5. Time Response-Based Design

5.1.1 Rules for Drawing the Root Locus Diagram

5.2.1 Proportional Control of Position

According to Chap. 11, the permanent magnet brushed direct current (DC) motor model is given as:

(5.12)

where θ(s) and I ^∗(s) stand for the position and the electric current commanded respectively. Suppose that the following proportional position controller is employed:

where θ _d(s) is the desired position and k _p is a constant known as the proportional gain. The closed-loop system can be represented as in Fig. 5.10, where it is concluded that the open-loop transfer function is given as:

(5.13)

Notice that the system type is 1, i.e., the steady state error is zero when the desired position is a step. Hence, the only design problem that remains is to choose k _p such that the closed-loop poles are assigned to the desired locations. Use of the root locus method to solve this problem is shown in the following. The root locus method forces the gain k _p to take values from 0 to + ∞. First, G(s)H(s) is rewritten as:

where the vectors and have been defined (see Fig. 5.11). The fundamental conditions for drawing the root locus are the angle condition (5.7) and the magnitude condition (5.6) which are expressed respectively as:

Rule 5 indicates that, on the real axis, the root locus only exists between the points s = 0 and s = −a. Furthermore, according to rule 1, the root locus begins (k _p = 0) at s = 0 and s = −a. On the other hand, according to rules 2 and 3, the branches starting at s = 0 and s = −a must approach some open-loop zero (no zero exists in this case) or some point at the infinity of the plane s as k _p tends to + ∞. Hence, the root locus has to move away from points s = 0 and s = −a, on the real axis, as k _p increases such that a breakaway point must exist somewhere between such points and then two branches appear to be moving away to infinity in plane s.

On the other hand, according to the angle condition, − (θ ₁ + θ ₂) = −180^∘ = −(α + θ ₁) in Fig. 5.11, it is concluded that α = θ ₂, i.e., both triangles t ₁ and t ₂ must be identical for any closed-loop pole s. Hence, both branches referred above, which are shown in Fig. 5.11, must be parallel to the imaginary axis. This means that the breakaway point is located at the middle point between s = 0 and s = −a, i.e., at (0 − a)∕2 = −a∕2. This can also be verified using rules 3 and 4. Finally, according to rule 6, both branches are symmetrical with respect to the real axis.

The magnitude condition is employed when the exact value of k _p that renders a specific point of the root locus an actual closed-loop pole is required. Notice, for instance, that lengths l ₁ and l ₂ grow as k _p tends to + ∞ to satisfy the magnitude condition:

(5.14)

which means that the closed-loop poles corresponding to large values of k _p tend to some point at infinity on the s plane. Hence, it is concluded: (i) Both closed-loop poles are real, negative, different, and approach the point s = −a∕2 when k _p > 0 is small; this means that the closed-loop system becomes faster because the slowest pole moves away from the origin, (ii) According to the magnitude condition, when:

both closed-loop poles are real, repeated, negative, and located at , i.e., the fastest response without oscillations is obtained, (iii) As increases, both poles move away from the real axis (one pole moves upward and the other downward) on the vertical line passing through ; this means that the closed-loop system becomes faster (because ω _n increases; see Sect. 3.​3) and more oscillatory (because the angle 90^∘− α decreases and damping, given as , decreases; see Sect. 3.​3).

The above discussion shows that it is not possible to achieve a closed-loop system response that is simultaneously fast and well damped. This is a direct consequence of the fact that closed-loop poles cannot be assigned at any arbitrary location of the s plane, as they can only be located on the thick straight line shown in Fig. 5.11. In the next example, it is shown that the introduction of an additional zero in the open-loop transfer function of the present example allows the closed loop poles to be assigned at any point on the s plane.

Example 5.1

The root locus diagram can also be plotted using MATLAB. Given a closed-loop system as that in Fig. 5.1, it suffices to use the command:

rlocus(GH)

where GH stands for the open-loop transfer function G(s)H(s). Consider, for instance, the closed-loop system in Fig. 5.10 when k = 2 and a = 8. The root locus diagram in Fig. 5.12 is plotted using the following commands:

k=2;

a=8;

gh=tf(k,[1 a 0]);

rlocus(gh);

rlocfind(gh)

The root locus in Fig. 5.12 is represented by the continuous lines. Notice that, contrary to the previous discussion, in the above commands the proportional gain k _p is not considered. This is because k _p is automatically increased from 0 to + ∞ by MATLAB when executing the command “rlocus()”. This must be taken into account when defining the open-loop transfer function to be used as an argument of this MATLAB command. In Fig. 5.12, each open-loop pole is represented by a symbol “×”, i.e., at s = 0 and s = −a = −8. Compare Figs. 5.12 and 5.11 to verify similarities between them. For instance, the horizontal and vertical lines in Fig. 5.12 intersect at s = −4, i.e., at s = −a∕2, as predicted in Fig. 5.11. The root locus diagram is plotted by MATLAB when executing the command

rlocus(gh);

alone. However, the additional use of the command:

rlocfind(gh)

allows the user to select any desired point on the root locus. When selected, this point and all points on the root locus corresponding to the same gain k _p are marked with a symbol “+ ”. For instance, in Fig. 5.12 the point − 4 + 3j was selected and, automatically, the point − 4 − 3j was also selected and marked with a “+ ” (only the horizontal line of this symbol is appreciable in Fig. 5.12 because of the particular geometry of this root locus). Also, MATLAB automatically sends the following text:

selected_point =

-3.9882 + 2.9969i

ans =

12.4907

This means that k _p = 12.4907 at s = −4 ± 3j and this can be verified using the magnitude condition in (5.14) as:

When using the command “rlocus()”, MATLAB automatically determines a maximal value of k _p to be considered to draw Fig. 5.12. However, the user can decide herself/himself the maximal value of k _p to be considered by using the command:

rlocus(gh,d);

where d is a vector containing all of the specific values of the gain k _p, which the user wants to be considered to plot the root locus.

Some closed-loop pole pairs are selected in Fig. 5.13. The time response of the closed-loop system Fig. 5.10 is presented in Fig. 5.14 when the corresponding closed-loop pole pairs are those indicated in Fig. 5.13. Recall that k _p increases as it passes from poles at “o” to poles at the square in Fig. 5.13. It is observed that the closed-loop system response becomes faster as k _p increases and overshoot is present once the closed-loop poles have imaginary parts that are different from zero. Moreover, overshoot is small if the imaginary parts are small. These observations corroborate the above root locus-based analysis.

The results in Figs. 5.13 and 5.14 were obtained by executing several times the following MATLAB code in an m-file:

k=2;

a=8;

kp=30; % 1 5 10 20 30

gh=tf(kp∗k,[1 a 0]);

figure(1)

rlocus(gh);

hold on

rlocus(gh,34);

axis([-10 2 -9 9])

hold on

M=feedback(gh,1,-1);

v=pole(M);

figure(1)

plot(real(v(1)),imag(v(1)),'bs')

plot(real(v(2)),imag(v(2)),'bs')

figure(2)

step(M,'b:',5)

hold on

5.2.2 Proportional–Derivative Control of Position

Consider again the motor model shown in (5.12), but now together with the following proportional–derivative (PD) controller:

where k _p is the proportional gain and k _d is a constant known as the derivative gain. Use of the Laplace transform yields:

The corresponding closed-loop block diagram is shown in Fig. 5.15. The open-loop transfer function is given as:

(5.15)

Notice that, now, k _d is the gain that varies from 0 to + ∞ to plot the root locus. First, G(s)H(s) is rewritten as:

(5.16)

where the vectors , , and have been defined. The angle and the magnitude conditions are expressed respectively as:

The root locus in this case can be obtained from the root locus obtained for the transfer function in (5.13) by simply taking into account that an additional zero at s = −c has been included.

According to rule 12, the branches of the root locus in Fig. 5.11 will bend toward the left as a consequence of the additional zero. This can be verified using rule 3 to find that, now, the root locus only has one branch whose asymptote forms a ± 180^∘ with the positive real axis. As shown in Fig. 5.16, the root locus has two different possibilities depending on the exact location of the zero at s = −c. If it is placed at the left of the pole at s = −a (as in Fig. 5.16a) then, according to rule 5, the root locus exists on two segments of the negative real axis: between the points s = 0 and s = −a, and to the left of zero at s = −c. Moreover, according to rules 2 and 3 one of the two root locus branches beginning (k _d = 0) at the open-loop poles located at s = 0 and s = −a must tend toward the zero at s = −c whereas the other branch must tend toward infinity in the s plane following the asymptote forming ± 180^∘ with the positive real axis. Hence, a breakaway point must exist between points s = 0 and s = −a. Furthermore, as the root locus is symmetrical with respect to the real axis (rule 6), these branches must describe two semi circumferences toward the left of the breakaway point, which join again in a break-in point located on the negative real axis to the left of the zero at s = −c. After that, one branch approaches the zero at s = −c and the other tends toward infinity along the negative real axis.

On the other hand, if the zero at s = −c is placed between the open-loop poles at s = 0 and s = −a, then, according to rule 5, the root locus exists on two segments on the negative real axis located between points s = −c and s = 0 and on the left of the pole at s = −a. Notice that, now, the branch beginning (k _d = 0) at s = 0 approaches the zero at s = −c, whereas the branch beginning (k _d = 0) at s = −a tends toward infinity along the negative real axis. Also notice that this is possible without the necessity for any branch to exist outside the real axis, as is shown in Fig. 5.16b.

From the study of the resulting root locus, it is possible to realize that the closed-loop system is stable for any k _p > 0 and k _d > 0 because both possibilities for the root locus that have been presented in Fig. 5.16 show that the closed-loop poles are always located on the left half-plane s, i.e., the closed-loop poles have a negative real part. Finally, according to Sect. 3.​8.​3, there always exist gains k _p and k _d allowing any values to be chosen for both ω _n and ζ. This means that it is always possible to assign both closed-loop poles at any desired point on the left half-plane s.

Notice that the open-loop zero at s = −c is also a zero of the closed-loop transfer function; hence, it also affects the closed-loop system transient response. This means that the transient response will not have the exact specifications computed using (3.​71).

Example 5.2

The following MATLAB code is executed in an m-file to simulate the closed-loop system in Fig. 5.15:

clc

a=7;

k=70;

c=4; %7 4 10

Polo_des=-20;

l3=abs(Polo_des)-c;

l1=abs(Polo_des);

l2=abs(Polo_des)-a;

kd=l1∗l2/(k∗l3);

kp=c∗kd;

Md=tf(20,[1 20]);

gm=tf(k,[1 a 0]);

PD=tf([kd kp],1);

M=feedback(gm∗PD,1,-1);

v=pole(M)

%{

figure(1)

rlocus(gm∗PD);

hold on

plot(real(v(1)),imag(v(1)),'kˆ');

plot(real(v(2)),imag(v(2)),'kˆ');

%}

figure(2)

step(Md∗2,'r--',M∗2,':',0.25)

hold on

It is assumed that a = 7 and k = 70. The desired response is that of a first-order system with time constant 0.05[s], i.e., with a real pole at s = −20. The idea is to propose different locations for the zero at s = −c to observe when the desired response is accomplished. The derivative gain k _d is computed using the magnitude condition where l ₁, l ₂, l ₃ are the magnitude of the vectors defined in the paragraph after (5.16). Then k _p = ck _d is computed. After that, it is possible to generate the closed-loop transfer function M and to simulate it together with the desired response. These results are shown in Fig. 5.17. Notice that the desired response is matched when c = a. The root locus diagram in this case is shown in Fig. 5.18 where the closed-loop poles are represented by a triangle up. One closed-loop pole is located at s = −20, as desired, and the other cancels with the zero at s = −c. This explains why the desired response is accomplished in this case.

When c = 4, the closed-loop poles are at s = −20, as desired, and s = −3.25. The latter pole is slow and does not cancel with zero at s = −c = −4; hence, its effects are important in the closed-loop transient response. This explains why, in this case, the response in Fig. 5.17 is slow. When c = 10, the closed-loop poles are located at s = −20 and s = −13, i.e., the pole at s = −13 cannot be cancelled by the zero at s = −c. This results in an overshoot, i.e., the desired response is not accomplished again. The reason for this overshoot is explained in Sect. 8.​1.​2, where it is stated that overshoot is unavoidable whenever all of the closed-loop poles are on the left of an open-loop zero despite all the closed-loop poles being real, i.e., as in the present case.

Example 5.3

The following MATLAB code is executed in an m-file to simulate the closed-loop system in Fig. 5.15:

clc

a=7;

k=70;

tr=0.05;

Mp=14;%

z=sqrt( log(Mp/100)ˆ2 /( log(Mp/100)ˆ2 + piˆ2 ) );

wn=1/(tr∗sqrt(1-zˆ2))∗(pi-atan(sqrt(1-zˆ2)/z));

s=-z∗wn+wn∗sqrt(1-zˆ2)∗j

kd=(2∗z∗wn-a)/k;

kp=wnˆ2/k;

den1=conv([1 -s],[1 -real(s)+j∗imag(s)]);

Md=tf(den1(3),den1);

gm=tf(k,[1 a 0]);

PD=tf([kd kp],1);

M=feedback(PD∗gm,1,-1);

v=pole(M)

%%{

figure(1)

rlocus(gm∗PD);

hold on

plot(real(v(1)),imag(v(1)),'kˆ');

plot(real(v(2)),imag(v(2)),'kˆ');

%}

figure(2)

step(Md∗2,'r--',M∗2,'-',0.25)

hold on

The desired specifications are a rise time of 0.05[s] and a 14% overshoot. This results in a pair of desired complex conjugate closed-loop poles located at s = −26.66 ± 42.59j. The controller gains k _p and k _d were computed using:

(see Sect. 3.​8.​3). This allows the corresponding root locus diagram to be plotted in Fig. 5.19. Notice that this diagram is similar to that depicted in Fig. 5.16a because c > a in this case. Although the desired closed-loop poles have been assigned, notice that the closed-loop zero at s = −c cannot be cancelled; hence, it will modify the actual closed-loop response with respect to the desired one. As a matter of fact, in Fig. 5.20, the desired response, with a dashed line, and the actual closed-loop response, with a continuous line, are presented. We conclude that the effects of the zero at s = −c are a shorter rise time and a larger overshoot.

5.2.3 Position Control Using a Lead Compensator

Consider again the DC motor model in (5.12) but, now, together with the following controller:

which is known as a lead compensator if c > d. This condition is introduced when it is desired to increase the damping in the system, i.e., when the closed-loop poles must be shifted to the left (see rules 11 and 12). The corresponding closed-loop block diagram is shown in Fig. 5.21. The open-loop transfer function is given as:

(5.17)

If d is chosen as:

(5.18)

then the close-loop transfer function is:

which means that:

(5.19)

Hence, if the expressions in (3.​71) are used to compute ζ and ω _n such that the desired rise time and overshoot are obtained, then (5.18) and (5.19) represent a simple tuning rule. Moreover, the formulas in (3.​71) for overshoot and rise time are exact in this case; thus, the drawback that a PD controller has in this respect is eliminated. The transient response in this case is rather similar to that obtained with the controller in Sect. 3.​8.​2, i.e., proportional position control plus velocity feedback.

5.2.4 Proportional–Integral Control of Velocity

Although proportional–integral (PI) velocity control has been studied in Sect. 3.​8.​4, in this part, some ideas are presented that are clearer when employing block diagrams and the root locus. According to Chap. 10, when the velocity ω(s) is the output, the permanent magnet brushed DC motor model is:

where the commanded electric current I ^∗(s) is the input and T _p(s) is an external torque disturbance. A PI velocity controller is given as:

where ω _d is the desired velocity, k _p is the proportional gain and the constant k _i is known as the integral gain. Using the Laplace transform:

Hence, the closed-loop block diagram is depicted in Fig. 5.22a. As this control system has two inputs, the superposition principle can be employed (see Sect. 3.​7) to write:

where G ₁(s) is the transfer function obtained when using ω _d(s) as the input and ω(s) as the output and assuming T _p(s) = 0, i.e., using the block diagram in Fig. 5.22b, whereas G ₂(s) is the transfer function when T _p(s) is the input and ω(s) is the output, assuming that ω _d(s) = 0, i.e., when using the block diagram in Fig. 5.22c. It is important to say that, in the case when T _p(s) is the input, ω(s) represents the velocity deviation produced by disturbance T _p(s).

Now, the problem of choosing the PI controller gains such that the transient response of the closed-loop system satisfies the desired specifications is studied. It is observed in Fig. 5.22b, that the open-loop transfer function is:

(5.20)

Notice that the system type is 1, which ensures that ω(t) = ω _d in a steady state if ω _d is a constant. This is one reason for using a PI controller in this example. Hence, the only problem that remains is to choose the controller gains k _p and k _i, such that the transient response satisfies the desired specifications. To this aim, notice that the open-loop transfer function shown in (5.20) is identical to the transfer function shown in (5.15), which corresponds to the PD control of position as only k _d has to be replaced by k _p. Thus, the root locus diagram corresponding to the PI velocity control is identical to both cases shown in Fig. 5.16 and the same conclusions are obtained:

1. i)

   There always exist some gains k _p and k _i allowing both closed-loop poles to be placed at any point on the left half-plane; hence, it is possible to tune the PI controller using a trial and error procedure (see Sect. 3.​8.​4).

    
2. ii)

   The open-loop zero located at s = −c is also a zero of the closed-loop transfer function G ₁(s), i.e., it also affects the closed-loop transient response. This means that the transient response does not have the specifications designed using (3.​71) to choose the closed-loop poles. This poses the following two possibilities.

    

1. 1.
   The problem pointed out at ii) can be eliminated if it is chosen:

   

   (5.21)

   because, in such a case and according to Fig. 5.22b and (5.1), the closed-loop transfer function is:

   

   This means that the closed-loop response is as that of a first-order system with a unit steady state gain and a time constant given as . Then, if a time constant τ is specified:

   

   (5.22)

   The conditions in (5.22) and (5.21) represent a simple tuning rule. Finally, the root locus diagram corresponding to this case (c = a, ) is shown in Fig. 5.23. This can be easily verified using rules 3 and 5. As there is only one open-loop pole, there is only one closed-loop pole, which must be real and tends toward one zero at infinity (on an asymptote forming − 180^∘ with the positive real axis) because there is no open-loop zero. Furthermore, the magnitude condition:

   

   where l ₁ is the distance from the desired closed-loop pole to the origin, establishes that the desired closed-loop pole located at is reached when:

   

   which corroborates (5.22). However, this tuning rule has a problem when considering the closed-loop response when an external disturbance T _p(s) appears. Consider the block diagram in Fig. 5.22c. If c = a is chosen, then the closed-loop transfer function is:

   

   (5.23)

   Using the final value theorem, the following is found:

   

   i.e., that the velocity deviation produced by the external torque disturbance with a constant value t _d vanishes in a steady state. This is the reason why a PI velocity controller is chosen. However, there is a problem. The poles of the transfer function in (5.23) determine how fast this velocity deviation vanishes. These poles are located at s = −k _p k and s = −a. Although one of these poles can be rendered as fast as desired just by choosing a suitable k _p, the other pole, at s = −a, cannot be modified and depends on the parameters of the DC motor. This has as a consequence that the velocity deviation due to the disturbance may approach zero very slowly, which is a serious drawback.

   

    
2. 2.
   Trying to solve the above problem, we might abandon the tuning rule in (5.22) and (5.21). As we now have c ≠ a, the closed-loop transfer function corresponding to the block diagram in Fig. 5.22c is:

   

   (5.24)

   Using the final value theorem, it can be easily verified that, in the case of a constant disturbance , the steady-state deviation is zero again: lim_t→∞ ω(t) = 0, because the transfer function in (5.24) has a zero at s = 0. On the other hand, poles of the transfer function in (5.24), which determine how fast the velocity deviation vanishes, are identical to the poles of the transfer function ; hence, they can be assigned using the root locus method from (5.20). As has been explained, the corresponding root locus is identical to both cases shown in Fig. 5.16 replacing k _d by k _p. It is important to stress that the transfer function in (5.24) has no zero at s = −c shown in the root locus diagram in Fig. 5.16. This means that none of the closed-loop poles obtained using the root locus method can cancelled with the zero at s = −c and the slowest pole will have the most significant effect on the time required for the velocity deviation due to the disturbance to vanish. With these ideas in mind, the following is concluded.

   • To render small the effect of the zero at s = −c on the transient response to a given velocity reference, one closed-loop pole must be placed close to the zero at s = −c. The other closed-loop pole (the fastest one) moves away to the left and tends toward infinity as the slow pole approaches to s = −c.

   • This means that the fast pole determines the transient response, i.e., the time constant, to a given velocity reference. Then, if it is desired to fix some finite value for the time constant (such that the fast pole is placed at a finite point on the negative real axis), the slow pole is always relatively far from zero at s = −c. This means that the transient response is always affected by both poles and the zero at s = −c. Hence, if c ≠ a, a tuning rule cannot be determined such that transient response to a given velocity reference satisfies the desired specifications. Notice that this is also true if the response to a reference is specified by two complex conjugate poles, as the zero at s = −c cannot be cancelled; hence, it has the effect of modifying the transient response with respect to that specified by the two complex conjugate poles.

   • It is more convenient to select c > a because the slowest closed-loop pole (that approaching s = −c) is shifted further to the left (it is faster) compared with the case when c < a is chosen.

   • This also implies that, if it is desired that the velocity deviation due to a disturbance vanishes faster, we must choose c ≠ a with c > 0 larger. According to c = k _i∕k _p, this means that a larger integral gain is obtained.

   Thus, although it is possible to render shorter the time it takes to the velocity deviation due to disturbance to vanish, it is concluded that the controller gains k _p y k _i cannot be exactly computed to ensure that, simultaneously, the desired transient response specifications to a given velocity reference are accomplished and that the disturbance effects vanish as fast as desired. This statement is verified experimentally in Chap. 10 and, because of this, in that chapter a modified PI velocity controller is introduced, solving this situation.

    

Anticipating its experimental use in Chap. 10, a tuning rule is proposed next for the case when c ≠ a. According to (5.20), the magnitude condition is:

where , , and . According to this tuning criterion, propose the slowest pole to be located at some known s = −p ₁, to fix an upper limit on the time required to render negligible the disturbance effect. On the other hand, according to the items listed above, c is proposed to be close to p ₁ such that p ₁ > c. Then, using the magnitude condition above, the following tuning rule is obtained:

(5.25)

According to the discussion above, the response to the reference of velocity is much faster than predicted by the pole at s = −p ₁. For comparison purposes this is very important as some velocity controllers are designed in Chap. 10 accomplishing simultaneously transient responses to a velocity reference and to an external disturbance that are determined by a pole at s = −p ₁. On the contrary, with the PI velocity controller studied in this section, if it is desired that the response to an external disturbance is faster (with a pole at s = −p ₁), the response to a desired reference of velocity must be much faster.

Example 5.4

The MATLAB/Simulink diagram corresponding to Fig. 5.22a is shown in Fig. 5.24. It is assumed that k = 70 and a = 7. The desired velocity ω _d is a step command with 2 as the magnitude. The disturbance T _p is a step applied at t = 2[s] with a magnitude 0.4 × 70. The PI controller block has k _p and k _i as proportional and integral gains.

The simulation in Fig. 5.24, was performed with the help of the following MATLAB code in an m-file:

a=7;

k=70;

Polo_des=-20;

c=15; %1 5 7 10 15

l1=abs(Polo_des)-c;

l2=abs(Polo_des);

l3=abs(Polo_des)-a;

kp=l2∗l3/(k∗l1);

ki=c∗kp;

PI=tf(kp∗[1 c],[1 0]);

gm=tf(k,[1 a]);

M=feedback(PI∗gm,1,-1);

figure(1)

gd=tf(20,[1 20]);

step(M∗2,'b:',gd∗2,'r--',0.2)

hold on

%%{

nn=length(InputOutput(:,2));

n=nn-1;

Ts=10/n;

t=0:Ts:10;

figure(2)

plot(t,InputOutput(:,2),'k:');

axis([-0.5 10 0 2.5])

xlabel('t [s]')

ylabel('w [rad/s]')

hold on

%}

In this code, the desired closed-loop response is specified to be as that of a first-order system with a time constant [s] when a desired step velocity is commanded. This desired response is shown in Fig. 5.25 with a dashed line. The design criterion is to find k _p and k _i such that the desired closed-loop pole is assigned at s = −20, provided that a ratio is given. This allows to the system response to be analyzed when the integral gain is enlarged and one closed-loop pole is at the desired location.

First, the location of zero at is fixed by defining the constant c. The proportional gain k _p is computed using the magnitude condition , where l ₁, l ₂, l ₃, are the magnitudes of vectors defined in the paragraph before (5.25), and k _i = ck _p is also computed. Then, the closed-loop transfer function is obtained and the closed-loop system response as well as the desired response can be simulated using the command “step()”. These signals are shown in Fig. 5.25. At this point, the simulation diagram in Fig. 5.24 is run and, with the help of the remaining code lines above, the corresponding results are shown in Fig. 5.26.

In Figs. 5.25 and 5.26 the following data are presented. (i) The line marked with “+” corresponds to c = 1, (ii) Continuous line: c = 5, (iii) Dashed line: c = 7, (iv) Dash–dot line: c = 10, and (v) Dotted line: c = 15. Notice that the disturbance effect is rejected very slowly when c = 1. In this case, the closed-loop poles are at s = −20 and s = −0.6842, which can be verified using the MATLAB command “pole(M)”. Recall that the transfer function in (5.24) has no zero to cancel the pole at s = −0.6842.² This explains the slow rejection of the disturbance effects in this case. Also notice that the response to the velocity reference is also slow. This is because the effect of the pole at s = −0.6842 is not suitably cancelled by the zero at s = −c = −1.

Notice that the disturbance rejection and the response to the reference improve as c increases. Moreover, the system response exactly matches the desired response when c = 7 = a. However, the disturbance effects are still important in this case. When c > 7 = a, the disturbance rejection further improves, but the response to the reference exhibits overshoot, i.e., it is no longer a first-order response. According to Sect. 8.​1.​2, this behavior appears, and it is unavoidable, when the closed-loop poles are on the left of an open-loop zero. In this respect, notice that the closed-loop poles are at s = −20 and s = −13 when c = 10 and at s = −20 and s = −39 when c = 15.

The simulation results in this example corroborate the system behavior predicted in the above discussion, i.e., the classical PI control has limitations when it is required to satisfy simultaneously a specified response to a reference and a specified performance for disturbance rejection.

5.2.5 Proportional–Integral–Derivative Control of Position

Consider again the DC motor model but, now, assuming the presence of an external disturbance:

together with the following proportional–integral–derivative controller:

where θ _d is the desired position and the constants k _p, k _d and k _i are known as the proportional, derivative, and integral gains respectively. Use of the Laplace transform, yields:

and the corresponding block diagram is shown in Fig. 5.27a. As this is a system with two inputs, the superposition principle (see Sect. 3.​7) can be used to write:

where G ₁(s) is the closed-loop transfer function when θ _d(s) is the input and θ(s) is the output with T _p(s) = 0, i.e., when the block diagram in Fig. 5.27b is used to find:

(5.26)

On the other hand, G ₂(s) is the closed-loop transfer function when T _p(s) is the input and θ(s) is the output with θ _d(s) = 0, i.e., when the block diagram in Fig. 5.27c is employed to write:

(5.27)

It is stressed that when T _p(s) is the input, then θ(s) stands for the position deviation (with respect to θ _d(s)) produced by the external disturbance. Using the final value theorem, it is found that:

the deviation of position produced by the constant external torque disturbance is zero in a steady state. Using the final value theorem again, it is not difficult to verify that, using (5.26), the final value of position θ is equal to the desired value θ _d when this is a constant. These are the main reasons for using a PID position controller.

The problem of choosing the controller gains for the PID controller such that the closed-loop response has the desired transient response specifications when a position reference θ _d is applied is studied in the following. To this aim, the three poles of the transfer function in (5.26) must be assigned at the desired locations on the left half-plane. This is accomplished by equating the characteristic polynomial of the transfer function in (5.26) to a polynomial with roots at the desired points s = p ₁, s = p ₂, s = p ₃:

It is clear that any values can be assigned to the coefficients of the characteristic polynomial by using suitable combinations of the controller gains k _p, k _d and k _i. This means that the three poles of the closed-loop system can be assigned at any desired location of the left half-plane. Any set of three poles satisfying this requirement can be written as:

Notice that σ ₁ = σ ₂ if ω ₁ > 0, i.e., when a pair of complex conjugate poles exists, but σ ₁ ≠ σ ₂ is possible when ω ₁ = 0. Hence:

Equating the coefficients and using the above data, the following tuning rule is obtained:

(5.28)

It is stressed that the three controller gains are positive. As a rule of thumb, if − p ₃ > 6|σ ₁| then the real and imaginary parts of the poles at p ₁ and p ₂ can be computed using (3.​71) such that the desired rise time t _r and overshoot M _p(%) are obtained. However, the system response presents some important differences with respect to these values because of the pair of zeros contained in the transfer function in (5.26). Although this is an important drawback of the tuning rule in (5.28), these values can be used as rough approximations of the required gains for the controller to perform fine adjustments afterward, by trial and error, until the desired transient response specifications are accomplished. To this aim, as the closed-loop system is third order, it is important to recall the stability rule obtained in Example 4.​12 of Sect. 4.​3. The possibility of adjusting the gains of a PID controller by trial and error (see Sect. 5.3) is one of the main reasons for the success of PID control in industry.

With the aim of finding a tuning rule allowing to exactly compute the gains of a PID controller, the root locus method is employed in the following. From the block diagram in Fig. 5.27b, it is concluded that the open-loop transfer function is given as:

(5.29)

for some constants α and β different from zero whose values are proposed as part of the design process. Then, from these values both k _p and k _i are computed. Notice that k _d is the gain that the method varies from 0 to + ∞ to plot the root locus diagram. It is clear that the system type is 2 (the motor is naturally provided with an integrator and another integrator is introduced by PID controller) and, hence, the steady-state error is zero if the reference of the position is constant.

Three possibilities for the corresponding root locus diagram are presented in Fig. 5.28. They depend on the values proposed for α and β. The reader is encouraged to use the rules introduced in Sect. 5.1.1 to corroborate these results. According to Fig. 5.28a, it is possible to design the control system such that two closed-loop poles are close to the zeros located at s = −α and s = −β. This would allow the closed-loop system to respond as a first-order system with one real and negative pole. One closed-loop pole may also be chosen to be close to the zero at s = −α such that the closed-loop system responds as a second-order system with complex conjugate poles and one real zero. Although the presence of this zero modifies the transient response, this is one way of specifying the transient response in terms of rise time and overshoot. This is the design criterion used in the following.

Now, the PD controller k _d(s + β) is designed for the plant , i.e., when the closed-loop system has the form presented in Fig. 5.29 and the open-loop transfer function is:

(5.31)

The closed-loop transfer function is, in this case:

Equating to a standard second-degree polynomial:

yields:

(5.32)

The values for ζ and ω _n can be computed using (3.​71) from the desired values for rise time and overshoot. The corresponding root locus diagram is shown in Fig. 5.30, which is identical to the case c > a depicted in Fig. 5.16, because only in this case can the PD control of the position produce complex conjugate closed-loop poles. From (5.31), it is found that the angle and the magnitude conditions establish:

(5.33)

where l ₁, l ₂, l ₃, θ ₁, θ ₂ and θ ₃ are defined in Fig. 5.30. When the factor is included in the open-loop transfer function shown in (5.31), the resulting open-loop transfer function becomes:

(5.34)

As α > 0 is small, the corresponding root locus has the shape shown in Fig. 5.31. The reader is encouraged to follow the rules in Sect. 5.1.1 to verify this result. The angle and the magnitude conditions are established in this case from (5.34) as:

(5.35)

where l ₁, l ₂, l ₃, θ ₁, θ ₂ and θ ₃ are defined as in Fig. 5.30, whereas l ₅ and θ ₅ are defined in Fig. 5.31. The reason for choosing α > 0 to be close to zero is to render l ₁ and l ₅ almost the same such that l ₅∕l ₁ ≈ 1. This also ensures that θ ₁ and θ ₅ are almost equal; hence, θ ₅ − θ ₁ ≈ 0. Then, the angle and the magnitude conditions in (5.33) and (5.35) are almost identical. This ensures that the closed-loop poles obtained when the open-loop transfer function is that presented in (5.31) are almost identical to the closed-loop poles obtained when the open-loop transfer function employed is that presented in (5.34). This ensures that the desired transient response specifications (rise time and overshoot) designed with the PD controller in (5.32) are also accomplished with the PID controller:

i.e., when the controller gains are chosen according to (5.30).

However, it is important to point out a drawback of this design approach: according to (5.30) a small α results in a small integral gain k _i. As a consequence, the deviation of the position produced by the external disturbance vanishes very slowly, which may be unacceptable in practice. This can also be explained from the root locus diagram in Fig. 5.31. Notice that a closed-loop pole (located, say, at s = −ε, ε > 0) approaches the zero at s = −α, which means that both cancel each other out in the transfer function shown in (5.26), i.e., the effect of none of them is observed in the transient response to a reference of the position. However, the zero at s = −α does not appear in the transfer function shown in (5.27). But the pole at s = −ε is still present in this transfer function; hence, it significantly affects the transient response: this slow pole (close to the origin) is responsible for a slow transient response when an external disturbance appears.

Despite this drawback, the traditional criteria for root locus-based design suggest proceeding as above when designing PID controllers. Moreover, it is interesting to say that these problems in traditional design methods remain without a solution despite the fact that they have been previously pointed out in some research works. See [10] for instance.

On the other hand, according to (5.30), a larger integral gain can be obtained (to achieve a faster disturbance rejection) by choosing a larger value for α. However, according to the previous discussion, this will result in a transient response to a reference of position that does not satisfy the desired specifications. Hence, it is concluded that there is no tuning rule allowing the gains of a PID controller for position to be computed exactly, simultaneously satisfying the desired transient response specifications to a given reference of the position and producing a satisfactory rejection of the effects of an external torque disturbance. These observations are experimentally verified in Chap. 11 where, given the described drawback, some new controllers solving these problems are designed and experimentally tested.

Finally, despite the above drawback, it is important to recall what was indicated just after (5.28): PID control is one of the most frequently employed controllers in industry because it can be tuned by trial and error (see Sect. 5.3). This has to be performed by taking into account the stability rule obtained in the Example 4.​12, Sect. 4.​3.

Example 5.5

The MATLAB/Simulink diagram corresponding to the block diagram in Fig. 5.27a is presented in Fig. 5.32. It is assumed that a = 7 and k = 70. The desired position θ _d is a step command applied at t = 0 with 2 as the magnitude, whereas the disturbance T _p is another step applied at t = 0.5[s] and 700 as the magnitude. The PID controller gains are k _p, k _d, and k _i, computed in the following MATLAB code executed in an m-file:

clc

a=7;

k=70;

tr=0.05;

Mp=14;%

z=sqrt( log(Mp/100)ˆ2 /( log(Mp/100)ˆ2 + piˆ2 ) );

wn=1/(tr∗sqrt(1-zˆ2))∗(pi-atan(sqrt(1-zˆ2)/z));

s=-z∗wn+wn∗sqrt(1-zˆ2)∗j

kd=(2∗z∗wn-a)/k;

beta=wnˆ2/(k∗kd)

den1=conv([1 -s],[1 -real(s)+j∗imag(s)]);

Md=tf(den1(3),den1);

gm=tf(k,[1 a 0]);

PD=tf(kd∗[1 beta],1);

alpha=20;%0.1 1 10 20

PI=tf([1 alpha],[1 0]);

M_pid=feedback(PI∗PD∗gm,1,-1);

pole(M_pid)

figure(2)

step(Md∗2,'r--',M_pid∗2,'k--',0.25)

hold on

kp=(alpha+beta)∗kd;

ki=alpha∗beta∗kd;

%%{

nn=length(InputOutput(:,2));

n=nn-1;

Ts=2/n;

t=0:Ts:2;

figure(3)

plot(t,InputOutput(:,2),'k--');

hold on

plot(t,InputOutput(:,1),'k:');

axis([0 1 0 3])

xlabel('t [s]')

ylabel('theta [rad]')

%}

The main idea is to propose different values for the design parameter α to see how the transient response is affected. Hence, the values for ζ and ω _n are first computed from the desired rise time t _r = 0.05[s] and overshoot M _p = 14%. This allows k _d and β to be computed using (5.32). Then, the desired time response, i.e., that dictated by the desired complex conjugate poles at s = −z ∗ wn ± wn ∗ sqrt(1 − z ²)j (the slowest dashed response in Fig. 5.33), and the actual closed-loop system response to the step reference are plotted on the same axes using the command “step(Md ∗ 2, ′r −−′, M_pid ∗ 2, ′k −−′, 0.25)”. The proportional and integral gains are computed using (5.30); hence, the simulation in Fig. 5.32 can be run. Once this is performed, the remaining commands in the code listed above can be used to draw Fig. 5.34.

Running the above code and the simulation in Fig. 5.32 several times, the responses are obtained when using different values for α. In Figs. 5.33 and 5.34: (i) Continuous lines stand for α = 0.1, (ii) Dash–dot: α = 1, (iii) Dotted: α = 10, and the fastest dashed line: α = 20.

It is observed that, when α = 0.1, the response is different from the desired one, i.e., the slowest dashed line in Fig. 5.33. This is because of the closed-loop zero at − β, located at s = −54.5225 in this case, which deviates the system response from the desired one (See also Fig. 5.20. It is remarked that the desired closed-loop poles are located at , as desired, in this case. Also notice that the disturbance rejection is so slow that it seems that a steady-state deviation might exist.

As α is chosen to be larger, the performance observed in Fig. 5.34 improves with respect to the disturbance rejection, but the price to be paid is that the response to the position reference is increasingly different from the desired one, as observed in Fig. 5.33.

Thus, the simulation results in this example corroborate the predictions stated in the above discussion in the present section.

5.2.6 Assigning the Desired Closed-Loop Poles

An example is presented in this section to show how the root locus method can be used to assign the desired closed-loop poles. Consider the following transfer function:

(5.36)

The gain k is varied by the method to take values from 0 to + ∞. First, G(s)H(s) is rewritten as:

where the vectors and have been defined. The fundamental conditions in the root locus method are the angle condition (5.7) and the magnitude condition (5.6), which are expressed respectively as:

Rule 5 indicates that the root locus on the real axis only exists between the points s = 35.7377 and s = −36.5040. Moreover, according to rule 1, the root locus begins (k = 0) at s = 35.7377 and s = −36.5040. On the other hand, according to rules 2 and 3, when k tends to + ∞ the branches starting at s = 35.7377 and s = −36.5040 must end at some open-loop zero (there is no zero in this case) or at some point at infinity on the s plane. Hence, the root locus must move away from the points s = 35.7377 and s = −36.5040 on the real axis as k increases and a breakaway point must exist somewhere between such points. Then, the branches must move toward infinity on the s plane.

On the other hand, according to the angle condition, to − (θ ₁ + θ ₂) = −180^∘ = −(α + θ ₁), and to Fig. 5.35, it can be observed that α = θ ₂; hence, the triangles t ₁ and t ₂ must be identical for any closed-loop pole s. Then, both branches, which are also shown in Fig. 5.35, must be parallel to the imaginary axis. This means that the breakaway point on the real axis is located at the middle point between s = 35.7377 and s = −36.5040, i.e., at (35.7377 − 36.5040)∕2 = −0.7663. This can also be verified using rules 3 and 4. Finally, according to rule 6, both branches are symmetrical with respect to the real axis.

The magnitude condition is employed when it is required to know the exact value of k rendering a specific point on the root locus an actual closed-loop pole. Notice, for instance, that both l ₁ and l ₂ must increase to satisfy the magnitude condition as k grows to + ∞:

which means that the closed-loop poles corresponding to large values of k tend toward some points at infinity on the s plane.

Now suppose that an additional pole is included in the transfer function in (5.36) to obtain:

(5.37)

Again, k is the gain that the method varies from 0 to + ∞ to plot the root locus diagram. First, G(s)H(s) is rewritten as:

where the vectors , , and have been defined. The angle condition (5.7) and the magnitude condition (5.6) are expressed respectively as:

The root locus diagram corresponding to this case can be obtained from the root locus diagram obtained for the transfer function in (5.36) by simply taking into account the additional pole at s = −71.7721.

According to the rule 11, the new pole is responsible for both branches of the root locus shown in Fig. 5.35 bending to the right, as can be observed in Fig. 5.36. This can be corroborated using rule 3 to find that, now, three branches exist, which tend toward the asymptotes at ± 180^∘ and ± 60^∘ with respect to the positive real axis.

On the other hand, the breakaway point between the points s = 35.7377 and s = −36.5040, which was located at (35.7377 − 36.5040)∕2 = −0.7663 in Fig. 5.35, is now shown in Fig. 5.36 shifted to the right of − 0.7663. The reason for this is explained in Fig. 5.37 where s′ and s stand for the points on the root locus which are very close to the breakaway point for the case of Figs. 5.35 and 5.36 respectively. As − (θ ₁ + θ ₂ + θ ₃) = −180^∘ must be satisfied in Fig. 5.36, whereas must be satisfied in Fig. 5.35, then both θ ₁ and θ ₂ must be smaller than and . This means that the point s must be shifted to the right of the point s′. Notice that this displacement of the breakaway point to the right of the point − 0.7663 is greater as the angle θ ₃ introduced by the pole at s = −71.7721 is greater, i.e., as this pole is placed farther to the right. In fact, it is observed in Fig. 5.36 that the breakaway point is located on the right half-plane.

This description of the root locus diagram in Fig. 5.36 allows us to conclude that there is always at least one closed-loop pole with a positive real part, i.e., the closed-loop system is unstable for any positive value of k. As k can be interpreted as the gain of a proportional controller, it is concluded that it is not possible to render the closed-loop system stable using any proportional controller and, thus, another controller must be proposed. According to rule 12 it is required to use a controller introducing an open-loop zero because this bends the root locus to the left accomplishing closed-loop stability.

Consider the following open-loop transfer function:

(5.38)

with b a positive constant. Again, k is the gain that the method varies from 0 to + ∞ to draw the root locus diagram. First, G(s)H(s) is rewritten as:

(5.39)

where the vectors , , , and have been defined. The angle condition (5.7) and the magnitude condition (5.6) are expressed respectively as:

(5.40)

(5.41)

Notice that, now, n = 3, m = 1, p ₁ = 35.7377, p ₂ = −36.5040, p ₃ = −71.7721, and z ₁ = −b. According to the rules 1, 2, and 3, one of the three branches starting (k = 0) at the points s = 35.7377, s = −36.5040, and s = −71.7721 (the open-loop poles) must end (k = +∞) at the open-loop zero s = −b. Hence, there are two root locus branches tending toward some points at infinity of the s plane. According to rule 3, the angles of the asymptotes toward which these branches tend are:

Moreover, according to rule 4, the point where these asymptotes cross the real axis can be shifted by using the zero at s = −b:

(5.42)

Notice that b must be positive, as, if this is not the case, the root locus would be pushed to the right half-plane producing unstable closed-loop poles. Also notice that the asymptotes move to the left (increasing stability) as the zero is located closer to the origin, i.e., as b tends toward zero. This means that the closed-loop poles cannot be placed arbitrarily to the left.

According to rule 5, the root locus now exists on two segments of the real axis. The locations of these segments depend on the value of b as shown in Figs. 5.38a and b. Furthermore, it is shown in Fig. 5.38c that in the case when b = 36.5040 is selected, the open-loop zero at s = −b cancels with the open-loop pole at s = −36.5040. This means that, in such a case, the root locus only exists in one segment on the real axis.

It is observed, in Fig. 5.38c, that only two closed-loop poles exist. It is observed, in Figs. 5.38a and b, that three closed-loop poles exist and one of them is on the right half-plane (closed-loop instability) if the gain k is too small. On the other hand, if k is too large, two complex conjugate poles exist with too large an imaginary part, i.e., producing a fast oscillation. Although in theory this can work because the three poles have negative real parts, a large k produces a number of practical problems such as noise amplification and power amplifier saturation, i.e., the control system may not work correctly in practice. Hence, a good design is that enabling the designer to locate as desired the closed-loop poles avoiding the use of values that are too large and values that are too small for the gain k.

Suppose that s = −25 ± j40 are the desired closed-loop complex conjugate poles. A procedure that is useful for computing the exact values required for both b and k is presented in the following. According to Fig. 5.39 the following angles are computed:

Using the angle condition (5.40), θ ₄ is computed and, also, b:

On the other hand, according to Fig. 5.39, the following lengths are computed:

and, finally, the magnitude condition (5.41) is employed to compute k:

The location of the third closed-loop pole can be found from the condition 1 + G(s)H(s) = 0 using G(s)H(s) given in (5.38) and b = 31.2463 and k = 0.0396, i.e.,

(5.43)

hence:

The roots of this polynomial are the closed-loop poles when b = 31.2463 and k = 0.0396. Using MATLAB, it is found that these roots are − 25.0037 + j39.9630, − 25.0037 − j39.9630 and − 22.5326. These poles are also shown in Fig. 5.38a using symbols “+ .” Notice that the desired complex conjugate poles have been successfully assigned and the third pole, which is real, is also located on the left half-plane, i.e., closed-loop stability has been accomplished. Recall that the factor k(s + b) represents a PD controller.

It is important to say that this control system is employed to regulate the output at a zero value. This implies that no matter what the system type, the desired output (i.e., zero) is reached. Hence, consideration with respect to the steady-state response is not necessary.

The closed-loop response (continuous line) is shown in Fig. 5.40 when the desired output is zero, r = 0 (see Fig. 5.1). In this simulation (5.38), b = 31.2463 and k = 0.0396 are employed. The dashed line represents the response of a system with the transfer function whose poles are located at s = −25 ± j40. Thus, this system represents what is known as a reference model, because its response possesses the desired transient response specifications. Both responses start from an initial output equal to 2.85. Notice that some differences exist between these responses, which are due to the zero at s = −31.2463 and the closed-loop pole at s = −22.5326, because they are not close enough to completely cancel out their effects.

5.2.7 Proportional–Integral–Derivative Control of an Unstable Plant

In this section, the use of the root locus method to select the gains of a PID controller for an open-loop unstable plant is presented. The open-loop transfer function is:

(5.44)

where k _d is the parameter that the method varies from 0 to + ∞ to draw the root locus, whereas and must be proposed. Notice that a PID controller is selected because it renders the system type 1, i.e., to ensure a zero steady-state error when the desired output is a constant. Notice that the PID controller introduces two open-loop zeros. There are four open-loop poles placed at:

Using rule 3 it is found that there are n − m = 4 − 2 = 2 branches of the root locus that tend toward infinity on the plane s following the asymptotes whose angles are given as:

Moreover, according to rule 4, the point on the real axis where these asymptotes intersect is:

where − σ _z1 < 0 and − σ _z2 < 0 are the real parts of both open-loop zeros introduced by the PID controller (these values have to be proposed). It is clear that σ _a moves to the left (which implies that the closed-loop system becomes more stable) if − σ _z1 < 0 and − σ _z2 < 0 are chosen to be close to zero. Notice that this is in agreement with rule 12. Thus, it is concluded that σ _a < −1100 is placed far to the left of the origin. Using these observations, in addition to rules 1, 2, 5, and 6, it is found that the root locus has the three possibilities depicted in Fig. 5.41.

Instead of assigning some desired closed-loop poles, the control objective is simply to render the closed-loop system stable. As previously stated, it is preferable to place both open-loop zeros close to the origin. This means that the possibility shown in Fig. 5.41a is to be used. On the other hand, two complex conjugate zeros would force two complex conjugate closed-loop poles to appear, which would result in a less damped closed-loop system. It is for this reason that the possibility in Fig. 5.41c is also avoided. Hence, searching for a root locus diagram such as that in Fig. 5.41a, the following values are proposed:

(5.45)

as this assigns two open-loop zeros at:

i.e., they are located between the open-loop poles at:

The facts that σ _a < −1100 is located far to the left of the origin and that the asymptotes form ± 90^∘ angles, ensure that a minimal k _d exists for all closed-loop poles to be placed on the left half-plane. Using (5.45) and Routh’s criterion, it is found that:

(5.46)

ensures that all of the closed-loop poles are located on the left half-plane and, hence, the closed-loop system is stable. This is performed as follows. From the condition 1 + G(s)H(s) = 0, i.e., using (5.44) and (5.45), the following characteristic polynomial is obtained:

(5.47)

To apply Routh’s criterion, the Table 5.1 is filled. Closed-loop stability is ensured if no sign changes exist in the first column of Table 5.1, i.e., if:

(5.48)

Table 5.1

Applying Routh’s criterion to the polynomial in (5.47)

s 4	1	a 2	a 0
s 3	a 3	a 1	0
s 2		a 0	0
s 1		0
s 0	a 0

Notice that the third condition in (5.48) is naturally satisfied and from the first and the last conditions it is found that:

(5.49)

From the second condition in (5.48) the following is found:

(5.50)

It is not difficult to find that the roots of the second-degree polynomial in (5.50) are k _d = 0.01 and k _d = −3.5 × 10⁻⁶. Moreover, it is possible to evaluate numerically to find that:

Thus, to simultaneously satisfy (5.49) and (5.50), i.e., to ensure closed-loop stability, (5.46) must be chosen.

Example 5.6

The root locus diagram for G(s)H(s) given in (5.44) under the conditions in (5.45) is shown in Fig. 5.42. This has been obtained using the following MATLAB code in an m-file.

clc

clear

gh=tf(11613700∗[1 31.24 31.24/0.8],[1 2739 -1250

 -3.536e6 0]);

k=0:0.00001:0.1;

figure(1)

rlocus(gh,k)

axis([-70 50 -10 10])

rlocfind(gh)

%%{

M=feedback(0.0226∗gh,1,-1)

A=[0 1 0 0;

0 0 1 0;

0 0 0 1;

-1.025e007 -4.664e006 -2.612e005 -2739];

B=[0;

0;

0;

1];

C=[1.025e007 8.2e006 2.625e005 0];

Me=ss(A,B,C,0);

figure(2)

initial(Me,'b-',[1/1.025e007 0 0 0])

axis([-0.1 3 -0.8 1.2])

%}

Using the command “rlocfind(gh)”, the closed-loop poles on the imaginary axis have been selected to find that k _d = 0.01, which corroborates the above findings, i.e., see (5.46). Using the command “pole(M),” all the closed-loop poles were found to be real, located at s = −2640, s = −75.6, s = −20.1, and s = −2.6 when k _d = 0.0226, i.e., the closed-loop system is stable. Notice that the open-loop zeros are located at s = −29.93 and s = −1.3, which was found using the command “zero(gh).” This means that all the closed-loop poles are located on the left of the open-loop zero at s = −1.3. Hence, according to Sect. 8.​1.​2, this means that overshoot is unavoidable despite the fact that all the closed-loop poles are real. This explains the large overshoot observed in Fig. 5.43, when a zero reference is commanded and the initial output is 1. Thus, in this example, it is not important to try to assign the closed-loop poles at some specific locations.

5.2.8 Control of a Ball and Beam System

In this section, a controller is designed for the ball and beam system that is built and experimentally tested in Chap. 14, Sect. 14.​9. Assume that a proportional controller is proposed with gain γ. First, one proceeds to study the possibility that the closed-loop system might be rendered stable for some positive gain γ. The other system parameters are positive too, but they cannot be changed. The corresponding block diagram is shown in Fig. 5.44. The open-loop transfer function is:

The stability of the closed-loop system is studied using Routh’s criterion. Thus, Table 5.2 is filled using the closed-loop characteristic polynomial s ⁴ + as ³ + γA _x kρ. Notice that one entry at the first column is zero; hence, the method suggests replacing it by a small ε > 0. Also notice that under this condition, there are two sign changes in the first column of Table 5.2 and this cannot be modified by adjusting γ > 0. Then, it is concluded that no proportional controller exists to render the closed-loop system stable. Thus, another controller must be proposed.

Table 5.2

Applying Routh’s criterion to the polynomial s ⁴ + as ³ + γA _x kρ

s 4	1	0	γA x kρ
s 3	a	0	0
s 2	0 ≈ ε	γA x kρ
s 1		0
s 0	γA x kρ

According to rule 12 in Sect. 5.1.1, a PD controller can render closed-loop stable system an open-loop unstable system because a PD controller introduces an open-loop zero. However, a PD controller amplifies noise because of the derivative action. This is clearly shown in Chap. 6 where a PD controller is proved to be a high-pass filter and noise is a high-frequency signal. One way of maintaining the stabilizing properties of a PD controller, but reducing the effects of noise is to use a lead compensator , i.e., a controller with the transfer function with c > δ > 0. It is for this reason that we study the possibility of stabilizing the ball and beam system using the block diagram in Fig. 5.45 in the following. The closed-loop transfer function is:

The closed-loop stability is studied using Routh’s criterion. Hence, the Table 5.3 is constructed using the closed-loop characteristic polynomial s ⁵ + (a + c)s ⁴ + acs ³ + γA _x kρs + γA _x kρδ. As all the parameters are positive, there are at least two sign changes in the first column of Table 5.3. Then, it is concluded that there is no lead compensator rendering the closed-loop system stable; thus, another control strategy must be proposed. Carefully analyzing Table 5.3, it is concluded that the problem is originated by the first entry corresponding to row s ², i.e., , which is negative. Notice that this is a consequence of the fact that the second entry in the row corresponding to s ⁴ is equal to zero. This is because s ² has a zero coefficient in the polynomial s ⁵ + (a + c)s ⁴ + acs ³ + γA _x kρs + γA _x kρδ. Thus, it is concluded that the closed-loop system may be rendered stable if the characteristic polynomial of the open-loop transfer function has a positive coefficient for s ². Next, it is shown that this is possible if two internal loops are employed, as shown in Fig. 5.46. Notice that a lead compensator is still considered. In this case, the open-loop transfer function is given as:

Table 5.3

Applying Routh’s criterion to polynomial s ⁵ + (a + c)s ⁴ + acs ³ + γA _x kρs + γA _x kρδ

s 5	1	ac	γA x kρ
s 4	a + c	0	γA x kρδ
s 3	ac
s 2		γA x kρδ
s 1		0
s 0	γA x kρδ

(5.51)

Notice that the system type is 2, i.e., the steady state error is zero if the reference is a step or a ramp. The reader may obtain the closed-loop transfer function to verify that the corresponding characteristic polynomial has degree 5 and all of its coefficients are positive (including the coefficient of s ²), as expected. According to the previous discussion, this means that there exists the possibility of finding positive values for γ, c, b (with c > b), k _v and α such that the closed-loop system is stable. These values are found in the following using the root locus method.

Suppose that the roots of the polynomial s ² + (a + k _v kA _θ)s + αkA _θ are complex conjugate, i.e., they can be written:

According to rule 3, there are n − m = 4 root locus branches tending toward infinity on the plane s along four asymptotes whose angles are given as:

Hence, using these results in addition to rules 1,2 and 5, it is concluded that the possibilities depicted in Fig. 5.50 exist. According to rule 4, the point where these asymptotes intersect with the real axis is given as:

This means that the four branches are pulled to the left; hence, the closed-loop system becomes more stable, if:

• A greater σ > 0 is chosen, i.e., if the system:

  

  (5.52)

  is damped enough, which is accomplished using a large enough value for k _v > 0.

• b > 0 Approaches to zero and c > 0 are large, i.e., if c > b. Note that this is in agreement with rules 11 and 12.

Following the second item above, the root locus shown in Fig. 5.47b is obtained. Notice that both branches starting at the poles located at s = 0 always remain on the right half-plane, i.e., the closed-loop system is unstable. The main reason for this behavior is that branches starting at the poles of the transfer function in (5.52) are pulled to the segment on the real axis between s = −c and s = −b. This is because both open-loop complex conjugate poles are too close to the segment between s = −c and s = −b. If these complex poles are placed far from such a segment, then the possibility exists for branches starting at s = 0 to be pulled to the segment between s = −c and s = −b to obtain the root locus diagram shown in Fig. 5.47c. This implies closed-loop stability for some of the small values of the loop gain. To achieve this, it is necessary to use large values for the coefficient αkA _θ (because this increases the distance to the origin of the above-mentioned open-loop complex conjugate poles), i.e., using a large value for α.

According to the previous discussion, large values must be used for k _v and α. However, these values are limited in practice by the system noise content; hence, k _v and α must be obtained using experimental tests. In fact, noise is also an important reason not to select real poles for the transfer function in (5.52): real poles produce a very damped system that requires a large value for k _v. According to the experimental tests reported in Chap. 14 the following values were selected:

Hence, the only values that remain to be determined are γ, c and b. Although these parameters can be determined such that the closed-loop poles are assigned at the desired locations (using the methodology presented in Sect. 5.2.6), in this case, it is preferred just to select them such that closed-loop stability is accomplished. To achieve this, we proceed as follows.

• b and c are proposed such that c > b > 0.

• MATLAB is employed to draw the root locus diagram.

• γ > 0 is chosen as that value placing all the closed-loop system poles on the left half-plane. This means that γ is the parameter used by the method to plot the root locus diagram.

• If such a γ > 0 does not exist, then we go back to the first step.

The root locus diagram shown in Fig. 5.48 is obtained following this procedure. It is observed that all the closed-loop poles, i.e., those indicated with a symbol “+ ,” have a negative real part if:

The simulation response of the closed-loop system when the reference is an unitary step is shown in Fig. 5.49. The graphical results in Figs. 5.48 and 5.49 were obtained by executing the following MATLAB code in an m-file:

5.2.9 Assigning the Desired Closed-Loop Poles for a Ball and Beam System

Consider again the control problem in the previous section, but now using the following system parameters:

(5.53)

Suppose³ that the transient response specifications are stated by requiring a rise time t _r = 1[s] and an overshoot M _p = 25%. The use of this information and:

allows us to find that a complex closed-loop pole must be located at:

(5.54)

According to the closed-loop block diagram shown in Fig. 5.46, the open-loop transfer function in (5.51) becomes:

(5.55)

Using these numerical values, it is found that the open-loop poles are located at:

Consider Fig. 5.50a. Proceed as in Sect. 5.2.6. The angle and the magnitude conditions establish that:

From these expressions, the following is found:

(5.56)

(5.57)

Also notice that:

(5.58)

(5.59)

(5.60)

(5.61)

From the second expression, the following is found:

(5.62)

The following design procedure is proposed:

1. 1.

   Compute θ ₁, θ ₂, θ ₅, θ ₆, using (5.59), (5.60), (5.61), and the difference θ ₃ − θ ₄ using (5.56).

    
2. 2.

   Propose some b and compute θ ₃ using (5.58). Using this and the value of the difference θ ₃ − θ ₄, compute θ ₄.

    
3. 3.

   Compute c and γ using (5.62) and (5.57) respectively.

    

Proposing b = 0.7, the above procedure yields c = 2.8658 and γ = 1.3531. The corresponding root locus diagram is depicted in Fig. 5.51 where the closed-loop poles are marked using the symbol “+ ”. Notice that the desired closed-loop pole at − 0.8765 + j1.9864 has been successfully assigned. In Fig. 5.52, we present the time response of the closed-loop system (continuous) when a unit step reference is applied. We also present, with a dashed line, the time response of the system:

(5.63)

where the values for ω _n and ζ ensure that the poles are identical to those defined in (5.54), i.e., the dashed line represents the desired response. An important difference between these time responses is observed. In particular, overshoot is almost twice its desired value. This difference may be explained by the fact that the closed-loop transfer function of the system in Fig. 5.46 has a zero⁴ at s = −b. This feature deviates the time response from its desired behavior represented by the dashed line, i.e., the second-order system without any zero defined in (5.63).

Thus, to solve this problem it is proposed to use the block diagram in Fig. 5.53. Notice that the only difference with respect to the block diagram in Fig. 5.46 is that the lead-compensator is now placed in the feedback path. The reason for this is that the resulting closed-loop transfer function now has no zero.⁵ Notice that the root locus diagram in this case is identical to that in Fig. 5.51 as the open-loop transfer function in this case G(s)H(s) is identical to that in (5.55) because b = 0.7, c = 2.8658, γ = 1.3531 again. Recall that the zero at s = −b is no longer closed-loop zero. The closed-loop time response for the block diagram in Fig. 5.53 is presented in Fig. 5.54 with a continuous line, whereas the desired response is represented by the dashed line. Notice that, again, an significant difference exists between the achieved response and the desired response. Analyzing the root locus diagram in Fig. 5.51, it is observed that a closed-loop real pole at ≈−0.95 is very close to the poles at − 0.8765 ± j1.9864, which determine the transient response. Hence, the effects of this real pole are not negligible, and they are important in the transient response: the closed-loop system behaves as a third-order system instead of a second-order system. This fact explains the lag observed in the response achieved in Fig. 5.54.

Hence, a remedy for this problem is to place this real pole further to the left. This may be accomplished by placing the zero at s = −b further to the left as depicted in Fig. 5.50b. Notice that in this case:

(5.64)

Thus, the design procedure proposed above is still valid, only requiring use of (5.64) instead of (5.58). Hence, proposing b = 1.4, the following is found: c = 5.1138 and γ = 2.1868. The root locus diagram for this case is shown in Fig. 5.55 where the closed-loop poles are marked with the symbol “+ .” It can be seen that the real pole is now placed at about − 3.5, far enough from the poles in (5.54). Moreover, the time response achieved in this case (continuous) is shown in Fig. 5.56 and compared with the desired response (dashed). We realize that both responses are very similar in this case, which corroborates the correctness of the design. This controller is tested experimentally in Chap. 14.

Finally, to further improve the transient response, in Figs. 5.57 and 5.58, the root locus diagram and the time response respectively are shown when using the controller gains b = 2, c = 11.7121 and γ = 4.6336. Note that the real pole is now placed at about − 10.5, i.e., it is far from the poles in (5.54). This results in a very small effect of this pole on the transient response, which, as shown in Fig. 5.58, is now very close to the desired response.