PLACING MINIMUM PHASE ZEROS TO SHAPE TRANSIENT RESPONSE:GENERALIZATION FROM CONTROL OF HYBRID POWER SYSTEMS

Bilal S. Salih ∗Department of Mech. and Aerospace Engineering

University of Central FloridaOrlando, Florida 32816

Email: Bilal.salih@knights.ucf.edu

Tuhin K. DasDepartment of Mech. and Aerospace Engineering

University of Central FloridaOrlando, Florida 32816

Email: tuhin.das@ucf.edu

ABSTRACTConservation of energy can be applied in designing control

of hybrid power systems to manage power demand and supply. Inpractice, it can be used for designing decentralized controllers.In this paper, this idea is analyzed in a generalized theoreticalframework. The problem is transformed to that of using minimumphase zeros to generate a specific type of transient response ad-mitted by dynamical systems. Here, the transient step response isshaped using an underlying conservation principle. In this paper,emphasis is placed on second order systems. However, the analy-sis can be extended to higher order transfer functions. Analyticalresults relating zero location to the matched/ mismatched areasof the transient response are established for a class of second or-der systems. A combination of feedback and feedforward actionsare shown to achieve the desired zero placement/addition andthe desired transient response. The proposed analysis promisesextension to nonlinear systems. Optimization studies also seemappropriate, especially for higher order transfer functions.

INTRODUCTIONHybrid Power Systems (HPS) are a combination of diver-

sified modes of energy generation technologies, both renew-able and non-renewable. HPS augment flexibility of powergeneration sources by optimizing the utilization, and balancingthe strengths and shortages of each source [1, 2]. Solid oxidefuel cells (SOFC) combined with gas turbines or wind turbines(WT )combined with solar dishes and battery storage are exam-ples of HPS. A thorough review of such systems and otherexamples can be found in [3, 4]. Strengthening HPS with anEnergy Storage Device (ESD) could improve efficiency signif-icantly. The integration of ESD in HPS ensures maintainingan instantaneous power demand when it exceeds the maximumpower supplied by the system [1]. Ultra-capacitor (UC) is domi-nantly used, solo or in combination with Li− ion battery, in order

∗Address all correspondence to this author.

to augment HPS performance and improve load following capa-bility. This is due to its ability of being charged or dischargedquickly without degradation or damaging the cell [5]. In orderto integrate UC in HPS, a control strategy and power manage-ment algorithm are required to safeguard the storage device frombeing overcharged or progressively discharged. Robust controlstrategies for a SOFC system with UC, based on nonlinear andH∞ control applications, were introduced in [6, 7]. The core ofthe aforementioned strategies is to have a central processor thatreceives locally sensed information and controls all the compo-nents of the HPS. Such a scheme is referred to as a central-ized power management control. Although a centralized controlapproach is easier to be developed, its implementation can en-counter practical limitations such as system scalability. Anotherdrawback is that any failure in the central processor will cruciallyaffect the functionality of the entire system [8]. In contrast, de-centralization solves the issue of scalability and provides a rem-edy for the practical limitations mentioned above. Research in-vestigations of decentralized control strategies of power systemscan be found in [9–11].

In prior work, addressed in [12], one of the authors of thisstudy proposed a decentralized power management for HPS. Thesystem consists of a power source SOFC connected in parallel toa storage device UC. While the load demand is being met bythe HPS, the proposed power management approach applies en-ergy conservation principles to assure that the total energy of theUC remains constant after the charge/discharge cycles. Individ-ual controllers for the power source and storage device are de-veloped as follows: The power source controller utilizes its owntransient response history to anticipate the energy deficit that isrecovered by the storage device, and accordingly, alter its ownoutput power. In the meantime, the second controller enablesthe UC to work as an energy buffer and track the load in thepresence of fluctuating load demand. The following research ismotivated by the observation that energy conservation principles

Proceedings of the ASME 2017 Dynamic Systems and Control Conference DSCC2017

October 11-13, 2017, Tysons, Virginia, USA

DSCC2017-5234

1 Copyright © 2017 ASME

can be applied to develop decentralized controllers of HPS. Inthis context, a theoretical framework study is conducted to fur-ther explore this idea. The characterizations of which a systemresponse should have to maintain energy conservation require-ments are investigated. In this regard, a formulation based onarea-matching is proposed. Fundamentally, the goal is to obtainthe overall transfer function a system should posses to gener-ate area-matching transient. Here, the dynamic response of thesystem will result in a shape for which the summation of areasabove and areas below a constant reference will be matched/ mis-matched. Therefore, the response of the first and second ordersystems will be examined. As a result, a modified second ordersystem model that assures area-matching will be formulated. Acombination of feedback and feedforward actions will be usedto achieve the desired transient response. Furthermore, the ne-cessity of adding/placing a proper minimum phase zero in sucha system will be confirmed. Analytical results relating the zerolocation to the aforementioned matched/mismatched areas willbe also presented. The proposed analysis can help generalizethe concept of area-matching transient to higher order linear andnonlinear systems and optimal control. In addition, this investi-gation can provide a path for more studies in the area of feedfor-ward controls and the affect of system zero placement/ addition.The direct application of our research is the power managementof HPS, as illustrated in the aforementioned discussion. The ap-plicability of this approach in the area of the rendezvous / pursuitproblems is also worth studying.

In this paper, theoretical analysis to demonstrate the conceptof area-matching transient is first presented. Then, the conditionsunder which a second order system can maintain area-matchingare obtained and discussed. Next, observations are addressed.Finally a conclusion is provided.

1 AREA-MATCHING TRANSIENTS1.1 Example Motivated by Energy Conservation

Based Control of HPSApplying conservation of energy in designing control of

HPS to manage power demand and supply, and to handle ESD,was proposed by one of the authors of this study in [12]. Theconcept of this idea is demonstrated in Fig. 1 and shows the loadfollowing mode response of a HPS. The response XRL repre-sents a step change in load demand that should be supplied bythe HPS, and XRS represents the corresponding response of theprimary power source of the system. The interval ts ≤ t ≤ t f rep-resents the transient region as the response XRS continues track-ing the changing load demand XRL, matching it at a steady state,t > t f . In order to avoid the fluctuation in delivered power dur-ing the transient interval, the ESD works as a buffer that deliv-ers the power demand and absorbs the extra energy provided bythe primary power source. Therefore, the primary power sourceshould provide enough energy to maintain the load demand andcompensate the energy removed from the ESD during the tran-

Time

Sy

ste

m R

esp

on

se

A b, i

A a, j

X RS

X RL A C

t s t f

FIGURE 1. AREA MATCHING TRANSIENT

sient response. In addition, the energy of the ESD should be pre-served in order from being overcharged while the load demand isbeing met. A detailed explanation is presented in Fig. 1. The un-shaded areas underneath XRL, denoted by Ab,i, represent the gapbetween the provided energy from the primary power source andthe load demand ; i.e. the amount of energy discharged from theESD. Similarly, the unshaded areas above XRL, denoted by Aa, j,represent the extra energy to be supplied by the primary powersource in order to maintain the load demand and charge the ESD.Hence, the condition

limt→∞

[∑i=1

Ab,i−η ∑j=1

Aa, j

]→ 0 (1)

will ensure that the energy of the ESD is preserved while theload demand is being met. The presence of system losses duringthe charge / discharge cycles is defined through the efficiencyη . Generalizing the result from Eqn. (1) for linear second ordersystem is addressed in the next section.

1.2 Generalized Problem FormulationIn this research study, our goal is to investigate which char-

acteristics, a transient response of a system should have in or-der to provide a behavior that can maintain energy conservationrequirements. Conceptually, the objective is to propose controlsolutions that can shape the response of the system in a form forwhich the difference between ∑

∞i=1 Ab,i and ∑

∞j=1 Aa, j equals a

targeted value η . Ideally when losses are absent, i.e. η = 1, thedifference equals zero because the above and below areas canceleach other. We refer to this approach by area-matching transientanalysis. Similar terminology, known as equal area criterion,can be found in literature [13–15]. However, the fundamentalof the two approaches is different. The equal area criterion is re-lated to the phase synchronization problem in AC power supplieswith mutable sources. On the other hand, the analysis of area-matching, proposed in this work, provides a control strategy tomodify system characteristics through controlled parameters asrequired. In order to generalize the discussion and formulate theproblem, let ARS, ARL, and AC be the area underneath the curve

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XRS, the area underneath the reference line XRL, and the com-mon area under XRL and XRS, respectively, as shown in Fig. 1.Consequently, expressions for ARS and ARL can be formulated asfollows:

ARL = AC +∞

∑i=1

Ab,i, ARS = AC +∞

∑j=1

Aa, j (2)

Upon adding AC to both sides of the desired Eqn. (1) and apply-ing the expressions from Eqn. (2),

AC +∑∞i=1 Ab,i = AC +η ∑

∞j=1 Aa, j

=⇒ ARL = [ARS−∑∞j=1 Aa, j]+η ∑

∞j=1 Aa, j

(3)

Eqn. (3) then can be simplified as:

ARS−ARL = [1−η ]∞

∑j=1

Aa, j (4)

The result from Eqn. (4) represents the condition that mustbe satisfied to ensure the applicability of the area-matching tran-sient which is addressed in the next section.

2 A CLASS OF SECOND ORDER SYSTEMS WITHMINIMUM PHASE ZEROSThe response of first order system does not exhibit the os-

cillatory behavior that is essential for area-matching transient.Thus, we begin by exploring the response of a pure second ordersystem that has only poles with no zeros as shown below:

X +2ζ ωnX +ωn2X = ωn

2u(t) (5)where ζ and ωn are the damping ratio and the natural frequency,respectively. However, Eqn. (4) can not be satisfied by a systemwith a pure second order dynamic as shown in the next calcu-lations. Eqn. (6) shows the under-damped step response of thesystem presented in Eqn. (5)

X(t) = 1− exp−σt[

cosωdt +σ

ωdsinωdt

](6)

where σ = ζ ωn and ωd = ωn√

1−ζ 2. Referring to Fig. 1, letXRS be the unit step response from Eqn. (6), that is XRS =X(t). Inaddition, let XRL be a unit step reference line, i.e. XRL = 1(t). Inorder to satisfy the condition from Eqn. (4) with the absence oflosses,η = 1, the ramp error eramp of Eqn. (6) should approachzero as follows:

ARS−ARL =−eramp = limt→∞

[∫ t

0XRSdt−

∫ t

0XRLdt

]→ 0 (7)

Substituting the expression of XRS from Eqn. (6) into Eqn. (7),and normalizing it with respect to XRL = 1(t), yields to the fol-lowing integral:

−eramp =∫

∞

0−exp−σt

[cosωdt +

σ

ωdsinωdt

]dt (8)

Solving the integral yields:

−eramp =−[

σ

ω2d +σ2 +

σ

ωd

ωd

ω2d +σ2

]=⇒ eramp =

2ζ

ωn(9)

The result from Eqn. (9) shows that, for a pure second order dy-namic, area-matching can be only ensured if ωn = ∞ or the sys-tem is undamped, i.e. ζ = 0. Such an undamped system which

exhibits a sinusoidal step response, is not appropriate for realis-tic applications such as power systems. In addition, for non idealcases of η < 1, Eqn. (4) requires that the ∑

∞j=1 Aa, j be greater

than ∑∞j=1 Ab,i, η =

∑∞j=1 Ab,i

∑∞j=1 Aa, j

. This can be achieved by havinga system with a negative eramp which is not feasible in systemswith a pure second order dynamic.2.1 Effect of Zero Placement

In the previous section, it was shown that a system with apure second order dynamic does not satisfy the condition of thearea-matching that is stated in Eqn. (4). Increasing the order ofthe system represents an alternate path to be explored ; how-ever, as a result, the system will be more complicated. One ofthe effective methods to modify response characteristics of sys-tems and at the same time preserve the order, is the minimumphase zeros placement approach. An additional zero with a cer-tain location can significantly influence the response of the sys-tem through increasing its overshoot. Moreover, the presence ofa proper zero in the system can substantially enhance the steadystate response and guarantee minimum or zero error in followingthe input signal. Therefore, a modified second order system withan additional zero is investigated.

Consider adding a zero with a value of α to Eqn. (5) as fol-low: X +2ζ ωnX +ωn

2X = ωn2u(t)+α u(t) (10)

To find the step response, Eqn. (10) is first transformed to thefrequency domain as shown below:

X(s)u(s)

= G(s) =ω2

n +αss2 +2ζ ωns+ω2

n(11)

The Laplace inverse of Eqn. (11) produces

X(t) = 1− exp−σt[

cosωdt +σ −α

ωdsinωdt

](12)

Applying Eqn. (12) into Eqn. (7) with XRS =X(t) and XRL = 1(t),leads to the following result:

−eramp =−[σ

ω2d +σ2 +

σ −α

ωd

ωd

ω2d +σ2 ] =

α−2ζ ωn

ω2n

(13)

The necessary condition of stability requires that the coefficient2ζ ωn to be positive. In Eqn. (13) setting α = 2ζ ωn returns a zeroeramp. Similarly, using 0 < α < 2ζ ωn and α > 2ζ ωn producea positive and negative eramp, respectively. Consequently, area-matching is feasible for a modified second order dynamic if aproper zero, α ≥ 2ζ ωn, is added to the system. Hence, for theideal case of η = 1, the proper value of α that ensures area-matching equals 2ζ ωn. However, for non ideal cases η ≤ 1, theanalysis is not explicit. Therefore, the analysis of finding the

location of the proper zero for a given value of [η =∑

∞j=1 Ab,i

∑∞j=1 Aa, j

]< 1is discussed in the next section.2.2 An Infinite Analysis to Relate the Efficiency η with

the Zero LocationThe condition to maintain area-matching with the presence

of η is stated in Eqn. (4). Utilizing the result of Eqn. (13) from

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0

t 1 t 2 t 3 t 4 t 5 t n-1 t nt n-2

A b, i

A a, 1

A a, 2

A a, j

t 0

A a, j-1

X RS

Time

Sy

ste

m E

rro

r

FIGURE 2. GEOMETRIC SERIES ANALYSIS.

the previous section, Eqn. (4) can be modified as follow:

ARS−ARL = [1−η ]∞

∑j

Aa, j =−eramp =α−2ζ ωn

ω2n

(14)

Similar to the ideal case, a proper zero α that can ensure the ap-plicability of area-matching must be found. Eqn. (14) includestwo unknowns that are required to obtain α . These are ∑

∞j=1 Aa, j

and η . Therefore, the method of finding α for a given η is ad-dressed below. Using this approach requires knowing the totalvalue of ∑

∞j=1 Aa, j when the dynamic response of the system con-

verges to a steady state.Referring to Fig. 2, one can conjecture that the areas of the

shown response with respect to the reference line of a step sig-nal follows a specific progression. Therefore, a geometric se-ries analysis is utilized to validate this resemblance argument andevaluate ∑

∞j=1 Aa, j. First, Eqn. (12) is rearranged as

X(t) = 1− exp−σtβ [sin(ωdt +φ)] (15)

where β =√

1+ γ2, γ = σ−α

ωdand φ is the phase angle . Then,

the step error of Eqn. (15) is

X(t) = exp−σtβ [sin(ωdt +φ)] (16)

Subsequently, Eqn. (16) is integrated for the interval tn →tn+1, n = 1,2, ... which yields:

Aa, j =−β

σ2 +ω2d

exp−σtn [σ sin(ωdtn +φ)+ωd cos(ωdtn +φ)]tn=2 jtn=2 j−1

(17)In order to utilize the result from Eqn. (17), integration inter-vals tn, n = 0,1,2, ... are evaluated through finding the roots ofEqn. (15) at sin(ωdt +φ) = 0 i.e. the roots at:

(ωdtn +φ) = nπ, n = 0,1,2, ... (18)which leads to:

tn =nπ−φ

ωd, n = 1,2, ... (19)

Thus, the values of areas Aa, j, j = 1,2,3, ... can be obtained us-ing:

Aa, j =−β ωdσ2+ω2

d

[exp−σ

(2 j)π−φ

ωd +exp−σ(2 j−1)π−φ

ωd

](20)

0 2 4 6 8 10 12 14 16 180

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time

Response X

RS

XRL

α=1.8 , η=0.5

α=0.6 , η=1

α=1 , η=0.7

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time

XRL

α=2.3 , η=0.25α

α

=1.8 , η=0.48

=1.4 , η=1

Re

sp

on

se

XR

S

(A)

(B)

FIGURE 3. RESPONSE WITH DIFFERENT VALUES OF α FOR:(A) ζ = 0.3, (B) ζ = 0.7.

Next, in order to evaluate ∑∞j=1 Aa, j using geometric series anal-

ysis, the ratio r between each two successive terms must be con-stant, which is indeed the case as shown in the result below:

r = Aa, j+1Aa, j

= exp−2π σωd , j = 1,2,3, ... (21)

The definition of geometric series is then used to find thevalue at which ∑

∞j=1 Aa, j converges as follows [16]:

∞

∑j=1

Aa, j =∞

∑j=1

Aa,1r( j−1) =Aa,1

1− r, | r |< 1. (22)

where r is defined in Eqn. (21), and Aa,1 can be found fromEqn. (20) as:

Aa,1 =−β ωdσ2+ω2

d

[exp−σ

2π−φ

ωd +exp−σπ−φ

ωd

](23)

Finally, by substituting the result from Eqn. (22) intoEqn. (14), the efficiency η can be expressed as:

η = 1− (1− r)[α − 2ζ ωn]

ω2n Aa,1

= 1+(1− r)eramp

Aa,1(24)

Although Eqn. (24) does not provide an explicit expression forα , it can be utilized numerically to define a range of values bywhich a proper zero for a desired η can be located.

3 OBSERVATIONS3.1 System Characteristics

Simulation results to validate the analysis of area-matchingare presented in Fig. 3 and Fig. 4. In all the simulation scenar-ios, the natural frequency was chosen to be ωn = 1rad/sec while

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0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

α

− e

ram

p

ηζ=0.3

ζ=0.7

ζ=0.9η

− e ramp

FIGURE 4. SYSTEM RAMP ERROR AND η VS. α with ωn = 1.

G (s)r y+

++

-D(s)

yd=1/s

F(s)= /b0 α

FIGURE 5. FEEDBACK AND FEEDFORWARD SCHEMATIC.

different values of the damping ratio ζ were selected. Fig. 3shows the response XRS using different values of system zero α .In Fig. 3 (A), the damping ratio is set to be ζ = 0.3. On the otherhand, in Fig. 3 (B), the damping ratio is set to be ζ = 0.7 whichprovides low oscillation behavior; thus cycling will be minimizedwhich is desired in realistic applications such as power systems.The curves with α = 0.6 and α = 1.4 represent the ideal case(η = 1) for ζ = 0.3 and ζ = 0.7, respectively, while the othercurves represent the non ideal case where η is less than unity.As seen from Fig. 3, by changing the value of α , the areas Aa,lwill be proportionally affected. Increasing α will increase the

areas Aa, j and decreases the areas Ab,i. Hence, η =∑

∞j=1 Ab,i

∑∞j=1 Aa, j

willbe affected. This observation is also confirmed in Fig. 4 whichpresents the relation between η and the system zero α . The con-dition, eramp ≤ 0 which is required to get η ≤ 1 is also confirmedin Fig. 4 where three different values of ζ = 0.3,0.7, and 0.9were used.

3.2 Realization with Feedback and Feedforward Con-trol

One of the main aspects of this research investigation was toanswer the following question : What kind of system can providea response that ensures the applicability of the area-matching ?This inquiry has been clarified in the aforementioned analysisand it has been shown that a system with a transfer function thatis equivalent to the one stated in Eqn. (11), achieves the task.

The transfer function of systems can take different forms of firstor second order or even higher. For instance, if the given transferfunction represents a first order system, then the order of the sys-tem has to be increased. On the other hand, if the given transferfunction represents a second order system, then the controller hasto modify the system without increasing its order. One approachmight be changing the type of the system from type one to typetwo. To illustrate this, without any loss of generality, considergiven a first order transfer function with a D.C. gain =1. Ideallywhen system η = 1, the area matching approach requires thateramp = 0 and this can be achieved by adding an integral blockin the feedback loop of the transfer function. However, chang-ing the type of the system will not provide a remedy to non idealcases where the eramp is required to be negative. Therefore, thecontrol strategy has to have the ability to place a proper zero, yetmaintain the required order of the system. Consequently, feed-forward and feedback actions should be implemented on the sys-tem as required to achieve the form of the transfer function fromEqn. (11). Fig. 5 shows an example to illustrate this idea. Con-sider given a strictly proper first order transfer function as shownin Eqn. (25):

T.F.= G(s) =b0

s+a0(25)

the order of the system has to be increased to a second order usingan integrator block. In addition, a proper zero has to be placedusing the feedforward loop. Hence, to have a transfer functionthat matches Eqn. (11), the gains F(s) and D(s) in Fig. 5 can beset to α

b0and 1

s , respectively, which produces

G(s) =G(s)[D(s)+F(s)]

1+D(s)G(s)=

b0 +αss2 +a0s+b0

(26)

with a0 = 2ζ ωn and b0 = ω2n . Similarly, for the case of being

given a strictly proper second order system with no zero, theonly modification would be to add a feedforward gain to placethe required zero and preserve the system order. It can be seenfrom the above example that the addition of zero is determinedby the controllers F(s) and D(s) , and not by the planet transferfunction G(s). Furthermore, it can be analyzed and shown that ifthe relative degree of the controller D(s) is (≥ 1), (e.g. the inte-gral controller 1

s ) then the additional zero will be introduced tothe system. However, when the relative degree of the controllerD(s) is zero (e.g. PI controller: cs+d

s ) the affect of the controllerF(s) is to change the location of the original zero without addingadditional zero to the system.3.3 Higher Order System Analysis

In this section, extending the analysis of area-matching to beapplied for higher order system is discussed. Consider the closeloop transfer function of the block diagram presented in Fig. 5

G(s) =G(s)[D(s)+F(s)]

1+D(s)G(s)(27)

Assuming that the D.C.gaino f G(s) = 1 then the eramp can beobtained as:

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eramp = lims→0

[1s2 −

G(s)s2

]s= lim

s→0

[1− G(s)

s

]= lim

s→0G′(s) (28)

where G′(s) = dds G(s). Consequently, a general outcome of this

analysis is that for any system with a D.C. gain of 1 , eramp canbe predicted by tacking the derivative of the closed loop transferfunction with respect to s and setting it to zero,

Next, consider given a higher order transfer function that isrational and strictly proper, and has a D.C. gain of 1.

G(s) =N(s)D(s)

=n(s)+ kns+ k0

d(s)+ kds+ k0(29)

By using the result from Eqn. (28), it can be shown that

eramp = lims→0

dds

(N(s)D(s)

)=

kd− kn

k0(30)

The result from Eqn. (30) can be compared with

eramp =2ζ ωn−α

ω2n

(31)

with kd = 2ζ ωn, kn = α , and k0 = ω2n . Eqn. (30) represents a

generalized result for rational-polynomial transfer functions, andshows that the location of a proper zero can be predicted by ex-amining the parameters kn, kd , and k0.

CONCLUSIONThe idea of shaping a transient response of dynamic sys-

tems using minimum phase zeros is explored. The goal is to de-termine the zero locations that satisfy a conservation principle.It is shown that such characteristic is infeasible for first orderand second order systems with no additional zero. The simplestdynamic system where this area matching can be achieved is asecond order system with an additional zero. In this regard, it isshown that a combination of feedback and feedforward actionscan place/add a proper zero at an appropriate location. Througha geometric series analysis, we analytically find a relation be-tween the zero location and the corresponding area mismatch. Inhybrid power systems, this refers to achieving a certain efficiencyof power transmission from the power source to the energy stor-age. Other potential applications could be in the rendezvous /pursuit problems. Future work in this would include an exten-sion and development of a rigorous theory for general transferfunctions, a part of which has been initiated in this work. Furtherdevelopment to nonlinear dynamical systems is also feasible.

REFERENCES[1] Barley, C. D., and Winn, C. B., 1996. “Optimal dispatch

strategy in remote hybrid power systems”. Solar Energy,58(4-6), pp. 165–179.

[2] Paska, J., Biczel, P., and Kłos, M., 2009. “Hybridpower systems–an effective way of utilising primary energysources”. Renewable Energy, 34(11), pp. 2414–2421.

[3] Barley, C. D., and Winn, C. B., 1997. “Remote hybridpower systems”. Advances in solar energy, 11.

[4] Nehrir, M., Wang, C., Strunz, K., Aki, H., Ramakumar,R., Bing, J., Miao, Z., and Salameh, Z., 2011. “A reviewof hybrid renewable/alternative energy systems for elec-tric power generation: Configurations, control, and appli-cations”. IEEE Transactions on Sustainable Energy, 2(4),pp. 392–403.

[5] Chu, A., and Braatz, P., 2002. “Comparison of commer-cial supercapacitors and high-power lithium-ion batteriesfor power-assist applications in hybrid electric vehicles: I.initial characterization”. Journal of power sources, 112(1),pp. 236–246.

[6] Das, T., Narayanan, S., and Mukherjee, R., 2010. “Steady-state and transient analysis of a steam-reformer based solidoxide fuel cell system”. Journal of Fuel Cell Science andTechnology, 7(1), p. 011022.

[7] Allag, T., and Das, T., 2012. “Robust control of solid oxidefuel cell ultracapacitor hybrid system”. IEEE transactionson control systems technology, 20(1), pp. 1–10.

[8] Vahidi, A., and Greenwell, W., 2007. “A decentralizedmodel predictive control approach to power managementof a fuel cell-ultracapacitor hybrid”. In American ControlConference, 2007. ACC’07, IEEE, pp. 5431–5437.

[9] Bakule, L., 2008. “Decentralized control: An overview”.Annual reviews in control, 32(1), pp. 87–98.

[10] Ugrinovskii*, V., and Pota, H. R., 2005. “Decentralizedcontrol of power systems via robust control of uncertainmarkov jump parameter systems”. International Journal ofControl, 78(9), pp. 662–677.

[11] Sendjaja, A. Y., and Kariwala, V., 2011. “Decentralizedcontrol of solid oxide fuel cells”. IEEE Transactions onIndustrial Informatics, 7(2), pp. 163–170.

[12] Madani, O., Bhattacharjee, A., and Das, T., 2016. “De-centralized power management in a hybrid fuel cell ultra-capacitor system”. IEEE Transactions on Control SystemsTechnology, 24(3), pp. 765–778.

[13] Llamas, A., and De La Ree, J., 1993. “Stability and thetransient energy method for the classroom”. In System The-ory, 1993. Proceedings SSST’93., Twenty-Fifth Southeast-ern Symposium on, IEEE, pp. 79–85.

[14] Amin, M., and Aqilah, N., 2013. “Power system transientstability analysis using matlab software”. PhD thesis, Uni-versiti Tun Hussein Onn Malaysia.

[15] Fang, L., and Ji-Lai, Y., 2009. “Transient stability analysiswith equal area criterion directly used to a non-equivalentgenerator pair”. In Power Engineering, Energy and Electri-cal Drives, 2009. POWERENG’09. International Confer-ence on, IEEE, pp. 386–389.

[16] Andrews, G. E., 1998. “The geometric series in calculus”.The American mathematical monthly, 105(1), pp. 36–40.

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