Consider the open-loop transfer function G(s) of a control system as 2. follows: G(s) = b) c) d) 2s +3 (s + 0.2)(s +0.6)(s² + 0.1s + 0.5) Plot the Bode diagram using MATLAB, incorporating grid lines. What are the phase margin, gain margin, gain crossover frequency, and phase crossover frequency of the system? You can use MATLAB to find them. Please list them explicitly in your answer sheet. Find the gain of the system. Plot the root locus graph for this system using MATLAB, treating the system's gain as the variable of interest. Use grid lines in the plot. Hint: Please ensure that your plots are printed clearly, large enough to draw on or read easily. For an enhanced root locus plot in part d, consider applying the following comments to set x- and y-limits: xlim (-2 0.2]) ; ylim ([-0.8 0.8]); Consider a self-balancing robot system represented by the following state-space equations, where K is the tunable system parameter: a) system G(s) = b) c) x(t) ) = [ { ¯ ] x(t) + [6] u(t) y(t) [06] x(t) Y(s) U(s) = Find the transfer function of the robot using the formula introduced in the course. A self-balancing robot Apply unity negative feedback (no additional controller) to the above robot system and treat K in the system as the variable of interest. i. Derive the transfer function H(S) to be employed in MATLAB for the root locus plot of the closed-loop system. ii. Plot the root locus of the closed-loop system using MATLAB. Show the default grids in the plot. iii. Tabulate the gain K versus poles for the gain range of 0 and 72 with a step of 3, using MATLAB. Consider the position control of the self-balancing robot as shown in Figure 1, where K/K₁ = 36 and the transfer function H(S) aligns with the expression derived in Part b.i. i. Plot the root locus of the control system in Figure 1 using MATLAB with respect to Ka, i.e., Kd is the variable of interest. Show the default grids in the plot. ii. Tabulate the gain Ka versus poles for the gain range of 0 and 150 using MATLAB. Implement a fine mesh for the gain values from 0 to 5 with a step of 0.1, and subsequently, a coarse mesh for the gain values from 5.5 to 150 with a step of 0.5. R(s) E(s) Кр SK d U(s) Y(s) H(s) Figure 1: Self-balancing robot position control

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O 1::09 O [Template] Ho... -> Homework For the system shown in figure below, Find the range of K for stable system. R K(s + 2) C s(s +5)(s² + 2s + 5) II

TIME DOMAIN MODELING AND RESPONSE FOR CONTROL SYSTEMS

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nk int m The spring-mass-system shown in the figure has the following parameters: spring constant k = 4 N/m; mass m 6 %3D kg and the constant n = 1.6. M is the corresponding mass-matrix of the system. V1 and V2 are the eigenvectors associated with the smallest and largest natural frequencies of the system, respectively. If V,TV, = 1 and V2 V2 = 1, then what is value of V,™MV2 (in kg)? Answer:

Determine the state-space representation. Let: x₁= €₁ (+)₁ X ₂ = ₁ (t), X₂ = 0₂ (t), X4 = + ₂Ct) Tit) D₁ It) k₁ M D₂Lt) B1 Ⓒ U₂ K₂ M B₂

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