System Identification of DragonFly UAV via Bayesian Estimation

The system identification of UAV, which is the estimation of the parameters of the equation of motion, is crucial for implementing the autopilot. The system identification especially with noisy data is challenging problem. In this report, we will estimate parameters of equation of motion for DragonFly UAV from Stanford Hybrid Systems Laboratory in Aero/Astro department via Baysian estimation. The basic idea here is that the most probable coefficients of the equation of motion, which is set to the unknown set, can be found by maximizing the probability of that unknown set given the data. Furthermore, from Bayesian rule, the conditional probability is proportional to the product of the likelihood function and the prior. By properly defining the likelihood function and the prior, we will be able to find the most probable unknown set of parameters. Identifying the system of UAV, the performance of the estimator will be evaluated by cross validation and in particularly the benefit of the Bayesian estimation will be highlighted by comparing it to the general least square method. 1 Bayesian System Identification The system identification is the first and crucial step for the design of the controller, simulation of the system and so on. Frequently it is necessary to analyze the flight data in the frequency domain to identify the UAV sytem. Our approach, in this project, uses the data from the flight test of the UAV in time domain thus does not require to excite every mode of a given system. This approach is beneficial in terms of the simpleness and computational efficiency as Bayesian estimator can be performed with less computational cost than frequency domain least square method. In addition, the theoretical derivation is not limited to the linear system thus it can be easily extended to the nonlinear system identification when the assumption imposed on the following theoretical background works well. Suppose the system, which is linear or nonlinear, is given. ẋ(t) = f(x; c) + u(t) + ω(t) (1) y(t) = x(t) + ν(t). (2) wherex, y ∈ R, c ∈ R are state variable, system output and unknown coefficient vector to be determined. Also the dynamic noise and measurement noise are assumed as white Gaussian noise which satisfies the following property. E [ω(t)] = 0, cov(ω(t)) = D̂, (3) E [ν(t)] = 0, cov(ν(t)) = σÎ . (4) whereω, ν ∈ R. Another assumption we will use through this report is that the full states were measured, thus ω(t) can represent both the dynamic noise and the measurement noise. From this assumption our unknown set is reduced to M = {c, D̂}. The purpose of this paper, which is to find out the most probable unknown set given the time series of data, can be achieved by choosing the unknown set that gives the peak value of posterior probability, Pps(M|Y). From Bayes’ theorem, Pps(M|Y) = P (Y|M)Ppr (M) ∫ P (Y|M) Ppr (M) dM . (5) The basic idea is to update the posterior of the unknown set by using the time sequence of data and replace the prior by the posterior probability distribution. The theoretical detail and the application in the case of the stochastic system driven by only white Gaussian noise is well provided in [1]. But the application of Bayesian inference to the practical system identification is meaningful task. The noise property has a key role to construct the probability density function for the state variable, that is time series of data, in our report. 1.1 Maximum Likelihood Estimation The midpoint Euler discretization scheme is used to construct the discrete system. xn+1 = xn + hf(x̃n; c) + hun + zn (6) yn = xn. (7) whereE [ zn, zn′ ] = hD̂δnn′ . The last equality came from the assumption about ω(t) which is previously mentioned. From this point, the state variable, xn, and the system output, yn, are set to be identical. The procedure toward the most probable unknown set is as follows. The likelihood function, P (Y|M) is obtained by transformation of the probability density function of zn to the function of the output, yn, along the given data. Adding initial guess for the prior distribution of the unknown set gives the posterior distribution of the unknown set which has to be maximized. The probability density function of zn along the given data is