Basin, M. (2019). Finite-and fixed-time convergent algorithms: Design
and convergence time estimation. Annual reviews in control,
48, 209-221.
1. Introduction
The objective of this review is to take a look at the history and
state-of-the-art of finite- and fixed-time convergent algorithms for
dynamic systems of various dimensions and relative degrees, with or
without disturbances.
2. Finite- and
fixed-time convergent regulators
2.1. Problem statement
Consider an -dimensional chain
of integrators where is a system state, is a scalar control
input, is a
disturbance satisfying certain conditions.
The control problem is to design a continuous control law, such that
the resulting closed-loop system is globally finite- or fixed-time
convergent to the origin in the sense of the following definitions, and
estimate the convergence (settling) time.
Definition 1: The control system is called
globallyfinite-time convergent to the origin, if for
any there
exists a time moment such
that the system state is equal to zero, , for all
.
Definition 2: The control system is called
fixed-time convergent to the origin, if there exists a time
moment such that the system state
is equal to
zero, , for all , starting from
any initial condition .
Discontinuous finite-time convergent regulators based on a
discontinuous feedback , or its multi-dimensional analogs are an efficient tool
to suppress bounded disturbances and drive the system states to the
origin for a finite time. Accordingly, in this subsection, the
disturbance is considered
uniformly bounded by a known constant .
2.2.1. Scalar systems
Here, we have the form The discontinuous control law is selected as The corresponding closed-loop system is finite-time convergent
to zero, if . The
convergence time can be estimated as
2.2.2. Two-dimensional systems
Here, we have the form The discontinuous control law, known as “twisting”algorithm is
assigned as The corresponding closed-loop system is finite-time convergent
to the origin, i.e, both state variables, and come to zero for a finite time if
.
If , the convergence
time would be no greater than If , the
total convergence time would be no greater than There is another class of finite-time convergent discontinuous
control algorithms for two-dimensional systems, called terminal sliding
mode control laws. terminal sliding mode control is a “nested
algorithm,”that is, the system state is first driven at a certain
one-dimensional sliding manifold in finite time and then reaches the
origin moving along this sliding manifold, again for a
finite time.
2.2.3. Multi-dimensional systems
apply a feedback discontinuous control in the form
To the best of the author’s knowledge, the finite-time
convergence conditions for the control gains have not been yet obtained
in a general multi-dimensional case.
The fixed upper bound for the convergence time is calculated
explicitly. A great advantage of the obtained convergence time estimate
consists in the fact that the desired convergence time can be directly
substituted into the given formulas to find the corresponding values of
control gains.
2.3.2. Two-dimensional systems
The idea of the designed control is to use the discontinuous term
to get to a certain
sliding manifold so that the equivalent sliding control includes the
terms where are odd
integers and ,
to drive the system state at the origin in fixed time.
2.4. Continuous
finite-time convergent regulators
Continuous finite-time convergent regulators based on a continuous
integral feedback including the term or its multi-dimensional analogs are an efficient tool to
suppress disturbances with bounded changing rates (derivatives), while
the disturbances themselves may be unbounded, and drive the system
states to the origin for a finite time. Accordingly, in this subsection,
the disturbance satisfies
the Lipschitz condition with a known constant or its derivative is uniformly bounded
by a known constant .
2.4.1. Scalar systems
Here, we have the form The continuous control law, known as
super-twisting algorithm is assigned as The corresponding closed-loop system is second-order
finite-time convergent to the origin, i.e, both the state variable and its derivative come to zero for a finite time,
if The last condition is further relaxed as is the base of natural
logarithms. The resulting closed-loop system can be also represented in
the form The indicated finite-time con- vergence conditions imply
finite-time convergence of the variable to zero as well. If , the convergence time can be
estimated as If is an arbitrary
known value, the convergence time can be estimated as
2.4.2. Multivariable systems
Consider a multivariable system and disturbance satisfying the Lipschitz condition The continuous control law directly generalizing the conventional super-twisting
algorithm to the multivariable case. The control law yields global
finite-time convergence to the origin for all states of the system ,
provided that . The resulting closed-loop system can also be
represented in the form If , the
convergence time can be estimated as If is an arbitrary
known value, the convergence time can be estimated as
2.4.3. Multi-dimensional
systems
In the absence of disturbances, , all the states of the system can be driven at the origin
by the continuous control law Control gains is are
assigned such that is a Hurwitz
polynomial. The corresponding convergence time can be estimated as The symmetric positive definite matrix satisfies a Lyapunov equation where is a symmetric positive definite matrix, and the matrix
is defined as If the disturbance is present and satisfies the
Lipschitz condition with a constant , all the states of the system can be
driven at the origin by the continuous control law It is impossible to estimate the convergence
time in this case, if the disturbance initial value, , is unknown.
If the disturbance is absent at the initial moment, , the convergence time can
be estimated as
2.5. Continuous
fixed-time convergent regulators
2.5.1. Scalar systems
A continuous control law providing fixed-time convergence of the
state of the scalar system is given as The corresponding closed-loop system is second-order
fixed-time convergent to the origin, i.e., both the state variable and its derivative come to zero for a fixed time
if
The resulting closed-loop system can also be represented in the form
The indicated fixed-time con- vergence conditions imply
fixed-time convergence of the variable to zero as well. If , the convergence time can be estimated as where . The
minimum value of is
reached for If is unknown,
the convergence time can be estimated as where is the
convergence time of the fixed-time convergent observer introduced in (
Section 3.5 ) and is the
estimate for the state variable , produced by the observer in the following section.
2.5.2. Multivariable systems
The continuous control law directly generalizing the scalar fixed-time convergent control
to the multivariable case. The control law yields fixed- time
convergence to the origin for all states of the system, provided that
. If , the
convergence time can be estimated as The minimum value of is reached for If is unknown,
the convergence time can be estimated as where is the
convergence time of the fixed-time convergent observer introduced in (
Section 3.5 ) and is the
estimate for the state variable , produced by the observer in the following section.
2.5.3. Multi-dimensional
systems
In the absence of disturbances, , all the states of the system can be driven at the origin
for a fixed time by the continuous control law Control gains is
are assigned such that is a
Hurwitz polynomial. The corresponding convergence time can be estimated
as The symmetric positive definite matrix satisfies a Lyapunov equation where is a symmetric positive definite matrix, and the matrix
is defined as The convergence time estimate does not depend on an initial
condition.
If the disturbance is
present and satisfies the Lipschitz condition with a constant , all the states of the system can be
driven at the origin by the continuous control law If the disturbance is absent at the initial moment, , the convergence time can
be estimated as If the disturbance initial value, , is unknown, the convergence
time can be estimated as
3.
Finite- and fixed-time convergent observers (differentiators)
3.1. Problem statement
Consider the following dynamic system where is the system state, is the measurable
variable (sensor output), is an external disturbance, and are known
functions.
Since only the scalar output can be measured, a finite-time or fixed-time convergent
differentiator is needed to reconstruct values of the output derivatives
, and
estimate values of all state components . This means
that the error system and is the
proposed estimate for the
th output derivative .
A finite-time convergent differentiator reconstructing the output
derivatives in finite time is
proposed in the non-recursive and recursive forms. The non-recursive
form is given by where is the
estimate for .
The finite-time convergent differentiator can also be represented in
the recursive form as
A smooth finite-time convergent differentiator reconstructing the
output derivatives in finite time is as follows: where Observer gains are assigned such that the matrix is defined as is Hurwitz.
The corresponding convergence time can be estimated The symmetric positive definite matrix satisfies a Lyapunov equation
A smooth fixed-time convergent differentiator reconstructing the
output derivatives , for the
system in fixed time is where Observer gains are assigned such that the matrix is defined as is Hurwitz. The corresponding convergence time can be estimated as The symmetric positive definite matrix satisfies a Lyapunov equation
3.5.
Smooth fixed-time convergent observers for super-twisting variables
3.5.1. Scalar case
Consider the following fixed-time observer for We have The estimates ,
and converge to the variables , and , respectively, for a fixed time
no greater than
3.5.2. Multivariable case
Consider the following fixed-time observer for the multivariable
super-twisting system: The estimates ,
and converge to the variables , and , respectively, for a fixed time
no greater than
