two new control signal approaches for obtaining the mras … · 2017-01-19 · the model reference...

International Journal of Automation and Computing 8(2), May 2011, 254-261

DOI: 10.1007/s11633-011-0580-6

Two New Control Signal Approaches for Obtaining the

MRAS-CDM and a Real-time Application

Omur Ocal1 Atilla Bir1 Bernd Tibken2

1Department of Electrical Engineering, Istanbul Technical University, Istanbul 34230, Turkey2Department of Electronic Engineering, University of Wuppertal, Wuppertal 42097, Germany

Abstract: The coefficient diagram method (CDM) is one of the most effective control design methods. It creates control systemsthat are very stable and robust with responses without the overshoot and small settling time. Furthermore, all control parameters ofthe control systems are changed by varying some adjustment parameters in CDM depending on the demands. The model referenceadaptive systems (MRAS) are the systems that follow and change the control parameters according to a given model reference system.There are several methods to combine the CDM with MRAS. One of these is to use the MRAS parameters as a gain of the CDMparameters. Another is to directly use the CDM parameters as the MRAS parameters. In the industrial applications, the systemparameters can be changed frequently, but if the controller, by self-tuning, recalculates and develops its own parameters continuously,the system becomes more robust. Also, if the poles of the controlled systems approach the jw axis, the response of the closed-loopMRAS becomes more and more insufficient. In order to obtain better results, CDM is combined with a self-tuning model referenceadaptive system. Systems controlled by a model reference adaptive controller give responses with small or without overshoot, havesmall settling times, and are more robust. Thus, in this paper, a hybrid combination of MRAS and CDM is developed and two differentcontrol structures of the control signal are investigated. The two methods are compared with MRAS and applied to real-time processcontrol systems.

Keywords: Coefficient diagram method (CDM), coefficient diagram method, adaptive control, process control, model referenceadaptive systems (MRAS).

1 Introduction

Today, control system designers try to apply differentcontrol algorithms in order to find the best controller pa-rameters to obtain the best solutions. Some of these meth-ods are very successful for special cases while unsuccessfulfor other general applications. In this paper, according tothis context, two effective controller algorithms are appliedsimultaneously and investigated to obtain the best resultsfor all purpose systems.

There are hybrid systems previously developed by usingthe model reference adaptive systems (MRAS) and coeffi-cient diagram method (CDM) methods simultaneously. In[1], sliding – mode control is combined with MRAS. The so-called SM-MRAS uses an estimator for the speed estimationin a sensorless vector-controlled induction machine drive.To obtain more robust control systems, neural network-based model reference adaptive systems are used to controlhigh performance motor drive and motion control systems.In [2], a permanent magnet synchronous motor (PMSM)drive application is described. Since the adaptive systemsare generally linear and neural networks are very good atestimating or controlling nonlinear systems, these kinds ofsolutions are frequently proposed. In terms of neural net-works, it is not necessary to know the system parameters,and this kind of approach can easily be applied to linear ornonlinear control systems. This is the reason why they areso popular worldwide. In [3], for the model reference adap-tive control, new output feedback adaptive control schemesare developed for linear multiple-input multiple-output sys-tems. This approach is known as an output feedback model

Manuscript received April 14, 2010; revised July 21, 2010

reference adaptive control mechanism. In [4], a sensorlessposition vector-control system based on MRAS is devel-oped to use for a low speed and high torque PMSM drive.In [5], a nonlinear adaptive state-feedback speed control ofa voltage-fed induction motor with varying parameters ispresented. Here, MRAS is used for correcting the rotor-resistance error adaptively.

Another common design method is known as the CDM.This was developed by Manabe et al.[6−10]. Using this de-sign method, it is possible to obtain systems that are verystable and robust, having small settling time without over-shoot. Furthermore, it can be easily applied to any kindof linear control system. If it is possible to combine theCDM with other controller design methods, it would bevery attractive. Similar to MRAS, there are some otherhybrid systems developed for CDM. In [11], CDM is com-bined with integral-proportional derivative acceleration (I-PDA), incorporating a feed-forward controller (FFC). Here,the I-PDA control system structure is given as an FFC,and the parameters of PIDA and I-PDA are designed byCDM. PIDA and I-PIDA are in the form of feedback con-troller (FBC) and FFC, respectively. In [12], proportional-integral (PI) controller is added to CDM as a feed-forwardcontroller. This method is similar to the method given in[11], but here acceleration and derivative parameters arenot present in the FBC controller. In [13, 14], to obtain themost robust system, a pole assignment method containingpole coloring is used in CDM. In [15, 16], the adaptive-CDMalgorithm is used for active queue management in congestednetworks. This is a very popular research area, especiallyin video streaming systems. The output of the adaptiveCDM-AQM controller system is satisfactory, the parameter

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identification is converged and the stability is guaranteed.In [17], the CDM and MRAS-CDM structure is briefly in-troduced, and a real-time process control application is in-vestigated. It is shown that the system outputs obtainedwith MRAS-CDM are much better for the poles near theorigin and also for the poles that are far away in the nega-tive s-plane.

In this paper, in addition to [17], using a different controlapproach, two design methods are developed for real timesystems that have poles far away in the s-plane. It is shownthat the designed new control systems give better resultsthan the MRAS.

2 Theory

Consider a system described by the model

dy

dt= −d0y + n0u (1)

where u is the control and y is the output signal. Assumethat we want to obtain a closed-loop system described by

dym

dt= −dmym + nmuc. (2)

2.1 CDM

In this section, only the structure, performance parame-ters, target characteristic polynomial of CDM and the de-sign procedure are given and discussed. The details of theCDM method can be found in [6−8].

The standard single-input single-output (SISO) block di-agram of the CDM is shown in Fig. 1. Here, R(s) is theinput reference; Y (s) is the output; U(s) is the control;Q(s) is the disturbance; E(s) is the error, and M(s) is themeasurement disturbance signal of the system.

Fig. 1 The standard block diagram of CDM

In Fig. 1, A(s), B(s), and F (s) are controller polynomi-als. The system to be controlled is represented by G(s)transfer function that is given by

G(s) =N(s)

D(s). (3)

It is assumed that deg D(s) > deg N(s). Also, it iseasy to show that

Y (s) =A(s)N(s)

P (s)Q(s) +

F (s)N(s)

P (s)R(s) −

B(s)N(s)

P (s)M(s)

U(s) =F (s)D(s)

P (s)R(s) −

B(s)N(s)

P (s)Q(s) −

B(s)D(s)

P (s)M(s)

where P (s) is the characteristic polynomial of the closed-loop system given by

P (s) = A(s)D(s) + B(s)N(s) =n∑

i=0

aisi. (4)

While obtaining the hybrid system, an equivalent block di-agram of CDM will be used, as shown in Fig. 2.

Fig. 2 Equivalent block diagram of CDM

2.1.1 Performance parameters and the target

characteristic polynomial

CDM needs some design parameters with respect to thecharacteristic polynomial coefficients, such as the equivalenttime constant τ , the stability indices γi, and the stabilitylimits γ∗

i . The relations between these parameters and thecoefficients of the characteristic polynomial ai are given in(5):

γi =a2

i

ai+1ai−1, i = 1, · · · , n − 1 (5a)

τ =a1

a0, γ∗

i =1

γi−1+

1

γi+1. (5b)

Using the relations in (5), it is possible to formulate thecharacteristic polynomial P (s) in terms of the design pa-rameters τ and γi as follows:

PT (s) = a0

[

n∑

i=2

(

i−1∏

j=1

1

γji−j

)

(τs)i

+ τs + 1

]

(6)

where PT (s) is the target characteristic polynomial. Notethat the coefficients ai of this polynomial can be expressedas

ai =τ i

i−1∏

j=1

γi−jj

a0. (7)

This is a relation between equivalent time constant τ andthe settling time Ts. Since τ = Ts/α, where α ∈ [2.5, 3].

2.1.2 CDM design procedure

Equating the closed-loop system characteristic polyno-mial found in Section 2.1 to the target characteristic poly-nomial given in Section 2.1.1, one obtains the following Dio-phantine equation

P (s) = A(s)D(s) + B(s)N(s) =n∑

i=0

aisi =

a0

[

n∑

i=2

(i−1∏

j=1

1

γji−j

)(τs)i

+ τs + 1

]

. (8)

256 International Journal of Automation and Computing 8(2), May 2011

If the polynomials are defined as

A(s) =

p∑

i=0

lisi and B(s) =

q∑

i=0

kisi

and then the equivalent Sylvester form equation is obtained

[C]r×r

[

liki

]

r×1

= [ai]r×1 . (9)

The coefficients of the matrix C and the parameters ai areknown values. Therefore, the coefficients of the controllerpolynomials A(s) and B(s) can be easily calculated.

The coefficient F is calculated by using the expression

F =

(

P (s)

N(s)

)∣

∣

∣

∣

s=0

. (10)

2.1.3 Stability analysis of CDM

In addition to Routh-Hurwitz criterion, CDM inserts theLipatov-Sokolov criterion[8].

According to Lipatov-Sokolov criterion, the stability andinstability conditions are shown as below[18,19] :

1) Necessary condition for instability

γi−1γi 6 1 for ∃i, i = 2, 3, · · · , n − 1

and

ai−1ai 6 ai−2ai+1 for ∃i, i = 2, 3, · · · , n − 1

2) Necessary condition for stabilityCondition 1.

γiγi−1 > 2.1505 for ∀i, i = 2, 3, · · · , n − 1

and

aiai−1 > 2.1505ai+1ai−2 for ∀i, i = 2, 3, · · · , n − 1

Condition 2.

γi > 1.1236γ∗

i for ∀i, i = 2, 3, · · · , n − 2

and with respect to ai

ai > 1.1236

(

ai+2ai−1

ai+1+ ai−2

ai+1

ai−1

)

for ∀i, i = 2, 3, · · · , n − 1.

To find a suitable desired (target) characteristic poly-nomial, these conditions can be used for choosing designparameters.

Details about the stability analysis of CDM are given in[6–8, 18, 19].

2.2 Hybrid MRAS-CDM configuration

The hybrid system configuration can be seen in Fig. 3.This system is controlled adaptively according to the modelreference output ym(t), and the control variables are appliedto the CDM controller as gain parameters.

According to the explanations given in Section 2 and us-ing (8)–(10) for the system (1), the controller polynomialsA(s) and B(s), the parameter F and the characteristic poly-nomial P (s) are calculated as

A = ℓ0 = a0τ (11a)

B = k0 =a0 − a0d0τ

n0(11b)

F =d0ℓ0 + k0n0

n0=

a0

n0(11c)

P (s) = a0τs + a0 = ℓ0s + (d0ℓ0 + n0k0). (11d)

The block diagram of the system can be seen in Fig. 4.

Fig. 3 Hybrid MRAS-CDM block diagram

Fig. 4 The block diagram of the system

Since according to the changes of parameter F , the con-troller will produce new parameters adaptively, the changeof F cannot affect the system after producing new controllerparameters. It is used only for the standard CDM structure.

Furthermore, uc(t) in Fig. 3 and r(t) in Fig. 4 are bothreference signals of the control systems. R(t) is the referencesignal for the standard CDM structure in Fig. 1. uc(t) is thereference signal of the MRAS-CDM structure.

According to these results, if a hybrid control systemstructure is applied, the parameters of the control systemcan be changed adaptively.

2.2.1 The first method to obtain the control signal

The controller is given by

u =a0

n0ℓ0θ1uc −

k0

ℓ0θ2y. (12)

As seen in (12), the controller has two free parameters, θ1

and θ2. In order to obtain the classical MRAS controlsignals given in the literature, one has to investigate thereferences[20, 21]. The stability analyses are given in [20, 22].For the non-linear system, the proposed adaptive controllerin [23] and the stability analysis in [24, 25] may be consid-ered as an alternative. If we assume that the input-output


relations of the system and the model reference system areidentical, then the following relations can be written

θ1 = θ01 =

nm

a0

a0n0ℓ0

(13a)

θ2 = θ02 =

dm − d0

n0

(

k0

ℓ0

) . (13b)

As a perfect model following system, the output of theclosed-loop system is obtained from (1) and (12) as

y =

(

a0

n0ℓ0

)

n0θ1

s + d0 + n0

(

k0

ℓ0

)

θ2

uc. (14)

To apply the Massachusetts Institute of Technology (MIT)rule[20], let the error be defined as

e = y − ym. (15)

If (2) and (14) are applied to (15), the sensitivity derivativesare obtained by taking partial derivatives with respect tocontroller parameters θ1 and θ2:

∂e

∂θ1=

(

a0

n0ℓ0

)

n0

s + d0 + n0

(

k0

ℓ0

)

θ2

uc (16)

∂e

∂θ2=

(

k0

ℓ0

)

n0

s + d0 + n0

(

k0

ℓ0

)

θ2

y. (17)

The following equation can be used in the approximation:

s + d0 + n0

(

k0

ℓ0

)

θ2∼= s + dm. (18)

Finally, the following equations are necessary in order toupdate the controller parameters[17]:

∂θ1

∂t= −γ

(

a0

n0ℓ0

)

n0

s + dm

uc

e (19a)

∂θ2

∂t= γ

(

k0

ℓ0

)

n0

s + dm

y

e. (19b)

Stability analysis. According to the Lyapunov stabil-ity analysis[20] , the time derivative of the positive definiteLyapunov function function has to be negative or

dV

dt< 0. (20)

By means of (1), (2), (12), (15), and the control signal givenin (12), the time derivative of the error is calculated as

de

dt= − dme −

(

n0k0

ℓ0θ2 + d0 − dm

)

y+

(

a0

ℓ0θ1 − nm

)

uc. (21)

If the positive definite Lyapunov function is chosen asfollows:

V (t, θ1, θ2) =1

2

(

e2 +ℓ0

n0k0γ

(

n0k0

ℓ0θ2 + d0 − dm

)2)

+

1

2

(

ℓ0a0γ

(

n0a0

ℓ0θ1 − nm

)2)

. (22)

The time derivative can be written as

dV

dt= e

de

dt+

1

γ

(

n0k0

ℓ0θ2 + d0 − dm

)

dθ2

dt+

1

γ

(

a0

ℓ0θ1 − nm

)

dθ1

dt=

−dme2 +1

γ

(

n0k0

ℓ0θ2 + d0 − dm

)(

dθ2

dt− γye

)

+

1

γ

(

a0

ℓ0θ1 − nm

)(

dθ1

dt+ γuce

)

.

(23)This suggests the adaptation law

dθ1

dt= −γuce (24a)

dθ2

dt= γye. (24b)

Thus asdV

dt= −dme2 (25)

dV

dt< 0 (26)

the system is asymptotically stable.2.2.2 The second method to obtain the control sig-

nal

The controller f with two free parameters m0 and f0 isapplied as

u =a0

n0

1

ℓ0uc −

k0

ℓ0y (27a)

u =a0

n0m0uc − k0m0y, for m0 =

1

ℓ0and k0m0 = f0.

(27b)If assuming that the input-output relations of the system

and the model system are identical, we obtain

ℓ0 = ℓ00 =nm

a0(28a)

k0 = k00 = a0

dm − d0

n0nm

. (28b)

This is again a perfect model following system. Using (1)and (27), the output of the closed-loop system can be cal-culated as

y =a0m0

s + d0 + n0f0uc. (29)

To apply the MIT rule[20], let the error be defined as

e = y − ym. (30)

If (2) and (29) are applied to (30), the sensitivity derivativesare obtained by taking partial derivatives with respect tocontroller parameters m0 and f0:

∂e

∂m0=

a0

s + d0 + n0f0uc (31)


∂e

∂f0=

n0m0

s + d0 + n0f0y. (32)

For the approximation, the following equation can be used:

s + d0 + n0f0∼= s + dm (33)

and finally, the following equations are necessary in orderto update the controller parameters:

∂m0

∂t= −γe

a0

s + d0 + n0f0uc (34a)

∂f0

∂t= γe

n0m0

s + d0 + n0f0y. (34b)

Stability Analysis. The stability of the system canbe examined by using the Lyapunov stability analysis onceagain.

The control signal and the error equation are given as

u =a0

n0m0uc − f0y (35)

de

dt= − dme − (n0f0 + d0 − dm) y+

(a0m0 − nm)uc. (36)

If the Lyapunov function is chosen as follows:

V (t, f0, m0) =1

2

(

e2 +1

n0γ(n0f0 + d0 − dm)2

)

+

1

2

(

1

a0γ(a0m0 − nm)2

)

. (37)

taking the time derivative

dV

dt= e

de

dt+

1

γ(n0f0 + d0 − dm)

df0

dt+

1

γ(a0m0 − nm)

dm0

dt=

−dme2 +1

γ(n0f0 + d0 − dm)

(

df0

dt− γye

)

+

1

γ(a0m0 − a0)

(

dm0

dt+ γuce

)

(38)the adaptation law can be suggested as

dm0

dt= −γuce (39a)

df0

dt= γye (39b)

thusdV

dt= −dme2 (40)

dV

dt< 0 (41)

and the system is asymptotically stable.In the first control design scheme in Section 2.2.1, the

parameters l0 and k0 introduced and combined in the hy-brid MRAS system are known as design parameters. But inthe second control design scheme in Section 2.2.2, the pa-rameters l0 and f0 are directly applied as control signals inMRAS, so that their values are unknown at the beginning.

Thus, if system parameters change according to the differ-ence between the model reference output and the output ofthe system, the system would automatically determine thevalues of the new parameters.

In the next section, the results are checked on a real-timeprocess control system and the simulated system outputsare evaluated.

3 A real-time process control applica-

tion

In this section, by using MRAS-CDM, the two controlsignal algorithms are applied to a process control systemand the results are compared with the MRAS outputs.

The model of the controlled process is given in Fig. 5.

Fig. 5 The model of the controlled process

The transfer function of the process can be expressed asfollows:

vo(s)

vi(s)=

R2

R1

(

1

R2Cs + 1

)

. (42)

To have unity gain, the ratio and the values of the systemparameters are chosen as R2/R1 = 1, R1 = R2 = 1MΩ,C = 1 µF, am = bm = nm = dm = 2, and γ = 1.

If these values are applied to the given equations, thefollowing transfer functions are obtained:

vo(s)

vi(s)=

1

s + 1(43)

and

Nm

Dm

=2

s + 2,

Bm

Am

=2

s + 2. (44)

In Figs. 6–8, the output signals for MRAS-CDM and MRASare given respectively. For both control signals, MRAS-CDM response is better than the MRAS control perfor-mance.

Furthermore, if the controller parameters are adjustedaccording to the plant characteristics, the system perfor-mance can be improved. But first under similar conditions,we must determine which one is the best among the threemethods.

Fig. 6 shows that this approach is satisfactory in con-trol applications. In fact, applications without overshootdynamics can also be achieved by adjusting the free param-eters of the MRAS-CDM controller[17]. We keep in mindthat in all cases the basic parameters are equal.


Fig. 6 The output signals. 1) The reference signal r(t); 2) The

output signal y(t) for MRAS-CDM using the first method for

the model control reference; 3) The output signal ym(t) of the

reference model


output signal y(t) for MRAS-CDM using the second method for

the model control reference; 3) The output signal ym(t) of the

reference model


output signal y(t) for MRAS controlled system; 3) The output

signal ym(t) of the reference model

As seen from Figs. 9–11, the control system obtained byusing the first control signal applying method is the best.The worst one is the system controlled by MRAS. The dy-namics of the MRAS and the MRAS-CDM control systemusing the second method are similar to a second order sys-tem. Since the control signal parameters are almost identi-cal, this is an expected result.

Fig. 9 Control signal u(t) for MRAS-CDM using the first

method

Fig. 10 Control signal u(t) for MRAS-CDM using the second

method

For both control signal applying methods, the control sig-nals u(t) are satisfactory. But for the system with MRAS,control signals become worse after the second period of thereference signal. The variations of the control parametersθ1, θ2, m0, and f0 are shown in Figs. 12–14.

Fig. 11 Control signal u(t) for the system using MRAS

As seen from Figs. 12 and 14, control parameter θ2 ofMRAS-CDM using the first method produces the signalwithout overshoot. This is the reason why the parame-ters of MRAS-CDM using the first method, and the secondmethod and MRAS are so different from each other.

In Fig. 15–17, the block diagrams are given in real-time applications. In Fig. 15, MultiQ-PCI denotes multiq-peripheral component interconnect, DAC denotes digital-to-analog converter, and ADC denotes analog-to-digital


converter. In Fig. 16, Q1 and Q2 are the controller param-eters in (19).

Fig. 12 Variation of the control parameter θ1 for (a) MRAS-

CDM using the first method and for (b) MRAS

Fig. 13 Variation of the control parameter θ2 for (a) MRAS-

CDM using the first method and for (b) MRAS

Fig. 14 MRAS-CDM using the second control signal method.

(a) Variation of the control parameter m0; (b) Variation of the

control parameter k0

Fig. 15 The block diagram of MRAS-CDM using the first con-

trol signal method

Fig. 16 The block diagram using MRAS control

Fig. 17 The block diagram of MRAS-CDM using the second

control signal method

4 Conclusions

In this paper, an MRAS-CDM control configuration,which consists of the hybrid combination of MRAS andCDM control strategies, has been analyzed, the method hasbeen applied to a real-time process control and the resultshave been discussed in detail. As demonstrated, the newcontrol design procedures and performances are better thanthe classical MRAS.

In this design procedure, the control parameters of theCDM are changed adaptively by estimating the control pa-rameters based on a given reference model. In classicalcontrol systems designed by CDM, the system parametersare calculated according to the special plant parameters.

If plant parameters change, the performance of the con-trol system diminishes and begins to give poor solutions. Incontrol systems controlled by MRAS-CDM, the control pa-rameters are changed adaptively according to the referencemodel.

In addition, if a control system is controlled by modelreference adaptive systems and the reference model param-eters are chosen near the jw axis, the performance of the sys-tem is unsatisfactory[17], because the closed-loop responseshave large overshoots. These kinds of problems are over-come by using the MRAS-CDM. This is true only for lim-ited changes of the plant parameters.

In the future, MRAS-CDM control method can be ap-plied to more complex real time systems. In general, it canbe said that systems like satellites and robots using adap-tive controls would achieve very successful results. The newdesign method is suggested to be applied to these kinds ofwork as an alternative.

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Omur Ocal graduated from OsmangaziUniversity, Eskisehir, Turkey in 2001. Hereceived the B. Sc. degree in electrical andelectronic engineering, M. Sc. degree at theControl and Automation Engineering Pro-gram from Istanbul Technical University(ITU), Istanbul, Turkey in 2004, and thePh. D. degree at the Control and Automa-tion Engineering Program in ITU in 2010.He worked as a research and teaching as-

sistant at the Computer Engineering Department of Kadir HasUniversity, Istanbul, Turkey from 2004 to 2007.

His research interests include linear control systems theoryand design, automatic control, digital control, real time control,adaptive control, robust control, industrial control, neural net-works, and fuzzy logic control.

E-mail: [email protected] (Corresponding author)

Atilla Bir received the B. Sc. andM. Sc. degrees in T. H. Karlsruhe-Nachrichtentechnik, Karlsruhe, Germanyin 1966. For one year, he worked as adesign engineer for Siemens, Karlsruhe,Germany. He received the Ph. D. degreeat the Electrical Faculty in IstanbulTechnical University (ITU), Istanbul,Turkey in 1975. He worked as a researchand teaching assistant in ITU from 1970 to

1980. Then, he became an assistant professor at the ElectricalFaculty of ITU, Istanbul, Turkey in 1980. He has worked asa professor in ITU since 1989. He was also department chairbetween 1994 and 1996 and the Control Department chair in1991–1994 and 1996–2004.

His research interests include linear control systems theoryand design, multivariable systems, robust control, optimal con-trol, non-linear control, stochastic control, industrial control, andhistory of science and technology.

E-mail: [email protected]

Bernd Tibken graduated from the Uni-versity of Hamburg, Germany in the win-ter semester 1979–1980 and in physics fromthe University of Hamburg in 1985. Hereceived the Ph. D. degree from the Tech-nical University of Hamburg-Harburg, andthe Habilitation degree from the Universityof Ulm, Germany in 1990 and 1996, respec-tively. Since 1999, he has been a full pro-fessor for automatic control in the Faculty

of Electrical, Information, and Media Engineering (Faculty E) ofthe University of Wuppertal, Germany. He received calls to fullprofessor positions at Technical University of Freiberg in 1999and the Helmut Schmidt University, Hamburg (former Univer-sity of the Armed Forces) in 2003. Since 2002, he has been thedean of Faculty E at the University of Wuppertal.

His research interest includes nonlinear control theory, espe-cially polynomial systems and the application of real algebra tocontrol problems.

E-mail: [email protected]

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