Version 3.0 of the Hybrid Equation Toolbox includes substantial improvements to most aspects of the MATLAB-based solver and plotting functions. All code written using v2.04 should continue to work on v3.0, but will need to be to modified to take advantage of some of the new features. This document describes how to make those change. Hybrid systems defined in Simulink can be left unmodified, but the plotting of solutions can be updated to take advantage of new tools.

Contents

Defining Hybrid Systems

In v2.04, a hybrid system was defined by creating four MATLAB function files C.m , f.m , D.m , and g.m anywhere on the MATLAB search path. Examples for the bouncing ball are shown at the end of this section. The only way that these four functions are grouped together, as far as MATLAB knows, is that they are all be passed together to HyEQsolver , e.g., HyEQsolver(@f, @g, @C, @D, ...) .

In v3.0, we introduced the HybridSystem class so that all of the data of a hybrid system is encapsulated in a single place. This has numerous benfits:

  • All of the data of hybrid system are defined in one place.
  • It is possible to have many hybrid systems defined on the MATLAB path without worrying about name conflicts.
  • It is no longer necessary to use global variables to define constants (see gamma and lambda , below).

For a detailed description of how to write HybridSystem classes, see How to Implement and Solve a Hybrid System . Now, you might not be interested in rewriting existing code but still want to take advantage of new features. As a half-way step, we have provided a method for creating a HybridSystem object from existing C.m , f.m , D.m , and g.m files: 

sys = HybridSystem(@f, @g, @C, @D);

The sys object can then be used just like any other HybridSystem . There are a couple of disadvantages of this approach, however. Namely, it misses out on the advantages associated with defining all of a hybrid system's data in a single file, listed above, and the computation of solutions is slower and harder to debug. For these reasons, we strongly reccomend writing new HybridSystem subclasses whenever writing new code.

Example Function Files for the Bouncing Ball Hybrid System

C.m : 

function in_C = C(x) % Flow set indicator function
    in_C = x(1) >= 0;
end

f.m : 

function xdot = f(x) % Flow map
    global gamma
    xdot = [x(2); gamma];
end

D.m : 

function in_D = D(x) % Jump set indicator function
    in_D = x1 <= 0 && x2 <= 0
end

g.m : 

function xplus = g(x) % Jump map
    global lambda
    xplus = [-x(1) ; -lambda*x2];
end

Solving Hybrid Systems

In v2.04, computing a solution required the data of the system, the initial condition, time horizons, jump/flow priority, and ODE solver options to HyEQsolver : 

% Initial conditions
x0 = [1; 0];

% Simulation horizon
TSPAN = [0 10];
JSPAN = [0 20];

% Set the behavior of the simulation in the intersection of C and D.
rule = 1;

% ODE solver options.
options = odeset('RelTol',1e-6, 'MaxStep', 0.1);

[t j x] = HyEQsolver(@f, @g, @C, @D, x0, TSPAN, JSPAN, rule, options);

For v3.0, the above process is split into two steps. First, you define a hybrid system, as we did for sys , above. Then call the sys.solve to comput the solution. The function sys.solve function takes as arguments x0 , tspan , and jspan , similar to HyEQsolver . Where it differs is in the (optional) fourth argument config , which is a HybridSolverConfig object used to define the jump/flow priorty and ODE solver options, as well as controlling how progress updates are displayed. To use the ODE option given above, simply pass change options = odeset('RelTol',1e-6, 'MaxStep', 0.1); to 

config = HybridSolverConfig('RelTol',1e-6, 'MaxStep', 0.1);

Then, to set jump priority in the intersection of C and D, call config.priority('jump'); (to set flow priority, this would be config.priority('flow'); ). Finally, to compute the solution, call 

sol = sys.solve(x0, tspan, jspan, config);

The resulting object is a HybridSolution object (which is a subclass of the HybridArc class). To extract the values of x, t, j in order to, say, continue using the v2.04 plotting functions, simply call 

t = sol.t;
j = sol.j;
x = sol.x;

Plotting Solutions

Creating plots with v3.0 is easier and more flexible. However, it requires that the data you want to plot is contained in a HybridArc object (or a HybridSolution object). If you are still using HyEQsolver to compute solutions, then your data is in the form of three variables, t, j, x . Thankfully, you can generate a HybridArc from this data by simply calling 

sol = HybridArc(t, j, x);

Then, the hybrid arc can be plotted using all of the tools described in Plotting Hybrid Arcs .

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