of 8
Convex Optimization-based Controller Design for Stochastic Nonlinear
Systems using Contraction Analysis
Hiroyasu Tsukamoto and Soon-Jo Chung
Abstract
— This paper presents an optimal feedback tracking
controller for a class of It
ˆ
o stochastic nonlinear systems,
the design of which involves recasting a nonlinear system
equation into a convex combination of multiple non-unique
State-Dependent Coefficient (SDC) models. Its feedback gain
and controller parameters are found by solving a convex
optimization problem to minimize an upper bound of the
steady-state tracking error. Multiple SDC parametrizations are
utilized to provide a design flexibility to mitigate the effects of
stochastic noise and to ensure that the system is controllable.
Incremental stability of this controller is studied using stochas-
tic contraction analysis and it is proven that the controlled
trajectory exponentially converges to the desired trajectory with
a non-vanishing error due to the linear matrix inequality state-
dependent algebraic Riccati equation constraint. A discrete-
time version of stochastic contraction analysis with respect
to a state- and time-dependent metric is also presented in
this paper. A simulation is performed to show the superiority
of the proposed optimal feedback controller compared to a
known exponentially-stabilizing nonlinear controller and a PID
controller.
I. I
NTRODUCTION
The problem of designing a controller for It
ˆ
o stochastic
nonlinear systems [1] is of significant importance in control
theory as many engineering systems are nonlinear and have
to deal with stochastic uncertainty to improve the controller
performance. The probability density function of stochastic
processes that are governed by It
ˆ
o stochastic differential
equations exhibits non-Gaussian behavior characterized by
the Fokker-Plank equation [1], [2], which makes the con-
troller design particularly difficult.
One approach to dealing with stochastic disturbances is to
ignore or model them as deterministic disturbances in order
to design a deterministic controller using feedback lineariza-
tion [3]–[5] or Control Lyapunov Functions (CLFs) [5]–[7].
For a nonlinear system with a polynomial or rational vector
field, the process of finding a CLF can be formulated as
sum of squares programming [8], [9]. The State-Dependent
Riccati Equation (SDRE) method [10], [11] can be used for
systems that are written in SDC linear structure.
These controllers, however, could violate control require-
ments for stochastically perturbed systems. The proposed
controller, in contrast, is based on the SDRE method and
guarantees exponential stability of It
ˆ
o stochastic nonlinear
systems with some bounded error.
There are some controllers specifically designed for
stochastic nonlinear systems as in this paper, one of which
The authors are with the Graduate Aerospace Laboratories (GALCIT),
California Institute of Technology, 1200 E California Blvd, Pasadena, CA,
USA. E-mail:
{
htsukamoto, sjchung
}
@caltech.edu
is a Lyapunov-like technique. This includes [12], which
provides sufficient conditions for the existence of stabilizing
feedback laws. In [13], a CLF-based controller design for
stochastic nonlinear systems with unknown covariance is
presented. A backstepping-based controller [14], [15] is
shown to be asymptotically stabilizing in probability for a
class of stochastic nonlinear systems in strict-feedback or
output-feedback form. A controller design for Hamiltonian
systems with stochastic disturbance is also well-studied [16].
An
H
control problem for a class of stochastic nonlinear
systems with both state- and disturbance-dependent noise
is discussed in [17]. In the context of optimal control, the
maximum principle for stochastic nonlinear optimal control
problems in the general case is studied in [18]. In [19], a
locally-optimal feedback control law for stochastic nonlinear
systems is derived by applying a Linear Quadratic Gaussian
(LQG) method iteratively to a linearized and discretized
dynamic model. For stochastic dynamic programs [1], [20],
some prior works have proposed different ways to overcome
the issues arising from the curse of dimensionality [21], [22].
Stochastic Model Predictive Control (SMPC) [23], [24] has
been derived to incorporate the probabilistic descriptions of
uncertainties into a stochastic optimal control problem.
Unlike these controllers developed for stochastic nonlinear
systems, the proposed feedback controller is different in
that it can be used for any It
ˆ
o stochastic nonlinear systems
written in SDC form and also is optimal in a sense that it
minimizes an upper bound of the steady-state tracking error
in a stochastic manner. This choice of the objective function
is more advantageous than integral objective functions as it
allows us to write the controller gain synthesis algorithm as
a convex optimization problem.
The tools to analyze incremental stability [25] of nonlinear
systems have been developed in response to emerging needs
in proving stability of trajectories instead of stability of
an equilibrium point. Contraction analysis [26], [27] is one
effective tool for studying incremental stability properties.
The stochastic version of contraction analysis is derived
in [28] with an application to the specialized context of
state-independent metrics. In [29], the incremental stability
of stochastic systems using more generalized metric func-
tions that are both state- and time-dependent is studied.
Contraction analysis for discrete-time and hybrid systems is
provided in [26], [30], [31] and its stochastic counterpart
is investigated in [32] with respect to a state-independent
metric. In this paper, we describe more general discrete-time
incremental contraction analysis with respect to a state- and
time-dependent metric.
The objective of this paper is to present a novel optimal
tracking control approach to a class of It
ˆ
o stochastic nonlin-
ear systems with guaranteed exponential stability, which in
turns results in a superior property of robustness, similar to
finite-gain
L
p
stability of a deterministic system. There are
three major contributions of this paper.
1) A convex optimization-based algorithm to select the
optimal feedback gain and controller parameters at each
time instant is proposed in Sec. IV for the control of It
ˆ
o
stochastic nonlinear systems. The design is based on non-
unique SDC forms and the control law is assumed to be
a simple state feedback. The main idea of this algorithm
lies in solving an optimization problem, the objective of
which is to minimize an upper bound of the steady-state
tracking error while satisfying the SDRE. As an attempt to
avoid computational complexity in real-time operation, the
original nonlinear optimization problem is reformulated as
a convex optimization problem, with the SDRE constraint
relaxed to an Linear Matrix Inequality (LMI) constraint with
some additional constraints [33]. The feedback gain and
controller parameters, including the coefficients of a convex
combination of multiple SDC parameterizations are selected
as a result of the convex optimization problem, which can
be solved using various computationally-efficient numerical
methods [33]–[35]. The superiority of this controller to the
prior work [36] and to a PID controller is shown using results
of a numerical simulation in Sec. V. Note that this con-
troller design is a significant improvement over the observer
design [29], whose optimization-cost function uses a linear
combination of controller parameters without accounting for
the contraction constraint.
2) Stability of the feedback tracking controller is studied
using stochastic incremental contraction analysis with a state-
and time-dependent Riemannian metric introduced in [29].
It is proved in Sec. III that the trajectory of the controlled
dynamics exponentially converges to the desired trajectory
with some known non-vanishing tracking error, which will
be minimized by solving the convex optimization problem
in Sec. IV. Multiple SDC parameterizations of the nonlinear
system provide a controller design flexibility to mitigate the
effect of stochastic noise while verifying that the system is
controllable.
3) A discrete-time version of stochastic contraction anal-
ysis with respect to a state- and time-dependent metric is
derived in Sec. II, which can be used for proving incremental
stability of discrete-time and hybrid stochastic nonlinear
systems, along with the known results for deterministic
systems [30], [31].
II. C
ONTRACTION
A
NALYSIS
In this section, we introduce some preliminaries that will
be used for the stability analysis in Sec. III. We also present
a new theorem for analyzing stochastic incremental stability
of discrete-time stochastic systems with respect to a state-
and time-dependent Riemannian metric.
A. Continuous-time Dynamical Systems
Consider the following continuous-time nonlinear system
and its virtual dynamics
̇
x
=
f
(
x
,
t
)
,
δ
̇
x
=
f
(
x
,
t
)
x
δ
x
.
(1)
Lemma 1:
The system (1) is contracting (i.e., all the
solution trajectories exponentially converge to a single tra-
jectory globally from any initial condition), if there exists a
uniformly positive definite metric
M
(
x
,
t
) =
Θ
(
x
,
t
)
T
Θ
(
x
,
t
)
,
M
(
x
,
t
)

m
I
,
x
,
t
, with a smooth coordinate transformation
of the virtual displacement
δ
z
=
Θ
(
x
,
t
)
δ
x
s.t.
f
x
T
M
(
x
,
t
)+
̇
M
(
x
,
t
)+
M
(
x
,
t
)
f
x
−
2
γ
M
(
x
,
t
)
,
x
,
t
(2)
where
γ
and
m
are some positive constants. If the system (1)
is contracting, then the following equation holds
δ
z
(
t
)
=
Θ
(
x
,
t
)
δ
x
(
t
)
‖≤‖
δ
z
(
0
)
e
γ
t
.
(3)
Proof:
See [26], [37].
Next, consider the continuous-time nonlinear system (1)
with a stochastic perturbation given by an It
ˆ
o stochastic
differential equation
dx
=
f
(
x
,
t
)
dt
+
G
(
x
,
t
)
dW
,
x
(
0
) =
x
0
.
(4)
We assume that the following conditions for existence and
uniqueness of a solution to (4) hold
L
1
>
0
,
t
,
x
1
,
x
2
R
n
s
.
t
f
(
x
1
,
t
)
f
(
x
2
,
t
)
+
G
(
x
1
,
t
)
G
(
x
2
,
t
)
F
L
1
x
1
x
2
L
2
>
0
,
t
,
x
1
R
n
s
.
t
f
(
x
1
,
t
)
2
+
G
(
x
1
,
t
)
2
F
L
2
(
1
+
x
1
2
)
(5)
where
G
:
R
n
×
R
R
n
×
d
is a matrix-valued function,
W
(
t
)
is a
d
-dimensional Wiener process, and
x
0
is a random
variable independent of
W
(
t
)
. Now, consider the following
two systems with trajectories
ξ
1
(
t
)
and
ξ
2
(
t
)
driven by two
independent Wiener processes
W
1
(
t
)
and
W
2
(
t
)
d
ξ
=
[
f
(
ξ
1
,
t
)
f
(
ξ
2
,
t
)
]
dt
+
[
G
1
(
ξ
1
,
t
)
0
0
G
2
(
ξ
2
,
t
)
][
dW
1
dW
2
]
(6)
where
ξ
(
t
) = [
ξ
1
(
t
)
T
ξ
2
(
t
)
T
]
T
R
2
n
. The following lemma
[29] analyzes stochastic incremental stability of the two
trajectories
ξ
1
(
t
)
and
ξ
2
(
t
)
with respect to each other in
the presence of stochastic noise. The trajectories of (4) are
parameterized as
x
(
0
,
t
) =
ξ
1
and
x
(
1
,
t
) =
ξ
2
, and
G
1
(
ξ
1
,
t
)
and
G
2
(
ξ
2
,
t
)
are defined as
G
(
x
(
0
,
t
)
,
t
) =
G
1
(
ξ
1
,
t
)
and
G
(
x
(
1
,
t
)
,
t
) =
G
2
(
ξ
2
,
t
)
.
Lemma 2:
Assume that the system (6) has the follow-
ing bounds, tr
(
G
i
(
ξ
i
,
t
)
T
M
(
ξ
i
,
t
)
G
i
(
ξ
i
,
t
)))
C
i
,
i
=
1
,
2,
m
x
=
sup
t
0
,
i
,
j
(
M
i j
)
x
, and
m
x
2
=
sup
t
0
,
i
,
j
2
(
M
i j
)
/
x
2
,
where
C
1
,
C
2
,
m
x
, and
m
x
2
are constant. Assume also
that (2) holds (i.e., contracting) for the deterministic sys-
tem (1). Consider the generalized squared length with
respect to a Riemannian metric
M
(
x
(
μ
,
t
)
,
t
)
defined
by
V
(
x
,
δ
x
,
t
) =
1
0
(
x
/
∂ μ
)
T
M
(
x
(
μ
,
t
)
,
t
)(
x
/
∂ μ
)
d
μ
s.t.
V
(
x
,
δ
x
,
t
)
m
ξ
1
ξ
2
2
2
. Then, the mean squared distance
between the two trajectories of the system (6) whose initial
conditions given by a probability distribution
p
(
a
0
,
b
0
)
are
independent of
W
1
(
t
)
and
W
2
(
t
)
satisfies the bound
E
[
ξ
1
(
t
)
ξ
2
(
t
)
2
]
1
m
(
C
2
γ
1
+
E
[
V
(
x
(
0
)
,
δ
x
(
0
)
,
0
)]
e
2
γ
t
)
(7)
where
γ
1
=
γ
((
g
2
1
+
g
2
2
)
/
2
m
)(
ε
m
x
+
m
x
2
/
2
)
>
0,
C
=
C
1
+
C
2
+ (
m
x
/
ε
)(
g
2
1
+
g
2
2
)
,
G
1
F
g
1
,
x
,
t
, and
G
2
F
g
2
,
x
,
t
.
E
[
·
]
denotes the expected value of the operator
and
γ
is the contraction rate for the deterministic system (1).
Proof:
See [29].
B. Discrete-time Nonlinear Dynamical Systems
We have a result similar to Lemma 1 for the following
discrete-time nonlinear system and its virtual dynamics
x
k
+
1
=
f
k
(
x
k
,
k
)
,
δ
x
k
+
1
=
f
k
(
x
k
,
k
)
x
k
δ
x
k
.
(8)
Lemma 3:
The system (8) is contracting if there ex-
ists a uniformly positive definite metric
M
k
(
x
k
,
k
) =
Θ
k
(
x
k
,
k
)
T
Θ
k
(
x
k
,
k
)
,
M
k
(
x
k
,
k
)

m
I
,
x
k
,
k
, with a smooth
coordinate transformation of the virtual displacement
δ
z
k
=
Θ
k
(
x
k
,
k
)
δ
x
k
s.t.
f
k
x
k
T
M
k
+
1
(
x
k
+
1
,
k
+
1
)
f
k
x
k

(
1
γ
d
)
M
k
(
x
k
,
k
)
,
x
k
,
k
(9)
where
γ
d
(
0
,
1
)
and
m
are some positive constants. If the
system (8) is contracting, then the following equation holds
δ
z
k
=
Θ
k
(
x
k
,
k
)
δ
x
k
‖≤‖
δ
z
0
(
1
γ
d
)
k
2
.
(10)
Proof:
See [26], [31], [37].
We now present a discrete-time version of Lemma 2,
which can be extensively used for proving the stability
of discrete-time and hybrid stochastic nonlinear systems,
along with the known results for deterministic systems [30],
[31]. Consider the discrete-time nonlinear system (8) with a
stochastic perturbation modeled by the stochastic difference
equation
x
k
+
1
=
f
k
(
x
k
,
k
)+
G
k
(
x
k
,
k
)
w
k
(11)
where
G
k
:
R
n
×
R
R
n
+
d
is a matrix-valued function and
w
k
is a
d
-dimensional sequence of zero mean uncorrelated
normalized Gaussian random variables. Consider following
two systems with trajectories
ξ
1
,
k
and
ξ
2
,
k
driven by two
independent stochastic perturbation
w
1
,
k
and
w
2
,
k
ξ
k
+
1
=
[
f
k
(
ξ
1
,
k
,
k
)
f
k
(
ξ
2
,
k
,
k
)
]
+
[
G
1
,
k
(
ξ
1
,
k
,
k
)
0
0
G
2
,
k
(
ξ
2
,
k
,
k
)
][
w
1
,
k
w
2
,
k
]
(12)
where
ξ
k
= [
ξ
T
1
,
k
ξ
T
2
,
k
]
T
R
2
n
. The following theorem ana-
lyzes stochastic incremental stability for discrete-time non-
linear systems. Theorem 1 is different from [38] in that the
form of the Lyapunov function is assumed and its Rieman-
nian metric is state- and time-dependent. The trajectories of
(11) are parameterized as
x
k
(
μ
=
0
) =
ξ
1
,
k
,
x
k
(
μ
=
1
) =
ξ
2
,
k
,
w
k
(
μ
=
0
) =
w
1
,
k
,
w
k
(
μ
=
1
) =
w
2
,
k
, and
G
1
,
k
(
ξ
1
,
k
,
k
)
and
G
2
,
k
(
ξ
2
,
k
,
k
)
are defined as
G
k
(
x
k
(
μ
=
0
)
,
k
) =
G
1
,
k
(
ξ
1
,
k
,
k
)
and
G
k
(
x
k
(
μ
=
1
)
,
k
) =
G
2
,
k
(
ξ
2
,
k
,
k
)
.
Theorem 1:
Assume that the system (12) has the fol-
lowing bounds,
M
k
(
x
k
,
k
)

mI
,
x
k
,
k
and Tr
(
G
T
1
,
k
G
1
,
k
+
G
T
2
,
k
G
2
,
k
)
C
d
/
m
,
ξ
1
,
k
,
ξ
2
,
k
,
k
, where
m
and
C
d
are
some positive constants and
G
i
,
k
=
G
i
,
k
(
ξ
i
,
k
,
k
)
,
i
=
1
,
2 for notation simplicity. Assume also that (9) holds
for the discrete-time deterministic system (8). Consider
the generalized squared length with respect to a Rie-
mannian metric
M
k
(
x
k
(
μ
)
,
k
)
defined by
v
k
(
x
k
,
δ
x
k
,
k
) =
1
0
(
x
k
/
∂ μ
)
T
M
k
(
x
k
(
μ
)
,
k
)(
x
k
/
∂ μ
)
d
μ
s.t.
v
(
x
k
,
δ
x
k
,
k
)
m
ξ
1
,
k
ξ
2
,
k
2
2
. Then the mean squared distance between
the two trajectories of the system (12) satisfies the bound
E
ε
0
[
ξ
1
,
k
ξ
2
,
k
2
]
1
m
(
̃
γ
k
d
v
0
+
C
d
1
̃
γ
k
d
1
̃
γ
d
)
.
(13)
where
̃
γ
d
=
1
γ
2
(
0
,
1
)
and
γ
2
(
0
,
1
)
is a positive con-
stant that satisfies 1
γ
2
(
m
/
m
)(
1
γ
d
)
with
γ
d
denoting
the contraction rate for the deterministic system (8). The
subscript
ε
0
means that
x
0
,
δ
x
0
, and
t
0
are fixed.
Proof:
Let
v
k
=
v
k
(
x
k
,
δ
x
k
,
k
)
and
M
k
=
M
k
(
x
k
,
k
)
for
notation simplicity. Using the bounds and the incremental
system (12), we have that
v
k
+
1
m
1
0
f
k
x
k
x
k
∂ μ
+
G
k
∂ μ
w
k
2
d
μ
(14)
m
m
(
1
γ
d
)
1
0
x
k
∂ μ
T
M
k
x
k
∂ μ
d
μ
+
m
1
0
(
2
x
k
∂ μ
T
f
k
x
k
T
G
k
∂ μ
w
k
+
w
T
k
G
k
∂ μ
T
G
k
∂ μ
w
k
)
d
μ
where
f
k
=
f
k
(
x
k
,
k
)
and
G
k
=
G
k
(
x
k
,
k
)
,
i
=
1
,
2 for notation
simplicity. Taking the conditional expected value of
(
14
)
when
x
k
,
δ
x
k
, and
k
are fixed, we have that
E
ε
k
[
v
k
+
1
]
γ
m
v
k
+
mE
ε
k
[
1
0
w
T
k
G
k
∂ μ
T
G
k
∂ μ
w
k
d
μ
]
γ
m
v
k
+
mE
ε
k
[
Tr
(
w
1
,
k
w
T
1
,
k
G
T
1
,
k
G
1
,
k
)]
+
mE
ε
k
[
Tr
(
w
2
,
k
w
T
2
,
k
G
T
2
,
k
G
2
,
k
)]
γ
m
v
k
+
m
(
Tr
(
G
T
1
,
k
G
1
,
k
)
+
Tr
(
G
T
2
,
k
G
2
,
k
))
.
(15)
where
γ
m
=
m
/
m
(
1
γ
d
)
and
x
k
,
δ
x
k
, and
k
are denoted as
ε
k
. Suppose that there exists
γ
2
(
0
,
1
)
s.t.
γ
m
1
γ
2
. Using
the property
E
ε
k
2
[
v
k
] =
E
ε
k
2
[
E
ε
k
1
[
v
k
]]
, we have that
E
ε
k
2
[
v
k
]
̃
γ
2
d
v
k
2
+
C
d
+
C
d
̃
γ
d
(16)
where
̃
γ
d
=
1
γ
2
. Continuing this operation yields
m
E
ε
0
[
ξ
1
,
k
ξ
2
,
k
2
]
̃
γ
k
d
v
0
+
C
d
k
1
i
=
0
̃
γ
i
d
=
̃
γ
k
d
v
0
+
C
d
1
̃
γ
k
d
1
̃
γ
d
(17)
which gives (13) by dividing (17) by
m
.
Remark 1:
Since
̃
γ
d
(
0
,
1
)
, we have
lim
k
E
ε
0
[
ξ
1
,
k
ξ
2
,
k
2
]
C
d
m
(
1
̃
γ
d
)
=
C
d
m
γ
2
.
(18)
III. S
TABILIZING
S
TOCHASTIC
F
EEDBACK
C
ONTROLLER
This section presents an exponentially stabilizing feedback
controller for It
ˆ
o stochastic nonlinear systems, which will
be optimized in Sec. IV. Note that this design is not for
finding an optimal control trajectory, which can be used as
desired values in the present control design. Also, we focus
on a controller design for continuous-time systems from
this section, although this approach can readily be used for
discrete-time and hybrid systems using Theorem 1 presented
earlier in Sec. II.
A. Problem Formulation
Consider the following input-affine It
ˆ
o stochastic nonlinear
system
dx
=
f
(
x
,
t
)
dt
+
B
(
x
,
t
)
udt
+
G
1
(
x
,
t
)
dW
1
(19)
dx
d
=
f
(
x
d
,
t
)
dt
+
B
(
x
d
,
t
)
u
d
dt
.
(20)
where
x
d
and
u
d
are the desired trajectory and desired input
respectively, which are deterministic as they are assumed to
be given.
Remark 2:
Since ̇
x
d
f
(
x
d
,
t
)
Im
B
(
x
d
,
t
)
must hold
for a feasible desired trajectory,
u
d
can be obtained as
u
d
=
B
(
x
d
,
t
)
+
(
̇
x
d
f
(
x
d
,
t
))
where
(
·
)
+
denotes the Moore-
Penrose inverse. When Ker
B
(
x
d
,
t
) =
{
0
}
, then
u
d
is
the unique solution to
B
(
x
d
,
t
)
u
d
=
̇
x
d
f
(
x
d
,
t
)
. When
Ker
B
(
x
d
,
t
)
6
=
{
0
}
, then
u
d
is a solution with the smallest
Euclidean norm. Also,
u
d
can be found by solving some
optimal control problem [39]. Further, general systems with
̇
x
=
f
(
x
,
u
)
can be transformed into an input-affine form by
treating ̇
u
as another input.
In SDC form, (19) and (20) can be expressed as
dx
=
A
(
ρ
,
x
,
t
)
xdt
+
B
(
x
,
t
)
udt
+
G
1
(
x
,
t
)
dW
1
(21)
dx
d
=
A
(
ρ
,
x
d
,
t
)
x
d
dt
+
B
(
x
d
,
t
)
u
d
dt
(22)
where
ρ
= (
ρ
1
,
···
,
ρ
s
1
)
are the coefficients of the convex
combination of SDC parameterizations, i.e.,
A
(
ρ
,
x
,
t
) =
s
1
i
=
1
ρ
i
A
i
(
x
,
t
)
. Writing the system dynamics (19) in SDC
form provides a design flexibility to mitigate effects of
stochastic noise while verifying that the system is control-
lable as shall be seen later on.
Remark 3:
Let
be the state set that is a bounded open
subset of some Euclidean space that contains the origin,
i.e., 0
R
n
. Under the assumptions
f
(
0
) =
0 and
f
(
x
)
is a continuously differentiable function of
x
on
, there
always exists at least one continuous nonlinear matrix-valued
function
A
(
x
)
on
s.t.
f
(
x
) =
A
(
x
)
x
, where
A
:
R
n
×
n
is
found by mathematical factorization and is nonunique when
n
>
1 [11].
B. Stabilizing Feedback Controller Design
A nonlinear feedback tracking controller is designed as
u
=
K
(
x
,
t
)(
x
x
d
)+
u
d
=
R
1
(
x
,
t
)
B
T
(
x
,
t
)
P
(
x
,
t
)(
x
x
d
)+
u
d
(23)
where
P
(
x
,
t
)
is a positive definite matrix which satisfies the
following equation
̇
P
(
x
,
t
) =
A
T
(
ρ
,
x
,
t
)
P
(
x
,
t
)+
P
(
x
,
t
)
A
(
ρ
,
x
,
t
)
+
2
α
P
(
x
,
t
)+
2
κ
P
2
(
x
,
t
)
(24)
P
(
x
,
t
)
B
(
x
,
t
)
R
1
(
x
,
t
)
B
T
(
x
,
t
)
P
(
x
,
t
)
.
Define
A
cl
(
ρ
,
q
,
t
)
,
A
(
ρ
,
q
,
t
)
, and
B
(
q
,
t
)
as
A
cl
(
ρ
,
q
,
t
) =
A
(
ρ
,
q
+
x
d
,
t
)
B
(
q
+
x
d
,
t
)
K
(
q
+
x
d
,
t
)
A
(
ρ
,
q
,
t
) =
A
(
ρ
,
q
+
x
d
,
t
)
A
(
ρ
,
x
d
,
t
)
(25)
B
(
q
,
t
) =
B
(
q
+
x
d
,
t
)
B
(
x
d
,
t
)
.
Substituting (23) into (21) yields
de
=
f
e
(
e
,
t
)
dt
+
G
1
(
e
+
x
d
,
t
)
dW
1
(26)
where
e
=
x
x
d
and
f
e
(
e
,
t
) =
A
cl
(
ρ
,
e
,
t
)
e
+
A
(
ρ
,
e
,
t
)
x
d
+
B
(
e
,
t
)
u
d
.
(27)
Let us define a virtual system for the deterministic system
̇
q
=
A
cl
(
ρ
,
e
,
t
)
q
+
A
(
ρ
,
q
,
t
)
x
d
+
B
(
q
,
t
)
u
d
=
f
v
(
q
,
t
)
.
(28)
Note that (28) has
e
and 0 as particular solutions. The virtual
dynamics of (28) is expressed as
δ
̇
q
=
A
cl
(
ρ
,
e
,
t
)
δ
q
+
(
Ax
d
+
Bu
d
)
q
δ
q
.
(29)
Using
f
v
(
q
,
t
)
, the virtual system for (26) with respect
q
is
defined as
dq
=
f
v
(
q
(
μ
,
t
)
,
t
)
dt
+
G
(
q
(
μ
,
t
)
,
t
)
dW
(30)
where
μ
[
0
,
1
]
is introduced to parameterize the trajectories
q
=
e
and
q
=
0, i.e.,
q
(
μ
=
0
,
t
) =
e
,
q
(
μ
=
1
,
t
) =
0,
G
(
q
(
0
,
t
)
,
t
) =
G
1
(
e
+
x
d
,
t
)
, and
G
(
q
(
1
,
t
)
,
t
) =
0
n
×
d
.
Remark 4:
(30) has
q
=
e
and
q
=
0 as particular solutions.
f
v
(
e
,
t
) =
f
e
(
e
,
t
)
and
G
(
e
,
t
) =
G
1
(
e
+
x
d
,
t
)
when
q
=
e
.
f
v
(
0
,
t
) =
A
(
ρ
,
0
,
t
)
x
d
+
B
(
0
,
t
)
u
d
=
0 and
G
(
0
,
t
) =
0
n
×
d
when
q
=
0.
Before analyzing the stability of the system with this con-
troller, let us introduce the following assumption.
Assumption 1:
There exist positive constants
p
,
p
,
p
x
,
p
x
2
,
r
,
r
,
δ
1
,
δ
2
,
β
, and
g
1
s.t.
p
≤‖
P
(
x
,
t
)
‖≤
p
,
r
≤‖
R
(
x
,
t
)
‖≤
r
,
p
x
=
sup
t
0
,
i
,
j
(
p
i j
)
x
,
p
x
2
=
sup
t
0
,
i
,
j
2
(
p
i j
)
/
x
2
,
(
Ax
d
)
/
q
δ
1
,
(
Bu
d
)
/
q
δ
2
,
β
≤ ‖
B
(
x
,
t
)
,
and
G
1
(
x
,
t
)
F
g
1
,
ρ
,
x
,
t
, where
p
i j
is the (
i
,
j
) com-
ponent of
P
(
x
,
t
)
.
Now, we introduce the following theorem that guarantees the
exponential stability of the controller (23).
Theorem 2:
Suppose that Assumption 1 is satisfied and
that there exists
α
1
>
0 s.t.
(
α
α
1
)
p
κ
1
p
κ
2
p
2
κ
p
2
+
α
g
(31)
α
α
1
0
(32)
where 2
α
g
=
g
2
1
(
p
x
ε
+
p
x
2
/
2
)
with
ε
being an arbitrary
positive constant. Then the mean squared distance between
the trajectory of the system with the controller (23) and
that of the desired system is exponentially bounded with the
bound
E
[
x
d
x
2
]
1
p
(
E
[
V
(
x
(
0
)
,
δ
q
(
0
)
,
0
)]
e
2
α
1
t
+
C
2
α
1
)
(33)
where
V
(
x
,
δ
q
,
t
) =
1
0
(
q
/
∂ μ
)
T
P
(
x
,
t
)(
q
/
∂ μ
)
d
μ
and
C
=
pg
2
1
+(
p
x
g
2
1
)
/
ε
.
Proof:
For notation simplicity, let
P
=
P
(
x
,
t
)
,
A
=
A
(
ρ
,
x
,
t
)
,
B
=
B
(
x
,
t
)
,
R
=
R
(
x
,
t
)
,
G
=
G
(
q
,
t
)
, and
φ
=
φ
(
ρ
,
q
,
t
) =
(
Ax
d
)
/
q
+
(
Bu
d
)
/
q
. Let us define an
infinitesimal differential generator as
L
[
V
(
x
,
δ
q
,
t
)] =
V
t
+
n
i
=
1
(
V
x
i
f
i
+
V
(
δ
q
i
)
f
v
q
δ
q
)
+
1
2
n
i
=
1
n
j
=
1
[
2
V
x
i
x
j
(
G
1
(
x
,
t
)
G
T
1
(
x
,
t
))
i j
+
2
2
V
x
i
(
δ
q
j
)
(
G
1
(
x
,
t
)
δ
G
T
(
q
,
t
))
i j
+
2
V
(
δ
q
i
)(
δ
q
j
)
(
δ
G
(
q
,
t
)
δ
G
T
(
q
,
t
))
i j
]
(34)
where
f
i
is the
i
th component of
f
. Substituting (24), (30),
and
V
into (34) yields
L
V
=
1
0
q
∂ μ
T
(
̇
P
+
A
T
P
+
PA
2
PB
T
R
1
B
T
P
+
φ
T
P
+
P
φ
)
q
∂ μ
d
μ
+
V
2
=
1
0
q
∂ μ
T
(
2
α
P
2
κ
P
2
PB
T
R
1
B
T
P
+
φ
T
P
+
P
φ
)
q
∂ μ
d
μ
+
V
2
.
(35)
where
V
2
is upper bounded by
V
2
[29] defined as
V
2
V
2
=
2
α
g
1
0
q
∂ μ
2
d
μ
+
C
.
(36)
Under Assumption 1, we can compute the following bounds
P
φ
+
φ
P
2
p
κ
1
,
PB
T
R
1
B
T
P
2
p
2
κ
2
(37)
where fixed constants
κ
1
and
κ
2
are defined as
κ
1
=
δ
1
+
δ
2
,
κ
2
=
β
2
2
r
.
(38)
Applying (31) and (36) to (35), we have that
L
V
2
1
0
q
∂ μ
T
(
α
P
+(
κ
1
p
κ
2
p
2
κ
p
2
)
I
)
q
∂ μ
d
μ
+
V
2
2
1
0
q
∂ μ
T
(
α
P
+(
α
α
1
)
p
I
)
q
∂ μ
d
μ
+
C
.
(39)
Then the condition (32) implies that
L
V
≤−
2
α
1
V
(
x
,
δ
q
,
t
)+
C
.
(40)
Therefore, (40) along with Lemma 2 completes the derivation
of (33).
IV. M
AIN
R
ESULT
: S
TOCHASTIC
O
PTIMAL
F
EEDBACK
C
ONTROLLER
The state feedback controller (23) is only a stabilizing
controller as we proved in Theorem 2. To make this con-
troller optimal in some sense, we formulate an optimization
problem which minimizes an upper bound of the steady-state
mean squared distance in (33) to find the optimal controller
parameters, i.e.,
minimize
p
p
+
c
1
1
p
(41)
subject to (24), (31), and (32)
where
c
1
=
p
x
/
ε
and we assume that
g
1
and
α
1
are given. We
propose one way to reformulate this non-convex optimization
problem as a convex optimization problem.
A. Constraint (24)
The constraint (24) can be relaxed to the following matrix
inequality
A
T
(
ρ
,
x
,
t
)
P
(
x
,
t
)+
P
(
x
,
t
)
A
(
ρ
,
x
,
t
)+
2
α
P
(
x
,
t
)
(42)
+
2
κ
P
2
(
x
,
t
)
P
(
x
,
t
)
B
(
x
,
t
)
R
1
(
x
,
t
)
B
T
(
x
,
t
)
P
(
x
,
t
)

0
.
Multiplying it by
Q
(
x
,
t
) =
P
1
(
x
,
t
)
from both sides, (42) is
equivalent to
Q
(
x
,
t
)
A
T
(
ρ
,
x
,
t
)+
A
(
ρ
,
x
,
t
)
Q
(
x
,
t
)+
2
α
Q
(
x
,
t
)
+
2
κ
I
B
(
x
,
t
)
R
1
(
x
,
t
)
B
T
(
x
,
t
)

0
.
(43)
Proposition 1:
The bilinear matrix inequality (43) can be
converted to the following linear matrix inequality (LMI) by
Shor’s relaxation in terms of variables
Q
,
Q
ρ
i
=
ρ
i
Q
,
ρ
i
, and
κ
, i.e.
s
1
i
=
1
A
i
Q
ρ
i
+
s
1
i
=
1
Q
ρ
i
A
T
i
+
2
α
Q
+
2
κ
I
BR
1
B
T

0
(44)
with the following constraints to ensure controllability and
Q
ρ
i
=
ρ
i
Q
Q

0
,
Q
ρ
i

0
,
s
1
i
=
1
Q
ρ
i
=
Q
,
sym
[
I Q
ρ
i
I Q
ρ
i
]

0
,
(45)
s
1
i
=
1
ρ
i
=
1
,
ρ
i
[
0
,
1
]
,
cc
k
(
ρ
,
x
)
<
0
,
i
,
k
=
1
,
···
,
n
c
where sym
(
·
)
is a symmetric part of a matrix and
cc
j
(
ρ
,
x
)
<
0
,
k
=
1
,
···
,
n
c
denotes
n
c
number of convex constraints
to maintain the controllability of the pair
(
A
,
B
)
. Then (44)
and (45) are convex constraints in terms of decision variables
assuming
α
is given.
Proof:
See [29].
B. Objective Function and Contraction Condition
Let us introduce a new variable
χ
defined as
χ
=
p
/
p
with
additional constraint [33]
1
p
I

Q

1
p
I
.
(46)
Using
χ
, the objective function of (41) is upper bounded as
follows
p
p
+
c
1
1
p
=
p
p
+
c
1
χ
p
χ
+
c
1
χ
2
p
(47)
where
χ
χ
2
is used to obtain the inequality. Defining
additional variables
ν
=
p
,
̃
(
·
) = (
·
)
p
, (46) is expressed as
I

̃
Q

χ
I
.
(48)
The constraints (44) and (45) can also be written in terms of
the new variables
s
1
i
=
1
A
i
̃
Q
ρ
i
+
s
1
i
=
1
̃
Q
ρ
i
A
T
i
+
2
α
̃
Q
+
2
̃
κ
I
ν
BR
1
B
T

0
(49)
where
̃
Q

0
,
̃
Q
ρ
i

0
,
s
1
i
=
1
̃
Q
ρ
i
=
̃
Q
,
sym
[
ν
I
̃
Q
̃
ρ
i
I
̃
Q
ρ
i
]

0
,
(50)
s
1
i
=
1
̃
ρ
i
=
ν
,
̃
ρ
i
[
0
,
ν
]
,
cc
k
(
̃
ρ
,
x
)
<
0
,
i
,
k
=
1
,
···
,
n
c
.
Finally, multiplying (31) by
p
/
p
2
, we have
(
α
α
1
)(
p
/
p
)
κ
1
(
p
/
p
)
2
κ
2
p
κ
p
+
α
g
(
p
/
p
2
)
. In terms of the new vari-
ables, this can be written as
(
α
α
1
)
χ
κ
1
χ
2
κ
2
ν
̃
κ
+
α
g
χ
2
ν
.
(51)
Assuming that
α
,
α
1
,
κ
1
,
κ
2
, and
α
g
are given, this is a
convex constraint as
a
2
/
b
is a convex function for
a
and
b
if
b
>
0. As a result, the optimization problem (41) with the
new objective function (47), which is an upper bound of the
steady-state tracking error, reduces to
minimize
χ
+
c
1
χ
2
ν
(52)
subject to (48), (49), (50), and (51)
.
Note that
χ
+ (
c
1
χ
2
)
/
ν
is a convex function in terms of
decision variables
χ
and
ν
>
0 assuming
c
1
is given.
Proposition 2:
The optimization problem (52) is convex
in terms of decision variables
̃
κ
>
0,
ν
>
0,
χ
>
0,
̃
Q

0,
̃
Q
ρ
i

0, and
̃
ρ
i
R
,
i
=
1
,
···
,
s
1
.
α
,
α
1
0 s.t.
α
α
1
0
and
R
(
x
,
t
)

0 have to be specified by a user. The optimal
solution provides an upper bound of the optimal steady-state
tracking error.
Remark 5:
(52) can be simplified further by selecting
some
ρ
that satisfies
i
ρ
i
=
1 and that preserves the con-
trollability
a priori
. Then, the constraints (49) and (50) are
reduced to one LMI constraint.
C. Variations of Convex Optimization Problem
1) Penalty Formulation:
Although (52) is convex, the
constraint (51) can be too conservative for the problem to
be feasible because of the fact that we consider an upper
bound of the steady-state tracking error. This motivate us to
introduce a penalty formulation with a new decision variable
ζ
that satisfies the following inequalities
ζ
≥−
(
α
α
1
)
χ
+
κ
1
χ
2
κ
2
ν
̃
κ
+
α
g
χ
2
ν
(53)
ζ
0
.
(54)
The penalty formulation of (52) is given as
minimize
χ
+
c
1
χ
2
ν
+
ζ
(55)
subject to (48), (49), (50), (53) and (54)
.
2) Input Constraints through the norm of K
(
x
,
t
)
:
Let us
consider the case when the input constraint can be relaxed
to a constraint
u
s
‖≤
u
max
. Then the sufficient condition for
the relaxed input constraint is
ν
R
1
B
T
‖‖
s
‖≤
u
max
λ
min
(
̃
Q
)
,
t
,
x
(56)
which is a convex constraint that can be implemented in (52)
without losing their convexity.
D. Summary for Stochastic Optimal Feedback Controller
The stochastic optimal feedback controller design is sum-
marized as follows.
Proposition 3:
An optimal stochastic nonlinear feedback
controller is designed as (23), where
P
(
x
,
t
)
is selected by
the convex optimization problem (52) or (55) with decision
variables
̃
κ
R
,
ν
R
,
χ
R
,
ζ
R
,
̃
Q
R
n
×
n
,
̃
Q
ρ
i
R
n
×
n
,
and
̃
ρ
i
R
,
i
=
1
,
···
,
s
1
.
Remark 6:
The constraints are state- and time-dependent,
which means that (52) or (55) has to be solved at each time
instant.
This controller synthesis algorithm provides a convex
optimization-based methodology for selecting decision vari-
ables that minimize an upper bound of the steady-state
tracking error. The controllability constraint at each step
can be expressed in a simple form thanks to the SDC
parametrization of the original dynamics. The convexity of
the optimization problem (52) or (55) allows us to solve
this problem using computationally-efficient numerical tech-
niques such as the polynomial-time interior point methods
[33]–[35]. In practice, the solution to the state-dependent
algebraic Riccati equation can also be obtained efficiently
by using various numerical techniques [40], [41].
V. N
UMERICAL
S
IMULATION
The performance of the controller in Proposition 3 is
evaluated using the following spacecraft attitude control
example.
A. Simulation Setup
We consider the spacecraft attitude dynamics given in
[42] with stochastic noise. The simulation parameters are
selected as
J
=
150
0
100
0
270
0
100
0
300
,
q
(
0
) = [
0
.
9
0
.
9 0
.
7
]
T
, ̇
q
(
0
) = [
0
.
6 0
.
7
0
.
5
]
T
, and
G
(
x
,
t
) =
0
.
2
×
1
1
1
n
×
d
with
d
=
1. The desired trajectories are defined as
q
1
d
=
0
10
20
30
40
50
60
70
80
90 100
-0.5
0
0.5
0
10
20
30
40
50
60
70
80
90 100
-0.5
0
0.5
0
10
20
30
40
50
60
70
80
90 100
-0.5
0
0.5
Fig. 1: Modified Rodrigues parameter
q
1
0
10
20
30
40
50
60
70
80
90 100
-0.5
0
0.5
0
10
20
30
40
50
60
70
80
90 100
-0.5
0
0.5
0
10
20
30
40
50
60
70
80
90 100
-0.5
0
0.5
Fig. 2: Modified Rodrigues parameter
q
2
0
10
20
30
40
50
60
70
80
90 100
-0.5
0
0.5
0
10
20
30
40
50
60
70
80
90 100
-0.5
0
0.5
0
10
20
30
40
50
60
70
80
90 100
-0.5
0
0.5
Fig. 3: Modified Rodrigues parameter
q
3
0
.
3 sin
(
2
π
(
0
.
1
)
t
)
,
q
2
d
=
0
.
2 sin
(
2
π
(
0
.
2
)
t
+
π
/
6
)
, and
q
3
d
=
0.
Parameters of the proposed controller are updated at every
time instant by solving the convex optimization problem
using the
cvx
toolbox in Matlab [43], [44]. Coefficients of the
convex combination of SDC matrices are selected
a priori
so that they preserve controllability.
For comparison, the same simulation is performed for a
PID controller and a nominal nonlinear controller [36] that
guarantees exponential stability. The P, I and D gains are
selected as
K
p
=
500
I
,
K
i
=
300
I
,
K
d
=
500
I
respectively.
K
r
=
100
I
and
Λ
=
1 are used for the controller [36].
Remark 7:
The parameters for each controller are selected
to achieve similar control effort for each controller to ensure
a fair comparison.
B. Simulation Results
Figures 1, 2, and 3 show the comparison between the
controlled and the desired modified Rodorigues parameters
q
for the proposed controller in Proposition 3, a controller
[36], and a PID controller. Figure 4 shows the steady-state
tracking error and the control effort for each controller, which
are computed by
x
x
d
2
2
and
T
0
u
2
2
dt
where
x
= [
q
̇
q
]
T
.
Note that
x
x
d
2
2
as
t
goes to infinity is the value we
attempt to minimize. In order to improve the visibility, the
simulation data of the tracking error is smoothed using a
moving average filter which takes the average of every 150
consecutive samples at each time instant. It is shown that the
controller in Proposition 3 achieves the smaller steady-state
tracking error than that of [36] and PID with smaller amount
of control effort as can be seen in Fig. 4.
VI. C
ONCLUSION
In this paper, we proposed a convex optimization-based
feedback tracking control approach for It
ˆ
o stochastic nonlin-
ear systems written using multiple SDC parameterizations. A
rigorous proof on stochastic incremental exponential stability
is provided using contraction analysis. We formulated a
convex program for obtaining the optimal feedback gain and
controller parameters, which minimize an upper bound of the
steady-state tracking error. It is shown in the simulation that
this controller outperforms a known exponentially-stabilizing
nonlinear controller and a PID controller. The SDC param-
eterization of the original dynamics allows us to guarantee
the controllability at each time instant through a simple con-
straint. We also derived discrete-time stochastic incremental
contraction analysis for state- and time-dependent metrics,
0
20
40
60
80 100
0
0.1
0.2
0.3
0.4
0.5
0
20
40
60
80 100
0
1
2
3
4
10
7
Fig. 4: Smoothed tracking error and control effort
which can be useful in proving the stability of discrete-
time and hybrid stochastic nonlinear systems. This theorem
implies a similar SDC-based design approach to nonlinear
optimal feedback controllers for discrete-time stochastic non-
linear systems.
A
CKNOWLEDGEMENT
This work was in part funded by the Jet Propulsion
Laboratory, California Institute of Technology and Raytheon
Company.
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