of 6
Sampled-Data Stabilization with Control Lyapunov Functions
via Quadratically Constrained Quadratic Programs
Andrew J. Taylor
1
, Victor D. Dorobantu
1
, Yisong Yue, Paulo Tabuada, and Aaron D. Ames
Abstract
—Controller design for nonlinear systems with Control
Lyapunov Function (CLF) based quadratic programs has re-
cently been successfully applied to a diverse set of difficult control
tasks. These existing formulations do not address the gap between
design with continuous time models and the discrete time sampled
implementation of the resulting controllers, often leading to poor
performance on hardware platforms. We propose an approach to
close this gap by synthesizing sampled-data counterparts to these
CLF-based controllers, specified as quadratically constrained
quadratic programs (QCQPs). Assuming feedback linearizability
and stable zero-dynamics of a system’s continuous time model,
we derive practical stability guarantees for the resulting sampled-
data system. We demonstrate improved performance of the pro-
posed approach over continuous time counterparts in simulation.
I. I
NTRODUCTION
Nonlinear control methods offer promising solutions to
many modern engineering applications. However, theoretically
sound controller designs often fail to achieve desired behaviors
when deployed on real systems. Thus, it is critical to under-
stand the discrepancies between theoretical design and practi-
cal implementation mathematically, and to design controllers
that close these gaps. Specifically, we address the challenges
in designing controllers with continuous time models and
realizing them with discrete time sampling implementations.
Feedback linearization is a powerful tool in nonlinear
control design, enabling the algorithmic synthesis of con-
trollers for a wide class of mechanical and electrical systems
[1]. Moreover, feedback linearization provides a constructive
method to find Control Lyapunov Functions (CLFs) [2] for
continuous time systems. This fact has been used to formulate
stabilizing controllers through Quadratic Programs (QPs) [3],
[4], seeing use in several applications such as robotics [5] and
autonomous vehicles [6]. Despite these successes, translating
these controllers to hardware platforms often requires addi-
tional effort to overcome the degradation of performance and
introduction of chatter caused by sample frequency limitations.
We propose an extension of the preceding nonlinear con-
troller designs to the
sampled-data
setting [7], in which
control inputs are specified at discrete sample times and held
constant between sequential sample times (referred to as a
zero-order hold). The resulting evolution of such systems
between sample times is described by discrete time models,
for which exact representations can rarely be derived, moti-
vating the synthesis of controllers with approximate discrete
1
Both authors contributed equally. A.J. Taylor, V.D. Dorobantu, Y. Yue,
and A.D. Ames are with the Department of Computing and Mathematical
Sciences, California Institute of Technology, Pasadena, CA 91125, USA,
{ajtaylor, vdoroban, yyue, ames}@caltech.edu
. P.
Tabuada is with the Department of Electrical Engineering, Univer-
sity of California at Los Angeles, Los Angeles, CA 90095, USA,
tabuada@ucla.edu
.
time models. The foundational work in [8], [9] established
a sampled-data framework for translating stability guarantees
for an approximate discrete time model to the exact discrete
time model. Resulting sampled-data synthesis methods [10],
[11], [12] specify controllers that often demonstrate improved
performance over continuous time designs [13], many using a
simple Euler approximate discrete time model [14], [10], but
optimization-based controllers synthesized using CLFs found
via feedback linearization have not yet been considered.
The relationship between feedback linearizability and sam-
pling has been thoroughly investigated [15], [16], [17], [18].
Much of this investigation has focused on whether feedback
linearizability of a system’s continuous time dynamics implies
feedback linearizability of the exact discrete time model of
the system (a fact that requires strict structure of the contin-
uous time dynamics). Even with Euler approximate discrete
time models, a continuous time feedback linearizable system
must be first expressed in appropriate coordinates before
sampling and approximating to ensure the approximate model
is feedback linearizable in the discrete time sense [17]. This
requirement is also seen with higher-order approximate models
obtained via Taylor expansion [19]. The work [20], [21],
[22] studies the zero-dynamics that arise due to sampling and
higher-order approximations, but not the impact of sampling
on the stability of existing continuous time zero-dynamics.
We make two main contributions in this work. First, we
formally integrate feedback linearization and zero-dynamics
with Euler approximate discrete time models for sampled-
data systems via the results in [8]. In particular, we demon-
strate that systems with feedback linearizable continuous time
dynamics and locally exponentially stable zero-dynamics can
be rendered practically-stable via a continuous time feedback
linearizing controller when the inputs to the system are im-
plemented with a zero order hold. The often local nature of
stability of zero-dynamics requires modification of the global
results in [8]. Second, we extend the preceding result to
optimization-based controllers using CLFs [4] synthesized via
feedback linearization. In Section V we propose a controller,
specified via a convex, quadratically constrained quadratic
program (QCQP), that replaces the standard affine constraint
on the time derivative of the CLF with a quadratic constraint
on the decrease of the CLF over a sample period (as approxi-
mated by the Euler discrete time model). We demonstrate the
improved performance of this controller over continuous time
CLF formulations with sample frequency limitations.
II. P
RELIMINARIES
Throughout this work, we will consider the nonlinear con-
trol system governed by the differential equation:
̇
x
=
f
(
x
) +
g
(
x
)
u
,
(1)
arXiv:2103.03937v1 [eess.SY] 5 Mar 2021
for state signal
x
and control input signal
u
taking values in
R
n
and
R
m
, respectively, drift dynamics
f
:
R
n
R
n
, and
actuation matrix function
g
:
R
n
R
n
×
m
. Consider an open
subset
Z ⊆
R
n
×
R
m
and its projection onto the state space
X
,
π
1
(
Z
)
R
n
. Assume there exists
T
max
R
++
such
that for every state-input pair
(
x
0
,
u
0
)
∈ Z
, there exists a
unique solution
φ
: [0
,T
max
]
R
n
satisfying:
̇
φ
(
t
) =
f
(
φ
(
t
)) +
g
(
φ
(
t
))
u
0
t
(0
,T
max
)
,
(2)
φ
(0) =
x
0
.
(3)
This enables the following reachable set definition:
D
,
{
x
R
n
| ∃
(
x
0
,
u
0
)
∈Z
, t
[0
,T
max
] s
.
t
.
x
=
φ
(
t
)
}
,
where
X ⊆D
. Given an
h
(0
,T
max
]
, we define a controller
k
:
X →
R
m
as
h
-admissible
if for any state
x
0
∈ X
, the
state-input pair
(
x
0
,
k
(
x
0
))
satisfies
(
x
0
,
k
(
x
0
))
∈Z
and the
corresponding solution
φ
satisfies
φ
(
t
)
∈X
for all
t
[0
,h
]
.
Remark
1
.
This requirement on
h
-admissible controllers en-
sures that in the sampled-data context, the evolution of the
system may be described by iterative solutions to (2)-(3). For
many systems, the assumption that an
h
-admissible controller
keeps the system’s state in the set
X
is relatively weak as
X
is defined to ensure the continued existence of solutions rather
than reflecting a task-specific set that must be kept invariant.
In many cases verifying
h
-admissibility of a controller may be
intractable, but the assumption of
h
-admissibility of a given
controller is often weak in practice.
Feedback linearization offers a tool for the synthesis of
stabilizing controllers for nonlinear continuous time systems,
and will serve an important role in constructing optimization-
based controllers for sampled-data nonlinear systems. We
make use of the following abbreviated definition, but more
details may be found in [1]:
Definition 1
(
Feedback Linearizability
)
.
The system (1) is
feedback linearizable
if there exist dimensions
k,γ
N
with
k
m
and
γ
n
, an open set
E ⊆
R
n
such that
D ⊆ E
, a
transformation
Φ
:
E →
R
n
that is a diffeomorphism between
E
and an open subset of
R
n
, a controller
k
fbl
:
R
k
R
m
,
a controllable pair
(
A
,
B
)
R
γ
×
γ
×
R
γ
×
k
, and a function
q
:
Φ
(
D
)
R
n
γ
satisfying:
D
Φ
(
x
)(
f
(
x
) +
g
(
x
)
k
fbl
(
x
,
v
)) =
[
A
η
+
Bv
q
(
ξ
)
]
,
(4)
for all states
x
∈ X
and auxiliary control inputs
v
R
k
,
where
η
R
γ
,
z
R
n
γ
, and
ξ
R
n
satisfy
(
η
,
z
) =
ξ
=
Φ
(
x
)
. Note that if
γ
=
n
, the system is full-state feedback
linearizable, and the function
q
does not appear in (4). The
corresponding system in
normal form
is governed by:
̇
ξ
,
[
̇
η
̇
z
]
=
[
f
η
(
ξ
)
q
(
ξ
)
]
+
[
g
η
(
ξ
)
0
n
γ
]
u
=
f
ξ
(
ξ
) +
g
ξ
(
ξ
)
u
,
(5)
for
normal state
signal
ξ
,
output
signal
η
,
zero-coordinate
signal
z
, and control input signal
u
, with
f
η
:
Φ
(
D
)
R
γ
and
g
η
:
Φ
(
D
)
R
γ
×
m
defined such that
f
ξ
:
Φ
(
D
)
R
n
and
g
ξ
:
Φ
(
D
)
R
n
×
m
satisfy:
D
Φ
(
Φ
1
(
ξ
))(
f
(
Φ
1
(
ξ
)) +
g
(
Φ
1
(
ξ
))
u
) =
f
ξ
(
ξ
) +
g
ξ
(
ξ
)
u
,
for all
ξ
Φ
(
D
)
and
u
R
m
.
Remark
2
.
As shown in [17], feedback linearizability of a
continuous time system does not guarantee feedback lineariz-
ability of the resulting sampled-data system, even when using
approximate discrete time models. In particular, this property
may be lost due to a change of coordinates. The preservation
of this property motivates studying the evolution of the normal
form system in the sampled-data context.
As we will consider the control design process for the
normal form system (5), it is useful to define the set:
Z
ξ
=
{
(
ξ
,
u
)
Φ
(
X
)
×
R
m
|
(
Φ
1
(
ξ
)
,
u
)
∈Z}
,
(6)
noting that for every state-input pair
(
ξ
0
,
u
0
)
∈ Z
ξ
, there
exists a unique solution
ψ
: [0
,T
max
]
R
n
satisfying:
̇
ψ
(
t
) =
f
ξ
(
ψ
(
t
)) +
g
ξ
(
ψ
(
t
))
u
0
t
(0
,T
max
)
,
(7)
ψ
(0) =
ξ
0
.
(8)
For
h
(0
,T
max
]
, a controller
k
:
Φ
(
X
)
R
m
is an
h
-
admissible controller if the corresponding controller
k
:
X →
R
m
given by
k
(
x
) =
k
(
Φ
(
x
))
for all
x
∈X
is
h
-admissible.
A controller
k
aux
:
Φ
(
X
)
R
m
is an
h
-admissible auxiliary
controller
if
k
:
Φ
(
X
)
R
m
given by:
k
(
ξ
) =
k
fbl
(
Φ
1
(
ξ
)
,
k
aux
(
ξ
))
,
(9)
is an
h
-admissible controller.
III. S
AMPLED
-D
ATA
C
ONTROL
This section provides a review of the sampled-data control
setting, in which inputs are applied to the system with a zero-
order hold. In this setting, the set of possible sample periods
is given by
I
= (0
,T
max
]
. Given a sample period
h
I
and
an
h
-admissible controller
k
:
Φ
(
X
)
R
m
, the normal state
and control input signals in (5) satisfy:
u
(
t
) =
k
(
ξ
(
t
k
))
t
[
t
k
,t
k
+1
)
,
(10)
with sample times satisfying
t
k
+1
t
k
=
h
for all
k
Z
+
. In
this setting, the evolution of the system over a sample period
is given by the
exact state discrete map
F
e,
x
h
:
Z → D
and
exact normal discrete map
F
e,
ξ
h
:
Z
ξ
Φ
(
D
)
, defined as:
F
e,
x
h
(
x
0
,
u
0
) =
x
0
+
h
0
[
f
(
φ
(
τ
)) +
g
(
φ
(
τ
))
u
0
] d
τ,
(11)
F
e,
ξ
h
(
ξ
0
,
u
0
) =
ξ
0
+
h
0
[
f
ξ
(
ψ
(
τ
)) +
g
ξ
(
ψ
(
τ
))
u
0
] d
τ,
(12)
for all state-input pairs
(
x
0
,
u
0
)
∈Z
and all normal state-input
pairs
(
ξ
0
,
u
0
)
∈Z
ξ
. The exact maps are related by:
F
e,
ξ
h
(
ξ
0
,
u
0
) =
Φ
(
F
e,
x
h
(
Φ
1
(
ξ
0
)
,
u
0
))
,
(13)
for all normal state-input pairs
(
ξ
0
,
u
0
)
∈Z
ξ
.
Remark
3
.
While an equivalence between the exact state
discrete map and exact normal discrete map is achieved via
the diffeomorphism
Φ
, it is useful to define both maps as the
notion of stability we consider for sampled-data systems is
defined for a particular exact map.
We call a family of controllers
{
k
h
:
Φ
(
X
)
R
m
|
h
I
}
a
family of admissible controllers
if there is an
h
I
such
that for each
h
(0
,h
)
,
k
h
is
h
-admissible. This enables the
following definition:
Definition 2
(
Exact Families
)
.
For a family of admissible
controllers
{
k
h
:
Φ
(
X
)
R
m
|
h
I
}
, we define the
exact
state family
{
(
k
h
Φ
,
F
e,
x
h
)
|
h
I
}
and
exact normal family
{
(
k
h
,
F
e,
ξ
h
)
|
h
I
}
of controller-map pairs.
For all
h
I
such that
k
h
is
h
-admissible, the recursion
ξ
k
+1
=
F
e,
ξ
h
(
ξ
k
,
k
h
(
ξ
k
))
Φ
(
X
)
is well-defined for all
ξ
0
Φ
(
X
)
and
k
Z
+
. In practice, closed-form expressions
for these maps are rarely obtainable, suggesting the use of
approximations in the control synthesis process. While there
are many approaches to approximating this map, we will use
the following approximation of the exact normal discrete map:
Definition 3
(
Euler Approximation Family
)
.
For every sample
period
h
I
, define the map
F
a,
ξ
h
:
Z
ξ
R
n
as:
F
a,
ξ
h
(
ξ
0
,
u
0
) =
ξ
0
+
h
(
f
ξ
(
ξ
0
) +
g
ξ
(
ξ
0
)
u
0
)
,
(14)
for all
(
ξ
0
,
u
0
)
∈ Z
ξ
. For a family of admissible controllers
{
k
h
:
Φ
(
X
)
R
m
|
h
I
}
, the corresponding
Euler
approximation family
of controller-map pairs is:
{
(
k
h
,
F
a,
ξ
h
)
|
h
I
}
.
(15)
The motivation behind this particular approximation is pre-
serving the strict feedback nature [23] of the normal form.
For
h
I
, we can also define
F
a,
η
h
:
Z
ξ
R
γ
and
F
a,
z
h
:
Φ
(
X
)
R
n
γ
such that for all
(
ξ
,
u
) = ((
η
,
z
)
,
u
)
∈Z
ξ
:
F
a,
ξ
h
(
ξ
,
u
) =
[
F
a,
η
h
(
ξ
,
u
)
F
a,
z
h
(
ξ
)
]
=
[
η
+
h
(
f
η
(
ξ
) +
g
η
(
ξ
)
u
)
z
+
h
q
(
ξ
)
]
.
Remark
4
.
Under the current assumptions, there may be an
h
I
such that the controller
k
h
is
h
-admissible but the
recursion
ξ
k
+1
=
F
a,
ξ
h
(
ξ
k
,
k
h
(
ξ
k
))
is not well-defined for all
ξ
0
Φ
(
X
)
and
k
Z
+
. This is due to the definition of this
map enabling
ξ
k
/
Φ
(
X
)
for some
k >
0
. While our results
do not need the recursion of the Euler approximation family be
well-defined, this can be achieved by extending the domains
of
f
ξ
,
g
ξ
, and
k
h
to
R
n
.
Defining class
K
(
K
) and
KL
(
KL
) comparison func-
tions as in [2], [8], the following definition characterizes how
accurately an approximate map captures the exact map:
Definition 4
(
One-Step Consistency
)
.
A family
{
(
k
h
,
F
h
) :
h
I
}
is
one-step consistent
with
{
(
k
h
,
F
e,
ξ
h
)
|
h
I
}
if, for
each compact set
K
Φ
(
X
)
, there exist a function
ρ
∈K
and
h
I
such that for all
ξ
K
and
h
(0
,h
)
, we have:
F
e,
ξ
h
(
ξ
,
k
h
(
ξ
))
F
h
(
ξ
,
k
h
(
ξ
))
‖≤
(
h
)
.
(16)
The following lemma (slightly modified from Lemma 1 in
[8], with proof in the appendix) relates the Euler approxima-
tion family and one-step consistency:
Lemma 1.
Suppose
f
ξ
and
g
ξ
are locally Lipschitz continuous
on
Φ
(
X
)
. Consider a family of admissible controllers
{
k
h
:
Φ
(
X
)
R
m
|
h
I
}
and suppose that for any compact set
K
Φ
(
X
)
there exist
h
I
and a bound
M
R
++
such
that for every sample time
h
(0
,h
)
, the controller
k
h
is
bounded by
M
on
K
. Then the family
{
(
k
h
,
F
a,
ξ
h
)
|
h
I
}
is one-step consistent with the family
{
(
k
h
,
F
e,
ξ
h
)
|
h
I
}
.
We note that if
f
,
g
, and
D
Φ
are locally Lipschitz continuous
on
X
, the first condition of Lemma 1 is met. We consider the
following stability property, defined for both the exact state
discrete map and the exact normal discrete map:
Definition 5
(
Practical Stability
)
.
Let
β
∈KL
and
N
R
n
be an open set containing the origin. A family
{
(
k
h
,
F
h
) :
h
I
}
is
(
β,N
)
-practically stable if for each
R
R
++
,
there exists an
h
I
such that for each sample period
h
(0
,h
)
, initial state
ζ
0
N
, and number of steps
k
Z
+
, the
recursion
ζ
k
+1
=
F
h
(
ζ
k
,
k
h
(
ζ
k
))
is well-defined and:
ζ
k
‖≤
β
(
ζ
0
,kh
) +
R.
(17)
The following lemma relates the practical stability of the
exact normal family and the exact state family. Importantly,
it justifies considering the sampled normal form dynamics,
which can be feedback linearized, rather than the sampled state
dynamics that may not be feedback linearizable.
Lemma 2.
Suppose that
0
n
∈X
and
Φ
(
0
n
) =
0
n
, and that
for any compact sets
K,K
R
n
,
Φ
and
Φ
1
are globally
Lipschitz continuous on
K
∩X
and
K
Φ
(
X
)
, respectively.
If the exact normal family
{
(
k
h
,
F
e,
ξ
h
)
|
h
I
}
is
(
β,N
)
-
practically stable, then there exist
β
∈KL
and a bounded
open set
N
R
n
with
0
n
N
such that the exact state
family
{
(
k
h
Φ
,
F
e,
x
h
)
|
h
I
}
is
(
β
,N
)
-practically stable.
A proof is provided in the appendix. The following class of
Lyapunov functions is useful in certifying practical stability:
Definition 6
(
Asymptotic Stability by Equi-Lipschitz Lyapunov
Functions
)
.
Consider a family of admissible controllers
{
k
h
:
Φ
(
X
)
R
m
|
h
I
}
. A family
{
(
k
h
,
F
h
)
|
h
I
}
is
asymptotically stable by equi-Lipschitz Lyapunov functions
if
for some open set
N
Φ
(
X
)
containing the origin and any
compact set
K
N
, there exist
h
I
, comparison functions
α
1
2
∈ K
and
α
3
∈ K
, a family
{
V
h
:
R
n
R
+
|
h
(0
,h
)
}
, and a Lipschitz constant
M
R
++
such that:
α
1
(
ξ
1
)
V
h
(
ξ
1
)
α
2
(
ξ
1
)
,
(18)
V
h
(
F
h
(
ξ
2
,
k
h
(
ξ
2
)))
V
h
(
ξ
2
)
≤−
3
(
ξ
2
)
,
(19)
|
V
h
(
ξ
3
)
V
h
(
ξ
4
)
|≤
M
ξ
3
ξ
4
,
(20)
for all
ξ
1
R
n
, normal states
ξ
2
N
and
ξ
3
,
ξ
4
K
, and
sample times
h
(0
,h
)
.
These functions serve to connect one-step consistency and
practical stability, given by the following result (a local variant
of Theorem 2 in [8], with proof in the appendix):
Lemma 3.
Consider a family of admissible controllers
{
k
h
:
Φ
(
X
)
R
m
|
h
I
}
. If the corresponding Euler approxi-
mation family
{
(
k
h
,
F
a,
ξ
h
)
|
h
I
}
is asymptotically stable by
equi-Lipschitz Lyapunov functions, then there exist
β
∈KL
and a bounded open set
U
Φ
(
X
)
with
0
n
U
such that
the family
{
(
k
h
,
F
e,
ξ
h
)
|
h
I
}
is
(
β,N
)
-practically stable
for any open set
N
U
with
0
n
N
.
IV. S
TABILIZATION
In this section we present our main results establishing
feedback linearization as a method for practically stabilizing
sampled-data nonlinear systems. The first result builds on [24]
to make a claim on the stabilizability of the output dynamics:
Lemma 4.
Given a feedback linearizable system satisfying
(4)
,
consider
K
R
k
×
γ
such that
A
cl
,
A
BK
is Hurwitz. Let
P
η
S
γ
++
solve the continuous time Lyapunov Equation:
A
>
cl
P
η
+
P
η
A
cl
=
Q
η
,
(21)
for some
Q
η
S
γ
++
. Define the function
V
η
:
R
γ
R
+
as
V
η
(
η
) =
η
>
P
η
η
for all
η
R
γ
. For any
c
(0
,
1)
, there
exists
h
η
I
such that for any
η
0
R
γ
,
ξ
= (
η
,
z
)
Φ
(
X
)
,
and
h
(0
,h
η
)
, there exists an input
u
R
m
such that:
λ
min
(
P
η
)
η
0
2
2
V
η
(
η
0
)
λ
max
(
P
η
)
η
0
2
2
,
(22)
V
η
(
F
a,
η
h
(
ξ
,
u
))
V
η
(
η
)
≤−
hcλ
min
(
Q
η
)
η
2
2
.
(23)
Proof.
The bounds in (22) follow from the definition of
V
η
.
Define the auxiliary controller
k
aux
((
η
,
z
)) =
K
η
for all
(
η
,
z
)
Φ
(
X
)
. For the controller
k
defined in (9), we have:
V
η
(
F
a,
η
h
(
ξ
,
k
(
ξ
)))
V
η
(
η
) =
V
η
((
I
γ
+
h
A
cl
)
η
)
V
η
(
η
)
=
h
η
>
(
A
>
cl
P
η
+
P
η
A
cl
+
h
A
>
cl
P
η
A
cl
)
η
≤−
h
(
λ
min
(
Q
η
)
max
(
A
>
cl
P
η
A
cl
))
η
2
2
,
for all
ξ
= (
η
,
z
)
Φ
(
X
)
and
h
I
. Picking
h
η
I
with:
h
η
(1
c
)
λ
min
(
Q
η
)
λ
max
(
A
>
cl
P
η
A
cl
)
,
(24)
implies that for all
h
(0
,h
η
]
and
ξ
Φ
(
X
)
, the input
k
(
ξ
)
satisfies (23).
We call the function
V
η
a discrete time Control Lyapunov
Function (CLF) for any Euler approximate model of the output
dynamics with
h
(0
,h
η
]
. For each
h
(0
,h
η
)
, define the
set-valued function
U
h
:
Φ
(
X
)
→P
(
R
m
)
as:
U
h
(
ξ
) =
{
u
R
m
|
(
ξ
,
u
)
∈Z
ξ
;
u
satisfies
(23)
for
h,
ξ
}
,
for all
ξ
Φ
(
X
)
. The next result connects these functions to
continuous time stability of the zero-dynamics, implying the
conditions of Lemma 3 are met for a wider class of controllers
than feedback linearizing controllers:
Theorem 1.
Let
V
η
and
h
η
be defined as in Lemma 4, and
assume that
q
is continuously differentiable and the zero-
dynamics system governed by the differential equation:
̇
z
=
q
(
0
γ
,
z
)
,
(25)
for zero-coordinate signal
z
is locally exponentially stable to
the origin. Let
{
k
h
:
Φ
(
X
)
R
m
|
h
I
}
be a family of
admissible controllers satisfying
k
h
(
ξ
)
∈ U
h
(
ξ
)
for all
h
(0
,h
η
]
and
ξ
Φ
(
X
)
. Then the family
{
(
k
h
,
F
a,
ξ
h
)
|
h
I
}
is asymptotically stable by equi-Lipschitz Lyapunov functions.
Proof.
The local exponential stability of (25) implies that for
any
Q
z
S
n
++
and
d
(0
,
1)
, there exist an open neighbor-
hood of the origin
N
R
n
γ
, an
h
z
I
, a
P
z
S
n
++
, and
a quadratic Lyapunov function
V
z
:
R
n
γ
R
+
defined as
V
z
(
z
) =
z
>
P
z
z
for all
z
R
n
γ
and satisfying:
λ
min
(
P
z
)
z
0
2
2
V
z
(
z
0
)
λ
max
(
P
z
)
z
0
2
2
,
(26)
V
z
(
F
a,
z
h
((
0
γ
,
z
)))
V
z
(
z
)
≤−
hdλ
min
(
Q
z
)
z
2
2
,
(27)
for all
z
0
R
n
γ
,
z
N
, and
h
(0
,h
z
)
. Construction of
V
z
follows the steps of Lemma 4 with the linearization of
q
at
the origin. Let
σ
R
++
be a coefficient to be specified later.
Define the composite Lyapunov function
V
:
R
n
R
+
as:
V
(
ξ
) =
σV
η
(
η
) +
V
z
(
z
)
,
(28)
for all
ξ
= (
η
,
z
)
R
n
. First, note that:
min
{
σλ
min
(
P
η
)
min
(
P
z
)
}‖
ξ
2
2
V
(
ξ
)
max
{
σλ
max
(
P
η
)
max
(
P
z
)
}
︷︷
,
μ
ξ
2
2
,
(29)
for all
ξ
R
n
. Second, note that:
‖∇
V
(
ξ
)
2
2 (
σλ
max
(
P
η
)
η
2
+
λ
max
(
P
z
)
z
2
)
2(
μ
ξ
2
+
μ
ξ
2
) = 4
μ
ξ
2
,
for all
ξ
= (
η
,
z
)
Φ
(
X
)
, implying that for any compact set
K
Φ
(
X
)
, we have:
|
V
(
ξ
1
)
V
(
ξ
2
)
|≤
4
μ
(
max
ξ
K
ξ
2
)
ξ
1
ξ
2
2
,
(30)
for all
ξ
1
,
ξ
2
K
. Third, define a bounded open set
N
ξ
R
n
with closure
cl(
N
ξ
)
Φ
(
X
)
(
R
γ
×
N
)
, let
L
q
R
++
be a global Lipschitz constant of
q
on
N
ξ
, and let
h
1
=
min
{
h
η
,h
z
}
. For all
ξ
= (
η
,
z
)
N
ξ
and
h
(0
,h
1
)
, note:
V
(
F
a,
ξ
h
(
ξ
,
k
h
(
ξ
)))
V
(
ξ
)
=
σ
(
V
η
(
F
a,
η
h
(
ξ
,
k
h
(
ξ
)))
V
η
(
η
)) +
V
z
(
F
a,
z
h
(
ξ
))
V
z
(
z
)
≤−
σhcλ
min
(
Q
η
)
η
2
2
+
V
z
(
F
a,
z
h
((
0
γ
,
z
)))
V
z
(
z
)
+
V
z
(
F
a,
z
h
((
η
,
z
)))
V
z
(
F
a,
z
h
((
0
γ
,
z
)))
≤−
σhcλ
min
(
Q
η
)
η
2
2
hdλ
min
(
Q
z
)
z
2
2
+ 2
h
z
>
P
z
(
q
((
η
,
z
))
q
((
0
γ
,
z
)))
+
h
2
(
q
((
η
,
z
))
>
P
z
q
((
η
,
z
))
q
((
0
γ
,
z
))
>
P
z
q
((
0
γ
,
z
)))
≤−
σhcλ
min
(
Q
η
)
η
2
2
hdλ
min
(
Q
z
)
z
2
2
+ 2
max
(
P
z
)
L
q
η
2
z
2
+
h
2
λ
max
(
P
z
)
L
q
ξ
2
2
=
h
[
η
2
z
2
]
>
[
ω
η
(
σ,h
)
ω
×
ω
×
ω
z
(
h
)
]
︷︷
,
σ
(
h
)
[
η
2
z
2
]
,
(31)
where
ω
η
(
σ,h
) =
σcλ
min
(
Q
η
)
max
(
P
z
)
L
q
,
ω
×
=
λ
max
(
P
z
)
L
q
, and
ω
z
(
h
) =
min
(
Q
z
)
max
(
P
z
)
L
q
. Pick
h
2
(0
,h
1
]
such that
h
2
< dλ
min
(
Q
z
)
×
and fix
σ
with:
σ >
ω
2
×
z
(
h
2
) +
h
2
λ
max
(
P
z
)
L
q
min
(
Q
η
)
,
to ensure that
σ
(
h
)
S
n
++
for all
h
[0
,h
2
]
. The
composition
λ
min
σ
is continuous and
R
++
-valued for all
h
[0
,h
2
]
as
σ
is an affine function. Therefore:
V
(
F
a,
ξ
h
(
ξ
,
k
h
(
ξ
)))
V
(
ξ
)
≤−
min
(
σ
(
h
))
ξ
2
2
≤−
h
(
min
h
[0
,h
2
]
λ
min
(
σ
(
h
))
)
ξ
2
2
,
(32)
for all
ξ
N
ξ
and
h
(0
,h
2
]
.
This result implies the practical stability of an exact fam-
ily built using feedback linearizing controllers. Specifically,
with the controller
k
used in Lemma 4, if there exists an
h
0
I
such that
k
is
h
0
-admissible, then the exact family
{
(
k
,
F
e,
ξ
h
)
|
h
I
}
is
(
β,N
)
-practically stable for some
β
∈KL
and open set
N
Φ
(
X
)
containing the origin.
V. O
PTIMIZATION
-B
ASED
S
AMPLED
-D
ATA
C
ONTROL
As motivated in [2, p. 6], the performance of a feedback
linearizing controller can be improved upon by optimizing
control inputs subject to stability constraints imposed via the
CLF
V
η
found in Lemma 4. We note that the existence of
the feedback linearizing controller ensures the function
V
η
is also a CLF for the continuous time output dynamics in
(5). For sample period
h
I
, continuous time design yields
a controller
k
qp
h
:
Φ
(
X
)
R
m
specified by the following
quadratic program (QP):
k
qp
h
(
ξ
) = argmin
u
R
m
u
2
2
(CLF-QP)
s
.
t
.
V
η
(
η
)
>
(
f
η
(
ξ
) +
g
η
(
ξ
)
u
)
≤−
λ
min
(
Q
η
)
η
2
2
,
for all
ξ
= (
η
,
z
)
Φ
(
X
)
. This controller often displays
degradation in performance with sample frequency limitations,
motivating the specification of a sampled-data controller. For
h
(0
,h
η
]
, using the Euler approximate model
F
a,
η
h
, consider
a controller
k
qcqp
h
:
Φ
(
X
)
R
m
specified by the following
quadratically constrained quadratic program (QCQP):
k
qcqp
h
(
ξ
) = argmin
u
R
m
u
2
2
(CLF-QCQP)
s
.
t
. V
η
(
F
a,
η
h
(
ξ
,
u
))
V
η
(
η
)
≤−
hcλ
min
(
Q
η
)
η
2
2
= argmin
u
R
m
u
2
2
s
.
t
.
u
>
Λ
h
(
ξ
)
u
+ 2
λ
h
(
ξ
)
>
u
+
l
h
(
ξ
)
0
,
for all
ξ
= (
η
,
z
)
Φ
(
X
)
where
Λ
h
:
Φ
(
X
)
S
m
+
,
λ
h
:
Φ
(
X
)
R
m
, and
l
h
:
Φ
(
X
)
R
are defined with
P
η
,
Q
η
,
and
c
from Lemma 4 as:
Λ
h
(
ξ
) =
h
g
η
(
ξ
)
>
P
η
g
η
(
ξ
)
,
(33)
λ
h
(
ξ
) =
g
η
(
ξ
)
>
P
η
(
η
+
h
f
η
(
ξ
))
,
(34)
l
h
(
ξ
) =
f
η
(
ξ
)
>
P
η
(2
η
+
h
f
η
(
ξ
)) +
min
(
Q
η
)
η
2
2
,
(35)
for all normal states
ξ
= (
η
,
z
)
Φ
(
X
)
. Note that for any
normal state
ξ
Φ
(
X
)
, the input
k
(
ξ
)
is in the feasible set
of the corresponding optimization problem, and as the feasible
set is closed and the
k
(
ξ
)
2
2
-sublevel set of the continuous
objective function is compact, there exists a minimizer in this
sublevel set. Moreover, since the objective function is strictly
convex and the feasible set is convex, this minimizer is unique.
For each
h
(
h
η
,T
max
]
, define
k
qcqp
h
:
Φ
(
X
)
R
m
arbitrarily. If
{
k
qcqp
h
|
h
I
}
is a family of admissible
controllers, then the exact family
{
(
k
qcqp
h
,
F
e,
ξ
h
)
|
h
I
}
is
(
β,N
)
-practically stable for some
β
∈ KL
and open set
N
Φ
(
X
)
by Theorem 1. This follows as the feasibility
of the feedback linearizing control input implies the family
{
(
k
qcqp
h
,
F
a,
ξ
h
)
|
h
I
}
is asymptotically stable by the same
Lyapunov functions as the family
{
(
k
,
F
a,
ξ
h
)
|
h
I
}
.
To illustrate the advantage of sampled-data design, consider
the following system with exponentially stable zero-dynamics:
̇
η
1
=
η
2
,
̇
η
2
= 10 sin(
η
1
) +
u,
̇
z
=
η
2
1
z,
(36)
0.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
0.00
1.00
2.00
CLF-QP
η
1
η
2
z
0.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
t
−0.25
0.00
0.25
0.50
0.75
1.00
CLF-QCQP
η
1
η
2
z
Fig. 1.
When inputs are applied with a zero-order hold, the controller
(CLF-QP) designed with continuous time models does not stabilize
the system (36) (Top), while the controller (CLF-QCQP) does (Bot-
tom). Simulation code listed at https://bit.ly/CLF-QCQP.
where
(
η
1
2
)
denote the output signal,
z
denotes the zero-
coordinate signal, and
u
denotes the control signal. For
K
=
[
1
/
2
3
/
2
]
,
Q
η
=
I
2
,
c
= 0
.
5
,
h
= 0
.
2
, and initial
condition
(1
,
0
,
1)
, the (CLF-QP) fails to stabilize the system,
while the (CLF-QCQP) achieves stability (see Figure 1).
VI. C
ONCLUSIONS
We presented an approach for sampled-data control synthe-
sis utilizing feedback linearization and CLFs. In particular, we
used the feedback linearizability of a system’s continuous time
model to yield a discrete time CLF for the Euler approximate
discrete time model. We specify a controller with this CLF
via a convex optimization problem that improves performance
over a continuous time counterpart. Future work will extend
this work to notions of safety and Control Barrier Functions.
P
ROOFS OF
L
EMMAS
Proof of Lemma 1.
Consider a compact set
K
Φ
(
X
)
and
corresponding
h
I
and
M
R
++
, and fix a sample period
h
(0
,h
)
. By assumption,
k
h
is bounded on
K
, and since
f
ξ
and
g
ξ
are continuous,
f
ξ
and
g
ξ
are also bounded on
K
,
implying there exists a bound
M
R
++
with:
f
ξ
(
ξ
2
) +
g
ξ
(
ξ
2
)
k
h
(
ξ
1
)
‖≤
M
,
for all normal states
ξ
1
,
ξ
2
K
. As
f
ξ
and
g
ξ
are locally
Lipschitz continuous over the compact set
K
, it follows that
f
ξ
and
g
ξ
are globally Lipschitz continuous over
K
. Therefore:
f
ξ
(
ξ
2
) +
g
ξ
(
ξ
2
)
k
h
(
ξ
1
)
(
f
ξ
(
ξ
1
) +
g
(
ξ
1
)
k
h
(
ξ
1
))
≤‖
f
ξ
(
ξ
2
)
f
ξ
(
ξ
1
)
+
g
ξ
(
ξ
2
)
g
ξ
(
ξ
1
)
‖‖
k
h
(
ξ
1
)
(
L
f
ξ
+
L
g
ξ
M
)
ξ
2
ξ
1
=
ρ
(
ξ
2
ξ
1
)
,
for all states
ξ
1
,
ξ
2
K
, where
L
f
ξ
,L
g
ξ
R
++
are Lipschitz
constants for
f
ξ
and
g
ξ
, respectively, and
ρ
∈ K
satisfies
ρ
(
r
) = (
L
f
ξ
+
L
g
ξ
M
)
r
for all
r
R
+
. The proof proceeds as
the proof of Lemma 1 in [8] by substituting
X
=
N
(
K,
)
Φ
(
X
)
, with proper containment implied for some

R
++
as
Φ
(
X
)
is open, and substituting
T
1
= min
{
h
,/M
}
.
Proof of Lemma 2.
Let
N
Φ
1
(
N
)
be a bounded open
set satisfying cl
(
N
)
⊆ X
. As cl
(
N
)
is compact and
Φ
is a