Chapter 14

# 14.6 Cauchy’s Mapping Theorem

Let’s summarize the above cases. Consider an arbitrarily closed contour [latex]\sigma[/latex] in the s-plane, traversed clockwise (CW), so that it does not go through any singularities of [latex]F(s)[/latex]. Let [latex]Z[/latex] be a number of zeros of [latex]F(s)[/latex] inside the [latex]\sigma[/latex] -contour, and [latex]P[/latex] be a number of poles of [latex]F(s)[/latex] inside the [latex]\sigma[/latex]-contour. Mapping the [latex]\sigma[/latex]-contour into the [latex]F(s)[/latex] plane will result in a closed [latex]\Gamma[/latex]-contour. Let [latex]N[/latex] be a number of clockwise (CW) encirclements of the resulting [latex]\Gamma[/latex]-contour around the origin of the [latex]F(s)[/latex]-plane. The total number of encirclements of the origin of [latex]F(s)[/latex]-plane through the above mapping is equal to:

[latex]N=Z-P[/latex] | Equation 14-8 |

**14.6.1 How Does Cauchy’s Mapping Theorem Apply to a Control System Stability?**

Consider a closed loop system:

The characteristic equation is:

[latex]1+ G(s)H(s)=0[/latex]

[latex]G(s)H(s)= \frac{N_{open}(s)}{D_{open}(s)}[/latex]

[latex]1+ \frac{N_{open}(s)}{D_{open}(s)}=0[/latex]

[latex]\frac{D_{open}(s)+N_{open}(s)}{D_{open}(s)}=0[/latex]

[latex]N_{char}(s)=D_{open}(s)+N_{open}(s)[/latex]

[latex]D_{char}(s)=D_{open}(s)[/latex]

Define a map (function) [latex]F(s)[/latex] such that it is described by a characteristic equation of the closed loop:

[latex]F(s)= \frac{N_{char}(S)}{D_{char}(s)}[/latex] | Equation 14-9 |

Note that the roots of the numerator of the map [latex]F(s)[/latex] are equivalent to closed loop poles and that the roots of the denominator of the map [latex]F(s)[/latex] are equivalent to open loop poles. Now consider taking a [latex]\sigma[/latex]-contour such that it encompasses all of the RHP, as shown:

Typically, the open loop pole locations are known, i.e. *P* is a known number of unstable open loop poles – we can count how many open loop poles are within this contour. The closed loop pole locations are unknown, i.e. *Z* is what we want to find out. However, from Cauchy’s Theorem:

[latex]Z=N+P[/latex]

The question then is, how to find *N*? If we can perform the mapping into [latex]F(s)[/latex]-plane, *N * can be simply counted. While the mapping into [latex]F(s)[/latex] (closed loop characteristic equation) is not simple, mapping into [latex]G(s)H(s)[/latex] is very simple – we will use a polar plot to do that. Note that:

[latex]F(s)= 1+ \frac{N_{open}(s)}{D_{open}(s)}=1+ G(s)H(s)[/latex]

Map [latex]F(s)[/latex] can then be obtained from the map [latex]G(s)H(s)[/latex] by a linear translation by [latex]-1[/latex]. Map [latex]G(s)H(s)[/latex] is easily obtained through frequency response. So, rather than watching the number *N * of CW encirclements of the origin of [latex]F(s)[/latex] plane, we will be watching the number *N* of CW encirclements of the (-1,j0) point in the [latex]G(s)H(s)[/latex]-plane.

Remember that *Z* represents the total number of zeros of the [latex]F(s)[/latex] map inside the chosen [latex]\sigma[/latex]-contour in the s-plane, i.e. in the RHP (unstable region). Since the map [latex]F(s)[/latex] was defined for the closed loop characteristic equation, its zeros represent **closed loop poles of the control system**. ** Z then represents the total number of unstable closed loop poles of the system**.

Since [latex]Z = N + P[/latex], **for the system to be stable, Z must be equal to zero**, i.e. *N + P = 0*, or *N = -P. *

We can then define the following stability criterion:

Take a closed [latex]\sigma[/latex] -contour in the s-plane in a CW direction so that it encompasses all of RHP. Obtain the Nyquist contour in the [latex]G(s)H(s)[/latex] plane through mapping (utilize frequency response – polar plots – to do so).

For stability, the Nyquist contour for the closed loop control system with *P* unstable open loop poles must encircle the (-1,j0) point in [latex]G(s)H(s)[/latex]-plane *P* times in CCW direction.

Typically, we are not interested in the **absolute system stability** (i.e. is the system stable or not?), but in the **relative system stability** (i.e. what is the range of gains [latex]K[/latex] for which this system is stable?):