Let's analyze and solve **(a)** as an example step-by-step, including MATLAB code for verification. --- ### **(a)** Given: \[ L(s) = \frac{(s+2)(s+6)}{s(s+1)(s+5)(s+10)} \] ### **Step 1: Identify Poles and Zeros** - **Zeros:** \( s = -2, -6 \) - **Poles:** \( s = 0, -1, -5, -10 \) ### **Step 2: Root Locus Start and End Points** - **Starts**: At open-loop poles (\( s = 0, -1, -5, -10 \)) - **Ends**: At open-loop zeros (\( s = -2, -6 \)) and infinity (since 4 poles, 2 zeros, so 2 loci end at infinity) ### **Step 3: Number of Asymptotes** - Number of poles \( n = 4 \), number of zeros \( m = 2 \) - Number of asymptotes \( = n - m = 2 \) ### **Step 4: Centroid and Asymptote Angles** - **Centroid:** \[ \text{Centroid} = \frac{\sum \text{Poles} - \sum \text{Zeros}}{n-m} \] \[ = \frac{(0) + (-1) + (-5) + (-10) - [(-2) + (-6)]}{2} = \frac{-16 + 8}{2} = \frac{-8}{2} = -4 \] - **Angles:** \[ \theta = \frac{(2q+1)180^\circ}{n-m}, \quad q=0,1 \] For \( n-m=2 \): \( q=0 \rightarrow 90^\circ \), \( q=1 \rightarrow 270^\circ \) (or \(-90^\circ\)) ### **Step 5: Departure and Arrival Angles** - **Departure Angle:** Only relevant for complex poles/zeros. Here, all real, so not applicable. ### **Step 6: Root Locus Sketch** - Draw the s-plane. - Mark zeros at \( s=-2, -6 \). - Mark poles at \( s=0, -1, -5, -10 \). - Sketch loci: From each pole, loci approach zeros or move to infinity along asymptotes from centroid at \( s = -4 \), at \( \pm 90^\circ \) angles. --- ## **MATLAB Code** ```matlab % Define numerator and denominator num = [1 8 12]; % (s+2)(s+6) = s^2 + 8s + 12 den = conv([1 0], conv([1 1], conv([1 5], [1 10]))); % s(s+1)(s+5)(s+10) % Root locus plot figure; rlocus(num, den); title('Root Locus of L(s) = (s+2)(s+6) / [s(s+1)(s+5)(s+10)]'); grid on; ``` --- ## **Summary Table for (a):** | Quantity | Value | |------------------|------------------------| | Poles | 0, -1, -5, -10 | | Zeros | -2, -6 | | Asymptote Number | 2 | | Centroid | -4 | | Asymptote Angles | 90°, 270° | --- **Repeat** similar steps for parts (b)-(e). Would you like a solution for the next part (b) as well?

Let's proceed step-by-step for each part (b) to (e), including hand sketches (conceptually described), asymptote details, departure angles, and MATLAB code for verification. --- ## (b) \(L(s) = \frac{1}{(s+2)(s+6)}\) ### **Step 1: Poles and Zeros** - **Zeros:** None - **Poles:** \( s = -2, -6 \) ### **Step 2: Root Locus Start and End Points** - Poles at \( -2 \) and \( -6 \) - Since no zeros, both loci go to infinity. ### **Step 3: Number of Asymptotes** - Number of poles \( n=2 \), zeros \( m=0 \) - Asymptotes: \( n - m = 2 \) ### **Step 4: Asymptote Centroid and Angles** - \[ \text{Centroid} = \frac{-2 + (-6)}{2} = \frac{-8}{2} = -4 \] - Angles: \( \pm 90^\circ \) (since \( n-m=2 \)) ### **Step 5: Sketch Description** - Poles at \( -2 \) and \( -6 \) - Loci on the real axis: Between \( -6 \) and \( -2 \), the locus exists. - From each pole, the locus extends to infinity along asymptotes at \( \pm 90^\circ \) crossing centroid at \( -4 \). --- ## MATLAB Code for (b): ```matlab % Part (b) num_b = 1; % numerator den_b = conv([1 2], [1 6]); % (s+2)(s+6) rlocus(num_b, den_b); title('Root Locus of 1/[(s+2)(s+6)]'); grid on; ``` --- ## (c) \(L(s) = \frac{s+1}{(s+3)(s+4)}\) ### **Step 1: Poles and Zeros** - **Zeros:** at \( s = -1 \) - **Poles:** at \( s = -3, -4 \) ### **Step 2: Poles and zeros summary** - Poles at \( -3, -4 \) - Zero at \( -1 \) ### **Step 3: Asymptotes** - \( n=2 \), \( m=1 \) - Asymptotes: \( 2 - 1=1 \) - **Centroid:** \[ \frac{-3 + (-4) - (-1)}{1} = \frac{-7 + 1}{1} = -6 \] - **Angles:** \[ \frac{(2q+1)180^\circ}{1} = 180^\circ \] Only one asymptote at \( 180^\circ \). ### **Step 4: Sketch description** - Zero at \( -1 \), poles at \( -3, -4 \). - One asymptote at \( 180^\circ \) crossing at \( s = -6 \). - Root locus: from poles, approaching the zero or going to infinity along the asymptote. --- ## MATLAB Code for (c): ```matlab % Part (c) num_c = [1 1]; % s+1 den_c = conv([1 3], [1 4]); rlocus(num_c, den_c); title('Root Locus of (s+1)/[(s+3)(s+4)]'); grid on; ``` --- ## (d) \(L(s) = \frac{5}{(s+1)(s+2)(s+3)}\) ### **Step 1: Poles and Zeros** - **Zero:** none - **Poles:** at \( -1, -2, -3 \) ### **Step 2: Asymptotes** - \( n=3 \), \( m=0 \) - Number of asymptotes: 3 - **Centroid:** \[ \frac{-1 + (-2) + (-3)}{3} = \frac{-6}{3} = -2 \] - **Angles:** \[ \frac{(2q+1)180^\circ}{3} = 60^\circ, 180^\circ, 300^\circ \] ### **Step 3: Sketch description** - Poles at \( -1, -2, -3 \) - Asymptotes at \( 60^\circ, 180^\circ, 300^\circ \) crossing at \( -2 \). --- ## MATLAB Code for (d): ```matlab % Part (d) num_d = 5; % numerator constant den_d = conv([1 1], conv([1 2], [1 3])); rlocus(num_d, den_d); title('Root Locus of 5/[(s+1)(s+2)(s+3)]'); grid on; ``` --- ## (e) \(L(s) = \frac{1}{(s+1)(s+2)}\) ### **Step 1: Poles and Zeros** - Poles at \( -1, -2 \) - No zeros ### **Step 2: Asymptotes** - \( n=2 \), \( m=0 \), asymptotes: 2 - **Centroid:** \[ \frac{-1 + (-2)}{2} = -1.5 \] - **Angles:** \(\pm 90^\circ\) ### **Step 3: Sketch** - Poles at \( -1, -2 \) - Loci on the real axis between the poles. - Asymptotes at \( \pm 90^\circ \) crossing at \( -1.5 \). --- ## MATLAB Code for (e): ```matlab % Part (e) num_e = 1; den_e = conv([1 1], [1 2]); rlocus(num_e, den_e); title('Root Locus of 1/[(s+1)(s+2)]'); grid on; ``` --- # **Summary:** | Part | Poles | Zeros | Asymptotes | Centroid | Asymptote Angles | Key Features | |--------|------------------------|----------------------|------------|---------|------------------|------------------------------------------------| | (a) | 0, -1, -5, -10 | -2, -6 | 2 | -4 | 90°, 270° | Poles and zeros, asymptotes crossing at -4 | | (b) | -2, -6 | None | 2 | -4 | 90°, 270° | Poles on real axis, loci extend to infinity | | (c) | -3, -4 | -1 | 1 | -6 | 180° | One zero, one asymptote at 180° | | (d) | -1, -2, -3 | None | 3 | -2 | 60°, 180°, 300° | Three asymptotes crossing at -2 | | (e) | -1, -2 | None | 2 | -1.5 | 90°, -90° | Poles on real axis, asymptotes at ±90° | --- **Note:** To verify your hand sketches, run the MATLAB code provided. The MATLAB plots will show the actual root locus, confirming asymptote directions, poles, zeros, and loci paths. --- **Would you like me to prepare a compiled document or further explanations?**