{"id":2118,"date":"2019-12-16T17:24:46","date_gmt":"2019-12-16T17:24:46","guid":{"rendered":"https:\/\/pressbooks.library.ryerson.ca\/controlsystems\/?post_type=chapter&p=2118"},"modified":"2021-01-14T16:25:21","modified_gmt":"2021-01-14T16:25:21","slug":"14-5-concept-of-mapping","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/controlsystems\/chapter\/14-5-concept-of-mapping\/","title":{"raw":"14.5 Concept of Mapping","rendered":"14.5 Concept of Mapping"},"content":{"raw":" \r\n\r\nConsider a map (function) [latex]F(s)[\/latex]:\r\n

[latex]F(s)=[\/latex] [latex] \\frac{s+5}{(s+1)(s+3)} [\/latex]<\/p>\r\nThe map (function) has [latex]3[\/latex] singularities: a zero at [latex]-5[\/latex], and two poles, at [latex]-1[\/latex] and [latex]-3[\/latex].\r\n\r\nConsider 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.\r\n\r\n14.5.1 Case 1<\/b>\r\n\r\nLet the [latex] \\sigma [\/latex]-contour be so chosen that Z = 0 and P = 0, i.e. there are no singularities of [latex]F(s)[\/latex] inside the [latex] \\sigma [\/latex] - contour. Note that the resulting [latex] \\Gamma [\/latex] -contour in the [latex]F(s)[\/latex]-plane does not encircle the origin of [latex]F(s)[\/latex]-plane, i.e. N = 0.\r\n\r\n \r\n\r\n\"\"\r\n\r\n \r\n\r\n14.5.2 Case 2<\/b>\r\n\r\nNext, let the [latex] \\sigma[\/latex] -contour be so chosen that [latex]Z =1[\/latex] and P = [latex]0[\/latex], i.e. there is one zero of [latex]F(s)[\/latex] inside the [latex] \\sigma[\/latex] -contour. Note that the resulting [latex] \\sigma[\/latex]-contour in the [latex]F(s)[\/latex]-plane encircles the origin of [latex]F(s)[\/latex]-plane once in a clockwise (CW) direction, i.e. N =[latex]+1[\/latex].\r\n\r\n \r\n\r\n\"\"\r\n\r\n \r\n\r\n14.5.3 Case 3<\/b>\r\n\r\nNext, let the [latex] \\sigma [\/latex]-contour be so chosen that [latex]Z =0[\/latex] and [latex]P = 1[\/latex], i.e. there is one pole of [latex]F(s)[\/latex] inside the [latex] \\sigma [\/latex]-contour. Note that the resulting [latex] \\Gamma [\/latex] -contour in the [latex]F(s)[\/latex]-plane encircles the origin of [latex] F(s)[\/latex]-plane once in a counter-clockwise (CCW) direction, i.e. [latex]N = -1[\/latex].\r\n\r\n \r\n\r\n\"\"\r\n\r\n \r\n\r\n14.5.4 Case 4<\/b>\r\n\r\nNext, let the [latex] \\sigma [\/latex]-contour be so chosen that [latex]Z = 1[\/latex] and [latex]P = 1[\/latex], i.e. there is one zero and one pole of [latex]F(s)[\/latex] inside the [latex] \\sigma [\/latex] -contour. Note that the resulting [latex] \\Gamma [\/latex]-contour in the [latex]F(s)[\/latex]-plane does not encircle the origin of [latex]F(s)[\/latex]-plane, i.e. [latex]N = 0.[\/latex]\r\n\r\n \r\n\r\n\"\"\r\n\r\n \r\n\r\n14.5.5 Case 5<\/b>\r\n\r\nNext, let the [latex] \\sigma [\/latex]-contour be so chosen that [latex]Z = 0[\/latex] and [latex]P = 2[\/latex], i.e. there are two poles of [latex]F(s)[\/latex] inside the [latex] \\sigma[\/latex]-contour. Note that the resulting [latex] \\Gamma [\/latex]-contour in the [latex]F(s)[\/latex]-plane encircles the origin of [latex]F(s)[\/latex]-plane twice in a counter-clockwise (CCW) direction, i.e. [latex]N = -2.[\/latex]\r\n\r\n \r\n\r\n\"\"\r\n\r\n \r\n\r\n14.5.6 Case 6<\/b>\r\n\r\nNext, let the [latex] \\sigma [\/latex]-contour be so chosen that [latex]Z = 1[\/latex] and [latex]P = 2[\/latex], i.e. there is [latex]1[\/latex]\u00a0 zero and [latex]2[\/latex] poles of [latex]F(s)[\/latex] inside the [latex] \\sigma [\/latex] -contour. The area of the origin may not be very visible, so see a zoomed version in Error! Reference source not found.<\/strong> Note that the resulting [latex] \\Gamma [\/latex]-contour in the [latex]F(s)[\/latex]-plane encircles the origin of [latex]F(s)[\/latex]-plane once in a counter-clockwise (CCW) direction, i.e. [latex]N = -1[\/latex].\r\n\r\n \r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n ","rendered":"

 <\/p>\n

Consider a map (function) [latex]F(s)[\/latex]:<\/p>\n

[latex]F(s)=[\/latex] [latex]\\frac{s+5}{(s+1)(s+3)}[\/latex]<\/p>\n

The map (function) has [latex]3[\/latex] singularities: a zero at [latex]-5[\/latex], and two poles, at [latex]-1[\/latex] and [latex]-3[\/latex].<\/p>\n

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.<\/p>\n

14.5.1 Case 1<\/b><\/p>\n

Let the [latex]\\sigma[\/latex]-contour be so chosen that Z = 0 and P = 0, i.e. there are no singularities of [latex]F(s)[\/latex] inside the [latex]\\sigma[\/latex] – contour. Note that the resulting [latex]\\Gamma[\/latex] -contour in the [latex]F(s)[\/latex]-plane does not encircle the origin of [latex]F(s)[\/latex]-plane, i.e. N = 0.<\/p>\n

 <\/p>\n

\"\"<\/p>\n

 <\/p>\n

14.5.2 Case 2<\/b><\/p>\n

Next, let the [latex]\\sigma[\/latex] -contour be so chosen that [latex]Z =1[\/latex] and P = [latex]0[\/latex], i.e. there is one zero of [latex]F(s)[\/latex] inside the [latex]\\sigma[\/latex] -contour. Note that the resulting [latex]\\sigma[\/latex]-contour in the [latex]F(s)[\/latex]-plane encircles the origin of [latex]F(s)[\/latex]-plane once in a clockwise (CW) direction, i.e. N =[latex]+1[\/latex].<\/p>\n

 <\/p>\n

\"\"<\/p>\n

 <\/p>\n

14.5.3 Case 3<\/b><\/p>\n

Next, let the [latex]\\sigma[\/latex]-contour be so chosen that [latex]Z =0[\/latex] and [latex]P = 1[\/latex], i.e. there is one pole of [latex]F(s)[\/latex] inside the [latex]\\sigma[\/latex]-contour. Note that the resulting [latex]\\Gamma[\/latex] -contour in the [latex]F(s)[\/latex]-plane encircles the origin of [latex]F(s)[\/latex]-plane once in a counter-clockwise (CCW) direction, i.e. [latex]N = -1[\/latex].<\/p>\n

 <\/p>\n

\"\"<\/p>\n

 <\/p>\n

14.5.4 Case 4<\/b><\/p>\n

Next, let the [latex]\\sigma[\/latex]-contour be so chosen that [latex]Z = 1[\/latex] and [latex]P = 1[\/latex], i.e. there is one zero and one pole of [latex]F(s)[\/latex] inside the [latex]\\sigma[\/latex] -contour. Note that the resulting [latex]\\Gamma[\/latex]-contour in the [latex]F(s)[\/latex]-plane does not encircle the origin of [latex]F(s)[\/latex]-plane, i.e. [latex]N = 0.[\/latex]<\/p>\n

 <\/p>\n

\"\"<\/p>\n

 <\/p>\n

14.5.5 Case 5<\/b><\/p>\n

Next, let the [latex]\\sigma[\/latex]-contour be so chosen that [latex]Z = 0[\/latex] and [latex]P = 2[\/latex], i.e. there are two poles of [latex]F(s)[\/latex] inside the [latex]\\sigma[\/latex]-contour. Note that the resulting [latex]\\Gamma[\/latex]-contour in the [latex]F(s)[\/latex]-plane encircles the origin of [latex]F(s)[\/latex]-plane twice in a counter-clockwise (CCW) direction, i.e. [latex]N = -2.[\/latex]<\/p>\n

 <\/p>\n

\"\"<\/p>\n

 <\/p>\n

14.5.6 Case 6<\/b><\/p>\n

Next, let the [latex]\\sigma[\/latex]-contour be so chosen that [latex]Z = 1[\/latex] and [latex]P = 2[\/latex], i.e. there is [latex]1[\/latex]\u00a0 zero and [latex]2[\/latex] poles of [latex]F(s)[\/latex] inside the [latex]\\sigma[\/latex] -contour. The area of the origin may not be very visible, so see a zoomed version in Error! Reference source not found.<\/strong> Note that the resulting [latex]\\Gamma[\/latex]-contour in the [latex]F(s)[\/latex]-plane encircles the origin of [latex]F(s)[\/latex]-plane once in a counter-clockwise (CCW) direction, i.e. [latex]N = -1[\/latex].<\/p>\n

 <\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

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