[SOLVED] AUVE Lab 1

Description

Statistical Signal Processing and Estimation Theory (SIP, EPICO)

1 Random variables

Please see the appendix “Matlab, Octave” of the book.

1.1A few univariate distributions

We consider a zero-mean unit-variance random variable, in the 3 cases below:

• X is normally distributed;
• X is uniformly distributed;
• X is driven by a Gaussian mixture whose both components have the same probability, the same variance; the means are ±m (necessarily, the variance of each component is 1 − m²); we take m = 0.

In all cases, with N = 100 and N = 4000:

1. Generate N ¹
2. Plot the normalized histogram of these N realizationsand plot the probability density function on the same figure.
3. Estimate the mean and the standard deviation (mean, std).
4. Plot the N realizations as an independent random signal.

1.2 Joint distribution

We consider a zero-mean bivariate normal random variable , where X₁ and X₂ have variances , and correlation coefficient ρ. Use σ₁ = 2, σ₂ = 5, ρ = 0,9.

• Generate N = 200 realizations of this random variable.
• Estimate the mean and the variance (mean, cov).
• plot these realizations together with the 91 % confidence ellipses.

2  Kalman filter

A DC motor is driven by the input voltage u(t). The angular position of the rotor θ(t) is measured with an incremental encoder (precision L = 512 angles per lap) which provides the measure y(t) of θ(t). Ω(t) is the angular velocity (Ω(t) = θ^˙(t)). To perform a velocity control, we have to estimate online θ(t) and Ω(t), from u(t) and y(t).

2.1 Input voltage

The input voltage u(t) is a zero-mean square wave with period ∆ = 100 ms, and peak-to-peak amplitude A = 0,1 V. This signal is sampled with sample time T_s = 1 ms. Create a MatLab function which provides this sampled input for a duration D: u = inputvoltage(D,A,Delta,Ts), where u is a column vector which contains the sampled input (square).

To generate N realizations of a zero-mean unit-variance random variable:

Gauss                                                                : x = randn(N,1);

Uniform                                                                 : x = 2*sqrt(3)*(rand(N,1)-0.5);

Symmetric Gaussian mixture : m=0.95; x=randn(N,1)*sqrt(1-m*m)+m; k=find(rand(N,1)>0.5); x(k) = x(k)-2*m;

CORO-IMARO AV, CORO-SIP STATES, CORO-EPICO STATES, 2020-2021, Lab work                                                                                                                                                  1/2

2.2  System modeling and simulation

With the state vector, we get the state-space representation:

The input-output signals are sampled with the sample time T_s, The input u(t) is constant between 2 sampling times. Thus, the continuous-time above can be sampled without any approximation using the step invariance method (or zero-order-hold method, “zoh”) ². For all integer n, and all function f, we note f_n = f(nT_s).

The measure y_n provided by the incremental encoder is a quantization of the actual angular position θ_n.³

1. Write a function [y,x] = simulate(u,G,T,Ts,L,x1), where y is a column matrix (same size as u) which contains the evolution of the output y_n, x is a 2-columns matrix which contains the evolution of the state vector; x1 is the initial state vector).
2. Test the simulator with G = 50 rad.s⁻¹.V⁻¹ et T = 20 ms.

2.3  Kalman filter

w_n is the quantization noise of the incremental encoder, r is its variance (y_n = θ_n+w_n). To take into account modeling errors, we assumes that the actual input is u_n + v_n, where v_n is a white noise with variance q, independent of w_n.

The goal is to estimate X_n. The motor is initially stopped, but there is no information on the initial angular position.

1. Write the equations of the Markov model of the system with input u_n and output y_n.
2. Propose a value of r (it is usual to model a quantization error as a uniform random variable).
3. Propose an initialization of the prior information X^ˆ_1/0 and propose a value for its variance P_1/0.
4. Write the Kalman filter⁴ and the stationary Kalman filter⁵, that is two functions xe = kal(y, u, G, T, Ts, L, x1 0, P1 0, q), where xe is a 2-columns matrix which contains the evolution of the state vector estimation.

2.4 Simulations

Compare the estimation of the position and the velocity given by both filters for different values of q (use the initialization θ^ˆ_1/0 = θ₁ ± 0.05) in the following cases:

• the model of the system is perfect: Gactual = Gfilter = 50 rad.s−1.V−1 Tactual = Tfilter = 20 ms                                                                                                                 −1 −1

Gactual = Gfilter = 50 rad.s .V • the model of the system is rough: T_actual = 20 ms

T_filter         = 25 ms

“actual” index if for the value used in the simulation, “filter” index is for the value used in the Kalman filter.

2

For a continuous-time system with the state space representation:

the step invariance sampling with sample time T_s writes (see MatLab function c2dm):

with

3

With L angles per lap, the precision is 2π/L rad. With MatLab: y = round(teta*L/2/pi)*2*pi/L

4

With time-varying Kalman gain, page 47 of the book.

5

With constant gain obtained with function dlqe, page 48 of the book.