Pachuca-Tulancingo, Km. 4.5 C.P. 42084, Pachuca, Hgo. MEXICO. â Institute of Information Theory and Automation. Academy of Sciences of the Czech Republic ...
Generalized Output Regulation for a Class of Nonlinear Systems via the Robust Control Approach ∗ † ∗ † L. E. RAMOS-VELASCO , S. CELIKOVSKÝ , V. LÓPEZ-MORALES AND V. KUCERA ∗ Centro de Investigación en Tecnologías de Información y Sistemas
Universidad Autónoma del Estado de Hidalgo Carr. Pachuca-Tulancingo, Km. 4.5 C.P. 42084, Pachuca, Hgo. MEXICO †
Institute of Information Theory and Automation Academy of Sciences of the Czech Republic. P.O. Box 18, 182 08 Prague CZECH REPUBLIC
Abstract: - We address the problem of generalized output regulation for nonlinear systems in the presence of unknown parameters in the full information case. We generalize the classical output regulation problem in order to expand the class of reference or disturbance signals. Under appropriate sufcient conditions, a state feedback regulator is built for a class of nonlinear systems where the term including the unknown parameter is assumed to satisfy a matching condition. Our study of the above problem, further referred to as the so-called generalized output regulation problem, combines the approach based on the well-known notion of the regulator equation with the classical concept of the invariant distribution and on the Lyapunov theory. Key-Words: - Nonlinear systems, output regulation, center manifold theory, Lyapunov function, disturbance rejection.
1 1
Introduction
systems have been presented in [8, 10] using full
A central problem in control theory and applica-
information" which includes the measurements of
tions is to design a control law to achieve asymptotic
exogenous signals as well as of the system state.
tracking with disturbance rejection in a nonlinear
The necessary and sufcient conditions for the exis-
system. When a class of reference inputs and dis-
tence of a local full information solution of the clas-
turbances are generated by an autonomous differen-
sical output regulation problem are given in [10, 8];
tial equation, this problem is called nonlinear output
they basically mean that the linearized system is sta-
regulation problem, or alternatively, nonlinear ser-
bilizable and there exists a certain invariant mani-
vomechanism problem [10]. The corresponding au-
fold. The classical output regulation via error feed-
tonomous differential equation is usually called as
back has been solved in [1, 9] by application of sys-
the exogeneous system and is supposed to be neu-
tem immersion technique.
traly stable. In the sequel, the above setting will be
parametrized by unknown constant parameters is
referred to as the classical output regulation prob-
treated as a special case of exogenous signals and
lem.
In other words, the classical output regula-
the solution, extended from the error feedback reg-
tion problem treats a possible unknown reference
ulation, is referred to as the structurally stable reg-
signal and/or disturbances generated by the known
ulation in [1]. Some of the recent results in the ro-
neutrally stable autonomous exosystem with possi-
bust control eld [13, 14, 18] are based on the center
ble unknown initial states.
manifold theory and the related nonlinear regulator
For linear systems the classical output regulation
The plant uncertainty
theory; in the adaptive control eld. However, the
For nonlin-
main limitation of the classical regulation scheme
ear systems, the problem was rst studied in [7],
is that a precise model of the system that generates
and solutions to the output regulation of nonlinear
all exogenous inputs must be available, to be repli-
was extensively studied in
[2, 3, 4].
cated in the control law. 1
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under
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This limitation becomes
immediately evident in the problem of rejecting a sinusoidal disturbances, not only of unknown am-
plitude and phase, but also of unknown frequency.
we outline the generalized nonlinear output regula-
Therefore, an alternative formulation would be to
tion problem which, as anticipated above, is based
require asymptotic tracking of known reference tra-
on the introduction of a driving signal to the exosys-
jectories in spite of unmodelled disturbances acting
tem. In Section 3, we state the assumptions neces-
on the exosystem. To unify this alternative formu-
sary to the well-posedness of the problem. In Sec-
lation with the classical output regulation concept,
tion 4 an adaptive controller solving the state feed-
the so-called generalized output regulation may
back generalized output regulation problem for a
be considered.
specic class of nonlinear systems including an un-
The generalized output regulation was rst posed
known vector parameter, is determined. We show
and solved in [17] for linear systems both continu-
how this specic problem is related to the standard
ous and discrete time in terms of necessary and suf-
problems of disturbance decoupling for nonlinear
cient geometric conditions involving the classical
systems which have been recently studied in [9, 12].
notions of disturbance decoupling. The correspond-
Finally, Section 5 draws conclusions and outlines
ing design procedure presented in [17] handles the
some future research.
unmodelled bounded disturbances generated by the known nonautonomous linear system driven by an unknown bounded reference signal. The question arises how the results established by [17] can be extended to the general case in which the plant is described by a nonlinear equation. The generalized output regulation problem resembles somehow the asymptotic model matching
2
Problem formulation
Following the linear concept of the generalized output regulation problem [17], the generalized output regulation problem for nonlinear systems can be introduced via the conguration provided by the master-slave block diagram of Figure 1. The task
problem (AMM) considered in [19], nevertheless,
! '
the crucial difference is that the input of the reference model in AMM is supposed to be known and
is then used in the corresponding feedback compensator. Therefore, AMM solution cannot be directly
$ ! 4 . 2 0 / 3
applied to the generalized output regulation problem, considering the exogeneous system driven by
" # y( ) *,+- . /10 / 76 2 % & 5
an unknown signal as the analogue of the reference model in AMM problem. In [15], we were able to characterize the solvability of the state feedback generalized output regulation problem for nonlinear systems in terms of
Figure 1:
Conguration of the output regulation
r 6≡ 0, generalized output r ≡ 0, classical output regulation.
schemes. (a) (b)
regulation.
the solvability of the regulator equation with the classical concept of the invariant distribution. The
of the controller is to generate u so that the tracking
state feedback generalized almost output regulation
error e is converging to zero for all initial conditions
problem for a class of nonlinear systems is solved
of both plant and exosystem and all external signals
in [16].
r(t)
from a suitable functional class.
The plants
The purpose of this paper is to point out that
(slaves) we consider in this paper are afne multi-
by combining the approach to classical output reg-
input multi-output (MIMO) nonlinear systems, de-
ulation presented in [9] with the design method for
scribed by the equations of the form
generalized output regulation via state feedback presented in [15], it is easy to address the problem of
x˙
generalized output regulation for a signicant class We establish a link between the two
approaches; as a matter of fact, in this work, an adaptive controller, including a particular Lyapunov function is determined by exploiting concepts of the classical nonlinear regulation theory. The paper is organized as follows. In Section 2
f (x) +
gi (x)ui +
i=1
of nonlinear systems, in the presence of unknown parameters.
=
m X
yi
l X
pi (x)wi
(1)
i=1
:= f (x) + g(x)u + p(x)w, = hi (x), i = 1, . . . , p
(2)
where (1) describes the plant with state x, dened
X of the p output y ∈ R ,
Rn ,
on a neighborhood
origin of
u ∈
subjected to the ef-
Rm and
input
fect of a disturbance represented by the vector elds
pi (x), i = 1, . . . , l. tor eld f (·), and the
It is assumed that the vec-
p(·) are smooth vector elds, while each component of hi (·) is a smooth function, with f (0) = 0, g(0) = 0 and h(0) = 0. We only consider reference outputs to be columns of
g(·)
RF I There exists a neighborhood U
the solution of the closed loop system
x˙ = f (x) + g(x)γ(x, w) + p(x)w w˙ = s(w) + d(w)r
tracked and perturbations to be rejected which both are generated by an unknown exosystem as follows.
lowing nonautonomous system with output
lim e(t) = 0.
(3)
s(·) and the columns of d(·) are smooth vector elds with s(0) = 0 and q(0) = 0, q, q(0) = 0, is smooth function, while r(t) is unknown external driving signal. Further, it is assumed that r(t) is Rρ limited to a functional subclass of L∞ where it holds for all solutions of (3) and some class-K functions α, β that
In [15] the following result is proved: Theorem 1 Consider the system given in (1)-(2). Let Assumption 1 be satised. Then, the generalized output regulation problem via state feedback regulator is solvable if
C k (k ≥ 2) mappings x = π(w), u = c(w), π : Rl → Rn , c : Rl → Rm , ρ : Rl → Rρ , dened locally in a neighborhood of 0 l the origin W ⊂ R , with π(0) = 0, c(0) = 0, ρ(0) = 0 satisfying the so-called generalized
(a) there exist
kw(t)k Rl ≤ α(kw(0)kRl ) + β(kr(t)kLRρ ). L∞ ∞
regulation equation
∂π(w) (s(w) + d(w)ρ(w)) = f (π(w)) ∂w +g(π(w))c(w) + p(π(w))w (7) 0 = h(π(w)) − q(w) (8)
Roughly speaking, the Assumption 1 restricts the class of exogenous inputs to those signals which do not decay to zero and do not tend to innity as time
r.
The controller is
to be designed so that the slave obeys the master,
C k (k ≥ 2) mapping u ˜(x, w), u ˜: × → Rm , u ˜(0, 0) = 0, dened locally 0 n l in a neighborhood of the origin U ⊂ R × R n and a regular involutive distribution ∆ in T R
(b) there exist a
Rn
namely the error signal
e(t) = y(t) − q(w(t)) converges asymptotically to zero as t
(6)
t→∞
Here,
goes to innity for any signal
(5)
and error (4) satises
Assumption 1 (Exosystem): Exosystem is the fol-
w˙ = s(w) + d(w)r(t), w ∈ R yref = q(w), r ∈ Rρ , d := [d1 | . . . |dρ ].
of
(0, 0) such that, for each initial condition on U and for any signal r (piecewise continuous),
and
l
⊂ Rn × Rl
(4)
→ ∞.
In the following statement we give a precise formulation of the control problem under consideration.
Rl
such that (b1 ) the linear approximation of
[f + g˜ u(x, 0)]
is
Hurwitz,
(b2 ) u ˜(π(w), w) = c(w), (b3 ) for all w , [f (x) + g(x)˜ u(x, w) (b4 )
+ p(x)w, ∆] ⊂ ∆ ⊂ ker dh, dπ(∆w ) ⊂ ∆ ∀w, ∆w = span [d1 (w), . . . , dρ (w)] ⊂ T Rl .
Denition 1 (State Feedback Generalized Output Regulation Problem (SFGORP)): Given the reference output yref generated by an exosystem (3), the SFGORP consists in nding a state feedback con-
u = γ(x, w) where γ(·, ·) is a C k (k ≥ 2) mapping, with γ(0, 0) = 0 such that:
troller
SF I The equilibrium x
3
Standing assumptions
In this section we consider a plant modelled by equations of the form
x˙ = f (x) +
gi (x)ui +
i=1
= 0 of +
x˙ = f (x) + g(x)γ(x, 0)
m X
l X
pi (x)wi
i=1
vj (x)θj
j=1
is asymptotically stable in the rst approximation.
m X
yi
= f (x) + g(x)u + p(x)w + v(x)θ, = hi (x), i = 1, . . . , p
(9) (10)
where
θ
is a vector of unknown parameters and the
same considerations presented in the previous sec-
parameter
θ, the state behavior.
tion with regard to system (1)-(2), apply.
u=u ˜(x, w) − k(x)θˆ
Our goal is to solve the SFGORP for the system (9)-(10) without knowing the parameter
θ.
In the
ˆ −θ φ(t) = θ(t)
(11)
ˆ and θ represent the current estimate and θ(t)
the exact value of the unknown parameter, respectively. The main assumptions needed to solve the problem are, for convenience, listed in the following and then briey justied:
The closed loop system (9)-(10) with (13), taking
m functions k(x) such that v(x) = g(x)k(x)
By the Assumption 3, there exists a regular involu-
(12)
T Rn such that
(b3 ) and (b4 ) hold.
u ˜(x, w)
with
u ˜(0, 0) = 0,
satisfying (b1 )
and (b2 ).
Let
exist by virtue of the Frobenius Theorem [10], such
∆ = span{ ∂ζn∂ +1 , · · · , ∂ζ∂n }. 1
(ζ 1 , ζ 2 )T with ζ 1
pact notation ζ and
= ζ 2 = (ζn1 +1 , . . . , ζn ),
c(w), ρ(w),
with
= π(w), u = π(0) = 0, c(0) = 0, ρ(0) = 0
satisfying (7)-(8).
ζ˙ 1
=
f˜1 (ζ) + g˜1 (ζ)˜ u(Φ−1 (ζ), w) +˜ p1 (ζ)w − v˜1 (ζ)φ
by (b3)
p¯(ζ 1 , w) − v˜1 (ζ 1 )φ f˜2 (ζ) + g˜2 (ζ)˜ u(Φ−1 (ζ), w)
:=
ζ˙ 2
=
(17)
yi
˜ i (ζ 1 ) i = 1, . . . , p. h
=
Further, let
tions show that
π ˜ (w) =
ular form. Assumption 3, namely the existence of a regular involutive distribution and, in turn, the exis-
set to zero, is established in Assumption 4. Finally Assumption 5 is standard in classical output regulation of nonlinear systems [10].
4
Straightforward computa-
π ˜ 1 (w) π ˜ 2 (w)
= Φ(π(w)),
c˜(w) = u ˜(Φ−1 (π(w)), w),
tence of the internal triangular decomposition (16)regulation problem when the unknown parameter is
with dim
π ˜ 1 = n1 , dim π ˜ 2 = n − n1
are solutions
to the following equations
∂π ˜ 1 (w) s(w) ∂w
=
f˜1 (˜ π (w)) + g˜1 (˜ π (w))˜ c(w) +˜ p1 (˜ π (w))w − v˜1 (˜ π (w))φ
by (b3)
:=
Regulator via state feedback
p¯(˜ π 1 (w), w) − v˜1 (˜ π 1 (w))φ
Although the main goal of paper is the design of an state feedback regulator, this section is devoted to briey discuss the solution when the state (x, w) is available. This preliminary discussion will make
(19)
∂π ˜ 2 (w) ∂w
s(w)
=
f˜2 (˜ π (w)) + g˜2 (˜ π (w))˜ c(w) +˜ p2 (˜ π (w))w − v˜2 (˜ π (w))φ
the presentation of the general solution more mean-
(20)
ingful. The rst step consists of choosing a control law to force, despite the presence of the unknown
(18)
π(w), c(w) be a solution of the reg-
parameter whose values enter to the plant in a partic-
(18). The state feedback controller, which solves the
(16)
+˜ p2 (ζ)w − v˜2 (ζ)φ
ulator equation (7)-(8). As specied in Assumption 2 we consider unknown
= (ζ1 , . . . , ζn1 ) ζ-
we have that in the
form
Assumption 5 (Solution of the regulator equation): There exist two smooth mappings x
Using the com-
coordinates the system (14)-(15) takes the following
Assumption 4 (Nominal regulator): There exists a mapping
∆ with dim∆ = n − n1 for some ζ = Φ(x) be new coordinates, which
tive distribution
that
Assumption 3 (Local decomposition): There exists a regular involutive distribution ∆ in
x˙ = f (x) + g(x)˜ u(x, w) − g(x)k(x)θˆ +p(x)w + v(x)θ (14) = f (x) + g(x)˜ u(x, w) + p(x)w − v(x)φ yi = hi (x), i = 1, . . . , p. (15)
n1 < n .
Assumption 2 (Matching condition): There exists a vector of
(13)
into account the Assumptions 2, has the form
following, we set
where
In view of this, we
consider the static feedback regulator
0
=
˜ π 1 (w)) − q(w). h(˜
(21)
Moreover, by Assumption 3
is stable. Then, the theory of stability [6, 11] assures the existence of a Lyapunov function V
∂π ˜ 1 (w) d(w) ≡ 0. ∂w
(22)
ζ˜ = ζ − π ˜ (w)
(23)
V (0, 0) = 0 (33) ˜ ˜ V (ζ, w) > 0 ∀ (ζ, w) ∈ U − (0, 0) (34) ˜ w) = ∂V F (ζ, ˜ w) + ∂V (s(w) V˙ (ζ, ∂w ∂ ζ˜ ˜ w) ∈ U (35) +d(w)r) ≤ 0 ∀ (ζ,
where π ˜ (w) is a solution of the system (19)-(20) and
ζ˜ = (ζ˜1 , ζ˜2 )T with ζ˜1 = (ζ˜1 , . . . , ζ˜n1 ) and 2 ˜ ζ = (ζ˜n1 +1 , . . . , ζ˜n ). Then the system (16)-(17) can be rewritten in the form
1 F 1 (ζ˜1 , w) G (ζ˜1 , w) + ˜ w) ˜ w) φ F 2 (ζ, G2 (ζ, ˜ w) + G(ζ, ˜ w)φ = F (ζ, (24) w˙ = s(w) + d(w)r (25) ˙ ζ˜ =
˜ w) = 0, R be the set of all points where V˙ (ζ, and M be the largest invariant set (with respect to the motion of the system (24)-(25)) in R. Let
Assumption 6 (Largest invariant): Let P be the following set
˜ w) : ζ˜ = 0} P = {(ζ, M ⊆ P.
F 1 (ζ˜1 , w) = p¯(ζ˜1 + π ˜ 1 (w), w) − p¯(˜ π 1 (w), w) (26)
˜2
˜ w) = f (ζ˜ + π F (ζ, ˜ (w)) + g˜ (ζ˜ + π ˜ (w)) · −1 ˜ u ˜(Φ (ζ + π ˜ (w)), w) + 2
2
+˜ p2 (ζ˜ + π ˜ (w))w − f˜2 (˜ π (w)) 2 2 −˜ g (˜ π (w))˜ c(w) − p˜ (˜ π (w))w
Now, we present the following theorem on the generalized output regulation problem via state feedback regulator. Theorem 2 Consider the system given in (9)-(10). Assume Assumptions 1-6. Then, the controller
u=u ˜(x, w) − k(x)θˆ
(27)
∂V ˜ w) φ˙ = − G(ζ, ∂ ζ˜
G (ζ˜1 , w) = −˜ v 1 (ζ˜1 + π ˜ 1 (w)) + v˜1 (˜ π 1 (w)) (28)
˜ w) = −˜ G (ζ, v 2 (ζ˜ + π ˜ (w)) + v˜2 (˜ π (w)) 2
u ˜(x, w)
(29)
solves the State Feedback General-
ized Output Regulator Problem for the system (9)-
˜=0 (10) considered with θ = 0, the equilibrium ζ of the system
" ˙ # ζ˜1 F 1 (ζ˜1 , w) = ˙ ˜ w) F 2 (ζ, ζ˜2 p¯(ζ˜1 , w) = ˜ + g˜2 (ζ)˜ ˜ u(Φ−1 (ζ, ˜ 0) f˜2 (ζ)
where
˜ w) G(ζ,
and
˜ w) V (ζ,
(38)
are the previously de-
ned functions, solves the State Feedback Generalized Output Regulation Problem. Proof: Assuming Assumptions 1-6 hold, we aim to show that Denition 1 is satised with
u ˜(x, w) − k(x)θˆ.
γ(x, w) =
Obviously, by Assumption 4 the
condition SF I of Denition 1 holds. To prove the condition RF I consider the Lyapunov function
˙ ζ˜ =
˜ φ, w) = V (ζ, ˜ w) + 1 φ2 W (ζ, 2 (30)
(39)
Taking into account (24)-(25)) and (38), the derivative of the function (39) is such that
is exponentially stable. Moreover, the equilibrium
w = 0 of the exosystem (25) is stable by Assump-
W (0, 0, 0) = 0 (40) ˜ φ, w) > 0 ∀ (ζ, ˜ w) ∈ U − (0, 0), ∀ φ W (ζ,
tion 1. The above properties imply [11] that the equilib-
˜ w) = (0, 0) of the system (ζ,
˙ ˜ w) ζ˜ = F (ζ, w˙ = s(w) + d(w)r
(37)
together with the update law
1
rium
(36)
with
where
Since
of a
neighborhood of the origin U such that
Further, let
denote
˜ w) (ζ,
(31) (32)
(41)
˜ φ, w) = ˙ (ζ, W
∂V ˜ w) + ∂V (s(w) F (ζ, ˜ ∂w ∂ζ ˜ w) ∈ U, ∀ φ +d(w)r) ≤ 0∀ (ζ, (42)
From (40)-(42), it can be deduced that the origin
[8] J. Huang and W.J. Rugh, Stabilization on zero-
of the closed loop system is stable. Moreover, by
error manifolds and the nonlinear servomech-
applying the invariance principle of LaSalle's The-
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orem [6, 11] it is possible to claim that the mo-
matic Control, Vol. 37, 1992, pp. 1009-1013.
tion of the system (24)-(25) originated in a point U , asymptotically converge to the largest invariant sub-
˜ φ, w) ˙ (ζ, set characterized by W
= 0, that is, due to
[9] A. Isidori, Nonlinear Control Systems, 3rd ed., New York: Springer-Verlag, 1995.
ζ˜ = 0. From (23), this implies, that every motion ζ(t), originated in U , asymptotically converges to the center manifold π ˜ (w), i.e. we can see from (4) and (8) that error e(t) tends to zero
[10] A. Isidori and C. I. Byrnes, Output regula-
as time tends to innity. In other words, the condi-
[11] H.K. Khalil, Nonlinear Systems, 2nd ed. New
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tion (RF I ) of Denition 1 holds.
/
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5
Conclusions
The solution of the problem of generalized regu-
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