LAEG31
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6 days ago %% Simulation of Bouncing Ball % % This example uses two models of a bouncing ball to show different % approaches to modeling hybrid dynamic systems with Zeno behavior. Zeno % behavior is informally characterized by an infinite number of events % occurring in a finite time interval for certain hybrid systems. As the % ball loses energy, the ball collides with the ground in successively % smaller intervals of time. % % Copyright 2009-2022 The MathWorks, Inc. %% Hybrid Dynamic Systems % A bouncing ball model is an example of a hybrid dynamic system. A hybrid % dynamic system is a system that involves both continuous dynamics and % discrete transitions where the system dynamics can change and the state % values can jump. The continuous dynamics of a bouncing ball are given by % these equations: % % $$ % \frac{d v}{d t} = -g, % $$ % % $$ % \frac{d x}{d t} = v, % $$ % % where $g$ is the acceleration due to gravity, $x(t)$ is the position of % the ball, and $v(t)$ is the velocity. The system has two % continuous states: the position $x$ and the velocity $v$. % % The hybrid system aspect of the model originates from the modeling of a % collision of the ball with the ground. If one assumes a partially % elastic collision with the ground, then the velocity before the % collision, $v^-$, and velocity after the collision, $v^+$, can be related % by the coefficient of restitution of the ball, $\kappa$, as follows: % % $$ % v^+ = -\kappa v^-, \qquad x = 0 % $$ % % The bouncing ball therefore displays a jump in a continuous state % (velocity) at the transition condition, $x=0$. The image shows a ball % thrown up with a velocity of 0 m/s from a height of 25 m. % % <<../sldemo_bounce_overview.png>> % %% Use Two Integrator Blocks to Model Bouncing Ball % % The |sldemo_bounce_two_integrators| model uses two Integrator blocks to % model a bouncing ball. The Integrator block on the left is the velocity % integrator modeling the first equation. The Integrator block on the right % is the position integrator. Open the Block Parameters dialog box for the % position integrator to see that the block has a % lower limit of zero. This condition represents the constraint that the % ball cannot go below the ground. % % The state port of the position integrator and the corresponding % comparison result are used to detect when the ball hits the ground and to % reset both integrators. The state port of the velocity integrator is % used for the calculation of $v^+$. % open_system('sldemo_bounce_two_integrators'); set_param('sldemo_bounce_two_integrators', 'ZeroCrossAlgorithm','Adaptive'); set_param('sldemo_bounce_two_integrators', 'StopTime','25'); warning('off', 'Simulink:Solver:ZeroCrossingNotBracketedDueToSmallSignalValues'); out=sim('sldemo_bounce_two_integrators', 'SaveTime', 'on', 'SaveState', 'on'); ti=out.tout; xi=out.xout;
%%
% To observe the Zeno behavior of the system, modify solver configuration
% parameters.
%
% # To open the Configuration Parameters dialog box, on the
% Modeling tab, under Setup, click Model Settings.
% # Select the Solver pane.
% # Set the Stop time to |25|.
% # Click the arrow next to Solver details to view additional solver
% parameters.
% # Under Zero-crossing options, set Algorithm to |Nonadaptive|.
%
% Simulate the model.
%
% As the ball hits the ground more frequently and loses energy, the
% simulation exceeds the default Number of consecutive zero crossings
% limit of |1000|.
%
% In the Configuration Parameters dialog box, set Algorithm to
% |Adaptive|. The adaptive algorithm introduces a sophisticated treatment
% for chattering behavior. You can now simulate the system beyond 20
% seconds. The chatter of the states between 21 seconds and
% 25 seconds is still large, and the software issues a warning around 20 seconds.
%% Use Second-Order Integrator Block to Model Bouncing Ball % % The |sldemo_bounce| model uses a single Second-Order Integrator block to % model a bouncing ball. In this model, the second equation $dx/dt=v$ is % internal to the Second-Order Integrator block. Open the % Second-Order Integrator block dialog box and see that $x$ has a lower % limit of zero. On the Attributes tab, select |Reinitialize dx/dt when x % reaches saturation|. This parameter allows you to reinitialize $dx/dt$ % ($v$ in the bouncing ball model) to a new value when $x$ reaches its % saturation limit. So, in the bouncing ball model, when the ball hits the % ground, its velocity can be set to a different value, such as to the % velocity after the impact. Note the loop for calculating the velocity % after a collision with the ground. To capture the velocity $v^-$ of the % ball just before the collision, the $dx/dt$ output port of the % Second-Order Integrator block and a Memory block are used. $v^-$ is then % used to calculate the rebound velocity $v^+$.
open_system('sldemo_bounce'); out=sim('sldemo_bounce', 'SaveTime', 'on', 'SaveState', 'on'); ts=out.tout; xs=out.xout;
%% % In the Configuration Parameters dialog box, go to the Solver pane. % % % In Simulation time, set Stop time to |25|. % Expand Solver details. In Zero-crossing options, set Algorithm to % |Nonadaptive| . % % Simulate the model. % % Note that the simulation encounters no problems. You can simulate the % model without experiencing excessive chatter after 20 seconds and without % setting Algorithm to |Adaptive|.
%% Compare Approaches to Modeling Bouncing Ball
%
% You can analytically calculate the exact time $t^$ when the ball settles
% down to the ground with zero velocity by summing the time required for
% each bounce. This time is the sum of an infinite geometric series given
% by:
%
% $$ t^ = \frac{1}{g}\left(v_0+ v_1\left(\frac{1+\kappa}{1-\kappa}\right)
% \right),\qquad v_1=\sqrt{v_0^2+2gx_0}, $$
%
% where $x_0$ and $v_0$ are initial conditions for position and velocity,
% respectively. The velocity and the position of the ball must be
% identically zero for $t>t^$. The figure shows results from both
% simulations near $t^$. The vertical red line in the plot is $t^$ for
% the given model parameters. For $t<t^$ and far away from $t^$, both
% models produce accurate and identical results. Only a magenta line from
% the second model is visible in the plot. However, the simulation results
% from the first model are inexact after $t^$. The plot continues to
% display excessive chattering behavior for $t>t^$. In contrast, the
% model that uses the Second-Order Integrator block settles to exactly
% zero for $t > t^$.
g = 9.81; v0 = 0; x0 = 25; k = 0.8; v1=sqrt(v0v0+2gx0); kfactor = (1+k)/(1-k); tzeno = (1/g)(v0+v1*kfactor); is = find(ts > 20.0, 1); ii = find(ti > 20.0, 1);
plot (ti(ii:end),xi(ii:end,1),'b'); hold on; plot (ts(is:end),xs(is:end,1),'m'); hold on; X = [tzeno, tzeno]; Y = [-1E-04 5.2E-03]; line('XData', X, 'YData', Y, 'Color','r'); axis([20 22 -1E-04 5.2E-03]); xlabel('Time'); ylabel('Position'); legend('Integrator','Second-Order Integrator', 't*');
%% % The model that uses the Second-Order Integrator block has superior % numerical characteristics compared to the first model because the second % differential equation $dx/dt=v$ is internal to the Second-Order % Integrator block. The block algorithms can leverage this relationship % between the two states and use heuristics to clamp down chattering % behavior for certain conditions. These heuristics become active when the % two states are no longer mutually consistent due to integration errors % and chattering behavior. You can thus use physical knowledge of the % system to prevent simulations getting stuck in a Zeno state for % certain classes of Zeno models.
bdclose all warning('on', 'Simulink:Solver:ZeroCrossingNotBracketedDueToSmallSignalValues');
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