Backstepping

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In control theory, backstepping is a technique developed circa 1990 by Petar V. Kokotovic and others[1][2] for designing stabilizing controls for a special class of nonlinear dynamical systems. These systems are built from subsystems that radiate out from an irreducible subsystem that can be stabilized using some other method. Because of this recursive structure, the designer can start the design process at the known-stable system and "back out" new controllers that progressively stabilize each outer subsystem. The process terminates when the final external control is reached. Hence, this process is known as backstepping.[3]

Backstepping approach[edit]

The backstepping approach provides a recursive method for stabilizing the origin of a system in strict-feedback form. That is, consider a system of the form[3]

\begin{cases} \dot{\mathbf{x}} = f_x(\mathbf{x}) + g_x(\mathbf{x}) z_1\\
\dot{z}_1 = f_1(\mathbf{x},z_1) + g_1(\mathbf{x},z_1) z_2\\
\dot{z}_2 = f_2(\mathbf{x},z_1,z_2) + g_2(\mathbf{x},z_1,z_2) z_3\\
\vdots\\
\dot{z}_i = f_i(\mathbf{x},z_1, z_2, \ldots, z_{i-1}, z_i) + g_i(\mathbf{x},z_1, z_2, \ldots, z_{i-1}, z_i) z_{i+1} \quad \text{ for } 1 \leq i < k-1\\
\vdots\\
\dot{z}_{k-1} = f_{k-1}(\mathbf{x},z_1, z_2, \ldots, z_{k-1}) + g_{k-1}(\mathbf{x},z_1, z_2, \ldots, z_{k-1}) z_k\\
\dot{z}_k = f_k(\mathbf{x},z_1, z_2, \ldots, z_{k-1}, z_k) + g_k(\mathbf{x},z_1, z_2, \dots, z_{k-1}, z_k) u\end{cases}

where

  • \mathbf{x} \in \mathbb{R}^n with n \geq 1,
  • z_1, z_2, \ldots, z_i, \ldots, z_{k-1}, z_k are scalars,
  • u is a scalar input to the system,
  • f_x, f_1, f_2, \ldots, f_i, \ldots, f_{k-1}, f_k vanish at the origin (i.e., f_i(0,0,\dots,0) = 0),
  • g_1, g_2, \ldots, g_i, \ldots, g_{k-1}, g_k are nonzero over the domain of interest (i.e., g_i(\mathbf{x},z_1,\ldots,z_k) \neq 0 for 1 \leq i \leq k).

Also assume that the subsystem

\dot{\mathbf{x}} = f_x(\mathbf{x}) + g_x(\mathbf{x}) u_x(\mathbf{x})

is stabilized to the origin (i.e.,  \mathbf{x} = \mathbf{0}\,) by some known control u_x(\mathbf{x}) such that u_x(\mathbf{0}) = 0. It is also assumed that a Lyapunov function V_x for this stable subsystem is known. That is, this \mathbf{x} subsystem is stabilized by some other method and backstepping extends its stability to the \textbf{z} shell around it.

In systems of this strict-feedback form around a stable \mathbf{x} subsystem,

  • The backstepping-designed control input u has its most immediate stabilizing impact on state z_n.
  • The state z_n then acts like a stabilizing control on the state z_{n-1} before it.
  • This process continues so that each state z_i is stabilized by the fictitious "control" z_{i+1}.

The backstepping approach determines how to stabilize the \mathbf{x} subsystem using z_1, and then proceeds with determining how to make the next state z_2 drive z_1 to the control required to stabilize \mathbf{x}. Hence, the process "steps backward" from \mathbf{x} out of the strict-feedback form system until the ultimate control u is designed.

Recursive Control Design Overview[edit]

  1. It is given that the smaller (i.e., lower-order) subsystem
    \dot{\mathbf{x}} = f_x(\mathbf{x}) + g_x(\mathbf{x}) u_x(\mathbf{x})
    is already stabilized to the origin by some control u_x(\mathbf{x}) where u_x(\mathbf{0}) = 0. That is, choice of u_x to stabilize this system must occur using some other method. It is also assumed that a Lyapunov function V_x for this stable subsystem is known. Backstepping provides a way to extend the controlled stability of this subsystem to the larger system.
  2. A control u_1(\mathbf{x},z_1) is designed so that the system
    \dot{z}_1 = f_1(\mathbf{x},z_1) + g_1(\mathbf{x},z_1) u_1(\mathbf{x},z_1)
    is stabilized so that z_1 follows the desired u_x control. The control design is based on the augmented Lyapunov function candidate
    V_1(\mathbf{x},z_1) = V_x(\mathbf{x}) + \frac{1}{2}( z_1 - u_x(\mathbf{x}) )^2
    The control u_1 can be picked to bound \dot{V}_1 away from zero.
  3. A control u_2(\mathbf{x},z_1,z_2) is designed so that the system
    \dot{z}_2 = f_2(\mathbf{x},z_1,z_2) + g_2(\mathbf{x},z_1,z_2) u_2(\mathbf{x},z_1,z_2)
    is stabilized so that z_2 follows the desired u_1 control. The control design is based on the augmented Lyapunov function candidate
    V_2(\mathbf{x},z_1,z_2) = V_1(\mathbf{x},z_1) + \frac{1}{2}( z_2 - u_1(\mathbf{x},z_1) )^2
    The control u_2 can be picked to bound \dot{V}_2 away from zero.
  4. This process continues until the actual u is known, and
    • The real control u stabilizes z_k to fictitious control u_{k-1}.
    • The fictitious control u_{k-1} stabilizes z_{k-1} to fictitious control u_{k-2}.
    • The fictitious control u_{k-2} stabilizes z_{k-2} to fictitious control u_{k-3}.
    • ...
    • The fictitious control u_2 stabilizes z_2 to fictitious control u_1.
    • The fictitious control u_1 stabilizes z_1 to fictitious control u_x.
    • The fictitious control u_x stabilizes \mathbf{x} to the origin.

This process is known as backstepping because it starts with the requirements on some internal subsystem for stability and progressively steps back out of the system, maintaining stability at each step. Because

  • f_i vanish at the origin for 0 \leq i \leq k,
  • g_i are nonzero for 1 \leq i \leq k,
  • the given control u_x has u_x(\mathbf{0}) = 0,

then the resulting system has an equilibrium at the origin (i.e., where  \mathbf{x}=\mathbf{0}\,, z_1=0, z_2=0, ..., z_{k-1}=0, and z_k=0) that is globally asymptotically stable.

Integrator Backstepping[edit]

Before describing the backstepping procedure for general strict-feedback form dynamical systems, it is convenient to discuss the approach for a smaller class of strict-feedback form systems. These systems connect a series of integrators to the input of a system with a known feedback-stabilizing control law, and so the stabilizing approach is known as integrator backstepping. With a small modification, the integrator backstepping approach can be extended to handle all strict-feedback form systems.

Single-integrator Equilibrium[edit]

Consider the dynamical system

\begin{cases}
\dot{\mathbf{x}} = f_x(\mathbf{x}) + g_x(\mathbf{x}) z_1\\
\dot{z}_1 = u_1
\end{cases}  (1)\,

where \mathbf{x} \in \mathbb{R}^n and z_1 is a scalar. This system is a cascade connection of an integrator with the \mathbf{x} subsystem (i.e., the input u enters an integrator, and the integral z_1 enters the \mathbf{x} subsystem).

We assume that f_x(\mathbf{0})=0, and so if u_1=0,  \mathbf{x} = \mathbf{0}\, and z_1 = 0, then

\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 (\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.

Single-integrator Backstepping[edit]

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 u_1(\mathbf{x},z_1) that ensures that the states (\mathbf{x}, z_1) return to (\mathbf{0},0) after the system is started from some arbitrary initial condition.

  • First, by assumption, the subsystem
\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 u_x(\mathbf{0}) = 0 has a Lyapunov function V_x(\mathbf{x}) > 0 such that
\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 W(\mathbf{x}) is a positive-definite function. That is, we assume that we have already shown that this existing simpler \mathbf{x} subsystem is stable (in the sense of Lyapunov). Roughly speaking, this notion of stability means that:
    • The function V_x is like a "generalized energy" of the \mathbf{x} subsystem. As the \mathbf{x} states of the system move away from the origin, the energy V_x(\mathbf{x}) also grows.
    • By showing that over time, the energy V_x(\mathbf{x}(t)) decays to zero, then the \mathbf{x} states must decay toward  \mathbf{x}=\mathbf{0}\,. That is, the origin  \mathbf{x}=\mathbf{0}\, will be a stable equilibrium of the system – the \mathbf{x} states will continuously approach the origin as time increases.
    • Saying that W(\mathbf{x}) is positive definite means that W(\mathbf{x}) > 0 everywhere except for  \mathbf{x}=\mathbf{0}\,, and W(\mathbf{0})=0.
    • The statement that \dot{V}_x \leq -W(\mathbf{x}) means that \dot{V}_x is bounded away from zero for all points except where  \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 (\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 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 \dot{\mathbf{x}} part of the larger (\mathbf{x},z_1) system, it becomes
\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
\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 \dot{\mathbf{x}} = F(\mathbf{x}) subsystem plus some error perturbation generated by the integrator.
  • We now can change variables from (\mathbf{x}, z_1) to (\mathbf{x}, e_1) by letting e_1 \triangleq z_1 - u_x(\mathbf{x}). So
\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 v_1 \triangleq u_1 - \dot{u}_x so that u_1 = v_1 + \dot{u}_x and
\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 v_1. By stabilizing the system at e_1 = 0, the state z_1 will track the desired control u_x which will result in stabilizing the inner \mathbf{x} subsystem.
  • From our existing Lyapunov function V_x, we define the augmented Lyapunov function candidate
V_1(\mathbf{x}, e_1) \triangleq V_x(\mathbf{x}) + \frac{1}{2} e_1^2
So
\dot{V}_1
= \dot{V}_x(\mathbf{x}) + \frac{1}{2}\left( 2 e_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
By distributing \partial V_x/\partial \mathbf{x}, we see that
\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 \dot{V}_1 \leq -W(\mathbf{x}) < 0 (i.e., to ensure stability of the supersystem), we pick the control law
v_1 = -\frac{\partial V_x}{\partial \mathbf{x}}g_x(\mathbf{x})- k_1 e_1
with k_1 > 0, and so
\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 e_1 through,
\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
So our candidate Lyapunov function V_1 is a true Lyapunov function, and our system is stable under this control law v_1 (which corresponds the control law u_1 because v_1 \triangleq u_1 - \dot{u}_x). Using the variables from the original coordinate system, the equivalent Lyapunov function
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 v_1 ultimately depends on all of our original state variables. In particular, the actual feedback-stabilizing control law
\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 \mathbf{x} and z_1 and functions f_x and g_x come from the system. The function u_x comes from our known-stable \dot{\mathbf{x}}=F(\mathbf{x}) subsystem. The gain parameter k_1 > 0 affects the convergence rate or our system. Under this control law, our system is stable at the origin (\mathbf{x},z_1)=(\mathbf{0},0).
Recall that 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 u_x. Not surprisingly, the control u_1 has a \dot{u}_x term that will be integrated to follow the stabilizing control law \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 u_1(\mathbf{x}, z_1) and has Lyapunov function V_1(\mathbf{x},z_1) with \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.

Motivating Example: Two-integrator Backstepping[edit]

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

\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 \mathbf{x} \in \mathbb{R}^n and z_1 and 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 u_2 enters through an integrator, and the output of that integrator enters the system in Equation (1) by its u_1 input).

By letting

  • \mathbf{y} \triangleq \begin{bmatrix} \mathbf{x} \\ z_1 \end{bmatrix}\,,
  • f_y(\mathbf{y}) \triangleq \begin{bmatrix} f_x(\mathbf{x}) + g_x(\mathbf{x}) z_1 \\ 0 \end{bmatrix}\,,
  • 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

\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 u_y(\mathbf{y}) \triangleq u_1(\mathbf{x},z_1) stabilizes the upper z_2-to-\mathbf{y} subsystem using the Lyapunov function 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 u_2 can be found using the same single-integrator procedure that was used to find u_1.

Many-integrator backstepping[edit]

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.

\dot{\mathbf{x}} = f_x(\mathbf{x}) + g_x(\mathbf{x}) u_x
that has scalar input u_x and output states \mathbf{x} = [x_1, x_2, \ldots, x_n]^{\text{T}} \in \mathbb{R}^n. Assume that
    • f_x(\mathbf{x}) = \mathbf{0} so that the zero-input (i.e., u_x = 0) system is stationary at the origin  \mathbf{x} = \mathbf{0}\,. In this case, the origin is called an equilibrium of the system.
    • The feedback control law u_x(\mathbf{x}) stabilizes the system at the equilibrium at the origin.
    • A Lyapunov function corresponding to this system is described by V_x(\mathbf{x}).
That is, if output states \mathbf{x} are fed back to the input u_x by the control law 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 u_x.
  • Next, connect an integrator to input u_x so that the augmented system has input u_1 (to the integrator) and output states \mathbf{x}. The resulting augmented dynamical system is
\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 z_1 and \mathbf{x} to input u_1 according to the control law
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 k_1 > 0, then the states z_1 and \mathbf{x} will return to z_1 = 0 and  \mathbf{x}=\mathbf{0}\, after a single perturbation. This subsystem is stabilized by feedback control law u_1, and the corresponding Lyapunov function from Equation (2) is
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 u_1, the Lyapunov function V_1 decays to zero as the states return to the origin.
  • Connect a new integrator to input u_1 so that the augmented system has input u_2 and output states \mathbf{x}. The resulting augmented dynamical system is
\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
\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 \mathbf{x}_1, f_1, and g_1, this system can also be expressed as
\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 z_1, z_2, and \mathbf{x} to input u_2 according to the control law
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 k_2 > 0, then the states z_1, z_2, and \mathbf{x} will return to z_1 = 0, z_2 = 0, and  \mathbf{x}=\mathbf{0}\, after a single perturbation. This subsystem is stabilized by feedback control law u_2, and the corresponding Lyapunov function is
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 u_2, the Lyapunov function V_2 decays to zero as the states return to the origin.
  • Connect an integrator to input u_2 so that the augmented system has input u_3 and output states \mathbf{x}. The resulting augmented dynamical system is
\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
\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 \mathbf{x}_1, f_1, and g_1 from the previous step, this system is also represented by
\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 \mathbf{x}_2, f_2, and g_2, this system can also be expressed as
\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 z_1, z_2, z_3, and \mathbf{x} to input u_3 according to the control law
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 k_3 > 0, then the states z_1, z_2, z_3, and \mathbf{x} will return to z_1 = 0, z_2 = 0, z_3 = 0, and  \mathbf{x}=\mathbf{0}\, after a single perturbation. This subsystem is stabilized by feedback control law u_3, and the corresponding Lyapunov function is
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 u_3, the Lyapunov function 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
\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
\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 (\mathbf{x},z_1) subsystem (i.e., with input z_2 and output \mathbf{x}) and iterating out from that inner subsystem until the ultimate feedback-stabilizing control u is known. At iteration i, the equivalent system is
\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
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 k_i > 0. The corresponding Lyapunov function is
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 u(\mathbf{x},z_1,z_2,\ldots,z_k) = u_k(\mathbf{x}_k) (i.e., ultimate control is found at final iteration i=k).

Hence, any system in this special many-integrator strict-feedback form can be feedback stabilized using a straightforward procedure that can even be automated (e.g., as part of an adaptive control algorithm).

Generic Backstepping[edit]

Systems in the special strict-feedback form have a recursive structure similar to the many-integrator system structure. Likewise, they are stabilized by stabilizing the smallest cascaded system and then backstepping to the next cascaded system and repeating the procedure. So it is critical to develop a single-step procedure; that procedure can be recursively applied to cover the many-step case. Fortunately, due to the requirements on the functions in the strict-feedback form, each single-step system can be rendered by feedback to a single-integrator system, and that single-integrator system can be stabilized using methods discussed above.

Single-step Procedure[edit]

Consider the simple strict-feedback system

\begin{cases}
\dot{\mathbf{x}} = f_x(\mathbf{x}) + g_x(\mathbf{x}) z_1\\
\dot{z}_1 = f_1(\mathbf{x}, z_1) + g_1(\mathbf{x}, z_1) u_1
\end{cases}  (6)\,

where

  • \mathbf{x} = [x_1, x_2, \ldots, x_n]^{\text{T}} \in \mathbb{R}^n,
  • z_1 and u_1 are scalars,
  • For all \mathbf{x} and z_1, g_1(\mathbf{x},z_1) \neq 0.

Rather than designing feedback-stabilizing control u_1 directly, introduce a new control u_{a1} (to be designed later) and use control law

u_1( \mathbf{x}, z_1 )
= 
\frac{ 1 }{ g_1( \mathbf{x}, z_1 ) }
\left( u_{a1} - f_1(\mathbf{x},z_1) \right)

which is possible because g_1 \neq 0. So the system in Equation (6) is

\begin{cases}
\dot{\mathbf{x}} = f_x(\mathbf{x}) + g_x(\mathbf{x}) z_1\\
\dot{z}_1 = f_1(\mathbf{x}, z_1) + g_1(\mathbf{x}, z_1) \overbrace{\frac{ 1 }{ g_1( \mathbf{x}, z_1 ) }
\left( u_{a1} - f_1(\mathbf{x},z_1) \right)}^{u_1(\mathbf{x}, z_1)}
\end{cases}

which simplifies to

\begin{cases}
\dot{\mathbf{x}} = f_x(\mathbf{x}) + g_x(\mathbf{x}) z_1\\
\dot{z}_1 = u_{a1}
\end{cases}

This new u_{a1}-to-\mathbf{x} system matches the single-integrator cascade system in Equation (1). Assuming that a feedback-stabilizing control law u_x(\mathbf{x}) and Lyapunov function V_x(\mathbf{x}) for the upper subsystem is known, the feedback-stabilizing control law from Equation (3) is

u_{a1}(\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 k_1 > 0. So the final feedback-stabilizing control law is

u_1(\mathbf{x},z_1) = \frac{1}{ g_1(\mathbf{x},z_1) } \left( \overbrace{-\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)}^{u_{a1}(\mathbf{x},z_1)} \, - \, f_1(\mathbf{x}, z_1) \right)  (7)\,

with gain k_1 > 0. The corresponding Lyapunov function from Equation (2) is

V_1(\mathbf{x},z_1) = V_x(\mathbf{x}) + \frac{1}{2} ( z_1 - u_x(\mathbf{x}) )^2  (8)\,

Because this strict-feedback system has a feedback-stabilizing control and a corresponding Lyapunov function, it can be cascaded as part of a larger strict-feedback system, and this procedure can be repeated to find the surrounding feedback-stabilizing control.

Many-step Procedure[edit]

As in many-integrator backstepping, the single-step procedure can be completed iteratively to stabilize an entire strict-feedback system. In each step,

  1. The smallest "unstabilized" single-step strict-feedback system is isolated.
  2. Feedback is used to convert the system into a single-integrator system.
  3. The resulting single-integrator system is stabilized.
  4. The stabilized system is used as the upper system in the next step.

That is, any strict-feedback system

\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 = f_1( \mathbf{x}, z_1 ) + g_1( \mathbf{x}, z_1 ) z_2\\
\dot{z}_2 = f_2( \mathbf{x}, z_1, z_2 ) + g_2( \mathbf{x}, z_1, z_2 ) z_3\\
\vdots\\
\dot{z}_i = f_i( \mathbf{x}, z_1, z_2, \ldots, z_i ) + g_i( \mathbf{x}, z_1, z_2, \ldots, z_i ) z_{i+1}\\
\vdots\\
\dot{z}_{k-2} = f_{k-2}( \mathbf{x}, z_1, z_2, \ldots z_{k-2} ) + g_{k-2}( \mathbf{x}, z_1, z_2, \ldots, z_{k-2} ) z_{k-1}\\
\dot{z}_{k-1} = f_{k-1}( \mathbf{x}, z_1, z_2, \ldots z_{k-2}, z_{k-1} ) + g_{k-1}( \mathbf{x}, z_1, z_2, \ldots, z_{k-2}, z_{k-1} ) z_k\\
\dot{z}_k = f_k( \mathbf{x}, z_1, z_2, \ldots z_{k-1}, z_k ) + g_k( \mathbf{x}, z_1, z_2, \ldots, z_{k-1}, z_k ) u
\end{cases}

has the recursive structure

\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 = f_1( \mathbf{x}, z_1 ) + g_1( \mathbf{x}, z_1 ) z_2
\end{cases}\\
\dot{z}_2 = f_2( \mathbf{x}, z_1, z_2 ) + g_2( \mathbf{x}, z_1, z_2 ) z_3
\end{cases}\\
\vdots\\
\end{cases}\\
\dot{z}_i = f_i( \mathbf{x}, z_1, z_2, \ldots, z_i ) + g_i( \mathbf{x}, z_1, z_2, \ldots, z_i ) z_{i+1}
\end{cases}\\
\vdots
\end{cases}\\
\dot{z}_{k-2} = f_{k-2}( \mathbf{x}, z_1, z_2, \ldots z_{k-2} ) + g_{k-2}( \mathbf{x}, z_1, z_2, \ldots, z_{k-2} ) z_{k-1}
\end{cases}\\
\dot{z}_{k-1} = f_{k-1}( \mathbf{x}, z_1, z_2, \ldots z_{k-2}, z_{k-1} ) + g_{k-1}( \mathbf{x}, z_1, z_2, \ldots, z_{k-2}, z_{k-1} ) z_k
\end{cases}\\
\dot{z}_k = f_k( \mathbf{x}, z_1, z_2, \ldots z_{k-1}, z_k ) + g_k( \mathbf{x}, z_1, z_2, \ldots, z_{k-1}, z_k ) u
\end{cases}

and can be feedback stabilized by finding the feedback-stabilizing control and Lyapunov function for the single-integrator (\mathbf{x},z_1) subsystem (i.e., with input z_2 and output \mathbf{x}) and iterating out from that inner subsystem until the ultimate feedback-stabilizing control u is known. At iteration i, the equivalent system is

\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-2}) z_{i-2} \\ f_{i-1}(\mathbf{x}_i) \end{bmatrix} }^{\triangleq \, f_{i-1}(\mathbf{x}_{i-1})}
+
\overbrace{ \begin{bmatrix} \mathbf{0}\\ g_{i-1}(\mathbf{x}_i)\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 = f_i(\mathbf{x}_i) + g_i(\mathbf{x}_i) u_i
\end{cases}

By Equation (7), the corresponding feedback-stabilizing control law is

u_i(\overbrace{\mathbf{x},z_1,z_2,\dots,z_i}^{\triangleq \, \mathbf{x}_i})
=
\frac{1}{g_i(\mathbf{x}_i)}
\left( \overbrace{-\frac{\partial V_{i-1}}{\partial \mathbf{x}_{i-1} }
g_{i-1}(\mathbf{x}_{i-1})
\, - \,
k_i\left( z_i \, - \, u_{i-1}(\mathbf{x}_{i-1}) \right)
\, + \,
\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) }^{\text{Single-integrator stabilizing control } u_{a\;\!i}(\mathbf{x}_i)}
\, - \,
f_i( \mathbf{x}_{i-1} )
\right)

with gain k_i > 0. By Equation (8), the corresponding Lyapunov function is

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 u(\mathbf{x},z_1,z_2,\ldots,z_k) = u_k(\mathbf{x}_k) (i.e., ultimate control is found at final iteration i=k). Hence, any strict-feedback system can be feedback stabilized using a straightforward procedure that can even be automated (e.g., as part of an adaptive control algorithm).

See also[edit]

References[edit]

  1. ^ Kokotovic, P.V. (1992). "The joy of feedback: nonlinear and adaptive". Control Systems Magazine, IEEE 12 (3): 7–17. doi:10.1109/37.165507. Retrieved 2008-04-13. 
  2. ^ Lozano, R.; Brogliato, B. (1992). "Adaptive control of robot manipulators with flexible joints". IEEE Transactions on Automatic Control, 37 (2): 174–181. doi:10.1109/9.121619. 
  3. ^ a b Khalil, H.K. (2002). Nonlinear Systems (3rd ed.). Upper Saddle River, NJ: Prentice Hall. ISBN 0-13-067389-7.