TDS10RaulFinal2

7/28/2019 TDS10RaulFinal2

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Tuning the leading roots of a second order DC

servomotor with proportional retarded control

R. Villafuerte and S. Mondie

Department of Applied mathematics of Papaloapan University (UNPA)Loma Bonita, Oax., M exico.

e-mail: [email protected], [email protected] of Automatic Control CINVESTAV-IPN

A.P. 14-740 07360, M exico DF.e-mail: [email protected].

Abstract: The stabilization of a second order system through a proportional delay controller insuringa specified closed loop exponential decay is studied. The analysis in the frequency domain allowsthe determination of the -stabilizability regions of the controller. The locus of the dominant roots isanalyzed in detail and the characterization of some key loci, including the maximum achievable decayis obtained. The tuning of a DC servomotor experimental setup illustrates the results.keyword:Second order system, -stable region, proportional retarded control, DC servomotor.

1. INTRODUCTION

The second order system of the form

(t) + 2(t) +2(t) = bu(t) (1)

where > 0, > 0 and b > 0, is the first choice modelfor a wide range of physical processes such as servomotors,linearized oscillators, etc.

The state space description given by

x(t) =

0 1

2 2

x(t) +

0b

u(t),

(t) = ( 1 0 )x(t),

is clearly controllable hence one can assign arbitrarily theclosed-loop spectrum with a full state feedback, a proportionalderivative feedback of the form

u(t) = k1(t)k2(t). (2)

Here, we want to avoid the design of an observer. We also wantto skip the use of time derivative measuring devices, digitalsuch as backward difference or analog such as tachometers,because of noise amplification.

An alternative is the use of the delayed output in the controlloop. The stabilizing effect of delays in feedback has beenstudied in depth in the case of second order systems Abdallahet al. (1993) as well as more general classes of oscillatorysystems Moreno and Michiels (2005), Suh and Bien (1979),Karafyllis (2008). An intuitive idea is that the substitution ofthe approximation of the derivative

(t) (t)(th)

h

for small h,

into (2) gives a control law of the form

u(t) = kp(t) + kr(th) (3)

where h is the delay, and

kp = k1 + k2/h, and kr = k2/h. (4)

Notice that h is now a design parameter that can be used fortuning.

A drawback of such a proportional-delay control strategy isthat the control design and response shaping tools available forachieving prescribed closed loop specifications of delay freesystems cannot be used. The main reason is that the analysisof the roots of the closed loop characteristic equation, the

quasipolynomialp(s,kp,kr,h) = s

2 + 2s +2 + bkpbkrehs (5)

with infinite number of roots, is now a more complex task.

As in the case of delay free systems, when the system has nozeros, the shaping of the response depends first on the distanceof the rightmost eigenvalue(s) of the closed loop system to theimaginary axis and second, on the nature of these dominantroots, real or complex, which define the oscillatory nature ofthe response.

This response shaping is closely related to the stabilizationwith prescribed exponential decay, named -stabilizability.This problem has been addressed in the framework of Linear

Matrix Inequalities, Mondie and Kharitonov (2005), Fridmanand Shaked (2003) but the results are usually poor because thesolution is very conservative, and when it exists, there is noinsight on which roots are the dominant ones, hence the finetuning of the response is not possible.

The aim of this contribution is to provide a detailed frequencydomain analysis of the -stabilizability the proportional-delayed control law (3) in closed loop with system (1). Themain definitions and tools for the analysis are introduced insection 2. In section 3, the -stabilizability boundaries andregions are determined and a complete analysis of the dominantroots for a given is performed in section 4. Furthermore,the maximum exponential decay strategy is characterized by

straightforward formulae in section 5. In section 6, an experi-mental setup of a DC-servomotor is employed to illustrate theresults. The contribution ends with some concluding remarks.

2. PRELIMINARY RESULTS

In this section, we introduce important properties and defini-tions.


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Lemma 1. Bellman and Cooke (1963), Datko (1978), For any R, there are only a finite number of roots with real part

greater than . Consider the quasi-polinomial (5) and let0 = sup

j=1,...,

Re{sj} : p(sj,kp,kr,h) = 0, sj C

. (6)

Then for any 0, there exists a constant L > 0 such that thesolution of (1)-(3) satisfies the inequality

|(t)| Leth, (7)

where is the initial condition and h = max[h,0] ().

Definition 2. We say that the triplet (kp,kr,h) -stabilizes sys-tem (1)-(3) if

0 , R+,

where 0 is given in (6).

As it is well known, the change of variable s (s) in thefrequency domain or y(t) = etx(t) in the time domain reducesthe analysis of the -stability to the stability of the transformedquasipolynomial

p(s,kp,kr,h) = s2 + 2()s + (2 +22+ bkp)

bkrehehs. (8)

Remark 3. The decay of the autonomous system (u 0) is .The analysis presented in this paper is restricted to the case ofclosed loop exponential decay > which corresponds to animproved system response.

The quasipolynomial (8) defines completely thestabilizabilityand roots dominance of system (1). As the roots behavior in the

complex plane is continuous with respect to continuous changesof the parameters Neimark (1949), loss of -stabilizabilitynecessarily occur when the quasipolynomial has roots either ats = 0 or at a pair of pure imaginary roots at s =j. Next, wecharacterize the root crossings of the imaginary axis which arecandidate loci for the stability/instability boundaries.

Proposition 4. Roots crossing loci of the quasipolynomial (8)of the imaginary axis at s = 0 satisfy

kr =2 +22+ bkp

beh. (9)

Proof. The result follows straightforwardly by substituting s = 0into (8):

p(0,kp,kr,h) =

2

+

2

2+ bkpbkre

h

= 0

Proposition 5. Roots crossing loci of the imaginary axis of the

quasipolynomial (8) at s = j occur at =1,2 for real

and positive 1,2 of the form

1,2 = 2

12

+ bkp ()2

(bkreh)24 ()

2 (2(12) + bkp). (10)

Proof. Setting

p(j,kp,kr,h) = 2 + 2j()

+ (2 +22+ bkp)bkrehejh = 0

and taking modulus yields422

+ bkp2 ()

2

+(+ bkp)2b2k2re

2h = 0

(11)where = 2 2+ 2. By introducing the change ofvariable = 2, we get a quadratic polynomial in

22+ bkp2 ()

2

+ (+ bkp)2b2k2re

2h = 0

whose roots are given by (10). Clearly, real roots of p corre-spond to the square root of positive 1,2.

Proposition 6. The parametric equations for root crossings ofthe imaginary axis at purely imaginary pairs are

h() =1

cot1

2 + (2 +22+ bkp)

2()

+ n

,n = 0,1 . . . , (12)

kr(,h) = 2()

beh sin(h). (13)

Proof. Substituting s = j and ejh = cos(h) j sin(h)into (8) yields

Re{p(j,kp,kr,h)}= 2 + (2 +22+ bkp)

bkrehcos(h) = 0, (14)

Im{p(j,kp,kr,h)}= 2()+ bkrehsin(h) = 0.

(15)

and the result follows by simple algebraic manipulations.

3. REGIONS OF -STABILIZABILITY

We now present graphical and analytical results concerning the-stabilizability properties of system (1).

The equation (9) and the parametric equations (12, 13) defineall boundaries corresponding to root crossings of the imaginaryaxis. The ones that delimit stabilizability regions in the

(kp,kr,h) tridimensional space are indeed the ones of interest. Itis possible to perform a formal analysis of the stability regionsof the quasipolynomial (8) which is close to the widely studiedform q(s) +p(s)esh where q(s) and p(s) are polynomials suchthat deg(p(s)) < deg(q(s)) (see Cooke and Van Den Driessche(1986), Michiels and Niculescu (2007), Cooke and Grossman(1982), Abdallah et al. (1993) among others). Although thequasipolynomial (8) is such that p(s) also depends on h it isclear that the methods can be extended. Here, given that thecase is quite simple we can characterize the stabilizabilityusing D-Partition techniques, Neimark (1949), by testing asingle point of each region by using, for example, the argumentprinciple.

The typical delay dependent stability/instability phenomenonof the stabilizability regions depicted in the bi-dimensionalspace (kr,h) for a fixed gain kp are sketched on Figure 1. Theparameter values are b = 1, = 2.3968, = 0.00055 andkp = 5. It appears that stabilizability can be achieved notonly in the first region but also in regions corresponding tolarger delays. However, the achievable decay decreases as hgrows and the stability region narrows significantly, resulting infragile controllers. The fact that he stabilizability regions inthe space (kr,h) grows as the gain kp does is shown on Figure 3.

The main region of interest for our analysis is shown on Figure3 where the level curves corresponding to different are

depicted in the bi-dimensional space (kr,h).One observe that for a givenstabilizability specification, thesame exponential decay is achieved at all points of the bound-aries. The upper boundary corresponds to loci of p(s,kp,kr,h)with at least one dominant real root at , while the lowerboundary corresponds to pairs of complex conjugate roots at j. We also observe that the level curves collapse into a


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Fig. 1. -stable region of system (1)-(3) (0,0.1), (0.1,0.5), (0.5,1), (1,2), (2,3.2).

Fig. 2. -stable region of system (1)-(3) for = 3.2 and kp [0,3].

Fig. 3. -stable region of system (1)-(3) (0,0.5), (0.5,1), (1,1.5), (1.5,2), (2,3.2).

single point that correspond to the unique maximum achievabledecay that will be characterized in section 5.

4. DOMINANT ROOTS ANALYSIS

As the system response is determined by the location of thedominant roots, we provide next the analytical characterizationof some key loci of the boundary for which the exponentialdecay is achieved.

4.1 Key loci on the boundary

Locus 1: Double root at The crossing of the branch corresponding to the root 1 withthe boundary (2 + 2 2+ bkp) bkre

h = 0 occurswhen 1 0. In this case there is a double root at hence

p(0,kp,kr,h) = 0 andd p(s,kp,kr,h)

ds

s=0

, therefore

(2 +22+ bkp) = bkreh (16)

and

kr =2()

hbeh. (17)

Combining (16) and (17), one obtains that

h =

2()

(2 +22+ bkp) . (18)

Thus, given > 0 and kp > 0 the parameters h and kr followfrom (18) and (17) respectively.

Locus 2: locus where 1 = 2This locus is when the imaginary part of the active root of thequasipolynomial turns from 1 to 2. One can see from (10)that 1 2. Clearly, the maximum possible value for1 occurswhen

(bkreh) =

4 ()2 (2(12) + bkp).

In this case,

1 = 2 =

2

(12

) + bkp ()2

and the corresponding delay h and gain kr can be obtained fromthe parametric equations (12, 13).

Locus 3: Pair of complex conjugate roots at + j2 anda real root at When the imaginary part of the active root is 2, then locus 3occurs when

(2 +22+ bkp)bkreh = 0

Then it follows that

2 = 2

12

+ bkp ()2

+(bkreh)24 ()2 (2(12) + bkp), (19)= 2

2 + 2222 +2 + bkp

(20)

and

2 =

2 (2 + 2222 +2 + bkp) (21)

The corresponding h and kr .follow from the parametric equa-tions (12, 13).

Locus 4: Simple root at - and a double real root at -As one can see on Figure 4, this locus has a special role: itcorresponds to the case where p(s,kp,kr,h) has a simple rootat - hence p(0,kp,kr,h) = 0, or

(2 +22+ bkp) = bkreh (22)

and a double real root at some hence p(0,kp,kr,h) = 0

andd p(s,kp,kr,h)

ds

s=0

, therefore

(2 +22+ bkp) = bkreh (23)

and

2() + hbkreh = 0 (24)


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Combining (22), (23) and (24) it follows that satisfies

(2 +22+ bkp)e

2()()

2+22 +bkp

= (2 +22+ bkp). (25)

This implicit expression corresponds to the crossing of anexponential and of a quadratic in the variable. It has a solutionat and at some = > 0 that can be computed by usingnumerical methods.

Thus, given > 0 and kp > 0 we obtain from (25) and theparameters h and kr follow from

h =2()

(2 +22+ bkp), (26)

kr = 2()

hbeh. (27)

Remark 7. The characterization of the above loci combinedwith the parametric equations allow to sketch straightforwardlythe first sigma-stability region as follows:Upper boundary: For the selected kp and , sketch in the (h,kr)plane

kr =2 +22+ bkp

beh,

where h

2()(2+22+bkp)

,h(2)

. Here h(.) and 2 are

defined in (13) and (21), respectively.

Lower boundary: For the selected kp and , sketch in the (h,kr)plane

kr() = 2()

beh sin(h())

with

h() =

=

1

cot1

2 + (2 +22+ bkp)

2()

, [,e];

1

cot1

2 + (2 +22+ bkp)

2()

+

, [e,2].

for [,2), > 0 and e = min{2,2 + 2 2+

bkp}

For a full picture of the dominant roots behavior, we presenton Figure 4 the evolution of the location of the dominant rootsas one travels on the -stabilizability boundary = 0.5 ofFigure 3 when the system parameters are b = 1, = 2.3968, = 0.00055 and the proportional gain is kp = 5.

5. MAXIMUM EXPONENTIAL DECAY STRATEGY

Next, we characterize the maximum achievable exponentialdecay for a fixed kp 0 and we compute the correspondingretarded gain kr and delay h

.

Lemma 8. Let the proportional gain of the controller kp 0 begiven. Then, the maximum exponential decay of the closed

loop system (1-3) that can be achieved by designing the pair(kr,h) is

= +2(12) + bkp. (28)

Moreover, the values of the delay gain kr and of the delay h

that stabilize the system (1-3) with the exponential decay are

h =2()

2 + ()22 + bkp, (29)

kr =2()

bheh

. (30)

Proof. The larger exponential decay occurs into the firststability region. This region collapses into a single point when increases. In this case, there is a triple root at , hencep(s,kp,kr,0) = 0 and its first and second derivative withrespect to are also null, namely,

(2 +22+ bkp) = bkreh, (31)

2() + hbkreh = 0 (32)

andh2bkre

h = 2. (33)

It follows from (31) and (32) that

h =2()

(2 +22+ bkp). (34)

Moreover, (31) and (33) imply

h2 =2

(2 +22+ bkp). (35)

Substituting (34) into (35) yields

()2 = (12)2 + bkpand (28) is obtained. Finally, (29) and (30) follow from (31) and(32).

6. PR TUNING OF AN EXPERIMENTAL DCSERVOMOTOR

In this section, we illustrate with the experimental setup of aDC servomotor the results of the previous theoretical analysis.

The servomechanism employed for the experiments consistsof a DC brushed motor controlled through a Copley Controlspower amplifier, model 413, configured in current mode. A BEIoptical encoder directly coupled to the motor shaft gives angu-lar position measurements. Resolution of the optical encoderis 2500 pulses per revolution. A Q8 card from Quanser Con-sulting endowed with inputs for optical encoders performs dataacquisition. The electronics associated to these inputs multiplyby 4 the encoder resolution. In this way, one motor turn cor-responds to 10 000-encoder pulses. A factor of 10,000 scalesdown the angular position measurements. The card also has 12bits digital-to-analog converters with an output voltage range of10 volts. All the programming is done using the MathworksMatLab Simulink environment under the Wincon real-time pro-gram. The software runs on a Personal Computer using an IntelCore 2 quad processor, and the Q8 board is allocated in a PCIslot inside this computer.

6.1 Servomechanism model

We consider the following second order model for the ser-vomechanism

Jq(t) + fq(t) = (t) = ku(t) (36)where q is the angular position, (t) the input torque, u thecontrol input voltage, J the motor and load inertia, f theviscous friction and k the amplifier gain. A motor, a poweramplifier and a position sensor compose the servomechanism.The power amplifier is set to current mode; therefore, theelectromagnetic torque is proportional to the input voltage


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Fig. 4. Root locus of the dominant roots of the quasipolynomial (5).

Fig. 5. DC Servomotor experimental setup.

applied to the amplifier. This approach works for DC and ACbrushless servomotors.

Observe that equation (36) can be written as

q(t) = aq(t) + bu(t) (37)

where a = f/J, b = k/J are positive parameters. In the abovedescribed platform, the estimated parameter values are a =0.45, b = 31. Notice that a preliminary proportional feedbackleads to the form (1) studied in the previous sections.

6.2 Experimental results

We now illustrate how the choice of the gains kr and kp and

of the delay h in the control law u(t) = kr(t h) kp(t)determines the system response. A proportional gain kp = 16such that the system operates in the region were the controlvoltage does not go to saturation is selected.

The maximum achievable exponential closed loop decay =22.4949 is obtained straightforwardly from the formula (28).The corresponding parameters (h,kr) = (0.0449,11.6526) fol-

low from equations (29) and (30). The system response and thecontrol signal are shown on Figure 6 and 7, respectively.

Fig. 6. Angular position q(t) for (h,kr,kp) =(0.0449,11.6526,16).

Next, we display on Figure 8 and 9 the system response andcontrol law for the loci 1,2,3 and 4. The corresponding param-eters h and kr are listed on Table 1. In all cases, the exponentialdecay is = 4.5 and the proportional gain is kp = 16. Thelocation of the roots on the boundary determines the shapeof the system response.

Remark 9. It appears that the above theoretical analysis leadsto a straightforward tuning formula for the proportional delaycontrol of second order systems which are the first electionmodel for a wide array of physical processes. The maximumachievable decay for a given proportional gain is given by (28)and the corresponding parameters are readily computed from(29) and (30). This seems to be an advantage compared with aproportional derivative controller that requires the tuning of a


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Fig. 7. Control signal for (h,kr,kp) = (0.0449,11.6526,16).

Locus h kr

1 0.01663 15.39

2 0.0683 4.574

3 0.0839 11.37

4 0.03931 13.92

5 0.0449 11.653

Table 1. Parameters (h,kr) of control (3) for =4.5 and kp = 16 given.

Fig. 8. Angular position q(t) for (h,kr,kp,) given en Table 1.

filter or of an observer along with the tuning of the control law.The delay physical implementation can be done with memoryelements.

7. CONCLUDING REMARKS

In this contribution a graphical and analytical study of thestabilization of a second order system through proportionaldelayed control law is performed. A comprehensive charac-terization of the regions and boundaries ofstabilization isperformed and a detailed analysis of the dominant roots of theclosed loop system is presented. This leads to the characteriza-tion of the maximum achievable exponential decay strategy forwhich simple formulae are available. An experimental setup ofa DC servomotor illustrates the results and the simplicity of theimplementation.

Fig. 9. Input forces with control (3) and parameter (h,kr,kp,)given in Table 1.

ACKNOWLEDGEMENTS

This work was supported by CONACyT, Project 61076 andScholar 206997. The authors thank Jose de Jesus Meza Serranofor his technical support.

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Cooke, K. and Grossman, Z. (1982). Discrete delay,dristributed delay and stability switches. Journal of math-ematical analysis and applications, 86:592-627.

Cooke, K. and Van Den Driessche, P. (1986). On zeroes of sometranscendental equations. Funkcialaj Ekvacioj, 29:77-90.

Datko, R. (1978). A procedure for determination of the ex-ponential stability of certain differential difference equation.Quart. Appl. Math., 1:279-292.

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Karafyllis, I. (2008). Robust global stabilization by means ofdiscrete-delay output feedback. Systems and Control Letters,987-995.

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