SYSC3600-L3

Feedback & Control Systems State-Space Representation Write the state-space representation of the system below. Let the output of the mechanical system is x3 (t). 1 N-s/m x₁ (t) M3 = 1kg 1 N/m М1 -0000 1kg > X3 (t) 1 N-s/m 1 N/m oooo x₂ (t) M₂ 1kg 4 1 N-s/m² -1 N-s/m →f(t)

Question 4 a) A control engineer has modelled the suspension system of a new model car using a 2nd order differential equation. Using Laplace Transform, the engineer has managed to work out the transfer function, which is given below: 0.001 s2 + 12s+81 What is the natural frequency, the damping ratio and the constant K of the system? Please show all calculations. b) The same engineer has studied the suspension system of a SUV vehicle and modelled it using again a 2nd order differential equation, which has resulted into the following 2nd order transfer function: Y(s) 32 U(s) 4s² +8s + 16 For this system first calculate the damping ratio and its natural frequency and state whether the system is underdamped, overdamped or critically damped. Then calculate the peak time, peak value, settling time and damped natural frequency of the system.

a) Suspension system of a car. Finding the transfer function F₁(s) = Y(s)/R(t) and F₂ (s) = Q(s)/R(t), consider the initial conditions equal to zero. car chassis www K₂ M₂ 1 Tire M₁ K₁ B₁ y(t)= output q(t) r(t)= input Where [r, q, y] are positions, [k1, k2] are spring constants. [B₁] coefficient of viscous friction, [M₁, M₂] masses. b) Find the answer in time q(t) of the previous system. With the following Ns values: M₁ = 1 kg, M₂ = 0 kg, k₁ = 4 N/m, k₂ = 0 N/m, B₁. = 1 Ns/m, considered m a unit step input, that is, U(s) = 1/s

For the system represented by the following block diagram, Find a) The Closed-loop Transfer function. b) Characteristic equation. c) Type and Order of the system. c) Time-domain Specifications ( Delay Time, Peak Time, Rise Time, Settling time, and Percentage overshoot). R(s) C(s) G(s) H(s) Where G(s) and H(s) are given as : 324 G(s) = s(s+6) H(S) : 1

11. Consider a system that can be modeled as shown. The input x in (t) is a prescribed motion at the right end of spring k 2. Find X(s) the system transfer function Xeq(s)* m k₂ ww Xin The values of the parameters are m= 30 kg, k ₁=700 N/m, k 2= 1300 N/m, and b=200 N- s/m. Write a MATLAB script file that: (a) calculates the natural frequency, damping ratio, and damped natural frequency for the system; and (b) uses the impulse command to find and plot the response of the system to a unit impulse input.

In this problem, you will have to first create a Python function called twobody_dynamics_first_order_EoMS. Given a time t and a state vector X, this function will return the derivatives of the state vector. Mathematically, this means you are computing X using some dynamics equation X = f(t, X). Once you have this function in Python, you can solve the differential equations it contains by using solve_ivp. The command will be similar to, but not necessarily exactly, what is shown below: solve_ivp(simple_pendulum_first_order_EoMS, t_span, initial_conditions, args=constants, rtol 1e-8, atol 1e-8) which integrates the differential equations of motion to give us solutions to the states (i.e., position and velocity of a satellite). In the above, t_span contains the initial time to and final time tƒ and it will compute the solution at every instant of time (you will define this later in Problem 1.3 below). The integration is done with initial state vector Xo which defines the initial position…

Example 1.1 Figure 1.5 shows the block diagram of a closed-loop flow control system. Identify the following elements: (a) the sensor, (b) the transducer, (c) the actuator, (d) the transmitter, (e) the controller, (f) the manipulated variable, and (g) the measured variable. In the figure above, the (a) sensor is labeled as the pressure cell. The (b) transducer is labeled as the converter. However, there are two converters: one for converting pressure to current, and another converting current to pressure for operating the actuator. The (c) actuator is labeled as the pneumatic valve. The (d) transmitter is labeled as the line driver. The (e) controller is labeled as the PLC. The (f) manipulated variable is labeled as the pressure developed by the fluid flowing through the orifice plate. The (g) measured variable is the flow rate of the liquid.

EXPERIMENT TITLE : PID CONTROLLER IN LEVEL CONTROL SYSTEM OBJECTIVE:To demonstrate the characteristic of P,PI and PID controller response on a level controller system. Based on the topic and objective given,please make a clear introduction.

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

Q2 The position of an elevator h(s) is controlled by means of lifting cables. A feedback control system is used to control the force applied by the cables to the elevator. The transfer function of the plant is, 1 Gp(s) = Controller Elevator hr(s) h(s) E(s) G.(s) GP(s) A unit step input is provided. If only proportional control is used, show that the position of the elevator oscillates about the reference value of 1. Find the period (a) of this oscillation. (b) Show that the addition of derivative action to the system can ensure a non- oscillatory response. Find a relation between the derivative and proportional gains that ensures the response is non-oscillatory. (c) When the proportional and derivative gains are set to Kp = 9 and Kd = 6, respectively, find the damping ratio and natural frequency of the system. Derive the time-domain response of the elevator for a unit step input and confirm that it is not oscillatory.

You are given a linear resonant actuator used for haptic vibration cues in phones - a device that might produce the vibration that happens when you get a notification. A basic model is shown on the left. For this problem, you'll need to derive the EOMS for the system (it will be a second order system) and then put it into a number of different system representation forms. Your tasks: karm Carm • I1 = Act • X2 = *1 • I3 = IP Actuator electromagnetic force Fact • I4 = 3 • U = FAct mp Fact kact Cact Xp mact Xact CHALOS CURRENT CONDUCTING PRECISION MICRODRIVES PRECISION HAPTIC™ NOVING MASS No 8 NEOUTMUN FLOCCUT BAGNET 0:0 AND VOICECORIS Z-AXIS LINEAR RESONANT ACTUATOR Credit Precision Microdrives. Ltd. FLYING LEADS FLEX CIRCENT NEOUT MALAET SELFDESVE CASA VERATING BASS ASSEMBLY A. Using Newton's Laws of Motion, derive equations of motion for the system using the variable names given in the figure on the left…

1) a) Derive the mathematical model for the system shown below. b) Find a state variable model (matrix form) for the system. b) Determine state matrix, input matrix, and output matrix, when f (t) is defined as the input and X2 is defined as output for the system. (Here, both of the X1 and x2 , are time-dependent functions) » f(t) X1 X2 3,000 N 1,000 N 4,000 30 kg 20 kg 200 유 N.s

The joint of a robot's shoulder can be modeled by the rotational mechanical system shown below. Calculate the transfer function G(s) = 0m(s)/Tm(s). Take bm = bL = 1 N-m.s/rad, k = 7.9 N/m, Im = 1 kg.m2, and j = 2.2 kg.m2. Upload your calculations in the next question. %3D %3D %3D Max arm inertia case Shoulder actuator model After normalizing the highest power of s in the denominator of the transfer function, what is the power of s in the denominator?

As the control engineer on a project you are given the following information: a) The free body diagram representation of a mechanical system with the various values for the damper, spring and mass. The transfer function for the system is also provided. 28 N/m -x(1) 1 G(s): - f(1) s? +s+5.6 5 kg 5 N-s/m b) The step response plot for the system. Step Response 0.14 0.12 0.1 0.08 0.06 0.04 0.02 10 Time (sec) Your task is to analyse the system and to confirm that the diagram is for the system in a) i.e. calculate all time responses and compare your answers with the data in the graph. apnadu

QUESTION 5 Consider the system whose block diagram is shown below. D(s) R(s) + + C(s) K G(s) G(8) R(s) is the reference input or target, and D(s) is the disturbance input. Gp(t) is the plant (house, motor, car,...) transfer function, Gc(t) is the compensator transfer function, and K is the compensator gain given by 1 G (s) G (s) = P s+10 1 3s+2 and K=5. Find the transfer function, C(s)/R(s). 3 s²+16s+14 8 3s²+32s+25 S 382+8+6 8(s+ 10) 3s² + 32s+25 3s+2 3s²+32s+25 5 3x²+32x+25

root locus electrical engineering  Don't overthink and reject. Complete the solution as per the given transfer function. No need of quadratic equation just simplify for the exact given transfer function.

penaulum shown in the figure, the nonlinear equations of motion are given by where g is gravity, L is the length of the pendulum, m is the mass attached at the end of the pendulum (we - 0. assume the rod is massless), and k is the coefficient of sin e friction at the pivot point. a) Obtain a state-variable representation of the system. b) Linearize the dynamics and obtain a state-space representation of the system around the equilibrium position @ = 180 degrees, ở = 0. Pvor point Llength c) A constant torque, in counter-clockwise direction, is applied as an input to the pendulum. Angular position is the measured output. Derive the transfer function from the torque to angular position for the linearized pendulum system obtained in part b). Massles rod m, mass

TIME DOMAIN MODELING AND RESPONSE FOR CONTROL SYSTEMS

1. Verify Eqs. 1 through 5. Figure 1: mass spring damper In class, we have studied mechanical systems of this type. Here, the main results of our in-class analysis are reviewed. The dynamic behavior of this system is deter- mined from the linear second-order ordinary differential equation: where (1) where r(t) is the displacement of the mass, m is the mass, b is the damping coefficient, and k is the spring stiffness. Equations like Eq. 1 are often written in the "standard form" ď²x dt2 r(t) = = tan-1 d²r dt2 m. M +25wn +wn²x = 0 (2) The variable wn is the natural frequency of the system and is the damping ratio. If the system is underdamped, i.e. < < 1, and it has initial conditions (0) = zot-o = 0, then the solution to Eq. 2 is given by: IO √1 x(1) T₁ = +b+kr = 0 dt 2π dr. dt ل لها -(wat sin (wat +) and is the damped natural frequency. In Figure 2, the normalized plot of the response of this system reveals some useful information. Note that the amount of time Ta between peaks is…

2- Using Matlab, what are the step response curves of the closed-loop system, as shown in fig.1. the feedback represents the second-order dynamic system. (fill in the following table) For=0.4 Wn 1 3 6 9 10 R(S) 0.1 0.3 0.6 0.9 1 For w 5 rad/sec 3 Settling time Peak response 2 Wn s(s+23wn) Settling time Peak response C(s) Discuss the follow Which parameters or w occur on the rise time of the response? Which parameter increases the speed of response? Which parameters can be decreases the response amplitude? Which parameter decreases the steady error state? fig.2

1. The equations of motion of this system are ÿ + 3y + 4y - 32 - 4Z = 0 Ż +52 +6Z-5ý - 6y = f(t) * = A + Bū y = Cx+Dū Put these equations into state variable form and express the model as a matrix vector equation if output of the system is y. Energy storage element m1 m2 k₁ k₂ State variable *1=ý x₂ = Ż x3 = y x4 =Z k₁ my D k₂ C₂H m₂

9 Reduce the following block-diagram into one block. (Describe the control system with one transfer function and write it.) * W2 W1 W3

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.

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.

Weight Ple Figure 4: Schematic of a pile driver f(t) m+bv=f(1) m by 3 X pile Figure 5: First-order model of the pile in soil 5. Figure 4 shows a pile driver that is used to drive piles into soil to provide foundation support for buildings. Two major components of a pile driver are a pile and a weight. The pile is in the form of a long rod going into the soil. The weight is to impact the pile at the top with an impulsive force, thus driving the pile into the soil. Figure 5 shows a first-order model of the pile in the soil. The impulsive force produced by the weight is f(t). The resistance of the soil is modeled via a viscous damping coefficient. b. The weight of the pile is balanced by the normal forces in the soil, so it does not need to be considered. In addition, the elasticity of the soil is ignored. Let z(t) and v(t) be the downward displacement and velocity of the pile, respectively. The equation of motion governing the pile velocity v(t) is dv (5) where m is the mass of the pile.…

A velocity of a vehicle is required to be controlled and maintained constant even if there are disturbances because of wind, or road surface variations. The forces that are applied on the vehicle are the engine force (u), damping/resistive force (b*v) that opposing the motion, and inertial force (m*a). A simplified model is shown in the free body diagram below. From the free body diagram, the ordinary differential equation of the vehicle is: m * dv(t)/ dt + bv(t) = u (t) Where: v (m/s) is the velocity of the vehicle, b [Ns/m] is the damping coefficient, m [kg] is the vehicle mass, u [N] is the engine force. Question: Assume that the vehicle initially starts from zero velocity and zero acceleration. Then, (Note that the velocity (v) is the output and the force (w) is the input to the system): A. Use Laplace transform of the differential equation to determine the transfer function of the system.

Task 4): If you are required to design a PID controller f for the Chassis, what model you should derive and what procedure you would follow and what PID design parameters you could suggest and why? m2 X2 k2 f m1 X1 ki Figure 4 ww

Discuss the process of solving engineering problems using MATLAB • Linear interpolation • Cubic-spline interpolation

The following EOMs describe the behavior of a dynamic system ( à = -K₁a + din b = K₁a - K₂b K₂b - K3c = ċ Assume K₁, K2, K3 are constant, and din is the input. A. How many states does this system of equations have, and why? B. Write the state space form of the system as x = Ax + Bu. You will have to define your state vector and input vector. Clearly identify the matrices. C. Assume the desired output of the system is the vector y that consists of a and K₂b as individual elements. Write the output equation in the form of y = Cx + Du. Clearly identify the matrices.

f Facebook (60) Binary operation (Mather x b Answered: Some states have x M Item shared with you: "Proble x 4 Problem Set 2-2_Axial Load- + A drive.google.com/file/d/1n5tC5BPOCKLLBIVXPMF8h9X3FCxLCHKu/view?ts=6153bb21 submissions. Do not attempt to submit others work as your own, rules agreed on course orientation will apply. 1.0 The post weighs 8 kN/m. Determine the internal normal force in the post as a function of x. 2 m 2.0 The rigid beam supports the load of 60 kN. Determine the displacement at B. Take E = 60 GPa, and ABC = 2 x 10-3 m2 60 kN -2 m- 4 m B 2 m 3 m D ENGG 410 | Fundamentals of Deformable Bodies 09:07 P Type here to search 28°C Partly sunny 29/09/2021

O 17161489 C Get Homework Help With b My Questions | bartleby O Mail - Castro Alvarez, Flavic X Content A learn-us-east-1-prod-fleet01-xythos.s3.amazonaws.com/5c1270dbb5a74/17161489?response-cache-control=private%2C%20max-age%3D21600&respon. ☆ ME 3022 - HW 19 1. (More practice from Monday) A 90° elbow in a horizontal pipe is used to direct a flow of oil (SG = 0.87) upward at a rate of 40 kg/s. The cross- sectional diameter of the elbow is a constant 20 cm. The elbow discharges at A into a large holding tank where the level of oil is 1.8 m above A. The 50 cm "weight" of the elbow and the oil in it is 50 kg. Determine (a) the gage pressure at the inlet of the elbow and (b) the anchoring force needed to hold the elbow stationary. Take the momentum-flux correction factor to be 1.03 40 kg/s at both the inlet and the outlet. Activar Windows Ve a Configuración para activar Windows. Esperando blackboard.ohio.edu. 17:07 P Escribe aquí para buscar a ) ENG 14/10/2020 近

11. A system is described by the following differential equation: [Section 2.3] d'x dx +2 + 3x = 1 dt with the initial conditions x(0) = 1, ¿(0) = -1. Show a block diagram of the system, giving its transfer function and all pertinent inputs and out- puts. (Hint: the initial conditions will show up as added inputs to an effective system with zero initial conditions.)

LESSON is Transfer Function: Mechanical System - Rotational Movement SUBJECT: FEEDBACK CONTROL SYSTEM Box the final answer