非線性重力波浪在緩坡底床上之淺化

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Asian Journal of Control, Vol. 6, No. 3, pp. 428-431, September 2004
-Brief Paper-
EXISTENCE ANALYSIS FOR LIMIT CYCLES OF
RELAY FEEDBACK SYSTEMS
Chong Lin, Qing-Guo Wang, and Tong Heng Lee
ABSTRACT
This paper is concerned with the problems on the existence of limit
cycles for SISO linear systems under relay feedback. A sufficient condition
is given based on the Brouwer’s fixed point theorem.
KeyWords: Relay feedback systems, limit cycles, Brouwer’s fixed point
theorem.
I. INTRODUCTION
Relay feedback systems have been widely
employed in a broad technological domain for many
decades [1]. Such a system may exhibit some special
behaviours such as non-uniqueness of solutions [2,3],
fast switches or chattering [4,5], sliding motion [6] and
chaos [7]. An important property is that the system
trajectories may tend to a periodic orbit. This periodic
orbit is often termed a limit cycle in the literature. This
property is very useful in modern control applications
such as automatic tuning of controllers and frequency
response estimation and identification [8~11]. Such
practical applications activate the intensive investigation
for limit cycle behaviors. However, most analysis work
have to be based on the assumption that a limit cycle
does exist, due to the difficulty of determining if it is
really the case. Many classical results based on the
describing function method are surveyed in [1]. The
describing function method gives an approximate
analysis for limit cycle behaviors for SISO systems. An
exact approach is reported in [12] and a necessary
condition is given. The drawback is to compute a period
parameter through numerical method and to determine it
really corresponds to a limit cycle. Another work [4]
considers the normalized state-space realization from a
given transfer function, and presents a sufficient
Manuscript received December 6, 2002, revised March 11,
2003, accepted September 1, 2003.
The authors are with Department of Electrical and
Computer Engineering, National University of Singapore, 10
Kent Ridge Crescent, Singapore 119260.
condition for the existence of a symmetric stable limit
cycle with chattering.
In this paper, we will study the problem of
existence of limit cycles for relay feedback systems with
hysteresis and time delay. We will present a sufficient
condition by using the Brouwer’s fixed point theorem.
This paper is organized as follows. In Section 2, the
considered system and problem are formulated. Section
3 establishes some useful lemmas and gives the main
result for the existence problem. This paper is concluded
in Section 4.
Notation:
and n denote, respectively, the field
of real numbers and the n-dimensional real Euclidean
space; I is the identity matrix; the superscript ‘T ’ stands
for the matrix transpose; ||  || denotes the Euclidean norm
for a vector, or the spectral norm for a matrix; f (t) : =
lim  0 + f (t)
II. PROBLEM FORMULATION
Consider a single-input single-output relay feedback system described by
x(t )  Ax(t )  bu(t   )
y(t )  cx(t )
(1)
u(t )  sgn d ( y(t )) ,
where x(t )  n , y(t ) 
and u(t ) 
are the state,
output and control input, respectively; A, b, c are constant
real matrices or vectors with appropriate dimensions;  > 0
and d > 0 stand for the time delay and hysteresis,
respectively; The symbol u(t) = sgnd (y(t)) stands for
C. Lin et al.: Existence Analysis for Limit Cycles of Relay Feedback Systems
 {1}

u (t )   {1}
{1 1}

if y(t )  d  or y (t )  d and u (t )  1
if y (t )  d  or y (t )  d and u (t )  1
if y(t )  d and u(t )  1 or y(t )  d and u(t )  1.
Define the switching planes:
S  { 
n
 c  d},
(3)
S  { 
n
 c  d},
(4)
and let
S  { 
429
(2)
Proof: We prove by contradiction. Suppose that a
solution  (t) to system (1) starting from the initial
condition  (0) will hit neither S+ nor S in a finite time.
The trajectory of  (t) is governed by
t
 (t )  e At  (0)   e A(t  z ) bu0 dz
0
n
 e At  (0)  (e At  I ) A1bu0 ,
  d  c  d}
Sd  { 
n
S d  { 
n
 c  d}
where u0 = 1 or 1. Since eAt  0 as t  +  (due to A
 c  d}
Thus, for some T > ,
For later reference, we introduce some concepts below.
An absolutely continuous function x(t) is called a
solution to system (1) if it satisfies the equations in (1)
almost everywhere. For a solution x(t) to system (1), we
say that its trajectory intersects S+ (or S) with a normal
switch, if it traverses the switching plane S+ (or S) from
S to S+d (or from S to Sd). The intersecting point is
called a traversing point. The instant corresponding to a
traversing point is called a traversing instant. We say
that a trajectory intersects S+ (or S) with normal
switches, if whenever the trajectory intersects S+ or (S),
it intersects with a normal switch. A set K is said to be
convex, if for any s1, s2K, we have  s1 + (1  ) s2K
holds for all [0,1]. A subset of n is compact if and
only if it is closed and bounded. In the next section, we
will study the existence of limit cycles for system (1)
using the following fixed point theorem (see, e.g., [13]).
Lemma 2.1. (Brouwer’s fixed point theorem) Let K be
a non-empty compact convex subset of n and let g:
K  K be a continuous mapping. Then, g has a fixed
point.
III. RESULTS
In this section, we first establish some useful
lemmas, and then give the main result for the existence
of limit cycles. Let G(s) = c(sI  A)1b. Then, we have
G(0) = cA1b if A is invertible.
Lemma 3.1. Suppose that A is Hurwitz and G(0) > d.
Then given an initial condition  (0) the system
trajectory starting from  (0) will intersect either S+ or
S in a finite time.
Hurwitz), we see that c (t )  cA1bu0 as t  .
c (t )  d  t  T  for u0  1 ,
c (t )  d  t  T  for u0  1 ,
which contradicts the control law of system (1). This
completes the proof.
■
Lemma 3.2. Suppose that A is Hurwitz. Let P = PT > 0
be the unique solution to the Lyapunov equation
AT P  PA   I ,
(5)
and define

|| P 2  || ,
(6)
 || P 2  ||  } .
(7)
1
max
||  ||  2 || Pb ||
  { 
n
1
Then,  is invariant for the solutions to system (1), i.e.,
any solution to system (1) starting from  will remain in
 for all t  0.
Proof: We prove the invariant property of  by
contradiction. Suppose there is a solution  (t) starting
from  (t0) such that  (t2) for some t2 > t0, i.e.,
 T (t2 ) P (t2 )   2 .
Noting  T(t0) P  (t0)  2, by continuity of  (t) there
exists a time t1[t0, t2) such that
 T (t1 ) P (t1 )   2 
 T (t ) P (t )   2  t  (t1 t2 ] .
This implies that for t(t1, t2], we have
(8)
Asian Journal of Control, Vol. 6, No. 3, September 2004
430
||  (t ) ||  2 || Pb || .
(9)
Now, let V(x) = xTPx. It follows that
0
e A( h t  z ) b(u ) dz
Since c (0) = d and c (h) = d, it follows that
  || x ||2  2 xT Pbu   || x ||2 2 || Pb || || x || .
It is seen that for || x || > 2 || Pb ||, we have V ( x )  0 . So,
dV ( (t ))
 0 , yielding
dt
t
 T (t ) P (t )  V ( (t ))  V ( (t1 ))   V ( ( z ))dz
t1
 V ( (t1 ))   2 ,
which contradicts (8). This completes the proof.
■
Remark 3.1. Under the conditions of Lemma 3.1, it is
sean that the region  is also a contraction region in the
sense that any trajectory of system (1) will eventually
enter . From Lemma 3.2, it is easy to know that there
exists a scalar a > 0 such that any solution  (t) to system
(1) starting from  (0)  satisfies
||  (t ) ||  a t  0.
(10)
Besides, under the conditions of Lemma 3.1, the sets
S+ and S are non-empty, where
S     S ,
(11)
S     S .
(12)
2d  (h)  0 ,
(15)
where
 (h)  c(e Ah  I )( (0)  A1bu )  2c(e A(ht )  I ) A1bu .
(16)
Now, suppose h  0. We show that this will lead to a
contradiction. Since h  0, we have || eAh I || < 1 and
|| eA(ht) I || < 1 in (6). Taking into account ||  (0) ||  a
(Remark 3.1), we obtain
||  (h) ||  1 || c || (||  (0) ||  || A1b ||)  21 || c || || A1b ||
 2d 
This contradicts (15). Hence, h > 0. This completes the
proof.
■
Remark 3.2. From the proof of Lemma 3.3, we see that
0 is independent of the solution  (t) and the time delay
. This means that any trajectory starting from S+ (or
S) will spend a time longer than 0 to intersect S (or
S+).
Next, we present our main result.
Theorem 3.1. Consider system (1). If
This is because any trajectory will traverse S+ and S in
the region .
The next lemma specifies a lower bound of the time
between any two traversing instants.
Lemma 3.3. Suppose that A is Hurwitz and G(0) > d.
Let h be the time for a trajectory of system (1) to go
from S+ (or S) to S (or S+). Then, we have h > 0,
where 0 > 0 is such that
|| e A  I ||  1    0 ,
(13)
with
1 
h  t
 e Ah ( (0)  A1bu )  A1bu  2e A(ht ) A1bu .
dV ( x)
 xT ( AT P  PA) x  2 xT Pbu
dt
from (9), for t(t1, t2], it holds
 (h)  e A( h t ) (t )  
2d
0.
|| c || (a  3 || A1b ||)
(14)
Proof: Without loss of generality, let  (0) S+ and the
trajectory of  (t) evolving from  (0) spends time h to
reach  (h). For some t[0, h] and u =  1, we have
t
 (t )  e At  (0)   e A(t  z ) bu dz
0
(i) G(0) > d ,
(ii)   0, and
(iii) A is Hurwitz ,
where 0 is as in Lemma 3.3, then system (1) has a limit
cycle.
Proof: From Lemma 3.1, any trajectory x(t) of system
(1) starting from x(0) will intersect, without loss of
generality, S+. Let the traversing point be x(t1)S+ (at
time t1). Since   0, by virtue of Lemma 3.3, after
switching, the trajectory of x(t) will intersect and
traverse S at some point x(t2) (at time t2 > t1 + 0), and
then intersect and traverse S+ again at some point x(t3)
(at time t3 > t2 + 0). Define the mapping : S+  S+
as
( x(t1 ))  x(t3 ).
(17)
Then  maps S+ into itself. The continuity of  can be
shown from the expressions of solutions to system (1).
The fact that S+ is non-empty is obvious and the
compactness of S+ can be easily verified from the
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C. Lin et al.: Existence Analysis for Limit Cycles of Relay Feedback Systems
definition for compact set. For the convexity, it is seen
from the following
1  2  S

c(1  (1   )2 )  d  1  (1   )2  S
 1
2

|| P (1  (1   )2 ) ||    1  (1   ) 2   ,
where [0,1]. Hence, by Lemma 2.1, there exists x*
S+ such that  (x*) = x*. This shows that x*(t) starting
from x* is a limit cycle. This completes the proof.
■
Remark 3.3. The result of Theorem 3.1 is for systems
of the form (1) with d > 0 and  > 0. For systems with
no delay and hysteresis (i.e., d = 0 and  = 0), fast
switches may occur, and there are possibly infinite (but
countable) number of switching times during a finite
time interval. (See Theorem 1 in [5].) In case of this
phenomenon, the analysis is very hard. As for the case
of d > 0 and  = 0, under the conditions of Theorem 3.1
and the well-posedness assumption, the result in
Theorem 3.1 is still valid (by removing item (ii)).
Remark 3.4. It is easy to show that under some
constraint, Theorem 3.1 can be made valid for systems
with any  > 0. For instance, if in the region  the time
duration between any two successive traversing instants
is greater than , then Theorem 3.1 still holds by
removing item (ii).
IV. CONCLUSION AND DISCUSSION
This paper gives a sufficient condition for the
existence of limit cycles of linear systems under relay
feedback. However, there are still lots of work need to
be done. To establish a more easy-checking condition is
a future work.
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