AUVE Lab 1 -Solved

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

1.1         A 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 = 0.95.

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

a)  Generate N realizations.1

b)  Plot the normalized histogram of these N realizationsand plot the probability density function on the same figure.

c)   Estimate the mean and the standard deviation (mean, std).

d)  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 Ts = 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 Ts, 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 fn = f(nTs).

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

a)  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 yn, x is a 2-columns matrix which contains the evolution of the state vector; x1 is the initial state vector).

b)  Test the simulator with G = 50 rad.s−1.V−1 et T = 20 ms.

2.3         Kalman filter
wn is the quantization noise of the incremental encoder, r is its variance (yn = θn+wn). To take into account modeling errors, we assumes that the actual input is un + vn, where vn is a white noise with variance q, independent of wn.

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

a)  Write the equations of the Markov model of the system with input un and output yn.

b)  Propose a value of r (it is usual to model a quantization error as a uniform random variable).

c)   Propose an initialization of the prior information Xˆ1/0 and propose a value for its variance P1/0.

d)  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 θˆ1/0 = 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: Tactual = 20 ms

                                                                               Tfilter         = 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 Ts 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.