xi$'s transient is the problem, make it vanish faster — is a trap, and the book dissects it with unusual care. Placing the eigenvalues of the $\xi$-subsystem at $-a$ gives $\|\xi(t)\| \le \gamma(a)\,\|\xi(0)\|e^{-at}$, but the transient bound $\gamma(a)$ grows polynomially in $a$ (even for the innocent-looking $\gamma(a) = a$: the peak of $ae^{-at}$ is $a/e$). Fast convergence is bought with tall transients — and a superlinear interconnection converts a tall enough transient into finite-time escape of $z$:
$\dot z = -z^3 + \xi_2^2z^3, \qquad \dot\xi_1 = \xi_2, \qquad \dot\xi_2 = -a^2\xi_1 - 2a\,\xi_2$
The $\xi$-pair is a critically damped oscillator with both eigenvalues at $-a$; from $\xi_1(0)=1, \xi_2(0)=0$ its velocity is $\xi_2(t) = -a^2te^{-at}$, peaking at $ae^{-1}$ — growing linearly in the decay rate. The $z$-equation solves exactly: $\frac{d}{dt}\big(\tfrac{1}{z^{2}}\big) = 2\big(1-\xi_2^2\big)$, so
By the peak time $t_p = 1/a$ the accumulated burn is $\int_0^{t_p}\xi_2^2 \approx a/4$, while the refund is only $t_p + z^{-2}(0) = a^{-1} + z^{-2}(0)$. For any $z(0)$, a large enough $a$ makes the ledger negative: $z$ escapes to infinity before $\xi$ has finished converging. The domain of attraction shrinks as the gain grows. Conclusion: growth restrictions on $\psi$ are not pedantry — they are the exact accounting of who wins the race between $\xi$'s transient and $z$'s dissipation. (Design consequence: either keep gains moderate — semiglobal results — or use designs, like forwarding with saturation, that never let the driven subsystem see the peak.)
requested decay rate $a$
Case study: the TORA benchmark (§4.4)
The book's running physical example is the Translational Oscillator with Rotating Actuator: a platform of mass $M$ on a spring, with an eccentric mass $m$ rotating on it. The motor torque $u$ is the only actuation, and it influences the platform only through the reaction force of the whirling eccentric mass — a physically literal cascade.
radius $e$, angle $\theta$
$u \to \dot x_4$driving subsystem ($\xi$)
$\dot z_2 = -z_1 + \varepsilon\sin z_3$driven subsystem ($z$)
into the platform; $\varepsilon$ set by the eccentricity (≈ 0.1)
With $z=(z_1,z_2,z_3)$, $\xi = x_4$ and the feedback linearizing $\dot\xi = v$, the cascade is $\dot z_1 = z_2,\; \dot z_2 = -z_1 + \varepsilon\sin z_3,\; \dot z_3 = \xi,\; \dot\xi = v$. The $z$-subsystem with $\xi$ frozen is a conservative oscillator with potential $\tfrac12(z_1-\varepsilon\sin z_3)^2$, so take
The interconnection into $z$ is $\psi = (0,0,\xi)$, so $L_\psi W = \big(k_1z_3 - (z_1-\varepsilon\sin z_3)\,\varepsilon\cos z_3\big)\xi$, and the passivating feedback $v = -L_\psi W + w$ yields the storage $V = W + \tfrac12\xi^2$ with $\dot V = \xi w$ — passive by construction. ZSD holds (working through $\xi \equiv 0, w \equiv 0$ forces $z_3 \to$ const, then $z_1 = \varepsilon \sin z_3$, then $z_2 = 0$, then the constraint forces $z_3 = 0$); the final loop $w = -k\xi$ gives GAS with the disk margin of an inverse optimal design. The nonlinearity $\varepsilon\sin z_3$ is never cancelled — only its passivity quotient is used.
7 · Composite Lyapunov functions and the cross-term (Ch. 5)
Everything so far needs a Lyapunov function for the whole cascade, not just its pieces. The naive composite $V = W(z) + U(\xi)$ fails for a simple reason: differentiate it.
The Chapter 5 remedy is to add a third piece with a precise job description: find $\Psi(z,\xi)$ — the cross-term — such that its derivative exactly absorbs the interconnection's work: $\dot\Psi = -L_\psi W$ along the cascade. Such a $\Psi$ is not guessed; it is integrated along the trajectories of the unforced cascade $(\tilde z, \tilde\xi)$ starting at $(z,\xi)$:
Why the integral converges: $\tilde\xi(s) \to 0$ exponentially, the driven trajectory $\tilde z(s)$ stays bounded, and $\partial W/\partial z$ is bounded along it — the integrand decays geometrically (this is Assumptions 5.1–5.2 territory: GS of the driven subsystem with a radially unbounded $W$; GAS of the driver). Why $V_0$ works: along trajectories, $\frac{d}{dt}\big(W + \Psi\big) = L_fW + L_\psi W - L_\psi W = L_fW \le 0$: the cross-term has surgically removed the indefinite term. Theorem 5.8 then certifies that $V_0$ is positive definite and radially unbounded — the total stored energy of the cascade counting the energy currently on loan to the driver.
Computing the cross-term
Two practical remarks the book is refreshingly honest about. Differentiability: $\Psi$ is continuous and p.d., but may fail to be $C^1$ (needed for $\dot\Psi$) unless extra growth/observability conditions hold; §5.3 develops relaxed constructions — e.g. the change of coordinates $\zeta = z + \int_0^\infty\!\!\ldots$ that folds the cross-term into a squared coordinate, $\Psi = \tfrac12(\zeta^2 - z^2)$-style, sidestepping smoothness. Computability: closed forms exist when the cascade solutions are explicit (linear driver, scalar $z$); otherwise $\Psi$ is a quadrature along trajectories — cheap enough to evaluate, hard to differentiate. This is the price of the method, and it shapes which systems the recursive designs of §8–9 target.
Cascade: $\dot x_2 = \psi(x_3) = x_3 + x_3^3, \qquad \dot x_3 = -x_3$ (the $x_3$-subsystem already stabilized by $u=-x_3$)
Motivation: $x_2$ is a pure integrator ($\dot x_2 = 0$ when $x_3 = 0$) — globally stable, but $\psi$ pumps $x_3$'s transient into it, so $V = \tfrac12x_2^2 + \tfrac12x_3^2$ has $\dot V = x_2(x_3+x_3^3)$ — sign-indefinite. Apply the construction with $W = \tfrac12x_2^2$, $U = \tfrac12x_3^2$.
Step 1 — the trajectory. $\tilde x_3(s) = x_3e^{-s}$ and $\tilde x_2(s) = x_2 + \int_0^s\psi(\tilde x_3)\,d\tau = x_2 + a(s)$ where $a(s) = x_3(1-e^{-s}) + \tfrac{x_3^3}{3}(1-e^{-3s})$.
Step 2 — the integral, done by a slick trick. Since $a'(s) = \psi(\tilde x_3(s))$, the integrand is $\tilde x_2(s)\,a'(s) = (x_2 + a)\,a'$:
Step 3 — assemble. $V_0 = \tfrac12x_2^2 + \Psi + \tfrac12x_3^2 = \tfrac12\big(x_2 + x_3 + \tfrac{x_3^3}{3}\big)^2 + \tfrac12x_3^2$: the cross-term folds into a shifted square $\zeta_2 = x_2 + x_3 + x_3^3/3$ — the relaxed change of coordinates in action. Indeed $\dot\zeta_2 = \dot x_2 + (1+x_3^2)\dot x_3 = (x_3 - x_3^2u) + (1+x_3^2)u = x_3 + u$, which vanishes exactly when $u = -x_3$: the coordinate is constant along the stabilized cascade, as the construction demands. This $\zeta_2$ and $V_0$ are precisely the objects the forwarding recursion (§9) manipulates.
8 · Backstepping: recursion down the feedback ladder (Ch. 6.1)
Strict-feedback systems are lower-triangular: the input reaches $x_1$ only through the chain $x_n \to x_{n-1} \to \cdots \to x_1$, with possibly destabilizing nonlinearities hanging off the states:
The output $x_1$ has relative degree $n$ — Door 1 of Section 4 is slammed shut. The book's move is to notice that the sub-system seen by treating $x_2$ as a control is only first-order, hence has relative degree one: a “virtual control.” Designing top-down, one integrator at a time, each step is a legal passivation (Theorem 2.47 applies), and the Lyapunov function is augmented by $\tfrac12(\text{new output})^2$ at each step. The name: on the block diagram, the design walks backwards, from the integrator farthest from $u$, toward the input.
$\dot x_1 = x_2 + \theta x_1^2, \quad \dot x_2 = x_3, \quad \dot x_3 = u, \qquad \theta \in [-1,1]$ uncertain
Two instabilities at once: the linear double integrator $(x_2,x_3)$ and the quadratic blow-up $\theta x_1^2$.
Step 1 — stabilize the $x_1$-subsystem with virtual control $x_2 = \alpha_1(x_1)$. Why not cancel? $\alpha_1 = -x_1 - \theta x_1^2$ would need the unknown $\theta$ — and cancellation of an uncertain term has no margin (§5). Instead, dominate: knowing only $|\theta| \le 1$, take $\alpha_1 = -x_1 - x_1^3$ and $V_1 = \tfrac12x_1^2$:
Step 2 — step back through the first integrator. Define $y_2 = x_2 - \alpha_1(x_1)$; differentiating and substituting $\dot x_1 = \alpha_1 + \theta x_1^2 + y_2$ gives $\dot y_2 = x_3 + (1+3x_1^2)(\alpha_1 + \theta x_1^2 + y_2)$. With $V_2 = V_1 + \tfrac12y_2^2$:
The structural miracle: every potentially positive term now rides on $y_2$ — the passivating output. The cancellation design would set the bracket to zero, but its $\theta$-term is unimplementable (book's eq. 6.1.10); brute-force domination works but “hardens” — the virtual gains balloon polynomially at each step (book's 6.1.11). The book's elegant exit: change the certificate, not the control. Use the modified first-stage function
Now the $\theta$-group appears twice with opposite signs — once as $\frac{\partial \tilde V_1}{\partial x_1}y_2 = (1+3x_1^2)(x_1+x_1^3-\theta x_1^2)y_2$ and once inside $\dot y_2$ — and cancels exactly in $\dot{\tilde V}_2$ (book's eq. 6.1.14), leaving
Choosing the virtual control $x_3 = \alpha_2 = -(2+3x_1^2)\,y_2$ kills the bracket to $-y_2^2$:
The virtual control $\alpha_2 = -(2+3x_1^2)y_2$ contains no $\theta$, uses less effort than brute domination, and carries a gain margin $[1,\infty)$ — the loop from $y_2$ is passive. (The certificate $\tilde V_1$ does depend on $\theta$ — that is allowed: certificates may know what controllers must not.)
Step 3 — repeat. With $y_3 = x_3 - \alpha_2$, the zero dynamics of $(x_1,x_2)$ are exactly the system just stabilized, so one more passivation gives $V_3 = \tilde V_2 + \tfrac12y_3^2$ and $u = \alpha_3$ making $\dot V_3 \le -(\text{p.d.})$; the only new work is dominating the $\theta$-dependence of $\dot\alpha_2$ (book's §6.1.1). GAS of the full third-order system, globally, for all $|\theta|\le 1$.
Second-order core of the book's example, $\dot x_1 = x_2 + \theta x_1^2,\ \dot x_2 = u$, under the $\theta$-free backstepping law $u = -(2+3x_1^2)\,y_2$, $y_2 = x_2 + x_1 + x_1^3$, simulated (RK4) for every slider value — the same controller stabilizes all $\theta\in[-1,1]$. Level sets of the $\theta$-dependent certificate $\tilde V_2 = \tilde V_1 + \tfrac12 y_2^2$ are drawn for the current $\theta$. Presentation only.
9 · Forwarding: recursion up the feedforward ladder (Ch. 6.2)
Strict-feedforward systems are upper-triangular — the mirror image: no feedback paths, the input enters the last integrator and nonlinearity-laden feedforward paths leak it upward:
Backstepping does not apply — the triangular structure points the wrong way, and the first state $x_1$ has relative degree 3. But the missing feedback paths are good news: solutions exist for all time (no finite escape), and the unforced system's instability is merely the drift of a triple integrator. Forwarding is the dual procedure: bottom-up, stabilize the last equation, then augment the Lyapunov function one state at a time using the cross-term construction of §7 — at each step the cross-term is a shifted square $\tfrac12(x + \phi)^2$.
Step 1 — stabilize the last equation. $\dot x_3 = u$ is a passive integrator: storage $V_3 = \tfrac12x_3^2$, feedback $\alpha_3 = -x_3 = -L_gV_3$.
Step 2 — augment by $x_2$ with the cross-term. With $u = \alpha_3$, the cascade is $\dot x_2 = \psi_2(x_3) = x_3 + x_3^3$, $\dot x_3 = -x_3$ — globally stable, and §7's worked example already built its certificate:
Restore the freedom $u = \alpha_3 + v_2$: because $\dot\zeta_2 = v_2$ (the shifted coordinate sees the input directly — one more reward of the cross-term) and $\dot V_2 = x_3\dot x_3 + \zeta_2\dot\zeta_2$,
Choose $v_2 = -(x_3+\zeta_2) = -L_gV_2 + L_gV_3$: then $\dot V_2 = -x_3^2 + x_3v_2 - v_2^2 \le -\tfrac12x_3^2 - \tfrac12v_2^2 < 0$ — GAS/LES of $(x_2,x_3)$, and the control is exactly of inverse-optimal form:
Step 3 — augment by $x_1$. With $u = \alpha_2$, the $(x_2,x_3)$ subsystem is GAS, and $\dot x_1 = \psi_1(x_2,x_3) = x_2 + x_3^2 + x_2\alpha_2$ feeds it forward. The cascade is again globally stable, so the cross-term construction gives
— no closed form this time (the book flags it: evaluate by quadrature or approximate analytically). Because $\zeta_1$ is constant along the $\alpha_2$-flow (that is what the construction built it for), the input coefficient of $\dot V_1$ is exactly the cross-term identity $L_gV_1 = L_gV_2 + \zeta_1\,L_g\zeta_1$, so $\dot V_1\big|_{\alpha_2} \le -\tfrac12(L_gV_2)^2$; choosing $v_1 = -L_gV_1 + L_gV_2$ yields $\dot V_1 \le -\tfrac12(L_gV_1)^2$ and the final law
Refinements
Chapter 6.2's second half deals with the two frictions of real feedforward systems. Weak minimum phase isn't needed, but growth is: the cross-term integral converges only if the driven subsystem dominates the interconnection; when the interconnection grows too fast, low-gain forwarding stabilizes a large (semiglobal) region with restrained gains. Saturation designs: inserting saturation functions into the $\alpha$'s keeps the control bounded for all states, decoupling “global boundedness” from “local convergence” — at the price of explicitly trading off the region of attraction against the control magnitude (§6.2.5–6.2.6).
10 · Interlaced systems: both keys, one lock (Ch. 6.3)
Real structures mix both triangularities. The book's introductory interlaced system feeds back and feeds forward:
— not strict feedback ($x_2x_3$ reaches up), not strict feedforward ($x_2^2$, $x_1x_2x_3$ loop back down). The design: start at the middle equation, $\dot x_2 = x_3 + x_2^2$, with $x_3$ as virtual control: $\alpha_1 = -x_2 - x_2^2$ gives $\dot x_2 = -x_2$, $V_1 = \tfrac12x_2^2$. Then one step of forwarding (the top equation sees only $(x_2,x_3)$): with $x_3 = \alpha_1 + v$, the shifted coordinate $\zeta_1 = x_1 + x_2 - \tfrac12x_2^2 - \tfrac13x_2^3$ satisfies $\dot\zeta_1 = (1-x_2^2)\,v$, so $V_2 = V_1 + \tfrac12\zeta_1^2$ and the extra feedback $v = -(1-x_2^2)\zeta_1 = -L_g(V_2-V_1)$ gives $\dot V_2 = -x_2^2 - (1-x_2^2)^2\zeta_1^2$. Then one step of backstepping through the $\dot x_3$ equation: $y = x_3 - \alpha_2$, $V_3 = V_2 + \tfrac12y^2$, and $u = \alpha_3$ makes $\dot V_3$ negative definite. Global stabilization by alternating the two engines — the sequence (forward, back) or (back, forward) is dictated entirely by where the nonlinearities live.
11 · The big picture
| structure | engine | certificate construction | obstacle removed | margin story |
|---|---|---|---|---|
| cascade | partial-state feedback + ISS | composite $W+\Psi+U$ (cross-term) | — | as good as the driver's design |
| strict feedback (lower-triangular) | backstepping (top-down) | $V \mathrel{+}= \tfrac12y^2$ per step; $\tilde V$-redesign against uncertainty | relative degree $r>1$ | domination ⟹ inverse optimality, sector/disk margins |
| strict feedforward (upper-triangular) | forwarding (bottom-up) | $V \mathrel{+}= \tfrac12(x+\phi)^2$ per step (cross-term) | minimum phase (not needed!) | $u=-L_gV$ at every step: disk $D(\tfrac12)$ |
| interlaced | alternating both | mix of the two augmentations | both | inherited from the engines used |
Where this sits in the larger theory
With Lyapunov theory: the book is an answer to the converse-theorem gap — existence of $V$ is replaced by construction procedures. With passivity: the storage functions of §3 are the Lyapunov functions of §§8–9; inverse optimality (§5) is the bridge that converts passivity shortages into disk margins. With ISS (Sontag): cascades are held together by ISS-Lyapunov functions; the cross-term is, in spirit, an ISS construction upgraded to a differentiable certificate. With adaptive control (§5.5): the parametric Lyapunov functions $V(x,\theta)$ are exactly the $\tilde V_1$-style modifications of §8 — the same bookkeeping that makes backstepping robust makes it adaptive. With what came after: energy-shaping/IDA-PBC designs, Krstić–Kanellakopoulos–Kokotović's nonrecursive backstepping monograph, and Freeman–Kokotović's robustness analysis all descend from the same program; the book itself points the reader to those complements.
What the method asks of you
The honest limitations, all stated in the book: the structure must be (approximately) triangular — genuinely non-triangular couplings fall outside; cross-terms rarely have closed forms, so quadrature (or symbolic/numerical differentiation of integrals) is part of the design loop; peaking forces semiglobal claims in some classes no matter the design; and quadratic augmentation hardens gains — the $\tilde V$-redesign and saturation tricks temper but do not abolish it.