def get_equation_io(self, eqn_sys: EqnSys):
# determine all signals present in the set of equations
all_signal_names = set(signal_names(eqn_sys.get_all_signals()))
# determine inputs
input_names = (signal_names(self.get_analog_inputs())
| self.assignments.keys()) & all_signal_names
inputs = self.get_signals(input_names)
# determine states
state_names = set(signal_names(eqn_sys.get_states()))
deriv_names = set(signal_names(eqn_sys.get_derivs()))
states = self.get_signals(state_names)
# determine outputs
output_names = (all_signal_names - input_names - state_names -
deriv_names) & self.signals.keys()
outputs = self.get_signals(output_names)
# determine sel_bits
sel_bit_names = set(signal_names(eqn_sys.get_sel_bits()))
sel_bits = self.get_signals(sel_bit_names)
# return result
return inputs, states, outputs, sel_bits
def main():
# equation display
print(EqnList([AnalogSignal('a') == AnalogSignal('b')]))
print()
# signal extraction
print(signal_names(EqnList([AnalogSignal('a') == AnalogSignal('b')]).get_all_signals()))
print(signal_names(EqnList([AnalogSignal('a') == Deriv(AnalogSignal('b'))]).get_derivs()))
print(signal_names(EqnList([AnalogSignal('a') == Deriv(AnalogSignal('b'))]).get_states()))
print(signal_names(EqnList([AnalogSignal('a') == EqnCase([1, 2], [DigitalSignal('s')])]).get_sel_bits()))
def make_array(self, expr: Array):
# make the output signal that will hold the results
output = Signal(name=next(self.namer), format_=expr.format_)
self.make_signal(output)
# compile the array elements and address to signals
elements = [self.expr_to_signal(element) for element in expr.elements]
address = self.expr_to_signal(expr.address)
# extra step for analog signals -- they must be aligned to the output format
if isinstance(expr.format_, RealFormat):
# rename the elements list to indicate these values are not aligned to the output format
non_aligned_elements = elements
# created an "aligned" version of each signal
elements = [Signal(name=next(self.namer), format_=expr.format_) for _ in range(len(expr))]
for element, non_aligned_element in zip(elements, non_aligned_elements):
self.make_signal(element)
self.make_assign(input_=non_aligned_element, output=element)
# now create the array
case_statment(gen=self, sel=address.name, var=output.name, values=signal_names(elements), default=0)
# and last return the output signal
return output
def make_concatenation(self, expr: Concatenate):
output = Signal(name=next(self.namer), format_=expr.format_)
self.make_signal(output)
inputs_ = [self.expr_to_signal(operand) for operand in expr.operands]
value = '{' + ', '.join(signal_names(inputs_)) + '}'
self.digital_assignment(signal=output, value=value)
return output
def make_bitwise_operator(self, expr: BitwiseOperator):
output = Signal(name=next(self.namer), format_=expr.format_)
self.make_signal(output)
inputs = [self.expr_to_signal(operand) for operand in expr.operands]
value = BITWISE_OP[type(expr)].join(signal_names(inputs))
self.digital_assignment(signal=output, value=value)
return output
def main():
from msdsl.eqn.deriv import Deriv
from msdsl.eqn.cases import eqn_case
from msdsl.expr.signals import DigitalSignal
model = MixedSignalModel('test', AnalogInput('x'), dt=1)
y = AnalogSignal('y')
z = model.add_signal(AnalogSignal('z', 10))
s = model.add_signal(DigitalSignal('s'))
eqn_sys = EqnSys(
[y == model.x + 1,
Deriv(z) == (y - z) * eqn_case([2, 3], [s])])
inputs, states, outputs, sel_bits = model.get_equation_io(eqn_sys)
print('inputs:', signal_names(inputs))
print('states:', signal_names(states))
print('outputs:', signal_names(outputs))
print('sel_bits:', signal_names(sel_bits))
def to_lds(self,
inputs: List[Signal] = None,
states: List[Signal] = None,
outputs: List[Signal] = None):
# set defaults
inputs = inputs if inputs is not None else []
states = states if states is not None else []
outputs = outputs if outputs is not None else []
# create list of derivatives of state variables
deriv_dict = {deriv.name: deriv for deriv in self.get_derivs()}
derivs = list(deriv_dict.values())
# sanity check: no repeated entries in inputs, states, derivatives, or outputs
assert len(set(signal_names(inputs))) == len(
inputs), 'Repeated entries in inputs.'
assert len(set(signal_names(outputs))) == len(
outputs), 'Repeated entries in outputs.'
assert len(set(signal_names(states))) == len(
states), 'Repeated entries in states.'
assert len(set(signal_names(derivs))) == len(
derivs), 'Repeated entries in derivatives.'
# sanity check: inputs, states, derivatives, and outputs should be disjoint
assert set(signal_names(inputs)).isdisjoint(
signal_names(outputs)), 'Inputs and outputs are not disjoint.'
assert set(signal_names(inputs)).isdisjoint(
signal_names(states)), 'Inputs and states are not disjoint.'
assert set(signal_names(inputs)).isdisjoint(signal_names(
derivs)), 'Inputs and state derivatives are not disjoint.'
assert set(signal_names(outputs)).isdisjoint(
signal_names(states)), 'Outputs and states are not disjoint.'
assert set(signal_names(outputs)).isdisjoint(signal_names(
derivs)), 'Outputs and state derivatives are not disjoint.'
assert set(signal_names(states)).isdisjoint(signal_names(
derivs)), 'States and state derivatives are not disjoint.'
# sanity check: signal names of derivatives should be the states
assert set(signal_names(states)) == set(
signal_names([deriv.signal for deriv in derivs]))
# create list of all internal signals, then use it to figure out what signals are completely internal
external_names = set(signal_names(inputs + outputs + states + derivs))
internal_name_set = set(signal_names(
self.get_all_signals())) - external_names
# indices of known and unknown variables
unknowns = list2dict(
list(internal_name_set) + signal_names(outputs) +
signal_names(derivs))
knowns = list2dict(signal_names(inputs) + signal_names(states))
# sanity checks
assert not (
len(self) > len(unknowns)
), f'System of equations is over-constrained with {len(self)} equations and {len(unknowns)} unknowns.'
assert not (
len(self) < len(unknowns)
), f'System of equations is under-constrained with {len(self)} equations and {len(unknowns)} unknowns.'
# build up matrices
U = np.zeros((len(self), len(unknowns)), dtype=float)
V = np.zeros((len(self), len(knowns)), dtype=float)
for row, eqn in enumerate(self):
# prepare equation for analysis
eqn = eqn.lhs - eqn.rhs
eqn = distribute_mult(eqn)
# extract coefficients of signals (note that some signals may be repeated - we deal with this in the next step
coeffs, others = extract_coeffs(eqn)
assert len(others) == 0, \
'The following terms are not yet handled: ['+ ', '.join(str(other) for other in others)+']'
# sum up all of the coefficients for each signal
for coeff, signal in coeffs:
if signal.name in unknowns:
U[row, unknowns[signal.name]] += +coeff
elif signal.name in knowns:
V[row, knowns[signal.name]] += -coeff
else:
raise Exception(
'Variable is not marked as known vs. unknown: ' +
signal.name)
# solve for unknowns in terms of knowns
M = np.linalg.solve(U, V)
# separate into A, B, C, D matrices
if len(states) > 0:
A = np.zeros((len(states), len(states)), dtype=float)
for row, out_state in enumerate(signal_names(states)):
for col, in_state in enumerate(signal_names(states)):
A[row, col] = M[unknowns[deriv_str(out_state)],
knowns[in_state]]
else:
A = None
if len(states) > 0 and len(inputs) > 0:
B = np.zeros((len(states), len(inputs)), dtype=float)
for row, out_state in enumerate(signal_names(states)):
for col, in_input in enumerate(signal_names(inputs)):
B[row, col] = M[unknowns[deriv_str(out_state)],
knowns[in_input]]
else:
B = None
if len(outputs) > 0 and len(states) > 0:
C = np.zeros((len(outputs), len(states)), dtype=float)
for row, out_output in enumerate(signal_names(outputs)):
for col, in_state in enumerate(signal_names(states)):
C[row, col] = M[unknowns[out_output], knowns[in_state]]
else:
C = None
if len(outputs) > 0 and len(inputs) > 0:
D = np.zeros((len(outputs), len(inputs)), dtype=float)
for row, out_output in enumerate(signal_names(outputs)):
for col, in_input in enumerate(signal_names(inputs)):
D[row, col] = M[unknowns[out_output], knowns[in_input]]
else:
D = None
return LDS(A=A, B=B, C=C, D=D)
def add_eqn_sys(self, eqns: List[ModelExpr], extra_outputs=None):
"""
Accepts a list of equations that can contain derivatives of analog state variables. The approach used is
to convert the system of differential equations into a standard-form linear dynamical system (reference:
http://ee263.stanford.edu/lectures/linsys.pdf). If there are several eqn_cases to be considered, we construct
a different LDS for each one. These LDS's are discretized using the given timestep by using the exponential
matrix function assuming piecewise-constant input (sometimes known as the zero-order hold approach).
:param eqns: List of equations.
:param extra_outputs: List of internal variables in the system of equations that should be bound to analog signals.
"""
# set defaults
extra_outputs = extra_outputs if extra_outputs is not None else []
# create object to hold system of equations
eqn_sys = EqnSys(eqns)
# analyze equation to find out knowns and unknowns
inputs, states, outputs, sel_bits = self.get_equation_io(eqn_sys)
# add the extra outputs as needed
for extra_output in extra_outputs:
if not isinstance(extra_output, Signal):
print('Skipping extra output ' + str(extra_output) +
' since it is not a Signal.')
elif extra_output.name in signal_names(outputs):
print('Skipping extra output ' + extra_output.name + \
' since it is already included by default in the outputs of the system of equations.')
else:
outputs.append(extra_output)
# initialize lists of matrices
collection = LdsCollection()
# iterate over all of the bit combinations
for k in range(2**len(sel_bits)):
# substitute values for this particular setting
sel_bit_settings = address_to_settings(k, sel_bits)
eqn_sys_k = eqn_sys.subst_case(sel_bit_settings)
# convert system of equations to a linear dynamical system
lds = eqn_sys_k.to_lds(inputs=inputs,
states=states,
outputs=outputs)
# discretize linear dynamical system
lds = lds.discretize(dt=self.dt)
# add to collection of LDS systems
collection.append(lds)
# construct address for selection
if len(sel_bits) > 0:
sel = concatenate(sel_bits)
else:
sel = None
# add the discrete-time equation
self.add_discrete_time_lds(collection=collection,
inputs=inputs,
states=states,
outputs=outputs,
sel=sel)