Backstepping

Consider the dynamical system

${\displaystyle {\begin{cases}{\dot {\mathbf {x} }}=f_{x}(\mathbf {x} )+g_{x}(\mathbf {x} )z_{1}\\{\dot {z}}_{1}=u_{1}\end{cases}}}$

(1)

where ${\displaystyle \mathbf {x} \in \mathbb {R} ^{n}}$ and ${\displaystyle z_{1}}$ is a scalar. This system is a cascade connection of an integrator with the x subsystem (i.e., the input u enters an integrator, and the integral ${\displaystyle z_{1}}$ enters the x subsystem).

We assume that ${\displaystyle f_{x}(\mathbf {0} )=0}$, and so if ${\displaystyle u_{1}=0}$, ${\displaystyle \mathbf {x} =\mathbf {0} \,}$ and ${\displaystyle z_{1}=0}$, then

${\displaystyle {\begin{cases}{\dot {\mathbf {x} }}=f_{x}(\underbrace {\mathbf {0} } _{\mathbf {x} })+(g_{x}(\underbrace {\mathbf {0} } _{\mathbf {x} }))(\underbrace {0} _{z_{1}})=0+(g_{x}(\mathbf {0} ))(0)=\mathbf {0} &\quad {\text{ (i.e., }}\mathbf {x} =\mathbf {0} {\text{ is stationary)}}\\{\dot {z}}_{1}=\overbrace {0} ^{u_{1}}&\quad {\text{ (i.e., }}z_{1}=0{\text{ is stationary)}}\end{cases}}}$

So the origin ${\displaystyle (\mathbf {x} ,z_{1})=(\mathbf {0} ,0)}$ is an equilibrium (i.e., a stationary point) of the system. If the system ever reaches the origin, it will remain there forever after.

In this example, backstepping is used to stabilize the single-integrator system in Equation (1) around its equilibrium at the origin. To be less precise, we wish to design a control law ${\displaystyle u_{1}(\mathbf {x} ,z_{1})}$ that ensures that the states ${\displaystyle (\mathbf {x} ,z_{1})}$ return to ${\displaystyle (\mathbf {0} ,0)}$ after the system is started from some arbitrary initial condition.

• First, by assumption, the subsystem

${\displaystyle {\dot {\mathbf {x} }}=F(\mathbf {x} )\qquad {\text{where}}\qquad F(\mathbf {x} )\triangleq f_{x}(\mathbf {x} )+g_{x}(\mathbf {x} )u_{x}(\mathbf {x} )}$

with ${\displaystyle u_{x}(\mathbf {0} )=0}$ has a Lyapunov function ${\displaystyle V_{x}(\mathbf {x} )>0}$ such that

${\displaystyle {\dot {V}}_{x}={\frac {\partial V_{x}}{\partial \mathbf {x} }}(f_{x}(\mathbf {x} )+g_{x}(\mathbf {x} )u_{x}(\mathbf {x} ))\leq -W(\mathbf {x} )}$

where ${\displaystyle W(\mathbf {x} )}$ is a positive-definite function. That is, we assume that we have already shown that this existing simpler x subsystem is stable (in the sense of Lyapunov). Roughly speaking, this notion of stability means that:

• • The function ${\displaystyle V_{x}}$ is like a "generalized energy" of the x subsystem. As the x states of the system move away from the origin, the energy ${\displaystyle V_{x}(\mathbf {x} )}$ also grows.
  • By showing that over time, the energy ${\displaystyle V_{x}(\mathbf {x} (t))}$ decays to zero, then the x states must decay toward ${\displaystyle \mathbf {x} =\mathbf {0} \,}$. That is, the origin ${\displaystyle \mathbf {x} =\mathbf {0} \,}$ will be a stable equilibrium of the system – the x states will continuously approach the origin as time increases.
  • Saying that ${\displaystyle W(\mathbf {x} )}$ is positive definite means that ${\displaystyle W(\mathbf {x} )>0}$ everywhere except for ${\displaystyle \mathbf {x} =\mathbf {0} \,}$, and ${\displaystyle W(\mathbf {0} )=0}$.
  • The statement that ${\displaystyle {\dot {V}}_{x}\leq -W(\mathbf {x} )}$ means that ${\displaystyle {\dot {V}}_{x}}$ is bounded away from zero for all points except where ${\displaystyle \mathbf {x} =\mathbf {0} \,}$. That is, so long as the system is not at its equilibrium at the origin, its "energy" will be decreasing.
  • Because the energy is always decaying, then the system must be stable; its trajectories must approach the origin.

Our task is to find a control u that makes our cascaded ${\displaystyle (\mathbf {x} ,z_{1})}$ system also stable. So we must find a new Lyapunov function candidate for this new system. That candidate will depend upon the control u, and by choosing the control properly, we can ensure that it is decaying everywhere as well.

• Next, by adding and subtracting ${\displaystyle g_{x}(\mathbf {x} )u_{x}(\mathbf {x} )}$ (i.e., we don't change the system in any way because we make no net effect) to the ${\displaystyle {\dot {\mathbf {x} }}}$ part of the larger ${\displaystyle (\mathbf {x} ,z_{1})}$ system, it becomes

${\displaystyle {\begin{cases}{\dot {\mathbf {x} }}=f_{x}(\mathbf {x} )+g_{x}(\mathbf {x} )z_{1}+{\mathord {\underbrace {\left(g_{x}(\mathbf {x} )u_{x}(\mathbf {x} )-g_{x}(\mathbf {x} )u_{x}(\mathbf {x} )\right)} _{0}}}\\{\dot {z}}_{1}=u_{1}\end{cases}}}$

which we can re-group to get

${\displaystyle {\begin{cases}{\dot {x}}={\mathord {\underbrace {\left(f_{x}(\mathbf {x} )+g_{x}(\mathbf {x} )u_{x}(\mathbf {x} )\right)} _{F(\mathbf {x} )}}}+g_{x}(\mathbf {x} )\underbrace {\left(z_{1}-u_{x}(\mathbf {x} )\right)} _{z_{1}{\text{ error tracking }}u_{x}}\\{\dot {z}}_{1}=u_{1}\end{cases}}}$

So our cascaded supersystem encapsulates the known-stable ${\displaystyle {\dot {\mathbf {x} }}=F(\mathbf {x} )}$ subsystem plus some error perturbation generated by the integrator.

• We now can change variables from ${\displaystyle (\mathbf {x} ,z_{1})}$ to ${\displaystyle (\mathbf {x} ,e_{1})}$ by letting ${\displaystyle e_{1}\triangleq z_{1}-u_{x}(\mathbf {x} )}$. So

${\displaystyle {\begin{cases}{\dot {\mathbf {x} }}=(f_{x}(\mathbf {x} )+g_{x}(\mathbf {x} )u_{x}(\mathbf {x} ))+g_{x}(\mathbf {x} )e_{1}\\{\dot {e}}_{1}=u_{1}-{\dot {u}}_{x}\end{cases}}}$

Additionally, we let ${\displaystyle v_{1}\triangleq u_{1}-{\dot {u}}_{x}}$ so that ${\displaystyle u_{1}=v_{1}+{\dot {u}}_{x}}$ and

${\displaystyle {\begin{cases}{\dot {\mathbf {x} }}=(f_{x}(\mathbf {x} )+g_{x}(\mathbf {x} )u_{x}(\mathbf {x} ))+g_{x}(\mathbf {x} )e_{1}\\{\dot {e}}_{1}=v_{1}\end{cases}}}$

We seek to stabilize this error system by feedback through the new control ${\displaystyle v_{1}}$. By stabilizing the system at ${\displaystyle e_{1}=0}$, the state ${\displaystyle z_{1}}$ will track the desired control ${\displaystyle u_{x}}$ which will result in stabilizing the inner x subsystem.

• From our existing Lyapunov function ${\displaystyle V_{x}}$, we define the augmented Lyapunov function candidate

${\displaystyle V_{1}(\mathbf {x} ,e_{1})\triangleq V_{x}(\mathbf {x} )+{\frac {1}{2}}e_{1}^{2}}$

So

${\displaystyle {\begin{aligned}{\dot {V}}_{1}&={\dot {V}}_{x}(\mathbf {x} )+{\frac {1}{2}}\left(2e_{1}{\dot {e}}_{1}\right)\\&={\dot {V}}_{x}(\mathbf {x} )+e_{1}{\dot {e}}_{1}\\&={\dot {V}}_{x}(\mathbf {x} )+e_{1}\overbrace {v_{1}} ^{{\dot {e}}_{1}}\\&=\overbrace {{\frac {\partial V_{x}}{\partial \mathbf {x} }}\underbrace {\dot {\mathbf {x} }} _{{\text{(i.e., }}{\frac {\operatorname {d} \mathbf {x} }{\operatorname {d} t}}{\text{)}}}} ^{{\dot {V}}_{x}{\text{ (i.e.,}}{\frac {\operatorname {d} V_{x}}{\operatorname {d} t}}{\text{)}}}+e_{1}v_{1}\\&=\overbrace {{\frac {\partial V_{x}}{\partial \mathbf {x} }}\underbrace {\left((f_{x}(\mathbf {x} )+g_{x}(\mathbf {x} )u_{x}(\mathbf {x} ))+g_{x}(\mathbf {x} )e_{1}\right)} _{\dot {\mathbf {x} }}} ^{{\dot {V}}_{x}}+e_{1}v_{1}\end{aligned}}}$

By distributing ${\displaystyle \partial V_{x}/\partial \mathbf {x} }$, we see that

${\displaystyle {\dot {V}}_{1}=\overbrace {{\frac {\partial V_{x}}{\partial \mathbf {x} }}(f_{x}(\mathbf {x} )+g_{x}(\mathbf {x} )u_{x}(\mathbf {x} ))} ^{{}\leq -W(\mathbf {x} )}+{\frac {\partial V_{x}}{\partial \mathbf {x} }}g_{x}(\mathbf {x} )e_{1}+e_{1}v_{1}\leq -W(\mathbf {x} )+{\frac {\partial V_{x}}{\partial \mathbf {x} }}g_{x}(\mathbf {x} )e_{1}+e_{1}v_{1}}$

To ensure that ${\displaystyle {\dot {V}}_{1}\leq -W(\mathbf {x} )<0}$ (i.e., to ensure stability of the supersystem), we pick the control law

${\displaystyle v_{1}=-{\frac {\partial V_{x}}{\partial \mathbf {x} }}g_{x}(\mathbf {x} )-k_{1}e_{1}}$

with ${\displaystyle k_{1}>0}$, and so

${\displaystyle {\dot {V}}_{1}=-W(\mathbf {x} )+{\frac {\partial V_{x}}{\partial \mathbf {x} }}g_{x}(\mathbf {x} )e_{1}+e_{1}\overbrace {\left(-{\frac {\partial V_{x}}{\partial \mathbf {x} }}g_{x}(\mathbf {x} )-k_{1}e_{1}\right)} ^{v_{1}}}$

After distributing the ${\displaystyle e_{1}}$ through,

${\displaystyle {\begin{aligned}{\dot {V}}_{1}&=-W(\mathbf {x} )+{\mathord {\overbrace {{\frac {\partial V_{x}}{\partial \mathbf {x} }}g_{x}(\mathbf {x} )e_{1}-e_{1}{\frac {\partial V_{x}}{\partial \mathbf {x} }}g_{x}(\mathbf {x} )} ^{0}}}-k_{1}e_{1}^{2}\\&=-W(\mathbf {x} )-k_{1}e_{1}^{2}\leq -W(\mathbf {x} )\\&<0\end{aligned}}}$

So our candidate Lyapunov function ${\displaystyle V_{1}}$ is a true Lyapunov function, and our system is stable under this control law ${\displaystyle v_{1}}$ (which corresponds the control law ${\displaystyle u_{1}}$ because ${\displaystyle v_{1}\triangleq u_{1}-{\dot {u}}_{x}}$). Using the variables from the original coordinate system, the equivalent Lyapunov function

${\displaystyle V_{1}(\mathbf {x} ,z_{1})\triangleq V_{x}(\mathbf {x} )+{\frac {1}{2}}(z_{1}-u_{x}(\mathbf {x} ))^{2}}$

(2)

As discussed below, this Lyapunov function will be used again when this procedure is applied iteratively to multiple-integrator problem.

• Our choice of control ${\displaystyle v_{1}}$ ultimately depends on all of our original state variables. In particular, the actual feedback-stabilizing control law

${\displaystyle \underbrace {u_{1}(\mathbf {x} ,z_{1})=v_{1}+{\dot {u}}_{x}} _{{\text{By definition of }}v_{1}}=\overbrace {-{\frac {\partial V_{x}}{\partial \mathbf {x} }}g_{x}(\mathbf {x} )-k_{1}(\underbrace {z_{1}-u_{x}(\mathbf {x} )} _{e_{1}})} ^{v_{1}}\,+\,\overbrace {{\frac {\partial u_{x}}{\partial \mathbf {x} }}(\underbrace {f_{x}(\mathbf {x} )+g_{x}(\mathbf {x} )z_{1}} _{{\dot {\mathbf {x} }}{\text{ (i.e., }}{\frac {\operatorname {d} \mathbf {x} }{\operatorname {d} t}}{\text{)}}})} ^{{\dot {u}}_{x}{\text{ (i.e., }}{\frac {\operatorname {d} u_{x}}{\operatorname {d} t}}{\text{)}}}}$

(3)

The states x and ${\displaystyle z_{1}}$ and functions ${\displaystyle f_{x}}$ and ${\displaystyle g_{x}}$ come from the system. The function ${\displaystyle u_{x}}$ comes from our known-stable ${\displaystyle {\dot {\mathbf {x} }}=F(\mathbf {x} )}$ subsystem. The gain parameter ${\displaystyle k_{1}>0}$ affects the convergence rate or our system. Under this control law, our system is stable at the origin ${\displaystyle (\mathbf {x} ,z_{1})=(\mathbf {0} ,0)}$.

Recall that ${\displaystyle u_{1}}$ in Equation (3) drives the input of an integrator that is connected to a subsystem that is feedback-stabilized by the control law ${\displaystyle u_{x}}$. Not surprisingly, the control ${\displaystyle u_{1}}$ has a ${\displaystyle {\dot {u}}_{x}}$ term that will be integrated to follow the stabilizing control law ${\displaystyle {\dot {u}}_{x}}$ plus some offset. The other terms provide damping to remove that offset and any other perturbation effects that would be magnified by the integrator.

So because this system is feedback stabilized by ${\displaystyle u_{1}(\mathbf {x} ,z_{1})}$ and has Lyapunov function ${\displaystyle V_{1}(\mathbf {x} ,z_{1})}$ with ${\displaystyle {\dot {V}}_{1}(\mathbf {x} ,z_{1})\leq -W(\mathbf {x} )<0}$, it can be used as the upper subsystem in another single-integrator cascade system.

Before discussing the recursive procedure for the general multiple-integrator case, it is instructive to study the recursion present in the two-integrator case. That is, consider the dynamical system

${\displaystyle {\begin{cases}{\dot {\mathbf {x} }}=f_{x}(\mathbf {x} )+g_{x}(\mathbf {x} )z_{1}\\{\dot {z}}_{1}=z_{2}\\{\dot {z}}_{2}=u_{2}\end{cases}}}$

(4)

where ${\displaystyle \mathbf {x} \in \mathbb {R} ^{n}}$ and ${\displaystyle z_{1}}$ and ${\displaystyle z_{2}}$ are scalars. This system is a cascade connection of the single-integrator system in Equation (1) with another integrator (i.e., the input ${\displaystyle u_{2}}$ enters through an integrator, and the output of that integrator enters the system in Equation (1) by its ${\displaystyle u_{1}}$ input).

By letting

• ${\displaystyle \mathbf {y} \triangleq {\begin{bmatrix}\mathbf {x} \\z_{1}\end{bmatrix}}\,}$,
• ${\displaystyle f_{y}(\mathbf {y} )\triangleq {\begin{bmatrix}f_{x}(\mathbf {x} )+g_{x}(\mathbf {x} )z_{1}\\0\end{bmatrix}}\,}$,
• ${\displaystyle g_{y}(\mathbf {y} )\triangleq {\begin{bmatrix}\mathbf {0} \\1\end{bmatrix}},\,}$

then the two-integrator system in Equation (4) becomes the single-integrator system

${\displaystyle {\begin{cases}{\dot {\mathbf {y} }}=f_{y}(\mathbf {y} )+g_{y}(\mathbf {y} )z_{2}&\quad {\text{( where this }}\mathbf {y} {\text{ subsystem is stabilized by }}z_{2}=u_{1}(\mathbf {x} ,z_{1}){\text{ )}}\\{\dot {z}}_{2}=u_{2}.\end{cases}}}$

(5)

By the single-integrator procedure, the control law ${\displaystyle u_{y}(\mathbf {y} )\triangleq u_{1}(\mathbf {x} ,z_{1})}$ stabilizes the upper ${\displaystyle z_{2}}$-to-y subsystem using the Lyapunov function ${\displaystyle V_{1}(\mathbf {x} ,z_{1})}$, and so Equation (5) is a new single-integrator system that is structurally equivalent to the single-integrator system in Equation (1). So a stabilizing control ${\displaystyle u_{2}}$ can be found using the same single-integrator procedure that was used to find ${\displaystyle u_{1}}$.

In the two-integrator case, the upper single-integrator subsystem was stabilized yielding a new single-integrator system that can be similarly stabilized. This recursive procedure can be extended to handle any finite number of integrators. This claim can be formally proved with mathematical induction. Here, a stabilized multiple-integrator system is built up from subsystems of already-stabilized multiple-integrator subsystems.

• First, consider the dynamical system

${\displaystyle {\dot {\mathbf {x} }}=f_{x}(\mathbf {x} )+g_{x}(\mathbf {x} )u_{x}}$

that has scalar input ${\displaystyle u_{x}}$ and output states ${\displaystyle \mathbf {x} =[x_{1},x_{2},\ldots ,x_{n}]^{\text{T}}\in \mathbb {R} ^{n}}$. Assume that

• • ${\displaystyle f_{x}(\mathbf {x} )=\mathbf {0} }$ so that the zero-input (i.e., ${\displaystyle u_{x}=0}$) system is stationary at the origin ${\displaystyle \mathbf {x} =\mathbf {0} \,}$. In this case, the origin is called an equilibrium of the system.
  • The feedback control law ${\displaystyle u_{x}(\mathbf {x} )}$ stabilizes the system at the equilibrium at the origin.
  • A Lyapunov function corresponding to this system is described by ${\displaystyle V_{x}(\mathbf {x} )}$.

That is, if output states x are fed back to the input ${\displaystyle u_{x}}$ by the control law ${\displaystyle u_{x}(\mathbf {x} )}$, then the output states (and the Lyapunov function) return to the origin after a single perturbation (e.g., after a nonzero initial condition or a sharp disturbance). This subsystem is stabilized by feedback control law ${\displaystyle u_{x}}$.

• Next, connect an integrator to input ${\displaystyle u_{x}}$ so that the augmented system has input ${\displaystyle u_{1}}$ (to the integrator) and output states x. The resulting augmented dynamical system is

${\displaystyle {\begin{cases}{\dot {\mathbf {x} }}=f_{x}(\mathbf {x} )+g_{x}(\mathbf {x} )z_{1}\\{\dot {z}}_{1}=u_{1}\end{cases}}}$

This "cascade" system matches the form in Equation (1), and so the single-integrator backstepping procedure leads to the stabilizing control law in Equation (3). That is, if we feed back states ${\displaystyle z_{1}}$ and x to input ${\displaystyle u_{1}}$ according to the control law

${\displaystyle u_{1}(\mathbf {x} ,z_{1})=-{\frac {\partial V_{x}}{\partial \mathbf {x} }}g_{x}(\mathbf {x} )-k_{1}(z_{1}-u_{x}(\mathbf {x} ))+{\frac {\partial u_{x}}{\partial \mathbf {x} }}(f_{x}(\mathbf {x} )+g_{x}(\mathbf {x} )z_{1})}$

with gain ${\displaystyle k_{1}>0}$, then the states ${\displaystyle z_{1}}$ and x will return to ${\displaystyle z_{1}=0}$ and ${\displaystyle \mathbf {x} =\mathbf {0} \,}$ after a single perturbation. This subsystem is stabilized by feedback control law ${\displaystyle u_{1}}$, and the corresponding Lyapunov function from Equation (2) is

${\displaystyle V_{1}(\mathbf {x} ,z_{1})=V_{x}(\mathbf {x} )+{\frac {1}{2}}(z_{1}-u_{x}(\mathbf {x} ))^{2}}$

That is, under feedback control law ${\displaystyle u_{1}}$, the Lyapunov function ${\displaystyle V_{1}}$ decays to zero as the states return to the origin.

• Connect a new integrator to input ${\displaystyle u_{1}}$ so that the augmented system has input ${\displaystyle u_{2}}$ and output states x. The resulting augmented dynamical system is

${\displaystyle {\begin{cases}{\dot {\mathbf {x} }}=f_{x}(\mathbf {x} )+g_{x}(\mathbf {x} )z_{1}\\{\dot {z}}_{1}=z_{2}\\{\dot {z}}_{2}=u_{2}\end{cases}}}$

which is equivalent to the single-integrator system

${\displaystyle {\begin{cases}\overbrace {\begin{bmatrix}{\dot {\mathbf {x} }}\\{\dot {z}}_{1}\end{bmatrix}} ^{\triangleq \,{\dot {\mathbf {x} }}_{1}}=\overbrace {\begin{bmatrix}f_{x}(\mathbf {x} )+g_{x}(\mathbf {x} )z_{1}\\0\end{bmatrix}} ^{\triangleq \,f_{1}(\mathbf {x} _{1})}+\overbrace {\begin{bmatrix}\mathbf {0} \\1\end{bmatrix}} ^{\triangleq \,g_{1}(\mathbf {x} _{1})}z_{2}&\qquad {\text{ ( by Lyapunov function }}V_{1},{\text{ subsystem stabilized by }}u_{1}({\textbf {x}}_{1}){\text{ )}}\\{\dot {z}}_{2}=u_{2}\end{cases}}}$

Using these definitions of ${\displaystyle \mathbf {x} _{1}}$, ${\displaystyle f_{1}}$, and ${\displaystyle g_{1}}$, this system can also be expressed as

${\displaystyle {\begin{cases}{\dot {\mathbf {x} }}_{1}=f_{1}(\mathbf {x} _{1})+g_{1}(\mathbf {x} _{1})z_{2}&\qquad {\text{ ( by Lyapunov function }}V_{1},{\text{ subsystem stabilized by }}u_{1}({\textbf {x}}_{1}){\text{ )}}\\{\dot {z}}_{2}=u_{2}\end{cases}}}$

This system matches the single-integrator structure of Equation (1), and so the single-integrator backstepping procedure can be applied again. That is, if we feed back states ${\displaystyle z_{1}}$, ${\displaystyle z_{2}}$, and x to input ${\displaystyle u_{2}}$ according to the control law

${\displaystyle u_{2}(\mathbf {x} ,z_{1},z_{2})=-{\frac {\partial V_{1}}{\partial \mathbf {x} _{1}}}g_{1}(\mathbf {x} _{1})-k_{2}(z_{2}-u_{1}(\mathbf {x} _{1}))+{\frac {\partial u_{1}}{\partial \mathbf {x} _{1}}}(f_{1}(\mathbf {x} _{1})+g_{1}(\mathbf {x} _{1})z_{2})}$

with gain ${\displaystyle k_{2}>0}$, then the states ${\displaystyle z_{1}}$, ${\displaystyle z_{2}}$, and x will return to ${\displaystyle z_{1}=0}$, ${\displaystyle z_{2}=0}$, and ${\displaystyle \mathbf {x} =\mathbf {0} \,}$ after a single perturbation. This subsystem is stabilized by feedback control law ${\displaystyle u_{2}}$, and the corresponding Lyapunov function is

${\displaystyle V_{2}(\mathbf {x} ,z_{1},z_{2})=V_{1}(\mathbf {x} _{1})+{\frac {1}{2}}(z_{2}-u_{1}(\mathbf {x} _{1}))^{2}}$

That is, under feedback control law ${\displaystyle u_{2}}$, the Lyapunov function ${\displaystyle V_{2}}$ decays to zero as the states return to the origin.

• Connect an integrator to input ${\displaystyle u_{2}}$ so that the augmented system has input ${\displaystyle u_{3}}$ and output states x. The resulting augmented dynamical system is

${\displaystyle {\begin{cases}{\dot {\mathbf {x} }}=f_{x}(\mathbf {x} )+g_{x}(\mathbf {x} )z_{1}\\{\dot {z}}_{1}=z_{2}\\{\dot {z}}_{2}=z_{3}\\{\dot {z}}_{3}=u_{3}\end{cases}}}$

which can be re-grouped as the single-integrator system

${\displaystyle {\begin{cases}\overbrace {\begin{bmatrix}{\dot {\mathbf {x} }}\\{\dot {z}}_{1}\\{\dot {z}}_{2}\end{bmatrix}} ^{\triangleq \,{\dot {\mathbf {x} }}_{2}}=\overbrace {\begin{bmatrix}f_{x}(\mathbf {x} )+g_{x}(\mathbf {x} )z_{2}\\z_{2}\\0\end{bmatrix}} ^{\triangleq \,f_{2}(\mathbf {x} _{2})}+\overbrace {\begin{bmatrix}\mathbf {0} \\0\\1\end{bmatrix}} ^{\triangleq \,g_{2}(\mathbf {x} _{2})}z_{3}&\qquad {\text{ ( by Lyapunov function }}V_{2},{\text{ subsystem stabilized by }}u_{2}({\textbf {x}}_{2}){\text{ )}}\\{\dot {z}}_{3}=u_{3}\end{cases}}}$

By the definitions of ${\displaystyle \mathbf {x} _{1}}$, ${\displaystyle f_{1}}$, and ${\displaystyle g_{1}}$ from the previous step, this system is also represented by

${\displaystyle {\begin{cases}\overbrace {\begin{bmatrix}{\dot {\mathbf {x} }}_{1}\\{\dot {z}}_{2}\end{bmatrix}} ^{{\dot {\mathbf {x} }}_{2}}=\overbrace {\begin{bmatrix}f_{1}(\mathbf {x} _{1})+g_{1}(\mathbf {x} _{1})z_{2}\\0\end{bmatrix}} ^{f_{2}(\mathbf {x} _{2})}+\overbrace {\begin{bmatrix}\mathbf {0} \\1\end{bmatrix}} ^{g_{2}(\mathbf {x} _{2})}z_{3}&\qquad {\text{ ( by Lyapunov function }}V_{2},{\text{ subsystem stabilized by }}u_{2}({\textbf {x}}_{2}){\text{ )}}\\{\dot {z}}_{3}=u_{3}\end{cases}}}$

Further, using these definitions of ${\displaystyle \mathbf {x} _{2}}$, ${\displaystyle f_{2}}$, and ${\displaystyle g_{2}}$, this system can also be expressed as

${\displaystyle {\begin{cases}{\dot {\mathbf {x} }}_{2}=f_{2}(\mathbf {x} _{2})+g_{2}(\mathbf {x} _{2})z_{3}&\qquad {\text{ ( by Lyapunov function }}V_{2},{\text{ subsystem stabilized by }}u_{2}({\textbf {x}}_{2}){\text{ )}}\\{\dot {z}}_{3}=u_{3}\end{cases}}}$

So the re-grouped system has the single-integrator structure of Equation (1), and so the single-integrator backstepping procedure can be applied again. That is, if we feed back states ${\displaystyle z_{1}}$, ${\displaystyle z_{2}}$, ${\displaystyle z_{3}}$, and x to input ${\displaystyle u_{3}}$ according to the control law

${\displaystyle u_{3}(\mathbf {x} ,z_{1},z_{2},z_{3})=-{\frac {\partial V_{2}}{\partial \mathbf {x} _{2}}}g_{2}(\mathbf {x} _{2})-k_{3}(z_{3}-u_{2}(\mathbf {x} _{2}))+{\frac {\partial u_{2}}{\partial \mathbf {x} _{2}}}(f_{2}(\mathbf {x} _{2})+g_{2}(\mathbf {x} _{2})z_{3})}$

with gain ${\displaystyle k_{3}>0}$, then the states ${\displaystyle z_{1}}$, ${\displaystyle z_{2}}$, ${\displaystyle z_{3}}$, and x will return to ${\displaystyle z_{1}=0}$, ${\displaystyle z_{2}=0}$, ${\displaystyle z_{3}=0}$, and ${\displaystyle \mathbf {x} =\mathbf {0} \,}$ after a single perturbation. This subsystem is stabilized by feedback control law ${\displaystyle u_{3}}$, and the corresponding Lyapunov function is

${\displaystyle V_{3}(\mathbf {x} ,z_{1},z_{2},z_{3})=V_{2}(\mathbf {x} _{2})+{\frac {1}{2}}(z_{3}-u_{2}(\mathbf {x} _{2}))^{2}}$

That is, under feedback control law ${\displaystyle u_{3}}$, the Lyapunov function ${\displaystyle V_{3}}$ decays to zero as the states return to the origin.

• This process can continue for each integrator added to the system, and hence any system of the form

${\displaystyle {\begin{cases}{\dot {\mathbf {x} }}=f_{x}(\mathbf {x} )+g_{x}(\mathbf {x} )z_{1}&\qquad {\text{ ( by Lyapunov function }}V_{x},{\text{ subsystem stabilized by }}u_{x}({\textbf {x}}){\text{ )}}\\{\dot {z}}_{1}=z_{2}\\{\dot {z}}_{2}=z_{3}\\\vdots \\{\dot {z}}_{i}=z_{i+1}\\\vdots \\{\dot {z}}_{k-2}=z_{k-1}\\{\dot {z}}_{k-1}=z_{k}\\{\dot {z}}_{k}=u\end{cases}}}$

has the recursive structure

${\displaystyle {\begin{cases}{\begin{cases}{\begin{cases}{\begin{cases}{\begin{cases}{\begin{cases}{\begin{cases}{\begin{cases}{\dot {\mathbf {x} }}=f_{x}(\mathbf {x} )+g_{x}(\mathbf {x} )z_{1}&\qquad {\text{ ( by Lyapunov function }}V_{x},{\text{ subsystem stabilized by }}u_{x}({\textbf {x}}){\text{ )}}\\{\dot {z}}_{1}=z_{2}\end{cases}}\\{\dot {z}}_{2}=z_{3}\end{cases}}\\\vdots \end{cases}}\\{\dot {z}}_{i}=z_{i+1}\end{cases}}\\\vdots \end{cases}}\\{\dot {z}}_{k-2}=z_{k-1}\end{cases}}\\{\dot {z}}_{k-1}=z_{k}\end{cases}}\\{\dot {z}}_{k}=u\end{cases}}}$

and can be feedback stabilized by finding the feedback-stabilizing control and Lyapunov function for the single-integrator ${\displaystyle (\mathbf {x} ,z_{1})}$ subsystem (i.e., with input ${\displaystyle z_{2}}$ and output x) and iterating out from that inner subsystem until the ultimate feedback-stabilizing control u is known. At iteration i, the equivalent system is

${\displaystyle {\begin{cases}\overbrace {\begin{bmatrix}{\dot {\mathbf {x} }}\\{\dot {z}}_{1}\\{\dot {z}}_{2}\\\vdots \\{\dot {z}}_{i-2}\\{\dot {z}}_{i-1}\end{bmatrix}} ^{\triangleq \,{\dot {\mathbf {x} }}_{i-1}}=\overbrace {\begin{bmatrix}f_{i-2}(\mathbf {x} _{i-2})+g_{i-2}(\mathbf {x} _{i-1})z_{i-2}\\0\end{bmatrix}} ^{\triangleq \,f_{i-1}(\mathbf {x} _{i-1})}+\overbrace {\begin{bmatrix}\mathbf {0} \\1\end{bmatrix}} ^{\triangleq \,g_{i-1}(\mathbf {x} _{i-1})}z_{i}&\quad {\text{ ( by Lyap. func. }}V_{i-1},{\text{ subsystem stabilized by }}u_{i-1}({\textbf {x}}_{i-1}){\text{ )}}\\{\dot {z}}_{i}=u_{i}\end{cases}}}$

The corresponding feedback-stabilizing control law is

${\displaystyle u_{i}(\overbrace {\mathbf {x} ,z_{1},z_{2},\dots ,z_{i}} ^{\triangleq \,\mathbf {x} _{i}})=-{\frac {\partial V_{i-1}}{\partial \mathbf {x} _{i-1}}}g_{i-1}(\mathbf {x} _{i-1})\,-\,k_{i}(z_{i}\,-\,u_{i-1}(\mathbf {x} _{i-1}))\,+\,{\frac {\partial u_{i-1}}{\partial \mathbf {x} _{i-1}}}(f_{i-1}(\mathbf {x} _{i-1})\,+\,g_{i-1}(\mathbf {x} _{i-1})z_{i})}$

with gain ${\displaystyle k_{i}>0}$. The corresponding Lyapunov function is

${\displaystyle V_{i}(\mathbf {x} _{i})=V_{i-1}(\mathbf {x} _{i-1})+{\frac {1}{2}}(z_{i}-u_{i-1}(\mathbf {x} _{i-1}))^{2}}$

By this construction, the ultimate control ${\displaystyle u(\mathbf {x} ,z_{1},z_{2},\ldots ,z_{k})=u_{k}(\mathbf {x} _{k})}$ (i.e., ultimate control is found at final iteration ${\displaystyle i=k}$).