Lecture 16 - Root Locus Based Controller Design Using MATLAB - Electrical Engineering (EE) PDF Download

Clear all your Electrical Engineering (EE) doubts with EduRev Join for Free

Join for Free

Lecture 16 - Root Locus Based Controller Design Using MATLAB, Control Systems

1 Root Locus Based Controller Design Using MATLAB 

In this lecture we will show how the MATLAB platform can be utilized to design a controller using root locus technique.
Consider the closed loop discrete control system as shown in Figure 1. Design a digital controller such that the closed loop system has zero steady state error to step input with a reasonable dynamic performance. Velocity error constant of the system should at least be 5.

Figure 1: A discrete time control system

G_p(s) = T = 0.1 sec

G_h0G_p(z) 

The MATLAB script to find out G_h0G_p(z) is as follows.

>> s=tf(’s’);

>> Gp=10/((s+1)*(s+2));

>> GhGp=c2d(Gp,0.1,’zoh’);

Figure 2: Root locus of the uncompensated system

The root locus of the uncompensated system (without controller) is shown in Figure 2 for which the MATLAB command is

>> rlocus(GhG_p)

Figure 3: Pole zero map of the uncompensated system

Pole zero map of the uncompensated system is shown in Figure 3 which can be generated using the MATLAB command

>> pzplot(GhG_p)

One of the design criteria is that the closed loop system should have a zero steady state error for unit step input. Thus a PI controller is required which has the following transfer function in z-domain when backward rectangular integration is used.

The parameter K_i can be designed using the velocity error constant requirement.

Above condition will be satisfied if K_i ≥ 1. Let us take K_i = 1. With K_i = 1, the characteristic equation becomes

(z − 1)(z − 0.9048)(z − 0.8187) + 0.004528(z + 0.9048) + 0.04528K_p(z − 1)(z + 0.9048) = 0

Now, we can plot the root locus of the compensated system with K_p as the variable parameter.

The MATLAB script to plot the root locus is as follows.

>> z=tf(’z’,0.1);
>> Gcomp=0.04528*(z-1)*(z+0.9048)/(z^3 - 2.724*z^2 + 2.469*z - 0.7367);
>> zero(Gcomp);
>> pole(Gcomp);
>> rlocus(Gcomp)

The zeros of the system are 1 and −0.9048 and the poles of the system are 1.0114 ± 0.1663i and 0.7013 respectively. The root locus plot is shown in Figure 4.

Figure 4: Root locus of the system with PI controller

Figure 5: Root locus of the system with PI controller

It is clear from the figure that the system is stable for a very small range of K_p. The stable portion of the root locus is zoomed in Figure 5. The figure shows that the stable range of K_p is 0.239 < K_p < 6.31. The best achievable overshoot is 45.5%, for K_p = 1, which is very high for any practical system. To improve the relative stability, we need to introduce D action. Let us modify the controller to a PID controller for which the transfer function in z-domain is given as below.

To satisfy velocity error constant, K_i ≥ 1. If we assume 15% overshoot (corresponding to ξ ≌ 0.5) and 2 sec settling time (corresponding to ω_n ≌ 4), the desired dominant poles can be calculated as,

Thus the closed loop poles in z-plane

The pole zero map including the poles of the PID controller is shown in Figure 6 where the red cross denotes the desired poles.

Figure 6: Pole zero map including poles of the PID controller

Let us denote the angle contribution starting from the zero to the right most pole as θ₁, θ₂, θ₃, θ₄ and θ₅ respectively. The angles can be calculated as θ₁ = 9.5^o, θ₂ = 20^o, θ₃ = 99.9^o, θ₄ = 115.7^o and θ₅ = 129.4^o.

Net angle contribution is A = 9.5^o − 20^o − 99.9^o − 115.7^o − 129.4^o = −355.5^o. Angle deficiency is −355.5^o + 180^o = −175.5^o

Thus the two zeros of PID controller must provide an angle of 175.5^o. Let us place the two zeros at the same location, z_pid.

Since the required angle by individual zero is 87.75^o, we can easily say that the zeros must lie on the left of the desired closed loop pole

The controller is then written as GD (z) =   The root locus of the compensated system (with PID controller) is shown in Figure 7. This figure shows that the desired closed loop pole corresponds to K = 4.33.

Thus the required controller is GD (z) =  If we compare the above

Figure 7: Root locus of compensated system

transfer function with the general PID controller, K_p and K_d can be computed as follows.

K_d/T = 0.5761 ∗ 4.33 ⇒ K_d = 0.2495
K_p + K_d/T = 4.33 ⇒ K_p = 1.835
K_iT − K_p − 2K_d/T = −1.518 ∗ 4.33 ⇒ K_i = 2.521

Note that the above Ki satisfies the constraint K_i ≥ 1. One should keep in mind that the design is based on second order dominant pole pair approximation. But, in practice, there will be other poles and zeros of the closed loop system which might not be insignificant compared to the desired poles. Thus the actual overshoot of the system may differ from the designed one.