DO NOT DO HOMEWORK UNTIL IT IS ASSIGNED. THE ASSIGNMENTS MAY CHANGE UNTIL ANNOUNCED

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1 EE 5322 Intelligent Control Systems Homewor for Spring 209 Updated: Tuesday, March 26, 209 DO NOT DO HOMEWORK UNTIL IT IS ASSIGNED. THE ASSIGNMENTS MAY CHANGE UNTIL ANNOUNCED For full credit, show all wor. Some problems require hand calculations. In those cases, do not use MATLAB except to chec your answers. It is OK to tal about the homewor beforehand. BUT, once you start writing the answers, MAKE SURE YOU WORK ALONE. The purpose of the Homewor is to evaluate you individually, not to evaluate a team. Cheating on the homewor will be severely punished. DO NOT find solutions online and hand them in as your own wor. This called plagiarism. Faculty who plagiarize the papers of others get fired. The next page must be signed and turned in at the front of ALL homewors submitted in this course.

2 EE 5322 Intelligent Control Spring 209 Homewor Pledge of Honor On all homewors in this class - YOU MUST WORK ALONE. Any cheating or collusion will be severely punished. It is very easy to compare your software code and determine if you wored together Or if you found code online written by someone else. It does not matter if you change the variable names. Please sign this form and include it as the first page of all of your submitted homewors Typed Name: Pledge of honor: "On my honor I have neither given nor received aid on this homewor. e-signature: 2

3 EE 5323 Homewor State Variable Systems, Computer Simulation 2. Simulate the van der Pol oscillator y" ( y ) y' y 0 using MATLAB for various ICs. Plot y(t) vs. t and also the phase plane plot y'(t) vs. y(t). Use y(0)=0., y'(0)= 0. a. For = b. For= Do MATLAB simulation of the Lorenz Attractor chaotic system. Run for 50 sec. with all initial states equal to 0.4. Plot states versus time, and also mae 3-D plot of x, x2, x3 using PLOT3(x,x2,x3). x ( x x x 2 ) x 3 bx3 xx use = 0, r= 28, b= 8/3. 2 rx x 2 x x 3. Consider the Voltera predator-prey system x xxx2. x x x x Simulate the system using MATLAB for various initial conditions. Tae ICs spaced in a uniform mesh in the box x=[-2,2], x2=[-2,2]. Mae one phase plane plot with all the trajectories on it. Plot phase plane on square [-5,5]x[-5,5]. 3

4 EE 5322 Homewor 2 Stoc Maret Time Series Analysis Download the file for this problem from the homewor assignment home page. The closing price for the NASDAQ tracing stoc NVDA is given as an Excel file. Note there are 254 trading days in the year. There are about 22 trading days in a month. Therefore, for trading on a monthly time scale, one considers a 20-day time window. This allows one to capture many motions of the stoc while not spending too much in broer s fees by churning the stoc. On-line trades now run about $5 per transaction.. a. Compute the 20 day MA. Plot on the same figure as the stoc closing price. b. Plot the stoc minus the 20 day MA. c. Compute and plot the 20 day moving sample variance. d. On the same figure, plot the stoc closing price, the 20 day MA, and the MA plus three times the 20 day standard deviation the MA minus three times the 20 day standard deviation. The last two lines are nown as the Bollinger Bands, after John Bollinger. 2. a. Compute and plot the 20 day moving sew. b. Compute and plot the 20 day moving urtosis. c. Can you use these statistics to find a leading indicator for movements in the stoc? i.e. how can we predict using statistics when the stoc is about to brea its trend (change its pattern)? 3. a. Compute and plot the overall autocorrelation. b. Compute and plot the overall autocovariance. 4. Any news about predicting movements in this stoc? Is it time to buy this stoc now? 4

5 EE 5322 Homewor 3 DFT Analysis Download the files for this problem from the homewor assignment home page.. Digital Speech Processing (The frequencies here are about /0 the actual values for ease of processing using MATLAB.) In speech, the vowels are characterized by three main frequencies nown as formants. The first two formants for each vowel in English are as follows: vowel Formant (Hz) Formant 2 (Hz) A 70 0 E I O U The data for this homewor contains an 8 sec speech signal that contains some vowels. The sampling period is msec= 0.00 sec. Chop the signal into eight bins of length sec. In each bin, do the FFT (using N= from Rabiner and Schafer, 978 a power of two). Determine which vowels occur and when. Finally, plot the DFT vs. time as a 3-D plot. 2. Machinery Monitoring An induction motor drive has a base rotation frequency of f 0 = 50 Hz, a frequency of 3 f 0 due to a three-bladed fan, and a component at 4 f 0 due to a 4: gearbox. When a certain pinion gear wears badly enough, a prominent frequency component of 277 Hz appears. Soon after that, the amplitude of the frequency component at 4 f 0 significantly increases due to the failure of a gear tooth. In the 6 sec data file, the sampling period is msec. Find out when the two anomaly failure events occur. Plot the DFT vs. time as a 3-D plot. Use moving average window for the DFT of length ½ sec. Use N= a power of two. 5

6 EE 5322 Homewor 4 Discrete Time Simulation, RLS. Discrete-Time System. A discrete time system is given by x Ax Bu x, 0 Write a MATLAB m file to simulate the system, i.e. to compute x for a given input u, initial condition x 0, and range of the time index =,2,,N. a. Simulate the system 0 0 x x u for u equal to the unit step and x 0 =0. Plot x vs. for 00 time samples. b. Simulate the same system but now add process noise so that x Ax Bu w. Tae the noise w as Gaussian with mean of zero and each component having covariance of 0., that is Q 0.I. Use MATLAB function randn. Plot x vs. for 00 time samples. 2. RLS System Identification Download the file for this problem from the homewor assignment home page. The input u and output y of a discrete time system are given in the data file. The system is of second order with a delay of d=2. a. Write a RLS program to identify the system transfer function. Tae measurement noise covariance constant at 0.. b. Plot the output y and the output of your identified system given the input u. They should be the same. 6

7 EE 5322 Intelligent Control- Exam Spring 209. This is a tae home exam. YOU MUST WORK ALONE. Any cheating or collusion will be severely punished. 2. To obtain full credit, show all your wor. No partial credit will be given without the supporting wor. 3. Please sign this form and include it as the first page of your submitted exam Typed Name: Pledge of honor: "On my honor I have neither given nor received aid on this examination." e-signature: 7

8 EE 5322 Exam I Mobile Robot Control & Potential Fields. Potential Field. Use MATLAB to mae a 3-D plot of the potential fields described below. You will need to use plot commands and maybe the mesh function. The wor area is a square from (0,0) to (3,3) in the (x,y) plane. The goal is at (0,0). There are obstacles at (3,4) and (8,5). Use a repulsive potential of Ki / r i for each obstacle, with ri the vector to the i-th obstacle. For the target use an attractive potential of Kr, T T with rt the vector to the target. Adjust the gains to get a decent plot. Plot the sum of the three potential fields in 3-D on the x,y-plane square from (0,0) to (3,3). 2. Potential Field Navigation. For the same scenario as in Problem, a mobile robot starts at (0,0). The front wheel steered mobile robot has dynamics x vt cos cos y vt cos sin vt sin L with (x,y) the position, the heading angle, v T the wheel speed, L the wheel base, and the steering angle. Set L= 2. a. Compute forces due to each obstacle and goal. Compute total force on the vehicle at point (x,y). b. Design a feedbac control system for force-field control. Draw your control system. c. Use MATLAB to simulate the nonlinear dynamics assuming a constant velocity v T and a steerable front wheel. The wheel should be steered so that the vehicle always goes downhill in the force field plot. Plot the resulting trajectory in the (x,y) plane. Use a square from (0,0) to (3,3). 3. Not Required for the Exam. You get Extra Credit if you do this problem- Swarm/Platoon/Formation. Do what you want to for this problem. The intent is to focus on some sort of swarm or platoon or formation behavior, not the full dynamics. Therefore, tae 5 vehicles each with the simple point mass (Newton s law) dynamics x Fx / m y Fy / m with (x,y) the position of the vehicle and Fx, F y the forces in the x and y direction respectively. The forces might be the sums of attractive forces to goals, repulsive forces from obstacles, and repulsive forces between the agents. Mae some sort of interesting plots or movies showing the leader going to a desired goal or moving along a prescribed trajectory and the followers staying close to him, or in a prescribed formation. Obstacle avoidance by a platoon or swarm is interesting. 8

9 EE 5322 Homewor 5 Lagrange s Equation and Some Representative Systems Problem - BALL BALANCER For Problem you will find the dynamics of the ball balancer. The inverted pendulum can be viewed as a two-degrees-of-freedom robot arm with a prismatic (e.g. extensible) joint followed by a revolute (e.g. rotational) joint. It has only one actuator-- on the prismatic lin. The ball balancing on a pivoted beam can be viewed as a robot arm with a revolute lin followed by a prismatic lin, also having only one actuator-- on the revolute lin. This is in some sense a dual system to the inverted pendulum. The ball balancer is representative of a large class of systems in industrial and military applications. The position of the ball is p, the angle of the beam is the torque input to the beam is f, the inertia of the beam is J, and the mass of the ball is m. p m J f g Ball Balancer a. Find the dynamics of the ball balancer by finding the inetic and potential energies and using Lagrange's equation. b. Write in Lagrange equation form. Problem 2- GANTRY CRANE The gantry crane is a load suspended by a wire rope from a moving trolley. The horizontal position of the load is p, the angle of the wire is the force input to the trolley is f, the mass of the trolley is M, and the mass of the load is m, and the length of the wire rope is L. Assume that the wire rope is stiff so that it does not flex or bend. 9

10 p f M L g m a. Find the dynamics of the gantry crane by finding the inetic and potential energies and using Lagrange's equation. b. Write the dynamics in the Lagrange equation form 0

11 EE 5322 Homewor 6 Adaptive Control, Robust Control. A system is given by 0 0 x Ax Bu x u, y Cx 0x 25 2 a. Find poles and transfer function b. Simulate the system with u(t)=unit step for 0 sec. 2. Write the system as y ay ay 2 u and assume the coefficients are unnown. Design an adaptive controller as described in class. Use desired trajectory of 5 u ( t), that is 5 times the unit step. Here are some hints - y T Write the system error dynamics as r f( x) v with f ( x) ay a2y a a2 W ( x) y Tune the unnown parameter vector using ˆ W F( x) r T. To simulate, tae states and 2 as the system dynamics. You will need to define states 3 and 4 to be the adaptive controller dynamics. 3. Write the system as y ay ay 2 u with unnown coefficients same as above values. Assume the estimated coefficients are aˆ ˆ, a2 20. Design a robust controller as described in class. Use desired trajectory of 5 u ( t), that is 5 times the unit step. Here are some hints - y ˆ T f ( x) aya2y a a2 W ( x) y Write the estimate ˆ ˆ ˆ ˆ ˆ y y Find the estimation error f ( x) f( x) fˆ ( x) a a aˆ aˆ Tae its nown norm bound as F( x) 5 y y The controller you will implement in MATLAB is f ˆ( x ) K rv v r v r F( x) r, r F( x) r, It is nondynamic and has no states. r r 2 2 y y

12 EE 5322 Homewor 7 - Consider the following training data set,,,,,,,. a- Use the feedforwardnet function in Matlab and train a perceptron to classify the data set into two classes. Plot points and decision boundaries. b- Without using the Matlab toolbox, program a single neuron with sigmoid activation function to do this classification problem. Plot the points and decision boundaries and compare the results to part a. 2- Consider the dynamical system where a- Approximate the function using an MLP neural networ and plot the function and the estimation on the same graph. b- Simulate the system response for exact and the approximation. Use different initial conditions. Compare the results. 2

DO NOT DO HOMEWORK UNTIL IT IS ASSIGNED. THE ASSIGNMENTS MAY CHANGE UNTIL ANNOUNCED.

EE 533 Homeworks Spring 07 Updated: Saturday, April 08, 07 DO NOT DO HOMEWORK UNTIL IT IS ASSIGNED. THE ASSIGNMENTS MAY CHANGE UNTIL ANNOUNCED. Some homework assignments refer to the textbooks: Slotine