A system can be approximately expressed as X₁ = −T₁×₁ +σx2 x2=-T2x2 + ku a. Given ₁ = 8,σ = 4, k = 0, find the range of T2 such that the system x = Ax is asymptotical stable. b. Given ₁ = 8, t₂ = −1, σ = k = 1, design a state-feedback controller, such that the system x = Ax+Bu is asymptotical stable. LG 11-7 IZ-N 13-

6. The figure below represents a time response of a control system. y(r) 0.63 What is it? a) Unit-step response of a Prototype First-Order System; b) Unit-Impulse Response of a Prototype First-Order System; c) Unit-Step Response of a Prototype Second-Order System; d) Unit-Impulse Response of a Prototype Second-Order System.

1) Consider the system below: Vehicle Controller Steering dynamics Desired Actual bearing angle bearing angle 50 1 K s2 + 10s + 50 s(s + 5) Figure 1: Simplified Block Diagram of a Self-Guiding Vehicle's Bearing Angle Control. • Find a K value that the system has minimum rise time and minimum overshoot. Let us call this proportional gain as Kopt Show each step while finding Kopt- Show the necessary graphical solutions. Simulate the system response with 3 different K values. (Kopt and two other K values close to Kopt) Show the system response (actual bearing angle) in a single graph for different K values. • Comment on the results.

The Gilles & Retzbach model of a distillation column, the system model includes the dynamics of a boiler, is driven by the inputs of steam flow and the flow rate of the vapour side stream, and the measurements are the temperature changes at two different locations along the column. The state space model is given by: x = 0 00 -30.3 0.00012 -6.02 0 0 0 -3.77 00 0 -2.80 0 0 Is the system?: a. unstable b. C. not unstable x+ 6.15 0 0 0 0 3.04 0 0.052 not asymptotically stable d. asymptotically stable -1 u y = 0 0 0 0 -7.3 0 0 -25.0 X

2. Consider the closed-loop system shown below. Determine the range of K for stability. Assume that K > 0. R(s) K S-2 (s + 1)(s² + 6s+25) C(s)

P6. The open loop transfer function of a unity feedback system is K(s+2) G(s) = s(s+3) (s²+2s+10) 1- Find the value of K so that the error steady state for the unit ramp input r(t)=t is less than or equal to 0.01.

P6. The open loop transfer function of a unity feedback system is K(s+2) G (s) = s(s+3)(s²+2s+10) 1- Find the value of K so that the error steady state for the unit ramp input r(t)=t is less than or equal to 0.01. 2-For the value of K found in part (1), use the Routh method to verify whether the closed loop system is stable.

3- Nise (4.4) A unity feedback control system has the following open-loop transfer function: G(s) = 45+¹ Find expressions for 4s+1 45² its time response when is subjected to unit impulse input.

QUESTION 1 A vertical vibrating system of 5 kg of mass and 500 N/m of spring stiffness is critically damped. The system is excited by a step input force f(t) = 50 N to generate an output vertical motion y(t), in metres, and t-is the time in seconds. 1.1. Determine the transfer function of the system 1.2. Provide an equivalent block diagram with a unitary negative feedback to control the motion y(t) 1.3. Using s-plane, locate the closed loop pole(s) and zero (s) of the system and provide the reasons of stability or non-stability of the system 1.4. Using the technique of partial fractions, establish the analytical expression of the time response of the vibrating system.

Q.4- A model for feedback control system employing both angular position and velocity feedback is shown in figure bellow .The equation describing d²0 do di the system is J di² - = • K.[V₁-K₁0. K₂0-Ka ((1) Summing V,(r) Motor 白口食 amplifier Velocity sensor K₂8 Position sensor Where J is the rotary inertia KandK are the position and velocity feedback gains, and K, is the gain between the input voltages to the motor torque produced. Derive expression for the closed loop system undamped natural frequency and damping ratio.