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Composite feedback control for a class of non-linear
singularly perturbed systems with fast actuators
a
JUHNG-PERNG SU & JER-GUANG HSIEH
a
a
Institute of Electrical Engineering, National Sun Yat-Sen University , Kaohsiung, Taiwan,
80424, Republic of China.
Published online: 17 Apr 2007.
To cite this article: JUHNG-PERNG SU & JER-GUANG HSIEH (1990) Composite feedback control for a class of nonlinear singularly perturbed systems with fast actuators, International Journal of Control, 52:3, 571-579, DOI:
10.1080/00207179008953553
To link to this article: http://dx.doi.org/10.1080/00207179008953553
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INT. J. CONTROL,
1990, VOL. 52, No.3, 571-579
Composite feedback control for a class of non-linear singularly
perturbed systems with fast actuators
JUHNG-PERNG
sur and JER-GUANG HSIEHt
A design methodology for a class of non-linear singularly perturbed systems with
Downloaded by [University of Leeds] at 00:38 16 January 2015
fast actuators is proposed using the compositecontrol technique. The origin of the
nominally unforced unperturbed system is assumed to be asymptotically stable and
a Lyapunov function for the nominal system is available. Then by appropriate
choice of the design manifold, a composite feedback control can be constructed to
guarantee the asymptotic stability of the origin of the closed-loop system.
1. Introduction
Most physical systems inherently contain both slow and fast dynamical phenomena. The singular perturbation method has been a successful analytic tool that directly
exploits the separation of system time scales, made explicit by the small singular
perturbation parameter a. The separation is useful both in understanding the time
scale property and in using it for separate analysis and design of slow and fast
subsystems.
Recently, the integral manifold approach, as a means for the control of non-linear
systems by the use of singular perturbation method, has been significantly developed
(Sobolev 1984, Kokotovic et al. 1986, Khorasani and Kokotovic 1986, Sharkey and
O'Reilly 1987, 1988, Marino and Kokotovic 1988). In this approach, the dynamical
behaviour of a non-linear singularly perturbed system is characterized by the rapid
approach of the fast system states to the designed attractive slow integral manifold.
The fundamental property of the integral manifold, characterized by the so-called
integral manifold condition, is that once the fast system states hit the integral manifold, they remain in the manifold thereafter.
For a particular class of non-linear singularly perturbed systems, i.e. systems with
fast actuators, Khorasani and Kokotovic (1986) have shown that, with an a-corrected
slow control, the design manifold can be approximated to the exact slow integral
manifold within 0(a 2 ) accuracy. For a more specific class of systems of the same type,
Sharkey and O'Reilly (1987) have pointed out that it is even possible to have the
design manifold as an exact slow integral manifold through an s-corrected slow
control.
In addition to being helpful in design procedures, the singular perturbation approach is also a useful tool for analytic investigation of robustness of the system
properties such as the stability robustness (Saberi and Khalil 1984). The stability
analysis for a general, non-linear, singularly perturbed system can be performed via
the Lyapunov's direct method. Under appropriate conditions, one can form a composite Lyapunov function for the overall system from the respective Lyapunov functions for the reduced-order slow and fast subsystems (Saberi and Khalil 1984).
Received 16 November 1989. Revised 16 January 1990.
t Institute of Electrical Engineering, National Sun Vat-Sen University, Kaohsiung, Taiwan
80424, Republic of China.
0020-7179/90
SJ.OO
© 1990 Taylor
& Francis Ltd.
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572
Juhng-Perng Su and Jer-Guang Hsieh
However, these conditions appear to be quite restrictive and only a small class of
non-linear singularly perturbed systems meet these conditions.
In this paper, an important class of non-linear singularly perturbed systems with
fast actuators is considered. If the origin of the nominally unforced unperturbed
system is asymptotically stable and if a Lyapunov function for the nominal system
is also available, then by appropriate choice of the design manifold a composite
feedback control can be constructed such that the design manifold becomes an exact
integral manifold, and the trajectories of the closed-loop system are steered along the
integral manifold to the origin. Our design logic is different from that of previous
work (Khorasani and Kokotovic 1986,Sharkey and O'Reilly 1987, 1988). First, using
the design manifold as a parameter, the reduced-order slow control is determined
from a set of algebraic equations, obtained by setting e = 0 in the fast subsystem
equations; an e-dependent corrective slow control is added to form an exact integral
manifold control, i.e. under the corrected slow control-the sum of the reducedorder slow control and the a-dependent corrective slow control-the design manifold
becomes an exact slow integral manifold. Second, for the fast subsystem design, an
explicit s-dependent fast control which is also parameterized by the design manifold
is constructed to drive the fast subsystem states to the exact design manifold. Finally,
by appropriate choice of the design manifold, the composite feedback control formed
by the sum of the corrected slow control and the s-dependent fast control, is shown
to render the closed-loop system asymptotically stable at the origin.
2. Slow integral manifold and exact design manifold control
Consider the following non-linear singularly perturbed system with fast actuators:
x = f(x) + G(x)z, x(O) = Xo
ei = g, (x, z) + eg2(x, z) + Q(x)u,
(2.1 a)
z(O) = Zo
(2.1 b)
where x E IR", z E IR m and u E IRP. The functions f, g" g2, G and Q are sufficiently,
many times, continuously differentiable functions of their arguments, and f(O) = 0,
G(O) = 0, g, (0, 0) = o. The parameter e > 0 represents the speed ratio of the slow versus
the fast phenomena of the system, x is the state of the slow plant (2.1 a) and z is the
state of the fast actuator (2.1 b). It is noted that the systems considered here are more
specific than those presented by Sharkey and O'Reilly (1987).
Throughout this paper, we will make the following assumptions.
Assumption I
rank Q(x) = m ~ p V X
E
IR"
Assumption 2
The equilibrium point x = 0 for the nominally unforced unperturbed system x =
f(x) is asymptotically stable and there exists a C'-function V(·): IR" -+ [0, 00), such
that
Lo(x) == VV(x)· f(x) < 0, V X E IR"
(2.2)
where Lo(x) indicates the rate of change of V(x) along the trajectories of the system
x=f(x).
Given any non-linear systems of the form (2.1) satisfying Assumptions I and 2,
Composite control of non-linear perturbed systems
573
our goal is to find a state feedback control u = u(x, z, e) that will steer the system
states along a design integral manifold to the equilibrium point at the origin.
Our feedback control u is composed of two parts:
u(x, z, e) = u,(x, e) + Uf(X, z, e)
(2.3)
where u,(x, e) and Uf(X, z, e) are called slow control and fast control, respectively, each
one of which also consists of two parts:
u,(x, e) = uo(x)
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Uf(X, z, e) =
+ eu.(x)
ut (x, z) + eu;(x, z)
(2.4 a)
(2.4 b)
where uo(x) and uc(x) are called the reduced-order and corrective slow controls,
respectively, and are to be designed on the design manifold. The two components
ui (x, z) and uf(x, z) of the fast control will be designed to steer the fast subsystem
states to the design manifold. As usual, they are designed to be inactive on the design
manifold.
The design manifold is obtained by setting e = 0 in (2.1 b) and by solving
gj (x, z) + Q(x) uo(x)
=0
(2.5)
for z. If the partial derivative [(ojoz)gj (x, z)] is non-singular for each (x, z) and uo(x)
satisfying (2.5), then by the implicit function theorem there exists a unique smooth
function h( • ) mapping an open neighbourhood B, c IR" of x onto an open neighbourhood B, c IR m of z such that
z = h(x, uo(x)) == ho(x)
(2.6)
is a solution of (2.5), and is the description of the design manifold. Conversely, if a
design manifold z = ho(x) is specified at the outset, then the corresponding reducedorder slow control uo(x) is determined from (2.5), i.e.
Q(x) uo(x)
=-g
j
(2.7)
(x, ho(x))
Define
(2.8)
e(t) '" z(t) - ho(x(t))
as the deviation of the fast states from the design manifold. In terms of the error e,
defined in (2.8), and x as new coordinates, the original system (2.1) can be written as
x = f(x) + G(x) ho(x) + G(x)e,
ee = gj (x, e + ho(x))
x(O) =
(2.9)
Xo
+ eg2(x, e + ho(x)) + Q(x) u
- { : x ho(X)] [f(x)
+ G(x) ho(x) + G(x) e],
e(O) = Zo - ho(xo)
(2.10)
The reduced-order subsystem is obtained from (2.9) by setting e = 0 in (2.10) and by
using (2.5). It is described by
x = f(x) + G(x) ho(x)
(2.11)
We already know that at s = 0, the graph z = ho(x) = h(x, uo(x)) = h(x, u,(x, 0)) defines
a slow design manifold. For sufficiently small s > 0, we propose a graph of the form
z = h(x, u,(x, e))
(2.12)
which is a regular perturbation of (2.6), be a manifold of the original system (2.1).
Juhng-Pemg Su and Jer-Guang Hsieh
574
Differentiating both sides of (2.12) with respect to t, we have
i= (dh)x= [ah
dx
ax
+ (ah)(~u,)Jx
au,
ax
(2.13)
Multiplying both sides of (2.13) by e and substituting for X, i, z and u from (2.1 a),
(2.1 b), (2.12), (2.3)and (2.4), respectively, we obtain an s-dependent partial differential
equation:
e(~~) [f(x) + G(x) h(x, u,(x, e))]
=g,(x, h(x, u,(x, e))) + eg2(x, h(x, u,(x, e)))
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+ Q(x) [uo(x) + suc(x) + ul(x, h(x, u,(x, e))) + sul(x, h(x, u,(x, e)))]
(2.14)
which is known as the integral manifold condition.
Definition I
A solution (2.12) of (2.14) is said to be a slow integral manifold of the singularly
perturbed system (2.1) in that, if (2.12) holds for some t = t*, it holds for all t ~ t".
Theorem I
If the corrective slow control ucfx) is designed to satisfy
Q(x) uc(x) = [dd ho(x)] [f(x) + G(x) ho(x)] - g2(X, ho(x))
x
(2.15)
and if the fast control (2.4 b) is designed to be inactive on the design manifold z =
ho(x), then the design manifold z = ho(x) is a slow integral manifold of the original
system (2.1) under the composite feedback control (2.3)and (2.4), where uo(x) satisfies
(2.7).
Proof
We want to show that the design manifold z = ho(x) satisfies the manifold condition (2.14). Replacing h(x, u,(x, e)) with
h(x, u,(x, 0)) = h(x, uo(x)) = ho(x)
in (2.14) and by (2.7) and (2.15), it is clear that under the assumption
ul (x, ho(x)) = uf(x, ho(x)) = 0
(2.16)
i.e. the fast control is inactive on the design manifold, the design manifold ho(x) does
satisfy the manifold condition (2.14) and thus is a slow integral manifold of (2.1).
o
3.
Design of fast control
In this section, a fast control will be constructed such that the magnitude of the
error Ile(t)ll, where I • II is the euclidean norm in IR", approaches zero as quickly as
required when t --+ 00; in other words, the fast states can be steered to the exact design
manifold z = ho(x) with a pre-specified rate.
Composite control of non-linear perturbed systems
575
Consider the error equation (2.10) which is repeated here for convenience:
se = gl (x, e + ho(x)) + Sg2(X, e + ho(x)) + Q(x) u
- {:x ho(X)] [f(x) + G(x) ho(x) + G(x) e], e(O) = zo - ho(xo)
(3.1)
Let A be any Hurwitz matrix, and let P be a positive definite matrix satisfying the
Lyapunov equation
(3.2)
Rewrite (3.1) as
ee =
Ae + [gl(X, z) - Ae] + Sg2(X, z) + Q(x)
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x [uo(x) + SUe (x) + ul (x, z) + SUf(x, z)]
-
e(O) =
S
Zo -
[:x ho(x)] [f(x) + G(x) ho(x) + G(x) e]
ho(x)
(3.3)
where z = e + ho(x). By (2.7) and (2.15), (3.3) can be written as
se = Ae + [gl (x, z) -
gl (x, ho(x)) - Ae]
+ S[g2(X, z) - g2(X, ho(x))]
+ Q(x)[ul(x, z) + suf(x, z)] - {:x ho(x)] G(x) e
e(O) = Zo - ho(x)
(3.4)
Theorem 2
Let the fast control ur(x, z, e] = ul (x, z) + sUf(x, z) be such that
Q(x) u/(x, z) = - [gl (x, z) - gl (x, ho(x)) - Ae]
(3.5)
Q(x) uf(x, z) =
(3.6)
-
{g2(X, z) - g2(X,ho(x)) - [:x ho(X)] G(x) e}
where e(t) = z(t) - ho(x(t)). Then, together with the corrected slow control described
by (2.7) and (2.15), the composite feedback control
u(x, z, s) =
Us(x,
s) + ur(x, z, s) = uo(x) + sue(x) + ul (x, z) + suf(x, z)
will render the system (3.3) asymptotically stable at the equilibrium manifold
z = ho(x).
Proof
Substituting (3.5) and (3.6) into (3.4), we obtain
se(t) = A e(t)
(3.7)
Thus, e = 0, or z = ho(x) is an equilibrium manifold for the subsystem (3.3), and is
0
asymptotically stable by the choice of the Hurwitz matrix A in (3.2).
In particular, if the number of the fast states is equal to the number of the inputs,
one can deduce the following corollary.
576
Juhng-Perng Su and Jer-Guang Hsieh
Corollary I
If m = p, by Assumption 1 Q(x) is invertible for all x E IR"; then under the composite
feedback control
u(x, z, e) = uo(x)
+ eu,(x) + ul (x, z) + euf(x, z)
where
uo(x}
= - Q(x} - [ g I (x, ho(x»
(3.8)
u,(x) = Q(X}-I {[ :x ho(x)] [f(x}
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ul(x, z)
= _Q(X)-I [gl (x, z) -
+ G(x) ho(x)] -
g2(X, ho(x»}
g[ (x, ho(x» - Ae]
ut(x, z) = - Q(X)-I [g2(X, z} - g2(X, ho(x» - (:x ho(X») G(x) e]
(3.9)
(3.IO)
(3.11)
and e = e(x, z) = z - ho(x), the equilibrium manifold e = 0, or z = ho(x), of the fast
subsystem (3.3) is asymptotically stable.
4.
Stability of the closed-loop system
The composite feedback control
u(x, z, e) = u,(x, e) + u((x, z, s) = uo(x)
+ eu,(x) + ul (x, z) + eut(x, z)
which is determined from (2.7), (2.15), (3.5) and (3.6), respectively, is parametrized by
the design manifold z = ho(x}. In this section, a design manifold will be specified to
guarantee the asymptotic stability of the origin of the closed-loop system.
Theorem 3
Consider the non-linear singularly perturbed system described by (2.1), or equivalently (2.10) and (2.1I). Let ho(x) be specified as
(4.1)
where
b;(x) = VV(x)· Gj(x),
1:;:;; i:;:;; m
and
G(x} = [G 1 (x)
...
Gm(x)]
G,(x) is the ith column vector of G(x}. Then, under the composite feedback
u(x, z, e)
= u,(x, e) + u((x, z, s) = uo(x) + euc(x) + ul (x, z) + eut(x, z)
which is determined from (2.7), (2.15), (3.5) and (3.6), respectively, (x, z) = (0, O) is an
asymptotically stable equilibrium point for the closed-loop system (2.1), or equivalently the closed-loop system (2.1 0) and (2.1 I}.
Composite control of non-linear perturbed systems
577
Proof
From the proof of Theorem 2, we know that under the composite feedback
control, the subsystem (2.l0) can be reduced into a stable subsystem (3.7). For the
system (2.9) and (3.7), consider the CI-function L: IR" x IRm --> [0,00), defined by
L{x, e) == V(x) + luT Pe
(4.2)
where V{x) is the CI-function satisfying Assumption 2, and P is the positive definite
matrix which solves (3.2). Let
a{x)== -Lo(x)
= -VV(x)·f(x) > 0
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with the design manifold z = ho{x) specified by (4.l) and (3.2), the rate of change of
L{x, e) along the closed-loop system (2.9) and (3.7) is
+ G{x) ho{x) + G{x) e] + eT P(Ee(t))
L(x{t), e{t)) = VV{x)· [f{x)
m
=
-a{x) +
L
[-lbf{x)
+ ej{t) b;{x) -lef(t)]
i= 1
= -
a{x) -
1
m
1
m
2: i~1
L
2
= - a(x) - -
j
[bf{x) - 2ej{t) b;{x) + ef(t)]
[bj(x) - ej(t)F
= 1
<0
This implies that (x, e) = (O, 0) is an asymptotically stable equilibrium point for the
closed-loop system (2.9) and (3.7). Therefore, the equilibrium point (x, z) = (O, 0) is
asymptotically stable for the closed-loop singularly perturbed system (2.l), or equivalently the closed-loop system (2.9) and (2.10).
0
s.
Example
To illustrate the design procedure, consider the following non-linear system with
high gain actuator, as shown in the Figure, where the non-linear block N is tan z,
Non-linear system with high gain actuator.
Pick
E
= 11K, where
K is the high amplifier gain of the actuator; then we obtain
x = sin x + (XZ + l)z ==f{x) + G(x) z,
TC
TC
--<x<2
2
d = -x - tan z -EZ + U ==gl(X, z) + Egz{X, z) + Q{x) u, z, U E 1R 1
(5.1)
(5.2)
578
Juhng-Perng Su and Jer-Guang Hsieh
Note that x =
°
is the only equilibrium point of :i: = f(x) = sin x for all
XEu={xERn:-~<x<~}
and V(x) = cos x is a Lyapunov function for the system :i: = sin x on U.
The design manifold for the system is chosen via (4.1) as
ho(x) = - tb(x) = !(x 2 + I) sin x
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where b(x) = VV(x)· G(x) = _(x 2 + I) sin x. In this example, since Q(x) = I, the composite feedback control laws are designed via (3.8), (3.9), (3.10) and (3.11), respectively.
The slow control is obtained as follows:
uo(x) = _Q(X)-I [gdx, ho(x»] = x + tan [!(x 2 + I) sin x]
uc(x) = Q(X)-I {[ :x ho(x)] [f(x)
+ G(x) ho(x)] - g2(X, ho(X»}
= [x + I(x 2 + 1)2] sin? x + !(x 2 + 1)[1 + !(x 2 + 1)2] sin x cos x
+!(x 2+I)sinx
Choose A
=-
2, then P
= t solves (3.3). The fast control is then obtained as
ul(x, z) = Q(X)-I [g, (x, z) - g, (x, ho(x» - Ae]
= tan [-!_(x 2 + I) sin x] - tan z + 2z - (x 2 + I) sin x
ul(x, z) = Q(X)-I [g2(X, z) - g2(X, ho(x» - (:x ho(x») G(x)e]
= [!(x 2 + I) sin x -
z]
2
- (x + I)[x sin x + !(x 2 + I) cos x] [z - !(x 2 + I) sin x]
With this feedback control, the equilibrium point (x, z) = (0, 0) of the closed-loop
system (5.1) and (5.2) is asymptotically stable, which can be asserted by using the
Lyapunov function
6. Conclusion
In this paper, an important class of non-linear singularly perturbed systems with
fast actuators, such as non-linear systems with high gain actuators, has been considered. If the origin of the nominally unforced unperturbed system is asymptotically
stable and if a Lyapunov function for the nominal system is also available, then by
appropriate choice of the design manifold, a composite feedback controller can be
constructed such that the design manifold becomes a slow integral manifold and the
asymptotic stability of the origin of the closed-loop system is guaranteed.
ACKNOWLEDGMENT
The authors are grateful to the reviewers for their comments and suggestions.
Composite control of non-linear perturbed systems
579
REFERENCES
and KOKOTOVIC, P. V., 1986, A corrective feedback design for nonlinear
systems with fast actuators. I.E.E.E. Transactions on Automatic Control, 31, 67-69.
KOKOTOVIC, P. V., KHALIL, H. K., and O'REILLY, 1., 1986, Singular Perturbation Methods in
Control: Analysis and Design (New York: Academic Press).
MARINO, R., and KOKOTOVIC, P. V., 1988, A geometric approach to nonlinear singularly
perturbed control systems. Automatica, 24, 31-41.
SABERI, A., and KHALIL, H., 1984, Quadratic-type Lyapunov functions for singularly perturbed
systems. I.E.E.E. Trallsactiolls 011 Automatic Control, 29, 542-550.
SHARKEY, P. M., and O'REILLY, J., 1987, Exact design manifold control of a class of nonlinear
singularly perturbed systems. J.E.E.E. Transactions 011 Automatic Control, 32, 933-935;
1988, Composite control of non-linear singularly perturbed systems: a geometric approach. Intemational Journal of Control, 48,2491-2506.
SOBOLEV, V. A., 1984, Integral manifolds and decomposition of singularly perturbed systems.
Systems and COlltrol Letters, 5, 169-179.
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KHORASANI, K.,