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# odedc

discrete/continuous ode solver

### Syntax

`yt = odedc(y0, nd, stdel, t0, t, f)`

### Arguments

y0

a real column vector (initial conditions), `y0=[y0c;y0d]` where `y0d` has `nd` components.

nd

an integer, dimension of `y0d`

stdel

a real vector with one or two entries, `stdel=[h, delta]` (with `delta=0` as default value).

t0

a real scalar (initial time).

t

a real (row) vector, instants where `yt` is calculated.

f

an external i.e. a function or a character string or a list with syntax: `yp = f(t,yc,yd,flag)`.

f

an external i.e. a function or a character string or a list with syntax: `yp = f(t,yc,yd,flag)`

a list

This form of external is used to pass parameters to the function. It must be as follows:

```list(f, p1, p2,...)
```

where the syntax of the function `f` is now

```yp = f(t, yc, yd, flag, p1, p2,...)
```

`f` still returns the function value as a function of `(t, yc, yd, flag, p1, p2,...)`, and `p1, p2,...` are function parameters.

a character string

it must refer to the name of a C or fortran routine, assuming that <`f_name`> is the given name.

• The Fortran calling sequence must be

`<f_name>(iflag, nc, nd, t, y, ydp)`

`double precision t, y(*), ydp(*)`

`integer iflag, nc, nd`

• The C syntax must be

```void <f_name> (int *iflag, int *nc, int *nd, double *t, double *y, double *ydp)```

In both Fortran and C cases, the input arguments are:

• `iflag` = `0` or `1`

• `nc` = number of continuous states `yc`

• `nd` = number of discrete states `yd`

• `t` = time

• `y` = `[yc; yd; param]`. param may be used to get extra arguments which have been given in the odedc call `(y = odedc([y0c; y0d], nd, stdel, t0, t, list('fexcd', param)))`

• As output `ydp`, the routine must compute `ydp[0:nc-1] = d/dt ( yc(t) )` for `iflag=0` and `ydp[0:nd-1] = yd(t+)` for `iflag=1`.

### Description

`y = odedc([y0c;y0d], nd, [h,delta], t0, t, f)` computes the solution of a mixed discrete/continuous system. The discrete system state `ydk` is embedded into a piecewise constant `yd(t)` time function as follows: `yd(t) = ydk` for `t` in `[tk = h*(delta+k) , tk+1 = h*(delta+k+1) )`.

The simulated equations are now, for t in [ tk , tk+1 ):

```dyc/dt = f(t, yc(t), yd(t), 0)
yc(t0) = y0c
```

and at instants `tk` the discrete variable `yd` is updated by

```yd(tk+) = f(yc(tk-), yd(tk-), 1)```.

Note that, using the definition of `yd(t)` the last equation gives

```ydk = f(tk, yc(tk-), yd(tk-1), 1)```

(`yc` is time-continuous: `yc(tk-) = yc(tk)`).

The calling parameters of `f` are fixed: `ycd = f(t,yc,yd,flag)`; this function must return either the derivative of the vector `yc` if `flag=0` or the update of `yd` if `flag=1`.

`ycd = dot(yc)` must be a vector with same dimension as `yc` if `flag=0` and `ycd = update(yd)` must be a vector with same dimension as `yd` if `flag=1`.

`t` is a vector of instants where the solution `y` is computed.

`y` is the vector `y = [y(t(1)), y(t(2)),...]`.

This function can be called with the same optional parameters as the `ode` function (provided `nd` and `stdel` are given in the syntax as second and third parameters). In particular integration flags, tolerances can be set. Optional parameters can be set by the `odeoptions` function.

### Examples

```//Linear system with switching input
deff('xdu = phis(t,x,u,flag)', 'if flag==0 then xdu=A*x+B*u; else xdu=1-u; end');
x0 = [1;1];
A = [-1,2;-2,-1];
B = [1;2];
u = 0;
nu = 1;
stdel = [1,0];
u0 = 0;
t = 0:0.05:10;
xu = odedc([x0;u0], nu, stdel, 0, t, phis);
x = xu(1:2,:);
u = xu(3,:);
nx = 2;
plot2d(t, x', [1:nx], '161')
plot2d2('onn', t', u', [nx+1:nx+nu], '000');
//Fortran external (see fydot2.f):
norm(xu - odedc([x0;u0], nu, stdel, 0, t, 'phis'), 1)```
```//Sampled feedback
//
//        |     xcdot=fc(t,xc,u)
//  (system)   |
//        |     y=hc(t,xc)
//
//
//        |     xd+=fd(xd,y)
//  (feedback) |
//        |     u=hd(t,xd)
//
deff('xcd = f(t,xc,xd,iflag)',...
['if iflag==0 then '
'  xcd=fc(t,xc,e(t)-hd(t,xd));'
'else '
'  xcd=fd(xd,hc(t,xc));'
'end']);
A = [-10,2,3;4,-10,6;7,8,-10];
B = [1;1;1];
C = [1,1,1];
Bd = [1;1];
Cd = [1,1];
deff('st = e(t)', 'st = sin(3*t)')
deff('xdot = fc(t,x,u)', 'xdot = A*x+B*u')
deff('y = hc(t,x)', 'y = C*x')
deff('xp = fd(x,y)', 'xp = Ad*x + Bd*y')
deff('u = hd(t,x)', 'u = Cd*x')
h = 0.1; t0=0; t=0:0.1:2;
x0c = [0;0;0];
x0d = [0;0];
nd = 2;
xcd = odedc([x0c;x0d], nd, h, t0, t, f);
norm(xcd - odedc([x0c;x0d], nd, h, t0, t, 'fcd1')) // Fast calculation (see fydot2.f)
plot2d([t',t',t'], xcd(1:3,:)');
scf(2);
plot2d2("gnn", [t',t'], xcd(4:5,:)');
scf(0);```

• ode — solveur d'équations différentielles ordinaires