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Root locus based design Effects of Addition of Poles and Zeros

Figure: Effect of Adding Poles

Figure: Effect of Adding Zero

1. Addition of poles pulls the root locus to the right 2. Additional zero pulls the root-locus to the left

Controller Design A compensator or controller placed in the forward path of a control system will modify the shape of the loci if it contains additional poles and zeros.

Characteristics One additional zero One additional zero and a pole at origin Two additional zeros and a additional pole at origin

Compensator PD PI PID

rlocus(num, den) sgrid(Z, Wn) [K, poles] = rlocfind(num, den)

Example A control system has the open-loop transfer function,

K ; k  1. s( s  2)( s  5) A PD compensator of the form Gc ( s)  K1 ( s  a) is to be introduced in the forward G f ( s) 

path to achieve the performance specification: Overshoot less than 5%, settling time less than 2seconds Determine the values of

K1 and a

to meet the specification.

Original Controller

For

  0.5,

P.O.  16.3% For

  0.7,

P.O.  4.6%

  0.7, corresponds to the controller gain of K  7.13 . Settling time condition is not met.

PD compensator design

When PD compensator is used, we actually add a zero in the open-loop transfer function. Potential locations of zero include: 1. At s  1 ,

2. At s  2 ,

3. At s  3

G( s) H ( s) 

K1 ( s  1) s( s  2)( s  5)

  0.7, K1  15, ts  3.9 sec, Here, the closed-loop pole in the root locus branch between 0 and -1 dominate the time response with oscillatory 2nd-order response superimposed.

G( s) H ( s) 

K1 ( s  2) s( s  2)( s  5)

  0.7, K1  12.8, ts  1.7 sec,

P.O.  4.1%

G( s) H ( s) 

K1 ( s  3) s( s  2)( s  5)

  0.7, K1  5.3, ts  3.1 sec, P.O.  5.3%

Summary Of the three compensators considered, only option 2 meets the performance specifications. The recommended compensator is therefore, Gc (s)  12.8(s  2) . The time-domain responses for the four conditions are shown in Figure below.

Realization of Compensator using Passive Components It is not possible to design isolated zero or pole at origin using passive components. In that case a pair of pole and zero is produced. Compensators may be of four types: Lead compensator, Lag compensator, Lag-lead compensator, and feedback compensator. [Cascade compensator]* A. Lead Compensator

Eo ( s) R2 R1Cs  1   Ei ( s ) R  R1 / Cs R1 R2Cs  R1  R2 2 R1  1/ Cs 

where   R1C and R2   1. R1  R2

s  zc s  1/ R1C s  1/    s  [( R1  R2 ) / R2 ] / R1C s  pc s  1/ 

;

zc  pc

The pole-zero configuration is shown in figure above on the right side. The zero frequency gain is cancelled by an amplifier of gain 1/  . B. Lag Compensator

Gc ( s ) 

R2  1/ Cs 1  R2Cs  R1  R2  1/ Cs 1  ( R1  R2 )Cs

s  1/ R2C R2 s  1/ R2C  R  R2 1/ R2C  1 s R1  R2 s   R2  / R2C R2  R1  R2  1 s  1/    s  1/ 



where,   R2C and



R1  R2 1 R2

The pole-zero configuration is shown in figure (a) above.

C. Lag-lead Compensator

 1  1  s  s   R1C1  R2C2   s  1/  1  s  1/  2   Gc ( s)    ; s  1/  1  s  1/  2   1 1 1  1 2  s    s  R1 R2C1C2  R1C1 R2C2 R2C1 

where,   1,   1. Comparing left and right side we get,

And,

R1C1  1; R2C2   2 ; R1R2C1C2 =1 2 . Therefore,   1. 1 1 1 1 1 1        . R1C1 R2C2 R2C1 1  2 1  2

The pole-zero configuration is shown in figure below. j

zc2

zc1 

pc2

pc1

Cascade Compensation in Time Domain Here the design specifications are converted to  and n of a complex conjugate pairof closed-loop poles based on the assumption that the system will be dominated by these two complex poles and therefore its dynamic behavior can be approximated by that of a second-order system. A compensator is designed so that closed-loop poles other than the dominant poles are located very close to the open-loop zeros or far away from the j axis so that they make negligible contribution to the system dynamics.

Lead Compensation Let us consider a unity feedback system with a forward path transfer function G f ( s) . Let, sd be the location of dominant complex closed-loop pole. If the angle condition at that point is not met for uncompensated system, a compensator has to be designed so that the compensated root locus passes through sd . Let the compensator has transfer function Gc ( s) . Applying angle condition we get Gc ( sd )G f ( sd )  Gc ( sd )  G f ( sd )  180 or, Gc ( sd )    180  G f ( sd )

For a given  there is no unique location for the pole-zero pair. Procedure for designing lead compensator is as follows: 1. From specification determine sd . 2. Draw the root-locus plot of uncompensated system and see whether only gain adjustment can yield the desired closed-loop poles. If not, calculate the angle deficiency,  . This angle will be contributed by the lead compensator.

1 T ;  1 3. Gc ( s )  K c 1 s T 4. Locate zc and pc so that the lead compensator will contribute necessary  . 5. Determine K of the compensated system from magnitude condition. s

If large error constant is required, cascade a lag network.

K . Compensate the system so as to meet the s ( s  1.5) transient response specifications: settling time  4 second. Peak overshoot for step input  20% .

Example 01 Let, G f ( s) 

This

specifications

2

imply

that,

  0.45

and

 M  e / 1   2 and t  4 /   s n  P 

The desired dominant roots lie at

sd  1  j 2.

   j 1   2  n n  

n  1 .

-line

sd 2



-19.8

-1.5

The angle contribution required for the lead compensator is,   180  (2 117  75)  129 . As  is large, a double lead network is appropriate. Each section of double lead network will then contribute an angle of 64.5 at sd . Let us now locate compensator zero at s  1.7. Join the compensator zero to sd and locate the compensator pole by making an angle of  . The pole is found to be at -19.8. The open-loop transfer function of the compensated system thus becomes, 8.30( s  1.7)2 G( s)  2 . s ( s  1.5)(s  19.8)2 Dominance of the closed-loop poles (-1 ± j2) is preserved.

K .   0.5, n  2. s( s  1)( s  4) The desired dominant closed-loop poles are located at, sd  1  j1.73 . The angle condition required from the lead compensator pole-zero pair is,   180  (120  90  30)  60 .

Example 02

G f ( s) 

Place a compensator zero close to the pole -1 at s = -1.2. Join the zero to sd and make an angle of 60 to the left of the line. The compensator pole will be found at -4.95. The open-loop transfer function of the transfer function becomes, K ( s  1.2) . G( s)  s( s  1)( s  4)(s  4.95) The gain K can be evaluated using magnitude condition at sd .

or,

zc  1/  pc  1/ 

K

sd sd  1 sd  4 sd  4.95 1  j1.73 j1.73 j1.73  3 j1.73  3.95 = j1.73  0.2 sd  1.2

K

1.99 1.73  3.463  4.312  29.527  30 1.741

   0.833  zc  0.2424      0.202 pc

We can select R1 , R2 , C to any suitable value.

Prob-2