Neural Network Adaptive Control for a Class of Nonlinear Uncertain Dynamical Systems With Asymptotic Stability Guarantees

Abstract

In this paper, a neuroadaptive control framework for continuous- and discrete-time nonlinear uncertain dynamical systems with input-to-state stable internal dynamics is developed. The proposed framework is Lyapunov based and unlike standard neural network (NN) controllers guaranteeing ultimate boundedness, the framework guarantees partial asymptotic stability of the closed-loop system, that is, asymptotic stability with respect to part of the closed-loop system states associated with the system plant states. The neuroadaptive controllers are constructed without requiring explicit knowledge of the system dynamics other than the assumption that the plant dynamics are continuously differentiable and that the approximation error of uncertain system nonlinearities lie in a small gain-type norm bounded conic sector. This allows us to merge robust control synthesis tools with NN adaptive control tools to guarantee system stability. Finally, two illustrative numerical examples are provided to demonstrate the efficacy of the proposed approach.

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Neural Network Adaptive Control for Nonlinear Uncertain Dynamical Systems
with Asymptotic Stability Guarantees
Tomohisa Hayakawa†, Wassim M. Haddad‡, and Naira Hovakimyan∗
†CREST, Japan Science and Technology Agency, Saitama, 332-0012, JAPAN
‡School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150
∗Aerospace and Ocean Engineering, Virginia Polytechnic Institute, Blacksburg, VA 24061
Abstract
A neuro adaptive control framework for nonlinear
uncertain dynamical systems with input-to-state stable
internal dynamics is developed. The proposed framework
is Lyapunov-based and unlike standard neural network con-
trollers guaranteeing ultimate boundedness, the framework
guarantees partial asymptotic stability of the closed-loop
system, that is, asymptotic stability with respect to part of
the closed-loop system states associated with the system
plant states. The neuro adaptive controllers are constructed
without requiring explicit knowledge of the system dynam-
ics other than the assumption that the plant dynamics are
continuously differentiable and that the approximation error
of uncertain system nonlinearities lie in a small gain-type
norm bounded conic sector. This allows us to merge robust
control synthesis tools with neural network adaptive control
tools to guarantee system stability. Finally, an illustrative
numerical example is provided to demonstrate the efficacy
of the proposed approach.
1.Introduction
One of the main motivation for developing neural network
adaptive control algorithms is their capability to approximate
alargeclassofcontinuousnonlinearmapsfromthecollective
action of very simple, autonomous processing units that are
connected in simple ways. These processing units involve
a weighted interconnection of fundamental elements called
neurons, which are functions consisting of a summing junc-
tion and a nonlinear operation involving an activation func-
tion. Inaddition, neuralnetworkshaveattractedattentiondue
to their inherently parallel and highly redundant processing
architecture that makes it possible to develop parallel weight
update laws. This parallelism makes it possible to effectively
update a neural network on line. Consequently, the use of
the neural networks for system identification and control of
complex highly uncertain dynamical systems has become an
active area of research [1–9].
Unlike adaptive controllers which guarantee asymptotic
stability of the closed-loop system states associated with the
system plant states, neural network adaptive controllers guar-
antee ultimate boundedness of the closed-loop system states
[10]. This fundamental difference between adaptive control
and neuro adaptive control can be traced back to the model-
ing and treatment of the system uncertainties. In particular,
adaptive control is based on constant, linearly parameterized
system uncertainty models of a known structure but unknown
variation [11–13], while neuro adaptive control is based on
This research was supported in part by the Air Force Office of
Scientific Research under Grants F49620-03-1-0178 and F49620-
03-1-0443.
the universal function approximation property, wherein any
continuous system uncertainty can be approximated arbitrar-
ily closely on a compact set using a neural network with ap-
propriate weights [14]. This system uncertainty parametriza-
tion makes it impossible to construct a system Lyapunov
function whose time derivative along the closed-loop system
trajectories is guaranteed to be negative definite. Instead, the
Lyapunov derivative can only be shown to be negative on a
sublevel set of the system Lyapunov function. This shows
that, in this sublevel set, the Lyapunov function will decrease
monotonically until the system trajectories enter a compact
set containing the desired system equilibrium point, and thus,
guaranteeing ultimate boundedness. This analysis is often
conservative since standard Lyapunov-like theorems used to
show ultimate boundedness of the closed-loop system states
provide only sufficient conditions, while neural network con-
trollers often achieve plant state convergence to a desired
equilibrium point.
In this paper, we develop a neuro adaptive control frame-
work for a class of nonlinear uncertain dynamical systems
which guarantees asymptotic stability of the closed-loop sys-
tem states associated with the system plant states, as well
as boundedness of the neural network weighting gains. The
proposed framework is Lyapunov-based and guarantees par-
tial asymptotic stability of the closed-loop system, that is,
Lyapunov stability of the overall closed-loop system states
and convergence of the plant states [15]. The neuro adaptive
controllers are constructed without requiring explicit knowl-
edge of the system dynamics other than the assumption that
the plant dynamics are continuously differentiable and that
the approximation error of uncertain system nonlinearities
lie in a small gain-type norm bounded conic sector. Further-
more, the proposed neuro control architecture is modular in
the sense that if a nominal linear design model is available,
then the neuro adaptive controller can be augmented to the
nominal design to account for system nonlinearities and sys-
tem uncertainty.
The notation used in this paper is fairly standard. Specif-
ically, (·)Tdenotes transpose, tr(·) denotes the trace opera-
tor, σmax(·) denotes the maximum singular value of a matrix,
vec(·) denotes the column stacking operator for a matrix, and
? · ? denotes the Euclidean vector norm.
2.Partial Stability
In this section we present partial stability theorems for
nonlinear dynamical systems. Specifically, consider the non-
linear interconnected dynamical system
˙ x1(t) = f1(x1(t),x2(t)),
˙ x2(t) = f2(x1(t),x2(t)),
x1(0) = x10,
x2(0) = x20,
t ∈ Ix0, (1)
(2)
2005 American Control Conference
June 8-10, 2005. Portland, OR, USA
0-7803-9098-9/05/$25.00 ©2005 AACC
WeC05.2
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where x1 ∈ D, D ⊆ Rn1is an open set such that 0 ∈ D,
x2 ∈ Rn2, f1 : D × Rn2→ Rn1is such that, for every
x2∈ Rn2, f1(0,x2) = 0 and f1(·,x2) is locally Lipschitz in
x1, f2 : D × Rn2→ Rn2is such that, for every x1 ∈ D,
f2(x1,·) is locally Lipschitz in x2, and Ix0 ? [0,τx0),
0 < τx0≤ ∞, is the maximal interval of existence for the
solution (x1(t),x2(t)), t ∈ Ix0, to (1), (2). Note that under
the above assumptions the solution (x1(t),x2(t)) to (1), (2)
exists and is unique over Ix0. The following definition intro-
duces several types of partial stability, that is, stability with
respect to x1, for the nonlinear dynamical system (1), (2).
For the following definition we assume that Ix0= [0,∞).
Definition 2.1. i) The nonlinear dynamical system (1), (2)
is Lyapunov stable with respect to x1uniformly in x20if, for
every ε > 0, there exists δ = δ(ε) > 0 such that ?x10?< δ
implies that ?x1(t)?< ε for all t ≥ 0 and for all x20∈ Rn2.
ii) The nonlinear dynamical system (1), (2) is asymptoti-
cally stable with respect to x1uniformly in x20if it is Lya-
punovstablewithrespecttox1uniformlyinx20andthereex-
ists δ > 0 such that ?x10?< δ implies that limt→∞x1(t) =
0 for all x20∈ Rn2.
iii) The nonlinear dynamical system (1), (2) is globally
asymptotically stable with respect to x1uniformly in x20if it
is Lyapunov stable with respect to x1uniformly in x20and
limt→∞x1(t) = 0 for all x10∈ Rn1and x20∈ Rn2.
Next, we present sufficient conditions for partial stability
of the nonlinear dynamical system (1), (2). For the follow-
ing result define˙V (x1,x2) ? V?(x1,x2)f(x1,x2), where
f(x1,x2) ? [fT
uously differentiable function V : D × Rn2→ R. Further-
more, we assume that the solution (x1(t),x2(t)) to (1), (2)
exists and is unique for all t ≥ 0. It is important to note that
unlike standard theory the existence of a Lyapunov function
V (x1,x2) satisfying the conditions in Theorem 2.1 below is
not sufficient to ensure that all solutions of (1), (2) starting
in D × Rn2can be extended to infinity since neither of the
states of (1), (2) serve as an independent variable. We do
note however that continuous differentiability of f1(·,·) and
f2(·,·) provides a sufficient condition for the existence and
uniqueness of solutions to (1), (2) for all t ≥ 0.
1(x1,x2),fT
2(x1,x2)]T, for a given contin-
Theorem 2.1 [15]. Consider the nonlinear dynamical sys-
tem (1), (2). Then the following statements hold:
i) If there exists a continuously differentiable function V :
D×Rn2→ R and class K functions α(·),β(·) such that
α(?x1?) ≤ V (x1,x2) ≤ β(?x1?),
(x1,x2) ∈ D × Rn2, (3)
(x1,x2) ∈ D × Rn2,
˙V (x1,x2) ≤ 0,
(4)
then the nonlinear dynamical system given by (1), (2) is
Lyapunov stable with respect to x1uniformly in x20.
ii) If there exists a continuously differentiable function V :
D × Rn2→ R and class K functions α(·),β(·),γ(·)
satisfying (3) and
˙V (x1,x2) ≤ −γ(?x1?),
(x1,x2) ∈ D × Rn2, (5)
then the nonlinear dynamical system given by (1), (2)
is asymptotically stable with respect to x1uniformly in
x20.
iii) If D = Rn1and there exists a continuously differen-
tiable function V : Rn1× Rn2→ R, a class K function
γ(·), and class K∞functions α(·),β(·) satisfying (3)
and (5), then the nonlinear dynamical system given by
(1), (2) is globally asymptotically stable with respect to
x1uniformly in x20.
By setting n1 = n and n2 = 0, Theorem 2.1 special-
izes to the case of nonlinear autonomous systems of the form
˙ x1(t) = f1(x1(t)). In this case, Lyapunov (respectively,
asymptotic) stability with respect to x1and Lyapunov (re-
spectively, asymptotic) stability with respect to x1uniformly
in x20are equivalent to the classical Lyapunov (respectively,
asymptotic) stability of nonlinear autonomous systems. In
particular, note that it follows from converse Lyapunov the-
ory that there exists a continuously differentiable function
V : D → R such that (3) and (5) hold if and only if V (·) is
such that V (0) = 0, V (x1) > 0, x1?= 0, V?(x1)f1(x1) < 0,
x1?= 0. In addition, if D = Rn1and there exist class K∞
functions α(·),β(·) and a continuously differentiable func-
tion V (·) such that (3) and (5) hold if and only if V (·) is such
that V (0) = 0, V (x1) > 0, x1 ?= 0, V?(x1)f1(x1) < 0,
x1 ?= 0, and V (x1) → ∞ as ? x1?→ ∞. Hence, in this
case, Theorem 2.1 collapses to the classical Lyapunov stabil-
ity theorem for autonomous systems.
In the case of time-invariant systems the Barbashin-
Krasovskii-LaSalle invariance theorem shows that bounded
systemtrajectoriesofanonlineardynamicalsystemapproach
the largest invariant set M characterized by the set of all
points in a compact set D of the state space where the Lya-
punov derivative identically vanishes. In the case of partially
stable systems, however, it is not generally clear on how to
define the set M since˙V (x1,x2) is a function of both x1
and x2. However, if˙V (x1,x2) ≤ −W(x1) ≤ 0, where
W : D → R is continuous and nonnegative definite, then a
set R ⊃ M can be defined as the set of points where W(x1)
identically vanishes; that is, R = {x1∈ D : W(x1) = 0}.
In this case, as shown in the next theorem, the partial system
trajectories x1(t) approach R as t tends to infinity.
Theorem 2.2 [15]. Consider the nonlinear dynamical sys-
tem (1), (2) and assume D × Rn2is a positive invariant set
with respect to (1), (2) and f1(·,·) is Lipschitz continuous in
x1uniformly in x2. Furthermore, assume there exist func-
tions V : D × Rn2→ R, W,W1,W2 : D → R such
that V (·,·) is continuously differentiable, W1(·) and W2(·)
are continuous and positive definite, W(·) is continuous and
nonnegative definite, and, for all (x1,x2) ∈ D × Rn2,
W1(x1) ≤ V (x1,x2) ≤ W2(x1),
˙V (x1,x2) ≤ −W(x1).
(6)
(7)
Then there exists D0⊆ D such that for all (x10,x20) ∈ D0×
Rn2, x1(t) → R ? {x1∈ D : W(x1) = 0} as t → ∞. If,
in addition, D = Rn1and W1(·) is radially unbounded, then
for all (x10,x20) ∈ Rn1× Rn2, x1(t) → R ? {x1∈ Rn1:
W(x1) = 0} as t → ∞.
Theorem 2.2 shows that the partial system trajectories
x1(t) approach R as t tends to infinity. However, since the
positive limit set of the partial trajectory x1(t) is a subset
of R, Theorem 2.2 is a weaker result than the standard in-
variance principle wherein one would conclude that the par-
tial trajectory x1(t) approaches the largest invariant set M
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contained in R. This is not true in general for partially sta-
ble systems since the positive limit set of a partial trajec-
tory x1(t), t ≥ 0, is not an invariant set. However, in the
case where f1(·,x2) is periodic, almost-periodic, or asymp-
totically independent of x2, then an invariance principle for
partially stable systems can be derived.
3.Stable Neuro Adaptive Control for Nonlinear Uncer-
tain Systems
In this section we consider the problem of characterizing
neural adaptive feedback control laws for nonlinear uncer-
tain dynamical systems. Specifically, consider the controlled
nonlinear uncertain dynamical system G given by
˙ x(t) = fx(x(t),z(t)) + G(x(t),z(t))u(t),
x(0) = x0, t ≥ 0,
z(0) = z0,
(8)
(9)
˙ z(t) = fz(x(t),z(t)),
where x(t) ∈ Rnx, t ≥ 0, and z(t) ∈ Rnz, t ≥ 0,
are the state vectors, u(t) ∈ Rm, t ≥ 0, is the control
input, fx : Rnx× Rnz→ Rnxsatisfies fx(0,z) = 0,
z ∈ Rnz, fx(·,z) is continuously differentiable on Rnx
for each z ∈ Rnz, fx(x,·) is continuous on Rnzfor each
x ∈ Rnx, fz : Rnx× Rnz→ Rnzsatisfies fz(x,0) = 0,
x ∈ Rnx, and G : Rnx× Rnz→ Rnx×m. The dynam-
ics (9) typically describe the internal dynamics of the system
G. The control input u(·) in (8) is restricted to the class of
admissible controls consisting of measurable functions such
that u(t) ∈ Rm, t ≥ 0. Furthermore, for the nonlinear un-
certain system G we assume that the required properties for
the existence and uniqueness of solutions are satisfied, that
is, fx(·,·), fz(·,·), G(·), and u(·) satisfy sufficient regularity
conditions such that (8), (9) has a unique solution forward in
time.
In this paper, we assume that fx(·,·) and fz(·,·) are un-
known functions, and fx(·,·) and G(·,·) are given by
fx(x,z) = Ax + ∆f(x,z),
G(x,z) = BGn(x,z),
(10)
(11)
where A ∈ Rnx×nxand B ∈ Rnx×mare known matri-
ces, Gn : Rnx× Rnz→ Rm×mis a known matrix func-
tion such that detGn(x,z) ?= 0, (x,z) ∈ Rnx× Rnz, and
∆f : Rnx× Rnz→ Rnxis an uncertain function belonging
to the uncertainty set F given by
F = {∆f : Rnx× Rnz→ Rnx: ∆f(0,·) = 0,
∆f(x,z) = Bδ(x,z), (x,z) ∈ Rnx× Rnz}, (12)
where δ : Rnx× Rnz→ Rmis an uncertain function such
that δ(·,z) is continuously differentiable on Rnxfor each
z ∈ Rnz, δ(x,·) is continuous on Rnzfor each x ∈ Rnx,
and δ(0,·) = 0. Furthermore, we assume that (9) is input-to-
state stable with x(t) viewed as the input. It is important to
note that since δ(x,z) is continuously differentiable in x and
δ(0,z) = 0, z ∈ Rnz, it follows that there exists a contin-
uous matrix function ∆ : Rnx× Rnz→ Rm×nxsuch that
δ(x,z) = ∆(x,z)x, (x,z) ∈ Rnx× Rnz. We assume that
the continuous matrix function ∆(·,·) can be approximated
over a compact set Dcx×Dcz⊂ Rnx×Rnzby a linear in the
parameters neural network up to a desired accuracy so that
coli(∆(x,z)) = WT
iσ(x,z)+εi(x,z), (x,z) ∈ Dcx×Dcz,
i = 1,···,nx,
(13)
ϕ(x)
x
Dcx
|slope| = γ−1
Figure 3.1: Visualization of function ϕ(·)
where coli(∆(·,·)) denotes the ith column of ∆(·,·), WT
Rm×s, i = 1,···,nx, are optimal unknown (constant)
weights that minimize the approximation error over Dcx×
Dcz, εi: Rnx× Rnz→ Rm, i = 1,···,nx, are modeling
errors such that σmax(Υ(x,z)) ≤ γ−1, (x,z) ∈ Rnx× Rnz,
where Υ(x,z) ? [ε1(x,z),···,εnx(x,z)] and γ > 0, and
σ : Rnx× Rnz→ Rsis a given basis function such that
0 ≤ σ(x,z) ≤ 1, (x,z) ∈ Rnx× Rnz.
Next, defining
i∈
ϕ(x,z) ? δ(x,z) − WT[x ⊗ σ(x,z)],
(14)
where WT
notes Kronecker product, it follows from (13) and Cauchy-
Schwartz inequality that
? [WT
1,···,WT
nx] ∈ Rm×nxsand ⊗ de-
ϕT(x,z)ϕ(x,z) = ?∆(x,z)x − WT(x ⊗ σ(x,z))?2
= ?∆(x,z)x − Σ(x,z)x?2
= ?Υ(x,z)x?2
≤ γ−2xTx,
(x,z) ∈ Dcx× Dcz, (15)
where Σ(x,z) ? [WT
where G does not possess internal dynamics (i.e., nz = 0),
(13) and (15) specialize to
1σ(x,z),···,WT
nxσ(x,z)]. In the case
coli(∆(x)) = WT
iσ(x)+εi(x),x ∈ Dcx, i = 1,···,nx,
(16)
and
ϕT(x)ϕ(x) ≤ γ−2xTx,x ∈ Dcx,
(17)
respectively. This corresponds to a nonlinear small gain-type
norm bounded uncertainty characterization for ϕ(·) (see Fig-
ure 3.1).
Theorem 3.1. Consider the nonlinear uncertain dynami-
cal system G given by (8) and (9) where fx(·,·) and G(·,·)
are given by (10) and (11), respectively, and ∆f(·,·) belongs
to F. Assume there exists a matrix K ∈ Rm×nxsuch that
As ? A + BK is asymptotically stable. Furthermore, for
a given γ > 0, assume there exist positive-definite matrices
P ∈ Rnx×nxand R ∈ Rnx×nxsuch that
0 = AT
sP + PAs+ γ−2PBBTP + Inx+ R.
(18)
In addition, assume that (9) is input-to-state stable with x(t)
viewed as the input. Finally, let Q ∈ Rm×mand Y ∈
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Rnxs×nxsbe positive definite. Then the neural adaptive feed-
back control law
u(t) = G−1
n(x(t),z(t))
?
Kx(t)
−ˆ WT(t)[x(t) ⊗ σ(x(t),z(t))]
?
,
(19)
whereˆ WT(t) ∈ Rm×nxs, t ≥ 0, and σ : Rnx× Rnz→ Rs
is a given basis function, with update law
˙ˆ WT(t) = QBTPx(t)[x(t) ⊗ σ(x(t),z(t))]TY,
ˆ WT(0) =ˆ WT
0, (20)
guarantees that there exists a positively invariant set Dα ⊂
Rnx× Rnz× Rm×nxssuch that (0,0,WT) ∈ Dα, where
WT∈ Rm×nxs, and the solution (x(t),z(t),ˆ WT(t)) ≡
(0,0,WT) of the closed-loop system given by (8), (9), (19),
(20) is Lyapunov stable and (x(t),z(t)) → (0,0) as t → ∞
for all ∆f(·,·) ∈ F and (x0,z0,ˆ WT
0) ∈ Dα.
Proof. First, note that with u(t), t ≥ 0, given by (19) it
follows from (8), (10), and (11) that
˙ x(t) = Ax(t) + ∆f(x(t),z(t)) + BKx(t)
−Bˆ WT(t)[x(t) ⊗ σ(x(t),z(t))],x(0) = x0,
t ≥ 0, (21)
or, equivalently, using (14),
˙ x(t) = Asx(t) + B
?
ϕ(x(t),z(t)) −˜ WT(t)
·[x(t) ⊗ σ(x(t),z(t))]
?
,x(0) = x0, t ≥ 0, (22)
where˜ WT(t) ?ˆ WT(t) − WT. To show Lyapunov stability
of the closed-loop system (9), (20), and (22) consider the
Lyapunov function candidate
V (x,z,˜ WT) = xTPx + trQ−1˜ WTY−1˜ W.
(23)
Note that (23) satisfies (3) and (6) with x1
[xT,(vec˜ W)T]T, x2
=
W1(x1) = W2(x1) = xTPx + tr˜ WQ−1 ˜ WT. Now, let-
ting x(t), t ≥ 0, denote the solution to (22) and using (15),
(18), and (20), it follows that the Lyapunov derivative along
the closed-loop system trajectories is given by
=
=
z, α(?x1?)=
β(?x1?)
˙V (x(t),z(t),˜ WT(t))
= 2xT(t)P
?
Asx(t) + B[ϕ(x(t),z(t))
−˜ WT(t)[x(t) ⊗ σ(x(t),z(t))]]
?
+2trQ−1˜ WT(t)Y−1 ˙ˆ W(t)
= −xT(t)(R + γ−2PBBTP + Inx)x(t)
?
−˜ WT(t)[x(t) ⊗ σ(x(t),z(t))]
+2xT(t)PBϕ(x(t),z(t))
?
+2tr˜ WT(t)?BTPx(t)[x(t) ⊗ σ(x(t),z(t))]T?T
Dcx× Rnz× Rm×nxs
˜Dα
x
˜ WT
×
0
Figure 3.2: Visualization of sets used in the proof of Theo-
rem 3.1
= −xT(t)Rx(t) − xT(t)(γ−2PBBTP + Inx)x(t)
+2xT(t)PBϕ(x(t),z(t))
≤ −xT(t)Rx(t)
−xT(t)[γ−1BTP + γInx]T[γ−1BTP + γInx]x(t)
≤ −xT(t)Rx(t)
≤ 0,t ≥ 0.
(24)
Next, let
˜Dα ?
?
(x,z,˜ WT) ∈ Rnx× Rnz× Rm×nxs:
V (x,z,˜ WT) ≤ α
?
,
(25)
where α is the maximum value such that
Dcx× Rnz× Rm×nxs(see Figure 3.2).
˙V (x(t),z(t),WT(t)) ≤ 0 for all (x(t),z(t),WT(t)) ∈˜Dα
and t ≥ 0, it follows that˜Dαis positively invariant. Fur-
thermore, it follows from Theorem 2.1 that the solution
(x(t),z(t),ˆ WT(t)) ≡ (0,0,WT) to (9), (20), and (22) is
Lyapunov stable with respect to x andˆ WT(uniformly in z0)
for all ∆f(·,·) ∈ F and (x0,z0,ˆ W0) ∈˜Dα. In addition,
since R > 0, it follows from Theorem 2.2 that x(t) → 0 as
t → ∞.
Next, since (9) is input-to-state stable with x(t) viewed as
the input, it follows from Theorem 1 of [16] that there exist
a continuously differentiable, radially unbounded positive-
definite function Vz : Rnz→ R and class K functions
γ1(·),γ2(·) such that
˜Dα
⊆
Now, since
V?
z(z)fz(x,z) ≤ −γ1(?z?),
Since ?x(t)? is bounded for all t ≥ 0, it follows that the set
given by
?z? ≥ γ2(?x?).
(26)
Dz?
?
z ∈ Rnz: Vz(z) ≤max
?z?=γ2(supt≥0?x(t)?)Vz(z)
?
(27)
is also positively invariant as long as1Dz⊂ Dcz. Now, since
˜Dαand Dzare positively invariant, it follows that
Dα ?
?
(x,z,ˆ WT) ∈ Rnx× Rnz× Rm×nxs:
(x,z,ˆ WT− WT) ∈˜Dα, z ∈ Dz
?
(28)
1See Remark 3.2.
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is also positively invariant. Furthermore, since (9) is input-
to-state stable with x(t) viewed as the input, it follows from
Lemma 4.7 of [10] that the solution (x(t),z(t),ˆ WT(t)) ≡
(0,0,WT) to (9), (20), and (22) is Lyapunov stable and
x(t) → 0 and z(t) → 0 as t → ∞ for all ∆f(·,·) ∈ F
and (x0,z0,ˆ W0) ∈ Dα.
?
Remark 3.1. Note that the conditions in Theorem 3.1
imply partial asymptotic stability, that is, the solution
(x(t),z(t),ˆ WT(t)) ≡ (0,0,WT) of the overall closed-loop
system is Lyapunov stable and (x(t),z(t)) → (0,0) as
t → ∞. Hence, it follows from (20) that
t → ∞.
˙ˆ WT(t) → 0 as
Remark 3.2. Since α(?x1?) used in the proof of Theo-
rem 3.1 is a class K∞function, the assumption Dz ⊂ Dcz
invoked in the proof of Theorem 3.1 is automatically sat-
isfied in the case where the neural network approximation
holds in Rnx× Rnz. Furthermore, in this case the control
law (19) ensures global asymptotic stability with respect to
x and z. However, the existence of a global neural net-
work approximator for an uncertain nonlinear map cannot
in general be established. Hence, as is common in the neu-
ral network literature, for a given arbitrarily large compact
set Dcx× Dcz ⊂ Rnx× Rnz, we assume that there ex-
ists an approximator for the unknown nonlinear map up to
a desired accuracy in the sense of (15). This assumption en-
sures that there exists a nontrivial Lyapunov level set such
that Dz ⊂ Dcz. In the case where ∆(·,·) is continuous
on Rnx× Rnz, it follows from the Stone-Weierstrass theo-
rem that ∆(·,·) can be approximated over an arbitrarily large
compact set Dcx×Dcz. In this case, our neuro adaptive con-
troller guarantees semiglobal partial asymptotic stability.
Remark 3.3. Note that the neuro adaptive controller (19)
and (20) can be constructed to guarantee partial asymptotic
stability using standard linear H∞theory. Specifically, it
follows from standard H∞theory [17] that ?G(s)?∞ < γ,
where G(s) = E(sIn−As)−1B and E is such that ETE =
Inx+ R, if and only if there exists a positive-definite matrix
P satisfying the bounded real Riccati equation (18). It is well
known that (18) has a positive-definite solution if and only if
the Hamiltonian matrix
?
−ETE
has no purely imaginary eigenvalues.
H =
As
γ−2BBT
−AT
s
?
,
(29)
It is important to note that the adaptive control law (19)
and (20) does not require the explicit knowledge of the opti-
mal weighting matrix W. Furthermore, no specific structure
on the nonlinear dynamics fx(x,z) and fz(x,z) is required
to apply Theorem 3.1. However, if (8) is in normal form
[18] with (9) being input-to-state stable with x viewed as the
input, then we can always construct a neuro adaptive con-
trol law without requiring knowledge of the system dynamics
fx(x,z) and fz(x,z). To see this, assume that the nonlinear
uncertain system G is generated by
q(ri)
i
(t) = fui(q(t),z(t)) +
m
?
j=1
t ≥ 0,
z(0) = z0,
Gs(i,j)(q(t),z(t))uj(t),
i = 1,···,m,
(30)
(31)
˙ z(t) = fz(q(t),z(t)),
where q = [q1,···,q(r1−1)
q0, q(ri)
i
denotes the rith derivative of qi, and ri denotes
the relative degree with respect to the output qi. Here we
assume that the square matrix function Gs(q,z) composed
of the entries Gs(i,j)(q,z), i,j = 1,···,m, is such that
detGs(q,z) ?= 0, (q,z) ∈ Rˆ r×Rnz, where ˆ r = r1+···+rm
is the (vector) relative degree of (30) and ˆ r = nx. Further-
more, we assume that fui(·,z) is continuously differentiable
on Rnxfor each z ∈ Rnz, fui(x,·) is continuous on Rnzfor
each x ∈ Rnx, and fui(0,·) = 0. In addition, we assume that
the dynamics given by (31) is input-to-stable with q viewed
as the input.
1
,···,qm,···,q(rm−1)
m
]T, q(0) =
Next, define xi ?
?
qi,···,q(ri−2)
i
?T
, i = 1,···,m,
xm+1 ?
?
q(r1−1)
1
,···,q(rm−1)
m
?T
, and x ?
?xT
1,···,
xT
m+1
?T, so that (30) can be described by (8) with
A =
0m×nx
?
Gs(x,z)
?
A0
?
,
∆f(x,z) =
?
0(nx−m)×1
fu(x,z)
?
,
G(x,z) =
0(nx−m)×m
?
,
(32)
where A0 ∈ R(nx−m)×nxis a known matrix of zeros
and ones capturing the multivariable controllable canoni-
cal form representation [19], fu : Rnx× Rnz→ Rnx
is an unknown function and satisfies fu(0,·) = 0, and
Gs : Rnx× Rnz→ Rm×m. Note that ∆f(·,·) ∈ F with
B = [0m×(nx−m),Im]Tand δ(x,z) = fu(x,z). In this case,
Gn(x,z) ≡ Gs(x,z). Furthermore, since A is in multivari-
able controllable canonical form, we can always construct K
such that A + BK is asymptotically stable.
4. Illustrative Numerical Example
In this section we present a numerical example to demon-
strate the utility of the proposed neuro adaptive control
framework for adaptive stabilization. Specifically, consider
the uncertain controlled Li´ enard system given by
¨ q(t) + c(q(t))˙ q(t) + k(q(t)) = bu(t),
q(0) = q0,
where c : R → R and k : R → R are unknown, contin-
uously differential functions. Note that with x1 = q and
x2= ˙ q, (33) can be written in state space form (8), (9), and
˙ q(0) = ˙ q0,t ≥ 0,
(33)
(15) with x = [x1,x2]T, z = Ø, A =
?
0
0
1
0
?
, ∆f(x) =
[0, −c(x1)x2−k(x1)]T, B = [0, b]T, and Gn(x) = 1. Here,
we assume that the unknown function ∆f(x) can be written
as ∆f(x) = Bδ(x), where δ(x) =
is an unknown, continuously differentiable function. Next,
let K =1
?
k1
k2
of k1and k2, it follows from Theorem 3.1 that if there exists
P > 0 satisfying (18), then the neuro adaptive feedback con-
troller (19) guarantees that x(t) → 0 as t → ∞. Specifically,
here we choose k1= −1, k2= −1, γ = 3, and R = I2, so
that P satisfying (18) is given by
1
b[−c(x1)x2− k(x1)]
b[k1,k2], where k1, k2are arbitrary scalars, so that
01
?
As= A+BK =
. Now, with the proper choice
P =
?
3.1586
1.0627
1.0627
2.3765
?
.
(34)
1305