EE 5327 Homeworks Fall 2010 Updated: Saturday, July 16, 2016 For full credit, show all work. Some problems require hand calculations. In those cases, do not use MATLAB except to check your answers. 1 EE 5327 Homework 1 Fall 2010 State Variable Systems, Computer Simulation, Recursive Maximum Likelihood Estimation 1. Discrete-Time System. A discrete time system is given by xk 1 Axk Bu k , x 0 Write a MATLAB m file to simulate the system, i.e. to compute x k for a given input u k , initial condition x 0 , and range of the time index k= 1,2,…,N. a. Simulate the system 1 0 0 x k 1 x k u k 0.89 1.8 1 for u k equal to the unit step and x 0 =0. Plot x k vs. k for 100 time samples. Find the period and percent overshoot. b. Simulate the same system but now add process noise so that xk 1 Axk Bu k wk . Take the noise wk as uniformly distributed between 0 and 0.2. Use MATLAB function rand. Plot x k vs. k for 100 time samples. 2. Recursive Maximum Likelihood Estimation a. Write a MATLAB program to recursively compute the ML estimate using eqs (1.4-7), (1.4-9). You can use MATLAB function lqe if you like. b. Use your program to solve the following problem. A polynomial-fit measurement system is given by z k 1 k k 2 X vk where the state X a b c T contains the unknown coefficients. Use your program to estimate the approximating coefficients for the NASDAQ stock data. Take the measurement noise as vk ~ N (0,1) . Plot the estimates aˆ , bˆ , cˆ versus k. k k k Plot the stock data and the polynomial estimate for it on the same graph. 2 EE 5327 Homework 2 Fall 2010 RLS 1. RLS System Identification The input u k and output y k 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. b. Make time plots of the parameter estimates as RLS is running on the data. c. Plot the output y k given in the file and the output of your identified system given the input u k . They should be the same. 2. Minimum Variance Prediction A system is given by yk 0.75 yk 1 uk d uk d 1 ek 0.9ek 1 a. Find the minimum variance predictor when d=1. Find the prediction error variance. b. Find the minimum variance predictor when d=2. Find the prediction error variance. c. For d=2, simulate your predictor and the original system for an input u_k (sinusoidal?anything) and small noise e_k that you generate. Plot both outputs on the same graph. Plot the output error, i.e. the difference between the two output signals. It should be small. 3 EE 5327 Homework 3 Fall 2010 Observer and Kalman Filter 1. DT Observer 2. Write a MATLAB m file to simulate a DT system plus a DT observer running at the same time. a. Design an observer for the system 1 0 0 x k 1 x k u k , x 0 =0 0.9409 .6596 3 z k [1 0]xk In eqs (2.1-6), (2.1-7) in the book use Q GGT I , R DDT 0.1 . Find the observer gain L and write down the observer equation. b. Simulate the system with observer for the system using u k equal to the unit step and x 0 =0. Use initial estimates of xˆ(0) [5 5]T . Plot the states and their estimates on the same graphs. i.e. x1 (k ), xˆ1 (k ) on one graph, and x2 (k ), xˆ 2 (k ) on another graph. Use 300 samples. 2. DT Kalman Filter Write a MATLAB m file to simulate a DT system xk 1 Axk Bu k Gwk z k Hx k vk plus DT Kalman Filter. Simulate the optimal time-varying DT Kalman Filter for the system 0 1 0 x k 1 x k u k wk , 0.9409 .6596 3 z k [1 0]xk vk . Take the process noise wk as a normal 2-vector (MATLAB function randn) with each component having zero mean and variance 0.1. Take measurement noise vk as normal (0,0.1). Use u k equal to the unit step and x 0 =0. Plot the states and their estimates on the same graphs. i.e. x1 (k ), xˆ1 (k ) on one graph, and x2 (k ), xˆ 2 (k ) on another graph. 4