ckbs_nonlinear: Unconstrained Nonlinear Transition Model Example and Test

ckbs_nonlinear: Unconstrained Nonlinear Transition Model Example and Test
Syntax
[ok] = vanderpol_ok(draw_plot)

draw_plot
If this optional argument is true, a plot is drawn showing the results. If draw_plot is not present or false, no plot is drawn.

ok
If the return value ok is true, the test passed, otherwise the test failed.

Reference
This example appears in many places; for example

Section 4.1.1 of Applying the unscented Kalman filter for nonlinear state estimation , R. Kandepu, B. Foss , L. Imsland, Journal of Process Control, vol. 18, pp. 753, 2008.
Section V of What Is the Ensemble Kalman Filter and How Well Does it Work? , S. Gillijns et-al. Proceedings of the 2006 American Control Conference, Minneapolis, June 14-16, 2006

Simulated State Vector
We use


 x1 (t)

and


 x2 (t)

to denote the oscillator position and velocity as a function of time. The deterministic ordinary differential equation for the Van der Pol oscillator is


 \[

\begin{array}{rcl}

      x1 '(t) & = & x2 (t)

      \\

      x2 '(t) & = & \mu [ 1 - x1(t)^2 ] x2 (t) - x1(t)

\end{array}

\]

The simulated state vector values are the solution to this ODE with nonlinear parameter


 \mu = 2

and initial conditions


 x1 (0) = 0


 x2 (0) = -5.

. The routine vanderpol_sim calculates these values.

Transition Model
We simulate imperfect knowledge in the dynamical system by using Euler's approximation for the ODE with a step size of


 \Delta t = .1

. In other words, given


 x_{k-1}

, the value of the state at time


 t_{k-1}

, we model

x_k

, the state value at time


 t_k = t_{k-1} + \Delta t

, by


 \[

\begin{array}{rcl}

      x_{1,k} & = & x_{1,k-1} + x_{2,k-1} \Delta t + w_{1,k}

      \\

      x_{2,k} & = & x_{2,k-1} +

      [ \mu ( 1 - x_{1,k-1}^2 ) x_{2,k-1} - x_{1,k-1} ] \Delta t

      + w_{2,k}

\end{array}

\]

where


 w_k \sim \B{N}( 0 , Q_k )

and the variance


 Q_k = \R{diag} ( 0.01, 0.01 )

. The routine vanderpol_g.m calculates the mean for

x_k

given


 x_{k-1}

. Note that the initial state estimate is calculated by this routine as


 \hat{x} = g_1 ( x_0 )

and has


 \hat{x}_1 = \hat{x}_2 = 0

and


 Q_1 = \R{diag} ( 100. , 100. )

.

Measurements
For this example, the measurements are noisy direct observations of position


 \[

\begin{array}{rcl}

      z_{1,k} & = & x_{1,k} + v_{1,k}

\end{array}

\]

where


 v_k \sim \B{N}( 0 , R_k )

and the measurement variance


 R_k = \R{diag} ( 0.01, 0.01 )

. The routine direct_h.m calculates the mean for

z_k

, given the value of

x_k

; i.e.


 x_{1,k}

.

Constraints
There are no constraints for this example so we define


 \[

      f_k ( x_k ) = - 1

\]

The routine no_f.m is used for these calculations.

Call Back Functions

vanderpol_sim	Van der Pol Oscillator Simulation (No Noise)
vanderpol_g.m	ckbs_nonlinear: Vanderpol Transition Model Mean Example
direct_h.m	ckbs_nonlinear: Example Direct Measurement Model
no_f.m	ckbs_nonlinear: Example of No Constraint

Source Code

 
function [ok] = vanderpol_ok(draw_plot)
    if nargin < 1
        draw_plot = false;
    end
    % ---------------------------------------------------------------
    % Simulation problem parameters
    max_itr = 20;           % Maximum number of optimizer iterations
    epsilon = 1e-4;         % Convergence criteria
    N       = 41;           % Number of measurement points (N > 1)
    T       = 4.;           % Total time
    ell     = 1;            % Number of constraints (never active)
    n       = 2;            % Number of components in the state vector (n = 2)
    m       = 1;            % Number of measurements at each time point (m <= n)
    xi      = [ 0. , -5.]'; % True initial value for the state at time zero
    x_in    = zeros(n, N);  % Initial sequence where optimization begins
    initial = [ 0. , -5.]'; % Estimate of initial state (at time index one)
    mu      = 2.0;          % ODE nonlinearity parameter
    sigma_r = 1.;           % Standard deviation of measurement noise
    sigma_q = .1;           % Standard deviation of the transition noise
    sigma_e = 10;           % Standard deviation of initial state estimate
    randn('seed', 4321);    % Random number generator seed
    % -------------------------------------------------------------------
    % Level of tracing during optimization
    if draw_plot
        level = 1;
    else
        level = 0;
    end
    % spacing between time points
    dt = T / (N - 1);
    % -------------------------------------------------------------------
    %  information used by vanderpol_g
    params.vanderpol_g_initial  = initial;
    params.vanderpol_g_dt       = dt;
    params.vanderpol_g_mu       = mu;
    % -------------------------------------------------------------------
    % global information used by direct_h
    %global direct_h_info
    params.direct_h_index = 1;
    % ---------------------------------------------------------------
    % Rest of the information required by ckbs_nonlinear
    %
    % Step size for fourth order Runge-Kutta method in vanderpol_sim
    % is the same as time between measurement points
    step    = dt;
    % Simulate the true values for the state vector
    x_true  = vanderpol_sim(mu, xi, N, step);
    time    = linspace(0., T, N);
    % Simulate the measurement values
    z       = x_true(1:m,:) + sigma_r * randn(m, N);
    % Inverse covariance of the measurement and transition noise
    rinv    = zeros(m, m, N);
    qinv    = zeros(n, n, N);
    for k = 1 : N
        rinv(:, :, k) = eye(m, m) / sigma_r^2;
        qinv(:, :, k) = eye(n, n) / sigma_q^2;
    end
    qinv(:, :, 1) = eye(n, n) / sigma_e^2;
    % ---------------------------------------------------------------
    % call back functions
    f_fun = @ no_f;
    g_fun = @ (k,x) vanderpol_g(k,x,params);
    h_fun = @(k,x) direct_h(k,x,params);
    % ---------------------------------------------------------------
    % call the optimizer
    [ x_out , u_out, info ] = ckbs_nonlinear( ...
        f_fun ,    ...
        g_fun ,    ...
        h_fun ,    ...
        max_itr ,  ...
        epsilon ,  ...
        x_in ,     ...
        z ,        ...
        qinv ,     ...
        rinv ,     ...
        level      ...
        );
    % ----------------------------------------------------------------------
    % Check that x_out is optimal
    % (this is an unconstrained case, so there is no need to check f).
    ok     = size(info, 1) <= max_itr;
    g_out  = zeros(n,   N);
    h_out  = zeros(m,   N);
    dg_out = zeros(n, n,   N);
    dh_out = zeros(m, n,   N);
    xk     = zeros(n, 1);
    for k = 1 : N
        xkm      = xk;
        %
        xk       = x_out(:, k);
        uk       = u_out(:, k);
        [gk, Gk] = g_fun(k, xkm);
        [hk, Hk] = h_fun(k, xk);
        %
        g_out(:, k) = gk - xk;
        h_out(:, k) = hk;
        dg_out(:,:, k) = Gk;
        dh_out(:,:, k) = Hk;
    end
    dx    = zeros(n, N);
    grad  = ckbs_sumsq_grad(dx, z, g_out, h_out, dg_out, dh_out, qinv, rinv);
    ok    = max( max( abs(grad) ) ) < epsilon;
    % ----------------------------------------------------------------------
    if draw_plot
        figure(1);
        clf
        hold on
        plot(time, x_true(1,:), 'b-');
        plot(time, x_out(1,:),  'g-');
        plot(time, z(1,:),      'ko');
        title('Position: blue=truth, green=estimate, o=data');
        hold off
        return
    end
end

Input File: example/nonlinear/vanderpol_ok.m