Backstepping - AbsoluteAstronomy.com

Control theory

Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...

, backstepping is a technique developed circa

Circa

Circa , usually abbreviated c. or ca. , means "approximately" in the English language, usually referring to a date...

1990 by Petar V. Kokotovic

Petar V. Kokotovic

Petar V. Kokotovic is a professor in the Department of Engineering at the University of California, Santa Barbara, USA. He has made contributions in the areas of adaptive control, singular perturbation techniques, and nonlinear control....

and others for designing stabilizing

Lyapunov stability

Various types of stability may be discussed for the solutions of differential equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Lyapunov...

controls for a special class of nonlinear dynamical system

Dynamical system

A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...

s. 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

Recursion

Recursion is the process of repeating items in a self-similar way. For instance, when the surfaces of two mirrors are exactly parallel with each other the nested images that occur are a form of infinite recursion. The term has a variety of meanings specific to a variety of disciplines ranging from...

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.

The backstepping approach provides a recursive

Recursion

method for stabilizing

Lyapunov stability

the origin

Origin (mathematics)

In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect...

of a system in strict-feedback form. That is, consider a system

Dynamical system

of the form

where

with ,
are scalar
Scalar (mathematics)
In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....

s,
is a scalar
Scalar (mathematics)
In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....

input to the system,
vanish
Vanish
Vanish may refer to:*Vanish , a toilet bowl cleaner by S.C. Johnson or a cloth stain remover by Reckitt Benckiser*"Vanish" an episode of the TV series Criss Angel Mindfreak*Vanishing, a type of magical effect...

at the origin
Origin (mathematics)
In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect...

(i.e., ),
are nonzero over the domain of interest (i.e., for ).

Also assume that the subsystem

is stabilized

Lyapunov stability

to the origin

Origin (mathematics)

(i.e.,

) by some known control

such that

. It is also assumed that a Lyapunov function

Lyapunov function

In the theory of ordinary differential equations , Lyapunov functions are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Named after the Russian mathematician Aleksandr Mikhailovich Lyapunov, Lyapunov functions are important to stability theory and control...

for this stable subsystem is known. That is, this

subsystem is stabilized by some other method and backstepping extends its stability to the

shell around it.

In systems of this strict-feedback form around a stable

subsystem,

The backstepping-designed control input has its most immediate stabilizing impact on state .
The state then acts like a stabilizing control on the state before it.
This process continues so that each state is stabilized by the fictitious "control" .

The backstepping approach determines how to stabilize the

subsystem using

, and then proceeds with determining how to make the next state

drive

to the control required to stabilize

. Hence, the process "steps backward" from

out of the strict-feedback form system until the ultimate control

is designed.

Recursive Control Design Overview

It is given that the smaller (i.e., lower-order) subsystem

is already stabilized to the origin by some control where . That is, choice of to stabilize this system must occur using some other method. It is also assumed that a Lyapunov function
Lyapunov function
In the theory of ordinary differential equations , Lyapunov functions are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Named after the Russian mathematician Aleksandr Mikhailovich Lyapunov, Lyapunov functions are important to stability theory and control...

for this stable subsystem is known. Backstepping provides a way to extend the controlled stability of this subsystem to the larger system.
A control is designed so that the system

is stabilized so that follows the desired control. The control design is based on the augmented Lyapunov function candidate

The control can be picked to bound away from zero.
A control is designed so that the system

is stabilized so that follows the desired control. The control design is based on the augmented Lyapunov function candidate

The control can be picked to bound away from zero.
This process continues until the actual is known, and
- The real control stabilizes to fictitious control .
- The fictitious control stabilizes to fictitious control .
- The fictitious control stabilizes to fictitious control .
- …
- The fictitious control stabilizes to fictitious control .
- The fictitious control stabilizes to fictitious control .
- The fictitious control stabilizes 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

vanish at the origin for ,
are nonzero for ,
the given control has ,

then the resulting system has an equilibrium at the origin (i.e., where

, … ,

, and

) that is globally asymptotically stable.

Integrator Backstepping

Before describing the backstepping procedure for general strict-feedback form dynamical system

Dynamical system

s, 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

Consider the dynamical system

Dynamical system

where

and

is a scalar. This system is a cascade connection of an integrator

Integrator

An integrator is a device to perform the mathematical operation known as integration, a fundamental operation in calculus.The integration function is often part of engineering, physics, mechanical, chemical and scientific calculations....

with the

subsystem (i.e., the input

enters an integrator, and the integral

Integral

Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

enters the

subsystem).

We assume that

, and so if

and

, then

So the origin

Origin (mathematics)

is an equilibrium (i.e., a stationary point

Stationary point

In mathematics, particularly in calculus, a stationary point is an input to a function where the derivative is zero : where the function "stops" increasing or decreasing ....

) of the system. If the system ever reaches the origin, it will remain there forever after.

Single-integrator Backstepping

In this example, backstepping is used to stabilize

Lyapunov stability

the single-integrator system in Equation (1) around its equilibrium at the origin. To be less precise, we wish to design a control law

that ensures that the states

return to

after the system is started from some arbitrary initial condition.

First, by assumption, the subsystem

with

has a Lyapunov function

Lyapunov function

such that

where

is a positive-definite function. That is, we assume that we have already shown that this existing simpler

subsystem is stable (in the sense of Lyapunov)
Lyapunov stability
Various types of stability may be discussed for the solutions of differential equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Lyapunov...

. Roughly speaking, this notion of stability means that:

The function is like a "generalized energy" of the subsystem. As the states of the system move away from the origin, the energy also grows.
By showing that over time, the energy decays to zero, then the states must decay toward . That is, the origin will be a stable equilibrium of the system – the states will continuously approach the origin as time increases.
Saying that is positive definite means that everywhere except for , and .
The statement that means that is bounded away from zero for all points except where . 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

that makes our cascaded

system also stable. So we must find a new Lyapunov function candidate for this new system. That candidate will depend upon the control

, and by choosing the control properly, we can ensure that it is decaying everywhere as well.

Next, by adding and subtracting (i.e., we don't change the system in any way because we make no net effect) to the part of the larger system, it becomes

which we can re-group to get

So our cascaded supersystem encapsulates the known-stable

subsystem plus some error perturbation generated by the integrator.

We now can change variables from to by letting . So

Additionally, we let

so that

and

We seek to stabilize this error system by feedback through the new control

. By stabilizing the system at

, the state

will track the desired control

which will result in stabilizing the inner

subsystem.

From our existing Lyapunov function , we define the augmented Lyapunov function candidate

By distributing

, we see that

To ensure that

(i.e., to ensure stability of the supersystem), we pick the control law

with

, and so

After distributing the

through,

So our candidate Lyapunov function

is a true Lyapunov function

Lyapunov function

, and our system is stable under this control law

(which corresponds the control law

because

). Using the variables from the original coordinate system, the equivalent Lyapunov function

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

Our choice of control ultimately depends on all of our original state variables. In particular, the actual feedback-stabilizing control law

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The states

and

and functions

and

come from the system. The function

comes from our known-stable

subsystem. The gain parameter

affects the convergence rate or our system. Under this control law, our system is stable

Lyapunov stability

at the origin

Recall that

in Equation (3) drives the input of an integrator that is connected to a subsystem that is feedback-stabilized by the control law

. Not surprisingly, the control

has a

term that will be integrated to follow the stabilizing control law

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

and has Lyapunov function

with

, it can be used as the upper subsystem in another single-integrator cascade system.

Motivating Example: Two-integrator Backstepping

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

Dynamical system

where

and

are scalars. This system is a cascade connection of the single-integrator system in Equation (1) with another integrator (i.e., the input

enters through an integrator, and the output of that integrator enters the system in Equation (1) by its

input).

By letting

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

By the single-integrator procedure, the control law

stabilizes the upper

-to-

subsystem using the Lyapunov function

, 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

can be found using the same single-integrator procedure that was used to find

Many-integrator backstepping

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

Mathematical induction

Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...

. Here, a stabilized multiple-integrator system is built up from subsystems of already-stabilized multiple-integrator subsystems.

First, consider the dynamical system
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...

that has scalar input

and output states

. Assume that

so that the zero-input (i.e., ) system is stationary
Stationary point
In mathematics, particularly in calculus, a stationary point is an input to a function where the derivative is zero : where the function "stops" increasing or decreasing ....

at the origin . In this case, the origin is called an equilibrium of the system.
The feedback control law stabilizes the system at the equilibrium at the origin.
A Lyapunov function
Lyapunov function
In the theory of ordinary differential equations , Lyapunov functions are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Named after the Russian mathematician Aleksandr Mikhailovich Lyapunov, Lyapunov functions are important to stability theory and control...

corresponding to this system is described by .

That is, if output states

are fed back to the input

by the control law

, 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

Next, connect an integrator
Integrator
An integrator is a device to perform the mathematical operation known as integration, a fundamental operation in calculus.The integration function is often part of engineering, physics, mechanical, chemical and scientific calculations....

to input so that the augmented system has input (to the integrator) and output states . The resulting augmented dynamical system is

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

and

to input

according to the control law

with gain

, then the states

and

will return to

and

after a single perturbation. This subsystem is stabilized by feedback control law

, and the corresponding Lyapunov function from Equation (2) is

That is, under feedback control law

, the Lyapunov function

decays to zero as the states return to the origin.

Connect a new integrator to input so that the augmented system has input and output states . The resulting augmented dynamical system is

which is equivalent to the single-integrator system

Using these definitions of

, and

, this system can also be expressed as

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

, and

to input

according to the control law

with gain

, then the states

, and

will return to

, and

after a single perturbation. This subsystem is stabilized by feedback control law

, and the corresponding Lyapunov function is

That is, under feedback control law

, the Lyapunov function

decays to zero as the states return to the origin.

Connect an integrator to input so that the augmented system has input and output states . The resulting augmented dynamical system is

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

By the definitions of

, and

from the previous step, this system is also represented by

Further, using these definitions of

, and

, this system can also be expressed as

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

, and

to input

according to the control law

with gain

, then the states

, and

will return to

, and

after a single perturbation. This subsystem is stabilized by feedback control law

, and the corresponding Lyapunov function is

That is, under feedback control law

, the Lyapunov function

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

has the recursive structure

and can be feedback stabilized by finding the feedback-stabilizing control and Lyapunov function for the single-integrator

subsystem (i.e., with input

and output

) and iterating out from that inner subsystem until the ultimate feedback-stabilizing control

is known. At iteration

, the equivalent system is

The corresponding feedback-stabilizing control law is

with gain

. The corresponding Lyapunov function is

By this construction, the ultimate control

(i.e., ultimate control is found at final iteration

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

Adaptive control

Adaptive control is the control method used by a controller which must adapt to a controlled system with parameters which vary, or are initially uncertain. For example, as an aircraft flies, its mass will slowly decrease as a result of fuel consumption; a control law is needed that adapts itself...

algorithm).

Generic Backstepping

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

Consider the simple strict-feedback system

Dynamical system

where

,
and are scalar
Scalar (mathematics)
In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....

s,
For all and , .

Rather than designing feedback-stabilizing control

directly, introduce a new control

(to be designed later) and use control law

which is possible because

. So the system in Equation (6) is

which simplifies to

This new

-to-

system matches the single-integrator cascade system in Equation (1). Assuming that a feedback-stabilizing control law

and Lyapunov function

Lyapunov function

for the upper subsystem is known, the feedback-stabilizing control law from Equation (3) is

with gain

. So the final feedback-stabilizing control law is

with gain

. The corresponding Lyapunov function from Equation (2) is

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

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

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

That is, any strict-feedback system

has the recursive structure

and can be feedback stabilized by finding the feedback-stabilizing control and Lyapunov function for the single-integrator

subsystem (i.e., with input

and output

) and iterating out from that inner subsystem until the ultimate feedback-stabilizing control

is known. At iteration

, the equivalent system is

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

with gain

. By Equation (8), the corresponding Lyapunov function is

By this construction, the ultimate control

(i.e., ultimate control is found at final iteration

).
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

Adaptive control

algorithm).

Recursive Control Design Overview

Integrator Backstepping

Single-integrator Equilibrium

Single-integrator Backstepping

Motivating Example: Two-integrator Backstepping

Many-integrator backstepping

Generic Backstepping

Single-step Procedure

Many-step Procedure

See also