Page 101 - Mechatronics with Experiments

CLOSED LOOP CONTROL 87
subplot(3,3,7) ; nichols(sys6); hold on; nichols(sys7); hold on;
nichols(sys8); hold on; grid off;
subplot(3,3,8) ; nichols(sys9); hold on; nichols(sys10); hold on;
nichols(sys11); hold on; grid off;

2.9.2 Stability Analysis in the Frequency Domain:
Nyquist Stability Criteria
Frequency domain methods are the most commonly used control system design methods in
practice. Therefore, it is important to be able to evaluate the stability of a dynamic system
in the frequency domain.
The stability of a linear time invariant dynamic system can be determined in the
frequency domain using the Nyquist stability criteria. Furthermore, relative stability can
be quantified (if stable, how far is the system from being unstable, or if unstable how far is
the system from being stable) using the gain margin and phase margin measures.
Consider the feedback control system shown in the Figure 2.17. The question is “how
can we determine if the closed loop system is stable” using frequency domain methods.

CLS poles : 1 + G(s)H(s) = 0 (2.90)
Are there any roots of this equation on the right-half of the s-plane?
The Nyquist stability criteria answers that question by using the frequency response
data of the loop transfer function, G(s)H(s)or G(s) if the sensor dynamics is included
in the loop transfer function. In the s-domain, we can find the roots of this equation. If
any roots are on the right-half s-plane, then the CLS is unstable. However, we would like
to determine if the CLS has poles on the RHP using frequency domain methods without
explicitly solving for the roots of the closed loop characteristic equation.
The Nyquist stability criteria is derived as a special case of the mapping theorem of
complex variables. Consider the mapping shown in Figure 2.33a. A contour C from the
1
s-plane is mapped to F(s)-plane by the function F(s). The closed contour C will be mapped
1
′
to another closed contour C on the F(s)-plane. The important point to note in this mapping
1
is the number of poles and zeros of the mapping function F(s) inside the C contour and
1
its relationship to the number of encirclements of the origin in the F(s)-plane by the C ′
1
contour. Notice that as a phasor from a zero inside C traverses clockwise (CW) over the
1
′
C ,the C will encircle the origin in the CW direction (Figure 2.33b). Similarly, as a phasor
1
1
′
from a pole inside C traverses clockwise (CW) over the C ,the C will encircle the origin
1
1
1
in the counter clockwise (CCW) direction (Figure 2.33c). If there are no poles or zeros of
′
F(s) inside the contour C , then the mapped contour C does not encircle the origin in the
1
1
′
F(s)-plane. If there are two poles inside the contour C , then the C contour encircles the
1 1
origin CCW two times (Figure 2.33d). If there are two zeros inside the contour C , then
1
′
the C contour encircles the origin CW two times.
1
In summary, as the mapping F(s) traverses for s variable values over the C contour
1
′
in CW direction, the corresponding contour in F(s)-plane, C , will encircle the origin in
1
the CW direction based on the following relationship:
N = Z − P (2.91)
where,
′
N is the number of CW encirclements of the origin by C ,
1
Z is the number of zeros of F(s)inside C ,
1
P is the number of poles of F(s)inside C .
1
Notice that encirclements of origin in CW direction are counted as positive, and the
encirclements of the origin in the CCW direction are counted as negative (Figure 2.33a–d).