Applied Mathematics and Computation 230 (2014) 57–64

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

Tolerable delay-margin improvement for systems withinput–output delays using dynamic delayed feedbackcontrollers

0096-3003/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.amc.2013.12.073

⇑ Corresponding author.E-mail addresses: dushmantakumardas29@gmail.com (D.K. Das), sandipg@nitrkl.ac.in (S. Ghosh), bidyadhar@nitrkl.ac.in (B. Subudhi).

Dushmanta Kumar Das ⇑, Sandip Ghosh, Bidyadhar SubudhiCenter of Industrial Electronics and Robotics, Department of Electrical Engineering, National Institute of Technology, Rourkela 769008, India

a r t i c l e i n f o

Keywords:Time-delay systemContinuous pole placement techniqueDynamic state feedback controller

a b s t r a c t

This paper presents investigations on a dynamic state feedback controller with state delaysthat improves tolerable delay margin for systems with input–output delays. Using an iter-ative pole placement technique for time-delay systems, the effect of introducing statedelay in the controller dynamics is studied. It is observed that such a controller improvesthe tolerable delay margins compared to its static or even simple dynamic counterpart.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

Time-delay is inherent to many feedback control systems owing to the fact that information takes finite time to get trans-ported. Often, delays appear in the feedback loop due to the time taken in (i) measuring outputs (ii) computing control ac-tions and (iii) actuating the plant. Such delays in the feedback loop are, in general, destabilizing [1]. However, it is alsopossible that purposeful use of artificial delays in the controller may improve stability of certain systems, e.g., (i) use ofan appropriate delay leads to chattering stability in a milling process [2], (ii) use of delay may yield better purchasingand stocking decisions in supply chain management [3]. Such stabilizing effect of delays is a motivation to many researchersto exploit the possibilities of using them with benefits.

This paper considers the problem of stabilizing systems with Input and Output (IO) delays as shown in Fig. 1. Time takenin measuring the output signal and thereby receiving at the controller is called as the output delay (ss), whereas the sendingtime for the control signal from the controller to the actuator is the input delay (sa). For such systems, if one uses a staticfeedback controller then the delay in the feedback loop may be represented as stotal ¼ sa þ ss [4].

For an illustration, consider a scalar system of the form

_xðtÞ ¼ axðtÞ þ uðt � saÞ; ð1Þ

It is well known that using a static state feedback controller of the form uðtÞ ¼ ksxðt � ssÞ, where ks is the control gain, system(1) can be stabilized till a sa þ ssð Þ < 1 [5]. However, if one uses an observer based controller of the form

_̂xðtÞ ¼ ax̂ðtÞ þ kx̂ðt � saÞ þ lx̂ðt � ssÞ � lxðt � ssÞ; ð2Þ

where x̂ðtÞ is the estimate of the state, l is the observer gain, then the scalar system (1) can be stabilized till asa < 1 andass < 1 [5], which is an improvement over the static feedback one. However, implementing such a controller is difficult since

Fig. 1. Feedback control system with input–output delays.

58 D.K. Das et al. / Applied Mathematics and Computation 230 (2014) 57–64

one has to obtain accurate information of the two delays, which is impractical specifically when these delays are uncertain ortime-varying.

From this perspective, it may be intuited that dynamic controller with delay might have stability improvement ability fortime-delay systems. Note that the inclusion of delay in such controllers is important in addition to the dynamicness. Since,similar to systems without time delays, simple dynamic controllers without time delay doesn’t have any stability improve-ment ability as compared to static controllers typically for state feedback case. The same has been experienced by presentauthors for several example cases.

From the above discussion, it may be perceived that dynamics and state delays combinedly in controllers may help inimproving the tolerable delay bound. Question that now arises is whether the controller dynamics, its state delays or bothof them contribute to this improvement. This paper attempts to address this question and proposes a dynamic feedback con-troller with state delays that improves tolerable delay bound in the feedback loop further. It is just to mention here that thiswork does not investigate the stabilizing ability of controller with time delays for systems that are not otherwise stabilizable,as it has been attempted in [6]. Rather, it looks into the possibility of tolerable delay margin improvement for systems thatare conventionally stabilizable.

2. Problem consideration

The plant dynamics with input delay is represented as:

_xpðtÞ ¼ ApxpðtÞ þ Bpupðt � saÞ; ð3Þ

where xpðtÞ 2 Rnp is the state, upðtÞ 2 Rmp is the control input; Ap and Bp are constant matrices of appropriate dimension. Forstabilizing such system, we consider the following controller types:

Type I: Simple dynamic controller

_xcðtÞ ¼ Ac0xcðtÞ þ Ccxpðt � ssÞ; upðtÞ ¼ xcðtÞ; ð4Þ

Type II: Dynamic controller with a state delay

_xcðtÞ ¼ Ac0xcðtÞ þ Ac1xcðt � s1Þ þ Ccxpðt � ssÞ; upðtÞ ¼ xcðtÞ; ð5Þ

Type III: Dynamic controller with two state delays

_xcðtÞ ¼ Ac0xcðtÞ þ Ac1xcðt � s1Þ þ Ac2xcðt � s2Þ þ Ccxpðt � ssÞ;upðtÞ ¼ xcðtÞ;

ð6Þ

where xcðtÞ 2 Rnc is the state of the dynamic controller and Ac0; Ac1; Ac2 and Cc are the controller matrices to be designed.Stabilization using Type I is of interest to study the effect of controller dynamics on improvement in tolerable delay

ranges whereas the same for Type II corresponds to the effect of both the dynamics and controller state delay. Comparisonof the stabilizing ability of Type III explores whether use of more than one delay in the controller states has any further effect.It may be noted that the controller of Type III is similar to the observer based controller. However, the delays s1 and s2 maytake different values other than the IO delays and may be chosen appropriately.

The closed-loop system for the Type-III controller, (3) along with (6), may be written as:

_nðtÞ ¼ AnðtÞ þ Bnðt � s1Þ þ Cnðt � s2Þ þ Dnðt � saÞ þ Enðt � ssÞ; ð7Þ

where

nðtÞ ¼xpðtÞxcðtÞ

� �; A ¼

Ap 00 Ac0

� �; B ¼

0 00 Ac1

� �;

C ¼0 00 Ac2

� �; D ¼

0 Bp

0 0

� �; E ¼

0 0Cc 0

� �

Note that, the closed loop system for other types of controllers are subset of the above.

D.K. Das et al. / Applied Mathematics and Computation 230 (2014) 57–64 59

3. Continuous pole-placement technique

For being infinite dimensional system, stability of time-delay systems are often addressed using computational ap-proaches, e.g., using frequency sweeping test [1], stability charts [7]. Stabilization, i.e. controller design guaranteeing stabil-ity is expectedly more complex. Lyapunov approaches lead to computationally efficient criteria in terms of Riccati equations[8] or Linear Matrix Inequalities [9,10], but they are indeed conservative so far. The finite spectrum assignment techniqueusing a predictor based controller [11] does not meet the present requirement. To this end, a numerical stabilization tech-nique is available, called as Continuous Pole Placement Technique (CPPT) using which stabilizing control gains may be ob-tained iteratively [5]. Since, there exists no closed form solution for stabilization of time-delay systems, we use thisnumerical procedure just to find the existence of, possibly superior, controllers. For completeness, we next present CPPTbriefly. Details on convergence and performance of the algorithm have been presented in [5].

Consider Cþr represents the closed right half plane of a vertical line through r 2 R and contains the poles with RðkÞP r.Further, C�r is complementary to Cþr . Note that, the number of poles for a time-delay system on Cþr is finite and computation-ally tractable [7]. Software packages are available that can be used readily for computing them, e.g., DDE-BIFTOOL [12]. Thistool has been used to compute the rate of synchronization in biological networks with time-delays [13], to design highertruncated predictor for linear systems [14] and to study flute-like instruments which are modelled as a delay dynamical sys-tem [15].

For stabilization, one requires to obtain a control gain (K) for which all the closed-loop poles are placed in C�0 . Now, let usdefine a K 2 Rn consisting of the n number of controller parameters and a sensitivity matrix S ¼ ½Si;j� 2 Rn�m, where Si;j ¼ @ki

@kj,

ki being the ith rightmost pole and kj the jth element of K. Then the CPPT with a slight modification to the one presented in [5]for obtaining a stabilizing controller is presented in the following.

Algorithm 1.

Step 1: Initialize controller parameters K. Set a �r for which the poles in Cþ�r will be regulated. Let the number of poles in Cþ�r be m andset a desired change in the regulated poles as dk 2 Rm.

Step 2: Update m; dk, and compute S.Step 3: Compute desired change in controller parameters as dK ¼ Sydk, where Sy is the Moore–Penrose inverse of S, and update K as

K ¼ K þ dK .Step 4: Check whether the rightmost pole is placed to the left-half plane. If not then go to Step 2 until a certain number of iter-

ations, else exit declaring stabilized or not.

Remark 1. Note that, m must be less than or at most equal to n. Therefore, a combination of �r and initial K may be chosen sothat the number of poles in Cþ�r is 6 n. Moreover, since mere stabilization is concerned in this paper, one may consider onlythe real parts of the poles for regulation, i.e., for a complex pole pair the number of controlled poles (real component) wouldbe one.

To this end, one need to compute @ki@kj

to obtain S. For the purpose, let ith root of the characteristic equation of (7) be ki andhence it satisfies

kiI � A� Be�kis1 � Ce�kis2 � De�kisa � Ee�kissn o

� v i ¼ 0 ð8Þ

where, v i is the corresponding eigenvector.Using a normalizing function for v i, one may write

f@ðv iÞ ¼ 0 ð9Þ

Taking derivative of (8) and (9) with respect to kj, one obtains

w1 w2

df T@ðv iÞdv i

0

" # @v i@kj

@ki@kj

24

35 ¼ w3v i

0

� �; ð10Þ

where

w1 ¼ kiI � A� Be�kis1 � Ce�kis2 � De�kisa � Ee�kiss ;

w2 ¼ I þ s1Be�kis1 þ s2Ce�kis2 þ saDe�kisa þ ssEe�kiss ;

w3 ¼ @ �A� Be�kis1 � Ce�kis2 � De�kisa � Ee�kiss� �

=@kj;

Finally, using (10), one can compute @ki@kj

and thereby obtain S. Before presenting the observations for different controllers, wefirst present a case to demonstrate the stabilizing ability of the CPPT. For implementation of Algorithm, we set (i) Minimumreal part part of eigenvalue as 20, (ii) Maximum number of eigen values to 10, (iii) Maximum number of newton iteration to

10 20 30 40 50 60−1.5

−1

−0.5

0

0.5

1

Fig. 2. Variations of real parts of rightmost poles.

60 D.K. Das et al. / Applied Mathematics and Computation 230 (2014) 57–64

12. in BIFTOOL. These parameter set are used for all the results presented in this paper. Consider stabilization of system (1)with a first order Type II controller for the choice sa ¼ ss ¼ s1 ¼ 0:50. Variations of real parts of some rightmost eigenvaluesis shown in Fig. 2. Initially one pole is in Cþ�r for a initial choice of K ¼ ½1� 0:7� 2�. We set �r ¼ �0:2 and correspondinglym ¼ 1. As the controller parameters evolve, the 2nd rightmost complex pole pair splits into two real ones at the 28 iteration.At 41 iteration m is updated to 2 and the system is stabilized at 66 iteration with a control gainK ¼ ½1:1779 �0:8958 �2:0736 �.

4. Stabilization of system (1)

This section presents stabilizing results obtained for the three controller types using CPPT for system (1) with a ¼ 1. Theapproach is rigorous but effective to study the problem in hand. First order controller dynamics has been used for the study.Moreover, the results are obtained in the delay parameter plane to examine their ability to improve tolerable delay values.

4.1. Simple dynamic controller (Type I)

The variation of maximum tolerable ss with respect to sa for Type I controller is shown in Fig. 3. It is found that the max-imum stotal is less than 1, i.e., the dynamic controller of the form (4) can stabilize the system up to the delay limit which isalso attainable using static feedback controllers.

4.2. Dynamic controller with a state delay (Type II)

Since the controller has more than two delays, the following subcases are considered.

0.2 0.4 0.6 0.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Fig. 3. Variation of maximum ss with respect to sa.

10−2

10−1

100

101

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2(1.74,1.17)

Fig. 4. Variation of sa ¼ ss with respect to s1.

D.K. Das et al. / Applied Mathematics and Computation 230 (2014) 57–64 61

4.2.1. Variation of sa ¼ ss with respect to s1

The behavior of tolerable sa ¼ ss with respect to variation in s1 is shown in Fig. 4. With increase in s1 from 0.04, tolerablesa (ss) also gradually increases from a initial value of less than 0:5 until s1 ¼ 1:74, the corresponding maximum value of sa

(ss) is 1.17, i.e. stotal ¼ 2:34. Note that, this stotal is larger than the one achievable using static feedback (and simple dynamicfeedback) controller and even using observer based controller of (stotal ¼ 2) [5]. However, the improvement ability getsstalled at s1 ¼ 1:74. This point may be referred to as a stalling point. At this point, the convergence of the system stateand controller state are shown in Fig. 5 and Fig. 6 with controller parametersAc0 ¼ �0:3247; Cc ¼ �3:9173; Ac1 ¼ �3:5925 and initial condition xpð0Þ ¼ 1; t 2 �2:34 0½ �; xcð0Þ ¼ 0 . Beyond this point,the tolerable sa first decreases abruptly and then remains at a value of ss ¼ sa � 0:5 for a while. It is also observed that thetolerable saðssÞ falls even below 0.5 with s1 > 30.

Hence, the use of dynamic controllers with a state delay s1 is beneficial compared to static controller or simple dynamiccontrollers provided s1 is chosen suitably (within a certain range). Moreover, the optimal value of s1 for which maximumtolerable delay obtained is highly fragile since a slight increase from this value decreases the tolerable delay valuesconsiderably.

4.2.2. Variation of sa with respect to ss for different values of s1

Next, consider the variation of ss with respect to sa for different choices of s1. Four different choices ofs1 ¼ 1:75; 1:74;1:70 and 1.50 (around the stalling point) are used for the study. The corresponding results are shown inFig. 7. For a chosen s1; stotal remains constant and s1 ¼ 1:74 is the optimal value for all combinations of sa and ss. It is alsoobserved that a single controller works for a particular s1. For example, the controller is as Ac0 ¼ �0:3247; Ac1 ¼ �3:9173and Cc ¼ �3:5925 for s1 ¼ 1:74. It may also be noted that unlike observer based controller [5] for which sa � 1 andss � 1, here the tolerable delay limit is only stotal 6 2:34 irrespective of individual values of the delays. Therefore, the resultin the previous subsection (Fig. 4) otherwise can be seen as the effect of s1 on stotal.

0 1000 2000 3000 4000−300

−250

−200

−150

−100

−50

0

50

100

Time

Sys

tem

sta

te x

p(t)

Fig. 5. Variation of system state xpðtÞ for Ac0 ¼ �0:3247; Ac1 ¼ �3:9173 and Cc ¼ �3:5925.

0 1000 2000 3000 4000−100

−50

0

50

100

150

200

250

300

Time

Con

trol

ler

stat

e u

p(t)

Fig. 6. Variation of controller state upðtÞ for Ac0 ¼ �0:3247; Ac1 ¼ �3:9173 and Cc ¼ �3:5925.

0.2 0.4 0.6 0.8 1

1.2

1.4

1.6

1.8

2

2.2 For τ1=1.74

For τ1=1.70

For τ1=1.50

For τ1=1.75

Fig. 7. Variation of ss with respect to sa for different s1.

0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.2

0.4

0.6

0.8

1

1.2

1.4

Fig. 8. Variation of sa with respect to s1 ¼ ss .

62 D.K. Das et al. / Applied Mathematics and Computation 230 (2014) 57–64

4.2.3. Variation of sa with respect to s1 ¼ ss

The previous result does not answer the question whether choosing s1 equal to sa or ss is advantageous or not. To studythis, consider the case that s1 ¼ ss. For this, variation of tolerable sa with respect to ss is presented in Fig. 8. Initially withincrease in ss until 0.21, sa increases to 1.31, but after that it decreases gradually. At ss ¼ 1:74; sa suddenly decreases to0.17. Note that, this value is same as obtained in the previous sub-cases and corroborates that there exists an optimal s1

1 1.5 2 2.5 31.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

(2.54,1.35)

Fig. 9. Variation of sa ¼ ss with respect to s2 for s1 equal to 1.74.

D.K. Das et al. / Applied Mathematics and Computation 230 (2014) 57–64 63

for which maximum tolerable delay may be obtained. Same as the previous cases, the maximum stotal is 2.34 at ss ¼ 1:74.Therefore, the choice of optimal s1 can be made independently. For a system of class (1) with a ¼ 1, it is s1 ¼ 1:74.

4.3. Dynamic controller with two state delays (Type-III)

We consider here obtaining variation of tolerable sa ¼ ss with respect to s2 for s1 ¼ 1:74 (the stalling point for Type IIcontroller). This result is shown in Fig. 9. It can be seen that for s2 � 1:89, the tolerable sa ¼ ss is 1.17, which is same as thatobtained using controller with a single delay. However, for 1:89 < s2 � 2:54, the stotal is larger compared to the single delaycase. However, beyond this value, the tolerable delay decreases significantly. The maximum tolerable delay is 2.70 fors2 ¼ 2:54.

5. Stabilization of a second order system

So far, we have studied the improvement abilities of dynamic state feedback controller with state delays for scalar sys-tems. To strengthen the observations further, a second order example is now considered for stabilization with a dynamiccontroller of Type II. The plant matrices are given by

Ap ¼0 11 �1

� �; Bp ¼

01

� �ð11Þ

Note that, the open-loop plant (11) has one real unstable eigenvalue at 0.6180. Following Theorem 7 of [16], (11) is stabil-izable using any LTI controller for stotal ¼ 2=0:6180 ¼ 3:2362. Now, we study the variation of stotal with respect to s1 using (5)(Type-II controller). The obtained maximum stotal is shown in Fig. 10. It can be seen that the tolerable delay (s) increases withrespect to s1 until s1 ¼ 2:7 (stalling point), but then decreases abruptly. At the stalling point, the total tolerable delay is 4.7,which is higher than that achievable using LTI controller (stotal ¼ 3:2362) [16]. The controller parameters corresponding tostotal ¼ 4:7 are obtained as k1 ¼ 0:85766; ac1 ¼ 0:347076; ac2 ¼ �1:852729; cc1 ¼ �1:75554 and cc2 ¼ �1:829. Therefore,

10−1

100

101

2

3

4

5(2.7,4.7)

Fig. 10. Variation of s with respect to s1.

64 D.K. Das et al. / Applied Mathematics and Computation 230 (2014) 57–64

this example also substantiates the observations made previously that dynamic controllers with state delays has tolerabledelay margin improvement ability.

6. Conclusion

Improving stabilizing ability of dynamic controllers with state delay is studied in this paper. Using CPPT technique, sta-bilizing results are obtained in the delay-parameter plane for some systems with input–output delay. The salient observa-tions made from the studies are:

1. Use of dynamic state feedback controller with state delays may improve tolerable delay margin compared to static orsimple dynamic feedback controllers.

2. Use of multiple delays may improve the tolerable delay margin further compared to using a single one.

Besides being the observations made, the work opens up a new set of problems involving the proposed dynamic controllers,for example, how to design such controllers using Lyapunov framework, how is the effect of such controllers on plant uncer-tainties, to mention a few.

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