~kdsch/ode2dsp

ref: ce5e508b6fb05322f476d3d2095e1226f9bbdd41 ode2dsp/ode2dsp/math.py -rw-r--r-- 9.6 KiB
ce5e508bKarl Schultheisz Implement stability report; fix some bugs 3 months ago
                                                                                
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from sympy import (
    Abs,
    Add,
    Basic,
    Dummy,
    Eq,
    Function,
    Matrix,
    pprint,
    pretty,
    re,
    solve,
    Subs,
    Symbol,
    symbols,
    Tuple,
    ZeroMatrix,
)
from typing import Any
from sympy.core.function import AppliedUndef
from sympy.logic import And
from sympy.integrals import laplace_transform, LaplaceTransform


def symbol_matrix(name, length):
    return Matrix(symbols(f"{name}_:{length}", real=True))


def function_matrix(name, length, *args):
    return Matrix(
        [func(*args) for func in symbols(f"{name}_:{length}", cls=Function, real=True)]
    )


class Report:
    def __init__(self, ode, fde, sfde, ode_stable, fde_stable, sfde_stable):
        self.ode = ode
        self.fde = fde
        self.sfde = sfde
        self.ode_stable = ode_stable
        self.fde_stable = fde_stable
        self.sfde_stable = sfde_stable

    @classmethod
    def from_spec(cls, ode, discretize, time, sample_freq, n, point):
        fde = FDE.from_ode(ode, discretize, time, sample_freq)
        solved_fde = SolvedFDE.from_fde(fde, n)
        return Report(
            ode,
            fde,
            solved_fde,
            ode.stable(point),
            fde.stable(point),
            solved_fde.stable(point),
        )

    def __str__(self):
        items = (
            ("ODE", self.ode),
            ("ODE stability condition", self.ode_stable),
            ("FDE", self.fde),
            ("FDE stability condition", self.fde_stable),
            ("Solved FDE", self.sfde),
            ("Solved FDE stability condition", self.sfde_stable),
        )

        def format(arg):
            label, value = arg
            return f"{label}:\n\n{pretty(value)}"

        return "\n\n\n".join(map(format, items))


class ODE(Basic):
    # Assumes state-space form:
    #
    #      d
    #    ----- state = expr(state, time)
    #    dtime
    #
    # See https://ccrma.stanford.edu/~jos/pasp/General_Nonlinear_ODE.html
    #
    def __init__(self, time: Symbol, inputs, state: Matrix, expr: Matrix):
        self.time = time
        self.inputs = inputs
        self.state = state
        self.expr = expr

    def _autonomize(self):
        return ODE(
            self.time,
            (),
            self.state,
            self.expr.subs([(input, 0) for input in self.inputs]),
        )

    def stable(self, operating_point: Matrix):
        ode = self._autonomize()
        return _ode_stability_condition(ode.expr, ode.state, ode.time, operating_point)

    def __str__(self):
        rep = Eq(self.state.diff(self.time), self.expr)
        return pretty(rep)


class FDE(Basic):
    # Assumes the form
    #
    #    0 = expr(state, time)
    #
    def __init__(self, time: Symbol, inputs, state: Matrix, expr: Matrix):
        self.time = time
        self.inputs = inputs
        self.state = state
        self.expr = expr

    @classmethod
    def from_ode(cls, ode: ODE, discretize, time: Symbol, sample_freq: Symbol):
        return FDE(
            time,
            ode.inputs.subs(ode.time, time),
            ode.state.subs(ode.time, time),
            discretize(ode.state, ode.expr, ode.time, time, sample_freq),
        )

    def _autonomize(self):
        return FDE(
            self.time,
            (),
            self.state,
            self.expr.subs([(input, 0) for input in self.inputs]),
        )

    def stable(self, operating_point: Matrix):
        fde = self._autonomize()
        return _fde_stability_condition(fde.expr - fde.state, fde.state, fde.time, operating_point)

    def __str__(self):
        rep = Eq(ZeroMatrix(*self.state.shape), self.expr)
        return pretty(rep)


class SolvedFDE(Basic):
    # Assumes the form
    #
    #    state = expr(state, time)
    #
    # which must be an explicit equation.
    #
    def __init__(self, time: Symbol, inputs, state: Matrix, expr: Matrix):
        self.time = time
        self.inputs = inputs
        self.state = state
        self.expr = expr

    @classmethod
    def from_fde(cls, fde: FDE, n):
        return SolvedFDE(
            fde.time,
            fde.inputs,
            fde.state,
            _solve_fde(fde.state, fde.expr, fde.time, n),
        )

    def _autonomize(self):
        return SolvedFDE(
            self.time,
            (),
            self.state,
            self.expr.subs([(input, 0) for input in self.inputs]),
        )

    def stable(self, operating_point: Matrix):
        sfde = self._autonomize()
        return _fde_stability_condition(
            sfde.expr,
            sfde.state,
            sfde.time,
            operating_point,
        )

    def __str__(self):
        rep = Eq(self.state, self.expr)
        return pretty(rep)


def backward_euler(y, f, t, k, f_s):
    return Matrix(
        [
            -f_s * _backward_difference(y[i], t, k) + f[i].subs(t, k)
            for i in range(f.shape[0])
        ]
    )


def trapezoidal(y, f, t, k, f_s):
    return Matrix(
        [
            -2 * f_s * _backward_difference(y[i], t, k)
            + f[i].subs(t, k)
            + f[i].subs(t, k - 1)
            for i in range(f.shape[0])
        ]
    )


def _backward_difference(y, t, k):
    return y.subs(t, k) - y.subs(t, k - 1)


def _newton(n, y, func, k):
    guess = y.subs(k, k - 1)
    return Newton(n, func, y, guess)


class Newton(Function):
    nargs = 4

    @classmethod
    def eval(cls, n, f, var, guess):
        if not n.is_Number:
            return None
        if n == 0:
            return guess
        if n == 1:
            return guess - (f / f.diff(var)).subs(var, guess)
        if n > 1:
            expr = guess
            for _ in range(n):
                expr = Newton(1, f, var, expr)

            return expr


def _solve_fde(y, f, k, n):
    def solution(i):
        try:
            return solve(f[i], y[i])[0]
        except (IndexError, NotImplementedError):
            return _newton(n[i], y[i], f[i], k)

    return Matrix([solution(i) for i in range(f.shape[0])])


# stability checking
#         ↓


def _linearize(func, state, point):
    """
    Scalar multi-variate function

           #x               ∂
    f(x) ≅  Σ  f(p[n]) +  ----- f |   (x[n] - p[n])
           n=0            ∂x[n]   |x=p

    Vector multi-variate function

              #x                  ∂
    f[m](x) ≅  Σ  f[m](p[n]) +  ----- f[m] |   (x[n] - p[n])
              n=0               ∂x[n]      |x=p
    """
    substs = [(state[n], point[n]) for n in range(state.shape[0])]
    return Matrix(
        [
            Add(
                func[m].subs(substs),
                *[
                    func[m].diff(state[n]).subs(substs) * (state[n] - point[n])
                    for n in range(state.shape[0])
                ],
            )
            for m in range(func.shape[0])
        ]
    )


def _block_dc(func, time):
    return Matrix(
        [
            eqn.func(*list(filter(lambda term: term.has(time), eqn.args)))
            for eqn in func.expand()
        ]
    )


def _reduce(state, func, time):
    func = list(func)
    targets = state[1:]

    while len(targets) > 0 and len(func) > 1:
        target = targets.pop()
        index, solution = _index_solution(func, target, time)
        func = _remove(func, index).subs(target, solution)

    assert len(targets) == 0
    assert len(func) == 1

    return func[0].simplify()


def _index_solution(func, target, time):
    for index, eqn in enumerate(func):
        if eqn.has(target.diff(time)):
            continue

        solutions = solve(eqn, target)
        if len(solutions) == 1:
            return index, solutions[0]

        assert len(solutions) == 0

    return None, None


def _remove(func, index):
    return Matrix(func[:index] + func[index + 1 :])


def _upper_symbol(expr):
    return Symbol(str(expr.func).upper())


def _laplace_transform(char_eqn, t, s):
    def _symbolized_laplace(lt):
        expr = lt.args[0]
        if expr.is_Derivative:
            return s ** expr.args[1][1] * _upper_symbol(expr.args[0])
        elif isinstance(expr, AppliedUndef):
            return _upper_symbol(expr)

        raise BaseException(f"unexpected laplace transform of {expr}")

    lt_result = laplace_transform(char_eqn, t, s)
    if hasattr(lt_result, "__getitem__"):
        raise BaseException(char_eqn)

    return lt_result.replace(
        lambda e: e.func == LaplaceTransform,
        _symbolized_laplace,
    )


def _ode_stable(char_eqn, t):
    s = Dummy("s")
    return And(*[re(root) < 0 for root in solve(_laplace_transform(char_eqn, t, s), s)])


def _ode_stability_condition(func, state, time, point):
    return _ode_stable(
        _reduce(
            state,
            _block_dc(_linearize(func, state, point), time) - state.diff(time),
            time,
        ),
        time,
    )
    # Try this version when trying to eliminate .has(state.diff(time)) check in _reduce.
    # return _ode_stable(_reduce(state, _block_dc(_linearize(func, state, point), time), time) - state.diff(time), time)


def _get_shift(signal, k):
    time = signal.args[0]
    d = Dummy("d")
    return solve(Eq(time, k + d), d)[0]


def _z_transform(expr, k, z):
    return expr.replace(
        lambda e: isinstance(e, AppliedUndef) and len(e.args) == 1 and e.has(k),
        lambda e: z ** _get_shift(e, k) * _upper_symbol(e),
    )


def _fde_stable(y_k, f_k_, k):
    z = Dummy("z")
    Y, F = _z_transform(Tuple(y_k, f_k_), k, z)
    return And(*[Abs(root) < 1 for root in solve(_reduce(Y, F, Dummy()), z)])


def _fde_stability_condition(f_k, y_k, k, point):
    state = Matrix(list(y_k) + list(y_k.subs(k, k - 1)))
    point_ = Matrix(list(point) + list(point))
    return _fde_stable(y_k, _block_dc(_linearize(f_k, state, point_), k), k)