EN530.678 Review.
System Models
Constraints
The configurations space a mechanical system is denoted by $Q$ and is assumed to be an $n$-dimensional manifold, locally isomorphic to $\mathbb{R}^n$. A configuration is denoted by $q\in Q$
$$ h_i(q) = 0 \quad (i = 1,\dots,k) $$$$ A^T(q) \dot{q} = 0 $$$$ \nabla_{q} h_{i}(q)=a_{i}(q) \quad (i=1, \dots, k) $$Dynamics
holonomic systems
$$ L(q, \dot{q})=\frac{1}{2} \dot{q}^{T} M(q) \dot{q}-V(q) $$$$ \frac{d}{d t} \nabla_{\dot{q}} L-\nabla_{q} L=f_{\text{ext}}(q, \dot{q})+B(q) u $$where $B(q)\in\mathbb{R}^{n\times m}$ is a matrix mapping from $m$ control inputs to the forces/torques acting on the generalized coordinates $q$.
$$ M(q) \ddot{q}+b(q, \dot{q})=B(q) u $$$$ \begin{equation}\label{bq} b(q, \dot{q})=\dot{M}(q) \dot{q}-\frac{1}{2} \nabla_{q}\left(\dot{q}^{T} M(q) \dot{q}\right)+\nabla_{q} V(q)-f_{\text{ext}}(q, \dot{q}) \end{equation} $$nonholonomic systems
$$ \frac{d}{d t} \nabla_{\dot{q}} L-\nabla_{q} L=A(q) \lambda+f_{\text{ext}}(q, \dot{q})+B(q) u $$where $\lambda\in\mathbb{R}^k$ is a vector of the Lagrange multipliers and the extra term $A(q)\lambda$ denotes the force that counters any motion in directions spanned by $A(q)$.
$$ \begin{gather*} M(q) \ddot{q}+b(q, \dot{q})=A(q)\lambda + B(q) u \\ A^T(q)\dot{q} = 0 \end{gather*} $$where $b(q)$ has the same form as $\eqref{bq}$.
$$ G^{T}(q)(K(q) \ddot{q}+n(q, \dot{q}))=G^{T}(q) S(q) u $$$$ \begin{gather*} \dot{q}=G(q) v \\ K(q) \dot{v}+n(q, v)=S(q) u \end{gather*} $$$$ \begin{gather*} K(q)=G^{T}(q) M(q) G(q)>0 \\ n(q, v)=G^{T} M(q) \dot{G}(q) v+G^{T}(q) b(q, G(q) v) \\ S(q)=G^{T}(q) B(q) \end{gather*} $$Stability
See Nonlinear System
Manifolds and Vector Fields
Manifolds
$$ \begin{gather*} \varphi^{\prime} \circ \varphi^{-1}\Big\vert_{\varphi\left(U \cap U^{\prime}\right)}: \varphi\left(U \cap U' \right) \to \varphi^{\prime}\left(U \cap U'\right) \\ \varphi \circ\left(\varphi^{\prime}\right)^{-1}\Big\vert_{\varphi^{\prime}\left(U \cap U^{\prime}\right)}: \varphi^{\prime}\left(U \cap U' \right) \to \varphi\left(U \cap U'\right) \end{gather*} $$are $C^\infty$
We call $M$ a differentiable $n$-manifold when:
- The set $M$ is covered by a collections of charts, that is, every point is represented in at least one chart
- $M$ has an atlas; that is, $M$ can be written as a union of compatible charts
Tangents
$$ c_1(0) = c_2(0) = m $$$$ \frac{d}{d t}\left(\varphi \circ c_{1}\right)\Big\vert_{t=0}=\frac{d}{d t}\left(\varphi \circ c_{2}\right)\Big\vert_{t=0} $$in some chart $\varphi$
A tangent vector $v$ to a manifold $M$ at point $m$ is an equivalent class of curves at $m$. The set of tangent vectors to $M$ at $m$ is a vector space, denoted by $T_m M$
$$ T M=\bigcup_{m \in M} T_{m} M $$Vector fileds
A vector field $X$ on $M$ is a map $X: M \to TM$ that assigns a vector $X(m)$ at the point $m\in M$.
An integral curve of $X$ with initial condition $m_0$ at $t=0$ is a map $c: (a,b) \to M$ s.t. $(a,b)$ is an open interval containing $0$, $c(0) = m_0$ and $c'(t) = X(c(t))$ for all $t\in (a,b)$.
The flow of $X$ is a collection of maps $\Phi_t: M\to M$ s.t. $t\to \Phi_t(m)$ is the integral curve of $X$ with initial condition $m$.
Lie bracket
Given two vector fields $g_1(x)$ and $g_2(x)$, do their flows commute? e.g. $\Phi_{t}^{g_{2}} \circ \Phi_{t}^{g_{1}}=\Phi_{t}^{g_{1}} \circ \Phi_{t}^{g_{2}}$
$$ \Phi_{t}^{-g_{2}} \circ \Phi_{t}^{-g_{1}} \circ \Phi_{t}^{g_{2}} \circ \Phi_{t}^{g_{1}}\left(x_{0}\right)=x_{0}+t^{2}\left[g_{1}, g_{2}\right]+O\left(t^{3}\right) $$$$ \left[g_{1}, g_{2}\right]=\frac{\partial g_{2}}{\partial x} g_{1}-\frac{\partial g_{1}}{\partial x} g_{2} $$$$ \left[g_{1}, g_{2}\right] \alpha=g_{1}\left(g_{2} \alpha\right)-g_{2}\left(g_{1} \alpha\right) $$A vector space $V$ with a bilinear operator $[\cdot, \cdot]: V\times V\to V$ is called a Lie algebra when satisfying:
- Skew-symmetry: $[v, w] = -[w, v]$ for all $v, w \in V$
- Jacobi identity: $[[v, w], z]+[[z, v], w]+[[w, z], v]=0$
Distributions and Controllability
Distributions
- Distributions determine possible directions of motion
- Controllability determines which states can be reached
Let $g_1(x), \dots, g_m(x)$ be linearly independent vector fields on $M$
$$ \Delta=\operatorname{span}\{g_{1}, \dots, g_{m}\} $$$$ \forall f(x), g(x) \in \Delta(x), \quad[f(x), g(x)] \in \Delta(x) $$A distribution $\Delta$ is regular if the dimension of $\Delta(x)$ does not vary with $x$
$$ L_{f} h_{i}=\frac{\partial h_{i}}{\partial x} f(x)=0, \quad i=1, \dots, n-k $$$$ \left\{q: h_{1}(x)=c_{1}, \dots, h_{n-k}(x)=c_{n-k}\right\} $$Frobenius Theorem: A regular distribution is integrable iff it is involutive
Controllability
reachable sets
$$ \begin{equation}\label{ncs} \quad \dot{x}=g_{0}(x)+\sum_{i=1}^{m} g_{i}(x) u_{i} \end{equation} $$A system is controllable if $\forall x_0, x_f \in \mathbb{R}^n$ there exists a time $T$ and $u:[0, T] \to U$ s.t $\eqref{ncs}$ satisfies $x(0)=x_0$ and $x(T)=x_f$
A system is small-time locally controllable (STLC) at $x_0$ if it can reach nearby points in arbitrary small times and stay near $x_0$
The reachable set $\mathcal{R}^V(x_0, T)$ is the set of states $x(T)$ for which there is a control $u:[0, T]\to U$ that steers the system from $x(0)$ to $x(T)$ without leaving an open set $V$ around $x_0$
$$ \mathcal{R}^{V}(x_{0}, \le T)=\bigcup_{0<\tau \le T} \mathcal{R}^{V}(x_{0}, \tau) $$controllability conditions
$$ \Omega \subset \mathcal{R}^{V}(x_{0}, \le T) $$NCS is STLC if for all neighborhood $V$ of $x_0$ and $T >0$, $\mathcal{R}^{V}(x_{0}, T)$ contains a neighborhood of $x_0$
STLC $\implies$ controllable $\implies$ LA
$$ \bar{\Delta}=\operatorname{span}\left\{v \in \bigcup_{k \ge 0} \Delta^{k}\right\} \text { with }\begin{cases} \Delta^{0}=\operatorname{span}\{g_{0}, g_{1}, \dots, g_{m}\} \\ \Delta^{k}=\Delta^{k-1}+\operatorname{span}\left\{[g_{j}, v],\quad j=0, \dots, m;\ \ v \in \Delta^{k-1}\right\} \end{cases} $$For driftless control systems ($g_0 = 0$), STLC $\iff$ controllable $\iff$ LA. The equivalence also holds when $g_0(x) \in \operatorname{span}\{g_{1}, \ldots, g_{m}\}$ (trivial drift)
$$ \dot{q} = \sum_{i=1}^m g_i(q) v_i $$$$ \begin{align*} &\dot{q}=\sum_{i=1}^{m} g_{i}(q) v_{i} \\ &\dot{v}_{i}=u_{i}, \quad i=1, \dots, m \end{align*} $$with state $x=(q,v)$ and controls $u$ is also controllable (vice versa)
$$ \dim \Delta^{k+1}=\dim \Delta^{k} $$- Completely nonholonomic: $\dim \Delta^k = n$
- Partially nonholonomic: $m < \dim \Delta^k < n$
- Holonomic: $\dim \Delta^k = m = n-k$
good and bad brackets
A bad bracket is a Lie bracket generated using an odd number of $g_0$ and even number of $g_i$ vectors. A good bracket is one that is not bag.
NCS $\eqref{ncs}$ is STLC at $x^*$ if
- $g_0(x^*) = 0$
- $U$ is open and its convex hull contains $0$
- LARC is satisfied using brackets of degree $k$
- Any bad bracket of degree $j \le k$ can be expressed as linear combinations of good brackets of degree $< j$
Stabilizability and Chained Forms
$$ \begin{equation}\label{ncs_simple} \dot{x} = f(x,u) \end{equation} $$the goal is to construct a control law $u=K(x)$ s.t.
- Stabilization: an equilibrium point $x_e$ is made asymptotically stable, or
- Tracking: a desired feasible trajectory $x_d(t)$ is asymptotically stable
The NCS can be locally smoothly stabilized at $x_e$ using $\delta u = K\delta x$ if the linearized system is controllable
$$ f: \mathbb{R}^{n} \times U \to \mathbb{R}^{n} $$contains some neighborhood of $x_e$
Nonholonomic mechanical systems cannot be stabilized at a point by smooth feedback, alternatives are
- Time-varying feedback $u = K(t,x)$
- Non-smooth feedback
Trajectory generation
Differential flatness
$$ y = h(x, u, \dot{u}, \dots, u^{(a)}) $$$$ \begin{gather*} x=\varphi(y, \dot{y}, \dots, y^{(b)}) \\ u=\alpha(y, \dot{y}, \dots, y^{(c)}) \end{gather*} $$The coordinates $y$ are called flat outputs
Trajectory generation
$$ \begin{gather*} \dot{x} = f(x,u) \\ x(0) = x_0,\quad x(T) = x_f \end{gather*} $$$$ \begin{align*} x(0) &=\varphi\big(y(0), \dot{y}(0), \dots, y^{(b)}(0)\big) \\ x(T) &=\varphi\big(y(T), \dot{y}(T), \dots, y^{(b)}(T)\big) \end{align*} $$$$ y(t) = A\lambda(t) $$$$ Y = A \Lambda $$$$ \begin{gather*} Y = \left[y(0), \dots, y^{(b)}(0), y(T), \dots, y^{(b)}(T)\right] \\ \Lambda = \left[\lambda(0), \dots, \lambda^{(b)}(0), \lambda(T), \dots, \lambda^{(b)}(T)\right] \end{gather*} $$It is necessary that $N\ge 2(b+1)$
Feedback Linearization
Input-output linearization
$$ \begin{align*} \dot{x} &=f(x)+G(x) u \\ y &=h(x) \end{align*} $$Input-output linearization is to use transformation of $u$ so that the input-output response between $v$ and $y$ is linear.
$$ u=a(x) + B(x)v $$where $B(x)$ is a nonsingular matrix, and $v$ is called virtual input
$$ \begin{gather*} u=a(x, \xi)+B(x, \xi) v \\ \dot{\xi}=c(x, \xi)+D(x, \xi) v \end{gather*} $$where $\xi$ is the compensator state
Static feedback
fully-actuated manipulator control
$$ \begin{equation}\label{fully-actuated} M(q) \ddot{q}+b(q, \dot{q})=u \end{equation} $$$$ y=q $$$$ v=\ddot{q}_{d}-k_{d}\left(\dot{q}-\dot{q}_{d}\right)-k_{p}\left(q-q_{d}\right) $$$$ \begin{gather*} \dot{z} = Az \\ z=\begin{pmatrix} q-q_{d} \\ \dot{q}-\dot{q}_{d} \end{pmatrix} \\ A=\begin{pmatrix} 0 & I \\ -k_p I & -k_d I \end{pmatrix} \end{gather*} $$where $A$ is Hurwitz. The virtual controls are mapped back to the original input $u$ using $\eqref{fully-actuated}$.
In general, whenever $\dim(Q) = \dim(U)$ one can always use nonlinear static feedback to achieve linearization.
partial feedback linearization
$$ M(q) \ddot{q}+b(q, \dot{q})=\begin{pmatrix} 0 \\ u \end{pmatrix} $$$$ \begin{bmatrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{bmatrix} \begin{bmatrix} \ddot{q_1} \\ \ddot{q_2} \end{bmatrix} + \begin{bmatrix} b_1 \\ b_2 \end{bmatrix} = \begin{bmatrix} 0 \\ u \end{bmatrix} $$$$ \bar{M}_{22}\ddot{q}_2 + \bar{b}_2 = u $$$$ \begin{gather*} \bar{M}_{22} = M_{22} - M_{21}M_{11}^{-1}M_{12} \\ \bar{b}_2 = b_2 - M_{21}M_{11}^{-1}b_1 \end{gather*} $$The rest is similar as the fully actuated control case.
$$ \eta = \begin{bmatrix} \eta_1 \\ \eta_2 \end{bmatrix} = \begin{bmatrix} q_1 \\ \dot{q}_1 \end{bmatrix} $$$$ \begin{gather*} \dot{\eta}_1 = \eta_2 \\ \dot{\eta}_2 = -M_{11}^{-1}(M_{12}(\ddot{q}_d - k_p z_1 - k_d z_2) + b_1) \end{gather*} $$$$ \begin{equation}\label{collocated} \begin{aligned} &\dot{z} = Az & &\text{linearized} \\ &\dot{\eta} = w(t, z,\eta) & &\text{non-linearized} \end{aligned} \end{equation} $$$$ \dot{\eta} = w(t, 0, \eta) $$Suppose $w(t, 0,\eta_0) = 0$ for $t\ge 0$, i.e. $(0, \eta_0)$ is the equilibrium of the full system $\eqref{collocated}$, and $A$ is Hurwitz. Then $(0, \eta_0)$ is locally stable if $\eta_0$ is locally stable for the zero dynamics. (respectively, locally asymptotically stable, unstable)
$$ \tilde{M}_{21} \ddot{q}_1 + \tilde{b}_2 = u $$$$ \begin{gather*} \tilde{M}_{21} = M_{21} = M_{22}M_{12}^\dagger M_{11} \\ \tilde{b}_2 = b_2 - M_{22}M_{12}^\dagger b_1 \end{gather*} $$$M_{12}^\dagger = M_{12}^T(M_{12} M_{12}^T)^{-1}$ is the right pseudo-inverse of $M_{12}$
The rest is similar.
Dynamic feedback
If a control system is differentially flat then it is dynamic feedback linearizable on an open dense set, with the dynamic feedback possibly depending explicitly on time
General case
SISO system
$$ \begin{gather*} \dot{x} = f(x) + g(x) u \\ y = h(x) \end{gather*} $$$$ \dot{y} = \nabla h^T [f(x) + g(x) u] = L_f h + u L_g h $$$$ u = \frac{1}{L_g h}(-L_f h + v) \\ \dot{y} = v $$$$ \begin{gather*} L_gL_f^i h = 0 \quad (i = 0, \dots, \gamma - 2) \\ L_gL_f^{\gamma -1} h \ne 0 \end{gather*} $$$$ \begin{gather*} u = \frac{1}{L_gL_f^{\gamma-1}h}(-L_f^\gamma h + v) \\ y^{(\gamma)} = v \end{gather*} $$i.e. the output becomes a $\gamma$-order linear system. $\gamma$ is called the strict relate degree
MIMO system
$$ \begin{gather*} \dot{x}=f(x)+g_{1}(x) u_{1}+g_{2}(x) u_{2} \\ y=\begin{bmatrix} y_{1} \\ y_{2} \end{bmatrix}=\begin{bmatrix} h_{1}(x) \\ h_{2}(x) \end{bmatrix} \end{gather*} $$$$ \begin{split} \begin{bmatrix} y_1^{(\gamma_1)} \\ y_2^{(\gamma_2)} \end{bmatrix} & = \begin{bmatrix} L_{g_{1}} L_{f}^{\gamma_{1}-1} h_{1} & L_{g_{2}} L_{f}^{\gamma_{1}-1} h_{1} \\ L_{g_{1}} L_{f}^{\gamma_{2}-1} h_{2} & L_{g_{2}} L_{f}^{\gamma_{2}-1} h_{2} \end{bmatrix} \begin{bmatrix} u_1 \\ u_2 \end{bmatrix} + \begin{bmatrix} L_f^{\gamma_1}h_1 \\ L_f^{\gamma_2}h \end{bmatrix} \\ &\triangleq G(x) u + H(x) \end{split} $$$$ L_{g_{j}} L_{f}^{k} h_{i}(x)=0, \quad (j=1,2,\ \ 0 \le k \le \gamma_{i}-2, \ \ i=1,2) $$$$ \begin{gather*} u = G^{-1}(x) [H(x) + v] \\ \begin{bmatrix} y_1^{(\gamma_1)} \\ y_2^{(\gamma_2)} \end{bmatrix} = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} \end{gather*} $$normal forms
$$ \Phi(x) = \begin{bmatrix} z \\ \eta \end{bmatrix} \triangleq \begin{bmatrix} h(x) \\ L_f h \\ \vdots \\ L_f^{\gamma-1}h \\ \eta_1(x) \\ \vdots \\ \eta_{n-\gamma}(x) \end{bmatrix} $$The last $n-\gamma$ coordinates $\eta$ are chosen so that the following conditions holds:
- $\Phi(x)$ is a diffeomorphism, i.e. a smooth map with smooth inverse. Equivalent to $\partial \Phi$ has full rank
- The dynamics of $\dot{\eta}$ is not directly affected by $u$, i.e. $L_g\eta_i = 0$. It means that $\eta$ are the internal dynamics $\dot{\eta} = w(t,z,\eta)$
is A.S. then the system is minimum phase, otherwise it is non-minimum phase
Backstepping
Backstepping is a nonlinear control design tool for underactuated systems.
Integrator Backstepping
$$ \begin{align} \dot{\eta} &= f(\eta) + g(\eta)\xi \label{integrator} \\ \dot{\xi} &= u \nonumber \end{align} $$where $[\eta^T, \xi]\in\mathbb{R}^{n+1}$ and $u\in\mathbb{R}$ is the control input. The functions $f,g$ are smooth in the domain that contains $0$ and $f(0)=0$. The goal is to design a controller which stabilizes the origin $(\eta, \xi)=(0,0)$
$$ \frac{\partial V_{0}}{\partial \eta}[f(\eta)+g(\eta) \phi(\eta)] \le-W(\eta) \quad (\forall \eta \in D) $$where $W(\eta)$ is P.D. Now using
$$ \require{mathtools} \DeclarePairedDelimiters\norm{\lVert}{\rVert} V(\eta, \xi) = V_0(\eta) + \frac{1}{2} \norm{\xi-\phi(\eta)}^2 $$$$ \begin{split} \dot{V} &=\frac{\partial V_{0}}{\partial \eta}(f+g\xi) + (\xi-\phi) (u - \frac{\partial \phi}{\partial \eta} \dot{\eta}) \\ & \le -W(\eta) + (\xi-\phi) (u - \frac{\partial \phi}{\partial \eta} \dot{\eta} + \frac{\partial V_{0}}{\partial \eta} g) \end{split} $$$$ u=\frac{\partial \phi}{\partial \eta}(f+g\xi)-\frac{\partial V_{0}}{\partial \eta} g-k(\xi-\phi) \quad (k> 0) $$then $\dot{V} < 0$. So that the origin $(\eta, z)=(0,0)$ is A.S. From $\phi(0)=0$ we can get the $(\eta,\xi)=(0,0)$ is A.S.
Block backstepping
$$ \begin{align} \dot{\eta}&=f(\eta)+G(\eta) \xi \label{block} \\ \dot{\xi}&=f_{a}(\eta, \xi)+G_{a}(\eta, \xi) u \nonumber \end{align} $$where $\eta\in\mathbb{R}^n$, $\xi\in\mathbb{R}^m$ and $u\in\mathbb{R}^m$ is the control input. The functions $f, f_a, G, G_a$ are smooth in the interested domain, $f(0) = f_a(0) = 0$ and $G_a$ is a non-singular $m\times m$ matrix.
$$ \frac{\partial V_{0}}{\partial \eta}[f(\eta)+G(\eta) \phi(\eta)] \le -W(\eta) \quad (\forall \eta \in D) $$$$ V(\eta, \xi) = V_0(\eta) + \frac{1}{2} \norm{\xi-\phi(\eta)}^2 $$$$ \begin{split} \dot{V}&=\frac{\partial V_{0}}{\partial \eta}(f+G \xi)+[\xi-\phi]^{T}\left[f_{a}+G_{a} u-\frac{\partial \phi}{\partial \eta}\dot{\eta}\right] \\ &\le -W(\eta) + [\xi-\phi]^{T}\left[f_{a}+G_{a} u-\frac{\partial \phi}{\partial \eta}\dot{\eta} + \left(\frac{\partial V_{0}}{\partial \eta} G\right)^T \right] \end{split} $$$$ u=G_{a}^{-1}\left[\frac{\partial \phi}{\partial \eta}(f+G \xi) - \left(\frac{\partial V_{0}}{\partial \eta} G\right)^{T} - f_a - k(\xi-\phi)\right] \quad (k>0) $$then $\dot{V} < 0$. So that the origin $(\eta, \xi)=(0,0)$ is A.S.
Lyapunov Redesign and Robust Backstepping
Uncertainty and Lyapunov redesign
$$ \begin{equation}\label{uncertain} \dot{x} = f(t,x) + G(t,x) [u + \delta(t,x,u)] \end{equation} $$where $x\in\mathbb{R}^n$ is the state and $u\in\mathbb{R}^p$ is the control. The functions $f,G,\delta$ are defined for $(x,u)\in D\times \mathbb{R}^p$ ($D$ contains the origin), piecewise continuous and Lipschitz in $x$ and $u$. Assume $f,G$ is know while $\delta$ is unknown.
When the uncertainty acts only along the control vector fields (the column of the matrix $G$) it is said to satisfy the matching condition, i.e. it matches the controls. $\eqref{uncertain}$ is in such form. Stabilizing controls for this case can be done by Lyapunov redesign. In the non-matching case, it is necessary to assume more restrictive assumptions about the bounds of $\delta$ and employ recursive techniques such as robust backstepping.
$$ \dot{x} = f(t,x) + G(t,x)u $$$$ \begin{gather*} \alpha_{1}(\norm{x}) \le V(t, x) \le \alpha_{2}(\norm{x}) \\ \partial_{t} V+\partial_{x} V \cdot[f(t, x)+G(t, x) \psi(t, x)] \le -\alpha_{3}(\norm{x}) \end{gather*} $$for all $x\in D$,$\alpha_i$ are strictly increasing and $\alpha_i(0)=0$
$$ \begin{equation}\label{bound} \norm{\delta(t, x, \psi(t, x)+v)} \le \rho(t, x)+k_{0}\norm{v} \qquad (0 \le k_{0}<1) \end{equation} $$where $\rho:[0,t_f]\times D\to \mathbb{R}$ is a non-negative continuous function and specifies the magnitude of the uncertainty. The idea behind Lyapunov redesign is to augment the nominal control law $\psi(t,x)$ with an extra term $v\in\mathbb{R}^p$ which suppresses the uncertainty so that the combined control $u=\psi(t,x)+v$ stabilizes the real system.
$$ \begin{split} \dot{V}&=\partial_{t} V+\partial_{x} V \cdot[f+G \psi]+\partial_{x} V \cdot G[v+\delta] \\ &\le -\alpha_{3}(\norm{x})+\partial_{x} V \cdot G[v+\delta] \end{split} $$$$ \dot{V}\le -\alpha_3 \norm{x} + w^T v + w^T \delta $$$$ w^{T} v+w^{T} \delta \le w^{T} v+\norm{w} (\rho+k_{0}\norm{v}) $$$$ \begin{gather*} v = -\eta(t,x)\frac{w}{\norm{w}} \\ \eta(t,x) \ge \frac{\rho(t,x)}{1-k_0} \end{gather*} $$$$ \begin{split} w^{T} v+w^{T} \delta &\le-\eta(x)\norm{w}+\norm{w}\left(\rho+k_{0} \eta(x)\right) \\ &=\norm{w}\left(\rho-\eta\left(1-k_{0}\right)\right) \\ &\le 0 \end{split} $$So that $\dot{V} \le 0$ for the whole system.
$$ \begin{align*} v &=-\eta(t, x) \frac{w}{\norm{w}} & &\text{if } \eta(t, x)\norm{w} \ge \epsilon \\ v &=-\eta(t, x)^{2} \frac{w}{\epsilon} & &\text{if } \eta(t, x)\norm{w}<\epsilon \end{align*} $$Robust backstepping
$$ \begin{align} \dot{\eta}&=f(\eta)+g(\eta) \xi+\delta_{\eta}(\eta, \xi) \label{robust-bs} \\ \dot{\xi}&=f_{a}(\eta, \xi)+g_{a}(\eta, \xi) u+\delta_{\xi}(\eta, \xi) \nonumber \end{align} $$$$ \begin{equation}\label{assume-1} \begin{aligned} \norm{\delta_{\eta}(\eta, \xi)}_{2} &\le a_{1}\norm{\eta}_{2} \\ |\delta_{\xi}(\eta, \xi)| &\le a_{2}\norm{\eta}_{2}+a_{3}|\xi| \end{aligned} \end{equation} $$for all $(\eta, \xi)$
$$ \frac{\partial V_{0}}{\partial \eta}\left[f(\eta)+g(\eta) \phi(\eta)+\delta_{\eta}(\eta, \xi)\right] \le -b\norm{\eta}^{2} $$$$ \begin{equation}\label{assume-2} |\phi(\eta)| \le a_{4}\norm{\eta} \qquad\norm*{\frac{\partial \phi}{\partial \eta}} \le a_{5} \end{equation} $$$$ V(\eta, \xi) = V_0(\eta) +\frac{1}{2}[\xi - \phi(\eta)]^2 $$$$ \dot{V}=\frac{\partial V_{0}}{\partial \eta}\left[f+g \phi+\delta_{\eta}\right]+\frac{\partial V_{0}}{\partial \eta} g(\xi-\phi)+(\xi-\phi)\left[f_{a}+g_{a} u+\delta_{\xi}-\frac{\partial \phi}{\partial \eta}\left(f+g \xi+\delta_{\eta}\right)\right] $$$$ u=\frac{1}{g_{a}}\left[\frac{\partial \phi}{\partial \eta}(f+g \xi)-\frac{\partial V_{0}}{\partial \eta} g-f_{a}-k(\xi-\phi)\right] \quad (k>0) $$$$ \dot{V} \le -b\norm{\eta}^{2}+(\xi-\phi)\left[\delta_{\xi}-\frac{\partial \phi}{\partial \eta} \delta_{\eta}\right]-k(\xi-\phi)^{2} $$$$ \begin{split} \dot{V} & \le-b\norm{\eta}^{2}+2 a_{6}|\xi-\phi|\norm{\eta}-\left(k-a_{3}\right)|\xi-\phi|^{2} \\ &=-\begin{bmatrix} \norm{\eta} \\ |\xi-\phi| \end{bmatrix}^T\begin{bmatrix} b & -a_{6} \\ -a_{6} & (k-a_{3}) \end{bmatrix}\begin{bmatrix} \norm{\eta} \\ |\xi-\phi| \end{bmatrix} \end{split} $$for some $a_6 > 0$. Taking $k\ge a_3 + a_6^2/b$ yields $\dot{V} \le 0$