Luo A.C.J. (Ed.) Dynamical Systems: Discontinuity, Stochasticity and Time-Delay

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12 Control of Systems with Time Delay 149

0 10 20 30 40 50 60 70 80

Control Index

0 10 20 30 40 50 60 70 80

x 10

−3

Switching Signal

Time (s)

Fig. 12.8 Switch signal (lower ﬁgure) and the control index (upper ﬁgure) of the hysteretic

switching algorithm for the closed loop response in Fig. 12.7

Table 12.2 Optimal PD feedback gains and jj

max

for the ﬁve sampled time delays of the periodic sys-

tem. note that the mapping step for the periodic

system is one period, while the mapping step for the

LTI system is only one time delay 

Time delay k

jj

max



D0.5498 3:6634 0:0990 0.0130



D0.6676 4:0000 0:8000 0.0083



D0.7854 2:8000 0:6000 0.0141



D0.9032 2:6000 0:6000 0.0213



D1.0210 1:8000 0:6000 0.0347

where x D .x; Px; Rx/

. The period of the system is T D . We select the parameters

to be " D 1; ı D 4,andN D 20 for all sampled time delays. The uncontrolled

system is parametrically unstable. Next, we show the closed-loop response of the

system under a switching PD control with time delay in the range [0.5498, 1.0210].

Five time delays are sampled from the interval, and their optimal gains are listed in

Table 12.2. The stability domains in the gain space are shown in Fig. 12.9. Note that

the optimal gains of all the controls with different time delays fall in the intersection

of the stability domains. Hence, it is possible to use the hysteretic algorithm to

switch among the predesigned controls and to keep the system stable all the time.

Another interesting phenomenon as shown in Fig. 12.9 andalsoin[13] is that the

stability domain in k

 k

gain space grows along the k

direction as time de-

lay increases.

Figure 12.10 shows the closed loop response of the system under the feedback

control with the ﬁrst four different time delays when the system true time delay is

150 B. Song and J.-Q. Sun

−6 −4 −2 0 2

−5

−4

−3

−2

−1

Fig. 12.9 Stability domains (lines) in the gain space and the optimal feedback gains (o) for the

periodic system with ﬁve different time delays 

(i D 1; 2; 3; 4; 5). The stability boundaries move

down along k

axis as time delay increases

0 10 20

−2

−1

Time (s)

, K=K

0 10 20

−2

−1

0 10 20

−2

−1

0 10 20

−2

−1

Time (s)

, K=K

Fig. 12.10 Response of the periodic system under PD feedback controls designed for a speciﬁc

gain when the system true time delay is 

and is assumed to be unknown

taken to be 

and is assumed to be unknown. As it can be seen from the ﬁgure,

when the control designed for the time delay that is close to 

is implemented, the

performance is better.

12 Control of Systems with Time Delay 151

0 5 10 15 20

−2

−1

Time (s)

τ=τ

, K=K

0 5 10 15 20

−2

−1

Time (s)

τ=τ

, Switch from K

Fig. 12.11 The closed loop response of the periodic system under the switched PD control when

the initial gain of the control is K

designed for 

while the system true time delay is 

(bottom),

as compared to the case when the gain is ﬁxed at K

(top)

0 5 10 15 20

Control Index

0 5 10 15 20

0.1

0.12

0.14

0.16

Switching Signal

Time (s)

Fig. 12.12 Switch signal (lower ﬁgure) and the control index (upper ﬁgure) of the hysteretic

switching algorithm for the closed loop response in Fig. 12.11

Next, we start with a control gain K

designed for 

. Figure 12.11 shows the

closed loop response. The hysteretic algorithm switches the gain to reduce the

switch signal .t/ as shown in Fig. 12.12.

152 B. Song and J.-Q. Sun

12.6.3 An Experimental Example

We experimentally study time-delayed feedback control of a rotary ﬂexible joint

made by Quanser. The test apparatus is shown in Fig. 12.13, which consists of a

rotary ﬂexible joint mounted on top of a rigid rotary platform. Two encoders are

used in the system. One measures the angular position of the platform, the other

measures the angular displacement of the ﬂexible joint relative to the platform. The

state equation of the system is of fourth order given by

x D Ax C bu; (12.36)

where

x D Œ; ˛;

; P˛

; (12.37)

A D

00 10

00 01

0 689:86 57:658 0

0 1359:2 57:658 0

; (12.38)

b D Œ0; 0; 107:39; 107:39

: (12.39)

 is the angular position of the platform, and ˛ is the angular position of the ﬂex-

ible joint relative to the platform. By examining the measured step response of the

open-loop system, we have found that the system has a time delay of 0:002 s. An

additional transport delay 0:008 s between the input and the output of the system is

digitally introduced, leading to a total time delay  D 0:01 s.

We consider the time-delayed control u Dkx.t  /. The closed-loop system

reads

x D Ax  bk x.t  /: (12.40)

Fig. 12.13 Experimental setup of the ﬂexible rotary joint by Quanser

12 Control of Systems with Time Delay 153

The mapping over one delay time step is constructed based on the solution given in

(12.28). Consider a region in the gain space

0<k

< 100; 50 < k

< 50; (12.41)

0<k

<5; 0<k

<5:

We look for optimal gains in this region. We initially discretize the domain by di-

viding each gain range into ten partitions. Successive reﬁnement of the grid leads to

the global minimum of jj

max

D 0:65128 in the domain (12.41) and the associated

optimal feedback gains

opt

D Œ16:256; 14:428; 1:089; 0:656: (12.42)

The optimal feedback gains reported in [13] under the same condition by means of

the mapping approach are k

opt

D Œ12:321; 15:432; 0:864; 0:452 with jj

max

0:6511. The difference between these two sets of gains is due to the different ap-

proximation schemes used to compute the delayed response and the mapping.

Figure 12.14 compares the closed-loop response of the rotary ﬂexible joint

system with the optimal feedback gains k

opt

and with the LQR controller de-

signed for the system without time-delay. The control performance of the feedback

controller designed for the system with time-delay is obviously much better

than that of the LQR control. For the completeness, we list the parameters

for the LQR control design here: Q D diag.Œ2; 500; 4; 500; 1; 1/, R D 2 and

k D Œ35:355; 39:574; 1:974; 0:873.

0 1 2 3 4 56

−1

−0.5

0.5

Time (s)

Joint Angle (θ+α)

Fig. 12.14 Comparison of the rotary ﬂexible joint response under the feedback control with

the optimal gains and the LQR controller designed for the system without time-delay. Solid line:

the feedback control with optimal gains. Dashed line: the LQR control for the system without

time-delay

154 B. Song and J.-Q. Sun

12.7 Concluding Remarks

We have presented a review of recent studies of analysis and control of dynamical

systems with time delay. Methods of solution including semi-discretization and con-

tinuous time approximationare reviewed, and their spectral properties are discussed.

Examples of supervisory control of systems with unknown time delay and experi-

mental validation are presented also.

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Chapter 13

Stability and Hopf Bifurcation Analysis

in Synaptically Coupled FHN Neurons

with Two Time Delays

Dejun Fan and Ling Hong

Abstract This chapter presents an investigation of stability and Hopf bifurcation

of the synaptically coupled nonidentical FHN neurons with two time delays.

By regarding the sum of the two delays as a parameter, it is shown that under

certain assumptions, the steady state of the model is absolutely stable; Under an-

other set of conditions, there is a critical value of the parameter, the steady state

is stable when the parameter is less than the critical value and unstable when the

parameter is greater than the critical value. Thus, oscillations via Hopf bifurcation

occur at the steady state when the parameter passes through the critical values.

Then, explicit formulas are derived by using the normal form method and center

manifold theory to determine the direction of the Hopf bifurcations and the stability

of the bifurcating periodic solutions.

13.1 Introduction

The FHN equation has been derived as a simplied model of Hodgkin–Huxley

(HH) equation by FitzHugh and Nagumo. They reduce a four-dimensional HH

equation [1] to a two-dimensional system called the FitzHugh–Nagumo (FHN)

model [2, 3] by extracting excitability of the dynamics of the behavior in the HH

equation. A complete topological and qualitative investigation of the FHN equa-

tion with a cubic nonlinearity has been done by Bautin [4], and a rich variety of

nonlinear phenomena is observed including a hard oscillation, separatrix loops,

bifurcations for equilibria and limit cycles. To understand information process-

ing in the brain, complex dynamics and bifurcations of oscillatory phenomena in

D. Fan (



)

MOE Key Laboratory for Strength and Vibration, School of Aerospace, Xi’an Jiaotong University,

Xi’an, Shaanxi 710049, People’s Republic of China

and

Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai, Shandong 264209,

People’s Republic of China

e-mail: fandejun@hitwh.edu.cn

A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay,

DOI 10.1007/978-1-4419-5754-2

13,

 Springer Science+Business Media, LLC 2010

157

158 D. Fan and L. Hong

coupled FHN neurons have been much investigated [5–7]. It is known that time

delays always occur in the signal transmission for real neurons. The observations

of a ﬁnite time delay in synaptic communication between neurons [8] have stimu-

lated some theoretical studies on coupled time-delay oscillators. Bifurcations and

synchronization have been investigated in coupled identical neurons with delayed

coupling [9, 10]. However, to our knowledge, there is little work in the literature

to deal with delay-coupled nonidentical neurons. For experimental relevance, the

questions of prime importance are about the effects of time delays on dynamical

behaviors.

Recently, Wang et al. [11] has studied the following model:

.t/

DV

.t/

C aV

.t/  W

.t/ C C

tanh.V

.t   //;

.t/

D V

.t/  b

.t/; (13.1)

.t/

DV

.t/

C aV

.t/  W

.t/ C C

tanh.V

.t   //;

.t/

D V

.t/  b

.t/;

where V

.t/ and V

.t/ represent the transmembrane voltage, W

.t/ and W

.t/

should model the time dependence of several physical quantities related to electrical

variables. Constants a; b

are positive,  represents time delay. In [11],

this nonidentical FHN neurons with time-delay coupling has been numerically in-

vestigated involving the effects of time delays and coupling strength on bifurcations

and synchronization.

In order to describe the model more reasonable, we introduce second time delay

in model (13.2), and for convenience, we rewrite this system as the following form:

.t/ Dx

.t/ C ax

.t/  x

.t/ C C

tanh.x

.t  

//;

.t/ D x

.t/  b

.t/; (13.2)

.t/ Dx

.t/ C ax

.t/  x

.t/ C C

tanh.x

.t  

//;

.t/ D x

.t/  b

.t/:

In this chapter, we regard the delay  D 

C 

as a parameter to investigate

the stability and bifurcation to the model (13.3). It is shown that under certain as-

sumptions the steady state of the model ia absolutely stable; Under another set

of conditions, there is a critical value of the parameter, the steady state is stable

when the parameter is less than the critical value and unstable when the parameter

is greater than the critical value. Speciﬁcally, there exists a sequence of values of ,

0<

<

< <

< 

13 Bifurcation Analysis in FHN Neurons with Delays 159

such that the zero equilibrium loses its stability when  passes through 

,anda

Hopf bifurcation occurs when  passes through each critical value 

. Thus, os-

cillations via Hopf bifurcation occur at the steady state when the parameter passes

through the critical value. Meanwhile, using the center manifold theory and normal

form method due to Hassard et al. [12], we derive the algorithm for determining

the direction of the Hopf bifurcations and the stability of the bifurcating periodic

solutions on the center manifold.

The rest of this chapter is organized as follows. In Sect. 13.2, we shall consider

the stability of the zero equilibrium and the existence of the local Hopf bifurcation.

In Sect. 13.3, the stability and direction of periodic solutions bifurcating from Hopf

bifurcations are investigated by using the normal form theory and the center mani-

fold theorem.

We would like to mention that there are several articles on the bifurcation for

neural network models with delays, we refer the readers to [13–23] and references

therein.

13.2 Stability Analysis

Obviously, the origin (0,0,0,0) is an equilibrium of system (13.3), linearizing it gives

.t/ D ax

.t/  x

.t/ C C

.t  

//;

.t/ D x

.t/  b

.t/; (13.3)

.t/ D ax

.t/  x

.t/ C C

.t  

//;

.t/ D x

.t/  b

.t/:

The characteristic equation associated with system (13.4)isgivenby



C A

C B

C CC D  E. C b

/. C b



D 0; (13.4)

where

A D b

C b

 2a;

B D b

 2a.b

C b

/ C a

C 2;

C D a

C b

/  2ab

C b

 2a; (13.5)

D D a

C 1  ab

 ab

;

E D C

;

 D 

C 

In this section, we ﬁrst study the distribution of roots of (13.4). Clearly,

i!.!>0/is a root of (13.4) if and only if ! satisﬁes

 B!

C D D EŒ.b

 !

/ cos ! C !.b

C b

/ sin !;

A!

C C! D EŒ!.b

C b

/ cos !  .b

 !

/ sin !: (13.6)