{"id":2135,"date":"2019-12-16T21:40:56","date_gmt":"2019-12-16T21:40:56","guid":{"rendered":"https:\/\/pressbooks.library.ryerson.ca\/controlsystems\/?post_type=chapter&p=2135"},"modified":"2021-01-14T16:34:18","modified_gmt":"2021-01-14T16:34:18","slug":"14-6-cauchys-mapping-theorem","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/chapter\/14-6-cauchys-mapping-theorem\/","title":{"raw":"14.6 Cauchy's Mapping Theorem","rendered":"14.6 Cauchy’s Mapping Theorem"},"content":{"raw":"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:\r\n\r\n\r\n\r\n
[latex] N=Z-P [\/latex]<\/td>\r\nEquation 14-8<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

14.6.1 How Does Cauchy's Mapping Theorem Apply to a Control System Stability?<\/b><\/p>\r\nConsider a closed loop system:\r\n

\"\"<\/p>\r\nThe characteristic equation is:\r\n

[latex] 1+ G(s)H(s)=0[\/latex]<\/p>\r\n

[latex]G(s)H(s)= \\frac{N_{open}(s)}{D_{open}(s)}[\/latex]<\/p>\r\n

[latex]1+ \\frac{N_{open}(s)}{D_{open}(s)}=0[\/latex]<\/p>\r\n

[latex] \\frac{D_{open}(s)+N_{open}(s)}{D_{open}(s)}=0[\/latex]<\/p>\r\n

[latex] N_{char}(s)=D_{open}(s)+N_{open}(s)[\/latex]<\/p>\r\n

[latex]D_{char}(s)=D_{open}(s)[\/latex]<\/p>\r\nDefine a map (function) [latex]F(s)[\/latex] such that it is described by a characteristic equation of the closed loop:\r\n\r\n\r\n\r\n
[latex]F(s)= \\frac{N_{char}(S)}{D_{char}(s)}[\/latex]<\/span><\/td>\r\nEquation 14-9<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNote 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:\r\n

\"\"<\/p>\r\nTypically, the open loop pole locations are known, i.e. P<\/em> 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<\/em> is what we want to find out. However, from Cauchy's Theorem:\r\n

[latex]Z=N+P[\/latex]<\/p>\r\nThe question then is, how to find N<\/em>? If we can perform the mapping into [latex]F(s)[\/latex]-plane, N\u00a0<\/em> 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:\r\n

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

\"\"<\/p>\r\n \r\n\r\nMap [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\u00a0 N\u00a0<\/em> of CW encirclements of the origin of [latex]F(s)[\/latex] plane, we will be watching the number N<\/i> of CW encirclements of the (-1,j0) point in the [latex]G(s)H(s)[\/latex]-plane.\r\n\r\nRemember that Z<\/i> 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<\/strong>. Z<\/em> then represents the total number of unstable closed loop poles of the system<\/strong>.\r\n\r\nSince [latex]Z = N + P[\/latex], for the system to be stable, Z must be equal to zero<\/strong>, i.e. N + P = 0<\/em>, or \u00a0N = -P. \u00a0<\/em>\r\n

\"\"<\/p>\r\nWe can then define the following stability criterion:\r\n\r\nTake 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).\r\n\r\nFor stability, the Nyquist contour for the closed loop control system with P<\/em> unstable open loop poles must encircle the (-1,j0) point in [latex]G(s)H(s)[\/latex]-plane P<\/em> times in CCW direction.\r\n\r\nTypically, we are not interested in the absolute system stability<\/strong> (i.e. is the system stable or not?), but in the relative system stability<\/strong> (i.e. what is the range of gains [latex]K[\/latex] for which this system is stable?):\r\n

\"\"<\/p>\r\n

<\/p>","rendered":"

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:<\/p>\n\n\n\n
[latex]N=Z-P[\/latex]<\/td>\nEquation 14-8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

14.6.1 How Does Cauchy’s Mapping Theorem Apply to a Control System Stability?<\/b><\/p>\n

Consider a closed loop system:<\/p>\n

\"\"<\/p>\n

The characteristic equation is:<\/p>\n

[latex]1+ G(s)H(s)=0[\/latex]<\/p>\n

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

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

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

[latex]N_{char}(s)=D_{open}(s)+N_{open}(s)[\/latex]<\/p>\n

[latex]D_{char}(s)=D_{open}(s)[\/latex]<\/p>\n

Define a map (function) [latex]F(s)[\/latex] such that it is described by a characteristic equation of the closed loop:<\/p>\n\n\n\n
[latex]F(s)= \\frac{N_{char}(S)}{D_{char}(s)}[\/latex]<\/span><\/td>\nEquation 14-9<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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:<\/p>\n

\"\"<\/p>\n

Typically, the open loop pole locations are known, i.e. P<\/em> 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<\/em> is what we want to find out. However, from Cauchy’s Theorem:<\/p>\n

[latex]Z=N+P[\/latex]<\/p>\n

The question then is, how to find N<\/em>? If we can perform the mapping into [latex]F(s)[\/latex]-plane, N\u00a0<\/em> 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:<\/p>\n

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

\"\"<\/p>\n

 <\/p>\n

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\u00a0 N\u00a0<\/em> of CW encirclements of the origin of [latex]F(s)[\/latex] plane, we will be watching the number N<\/i> of CW encirclements of the (-1,j0) point in the [latex]G(s)H(s)[\/latex]-plane.<\/p>\n

Remember that Z<\/i> 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<\/strong>. Z<\/em> then represents the total number of unstable closed loop poles of the system<\/strong>.<\/p>\n

Since [latex]Z = N + P[\/latex], for the system to be stable, Z must be equal to zero<\/strong>, i.e. N + P = 0<\/em>, or \u00a0N = -P. \u00a0<\/em><\/p>\n

\"\"<\/p>\n

We can then define the following stability criterion:<\/p>\n

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).<\/p>\n

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

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

\"\"<\/p>\n

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