Break in and break away points on Root Locus

Hi,

I'm busy developing a controller, but for some reason my plot does not cut through the real axis and I'm not able to find the locations where the damping ratio is 0.59.

Herewith the code, block diagram and RLocus plot:

G=tf([0.151 0.1774], [1 0.739 0.921 0.25])

rlocus(G)

Any assistance will be appreciated.

Thanks

Niel

Answer 1

[Sam Chak]

Hi @Niel

Previously in this post, it was shown that we must at least place a compensating pole at the origin so that the root locus intersects the desired damping ratio grid.

In fact, we can place additional compensating poles and zeros at appropriate locations in the s-plane so that the root locus takes on a cross-shaped form. This is ideal in root locus design because it allows us to theoretically vary the feedback gain parameter k from just above 0 to ∞, achieving the desired damping ratio ζ and natural frequency without destabilizing the system.

This approach only works if the zeros and poles of the plant, , have no positive real parts and are not located near the imaginary axis of the s-plane. While this approach is effective, it is also necessary to ensure that the transfer function of the compensator, , is proper, which in turn limits the overall effectiveness of such method for other types of plants.

%% Plant
G   = tf([0.151 0.1774], [1 0.739 0.921 0.25])

G =
 
          0.151 s + 0.1774
  --------------------------------
  s^3 + 0.739 s^2 + 0.921 s + 0.25
 
Continuous-time transfer function.

% check if zero of G has positive real part
zG  = zero(G)

zG = -1.1748

% check if poles of G have negative real parts
pG  = pole(G)

pG = 
  -0.2107 + 0.8618i
  -0.2107 - 0.8618i
  -0.3176 + 0.0000i

%% Pole–Zero compensator
zeta= 0.59;                 % desired damping ratio
wn  = 1;                    % desired natural frequency
zC  = pG;                   % desired zeros of Compensator
pC  = [zG, -2*zeta*wn, 0];  % desired poles of Compensator (minimally put a pole at the origin)
C   = tf(zpk(zC, pC, 1))    % check if Compensator TF is proper

C =
 
  s^3 + 0.739 s^2 + 0.921 s + 0.25
  --------------------------------
     s^3 + 2.355 s^2 + 1.386 s
 
Continuous-time transfer function.

%% Check if the Root Locus takes on a "Cross-shaped form"
figure
rlp = rlocusplot(C*G);
axis([-1.4, 0.2, -1.5, 1.5])
rlp.AxesStyle.GridVisible       = "on";
rlp.AxesStyle.GridDampingSpec   = zeta;     % grid of desired damping ratio
rlp.AxesStyle.GridFrequencySpec = wn;       % grid of desired natural frequency

%% Closed-loop system
% if root locus takes on a "Cross-shaped form", tune feedback gain until zeta = 0.59.
k   = 6.62251655629139;     % feedback gain
Gcl = feedback(k*C*G, 1)    % actual closed-loop

Gcl =
 
              s^4 + 1.914 s^3 + 1.789 s^2 + 1.332 s + 0.2937
  ----------------------------------------------------------------------
  s^6 + 3.094 s^5 + 5.048 s^4 + 5.357 s^3 + 3.655 s^2 + 1.679 s + 0.2937
 
Continuous-time transfer function.

Gcl = minreal(Gcl)          % perform pole-zero cancellation

Gcl =
 
         1
  ----------------
  s^2 + 1.18 s + 1
 
Continuous-time transfer function.

[wn, zeta] = damp(Gcl)      % check if desired wn and zeta are achieved

wn = 2×1

1.0000 1.0000

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zeta = 2×1

0.5900 0.5900

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Scl = stepinfo(Gcl)

Scl = struct with fields:
         RiseTime: 1.8314
    TransientTime: 5.9235
     SettlingTime: 5.9235
      SettlingMin: 0.9047
      SettlingMax: 1.1007
        Overshoot: 10.0686
       Undershoot: 0
             Peak: 1.1007
         PeakTime: 3.9027

%% Simulation
figure
step(Gcl, 2*round(Scl.SettlingTime)), grid on

[Niel]

Thank you Sam this has been extremely helpful, much appreciated.

[Sam Chak]

You're welcome, @Niel.

Answer 2

[Sulaymon Eshkabilov]

IS it what you're looking for:

% Transfer Function:
SYS = tf([0.151 0.1774], [1 0.739 0.921 0.25]);
% Plot the RLocus:
figure
rlocus(SYS)
title('Root Locus of the Transfer Function')
grid on

●

% Compute and display the damping ratio and natural frequency:
[wn zeta] = damp(SYS);
fprintf('Natural Frequencies: %f \n', wn)

Natural Frequencies: 0.317596 
Natural Frequencies: 0.887223 
Natural Frequencies: 0.887223 

fprintf('Damping Ratios: %f \n', zeta);

Damping Ratios: 1.000000 
Damping Ratios: 0.237485 
Damping Ratios: 0.237485 

Answer 3

[Sam Chak]

Hi @Niel

The reason is simply that the locus never intersects the 0.59 damping ratio grid line, regardless of how the feedback gain parameter is varied from 0⁺ to 6 (within the stable region). Looking at the blue and green root locus branches, it starts with a damping ratio of 0.237. As the feedback gain parameter is increased, the damping ratio continues to decrease, resulting in an oscillatory step response. This is certainly undesirable.

Since you have not mentioned any performance specifications other than the desired damping ratio of 0.59, if you use the graphical root locus design approach, minimally, you will need to add a pole at the origin to provide the flexibility in varying the feedback gain parameter until the locus intersects the 0.59 damping ratio grid line in the 2nd figure.

However, please note that the feedback gain will also intersect the red and green root locus branches (low damping ratio paths). Because these branches are closer to the imaginary axis, they can significantly influence the system's transient response in the step response (see the 3rd figure).

G       = tf([0.151 0.1774], [1 0.739 0.921 0.25])

G =
 
          0.151 s + 0.1774
  --------------------------------
  s^3 + 0.739 s^2 + 0.921 s + 0.25
 
Continuous-time transfer function.

%% Step 1: Analysis
damp(G)

                                                                       
         Pole              Damping       Frequency      Time Constant  
                                       (rad/seconds)      (seconds)    
                                                                       
 -3.18e-01                 1.00e+00       3.18e-01         3.15e+00    
 -2.11e-01 + 8.62e-01i     2.37e-01       8.87e-01         4.75e+00    
 -2.11e-01 - 8.62e-01i     2.37e-01       8.87e-01         4.75e+00    

[r1, k1] = rlocus(G);
display('Poles near the imaginary axis'), display(r1(:,30))

Poles near the imaginary axis
  -0.0088 + 1.3423i
  -0.0088 - 1.3423i
  -0.7214 + 0.0000i

display('Corresponding feedback gain value'), display(k1(30))

Corresponding feedback gain value
    5.9181

figure
rlp1    = rlocusplot(G);
zeta1   = [0.1  0.2  0.3 0.59 0.9];
wn1     = [  1    2    3    4];
ylim([-4  4])
rlp1.AxesStyle.GridVisible       = "on";
rlp1.AxesStyle.GridDampingSpec   = zeta1;
rlp1.AxesStyle.GridFrequencySpec = wn1;

●

%% Step 2: Design – Pure graphical Root Locus method (no advanced control theory)
C   = tf(1, [1 0])  % add a pole at the origin

C =
 
  1
  -
  s
 
Continuous-time transfer function.

figure
rlp2    = rlocusplot(C*G);
zeta2   = [0.07, 0.59];
wn2     = 1;
axis([-2, 1, -1.5, 1.5])
rlp2.AxesStyle.GridVisible       = "on";
rlp2.AxesStyle.GridDampingSpec   = zeta2;
rlp2.AxesStyle.GridFrequencySpec = wn2;

[r2, k2] = rlocus(C*G);
display(r2(:,24))

  -0.3429 + 0.4504i
  -0.0266 + 0.7505i
  -0.0266 - 0.7505i
  -0.3429 - 0.4504i

display(k2(24))      % approx. limit of k before Closed-loop becomes unstable

    1.0188

k         = 0.9205  % tune the feedback gain until you see the damping ratio of 0.59

k = 0.9205

Gcl       = feedback(k*C*G, 1);
[wn zeta] = damp(Gcl)

wn = 4×1

0.5372 0.5372 0.7522 0.7522

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zeta = 4×1

0.5900 0.5900 0.0698 0.0698

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%% Step 3: Simulation
figure
Scl = stepinfo(Gcl)

Scl = struct with fields:
         RiseTime: 2.1835
    TransientTime: 65.5187
     SettlingTime: 65.5187
      SettlingMin: 0.6135
      SettlingMax: 1.6276
        Overshoot: 62.7612
       Undershoot: 0
             Peak: 1.6276
         PeakTime: 6.1020

step(Gcl), grid on