Creating custom acausal components¶
This tutorial assumes that the reader is familiar with the potential/flow (also known as through/across) variable approach for acausal modeling. Briefly:
- Potential:
- Physically: a quantity that is measured using a sensor placed across the component.
- Mathematically: a quantity that is the same at all ports connected at a node.
- Examples: voltage, pressure, temperature difference, relative position/speed/acceleration.
- Flow:
- Physically: a quantity that is measured using a sensor placed inline, in series with the component.
- Mathematically: a quantity that sums to zero for all ports connected at a node.
- Examples: current, mass flow, heat flow, force, torque.
From here on, acausal components will be referred to as 'components', so as not to be confused with 'blocks', which we use for causal modeling.
Components in jaxonomy are not used directly for simulation. Components are first connected together to form an AcausalDiagram, which is compiled by AcausalCompiler to create an AcausalSystem. AcausalSystem is a subclass of LeafSystem, and hence the AcausalSystem is used directly in simulation. AcausalSystem contains the index-1 DAEs, events, etc. needed by the simulator to advance the acausal model in time.
The equations of components are all collected together to form the 'system of equations' for the acausal model. The AcausalCompiler requires that symbols of these equations all use some common sympy.Symbol objects when defining symbols and equations. For this reason, it is required that all the components of an AcausalDiagram share the same 'equation environment'. This is achieved by creating an instance of EqnEnv class by writting ev = EqnEnv(), and passing ev to initializers and class methods where necessary. ev.t is the symbol for 'time' used in all equations, this is one reason why ev is passed in almost everywhere. ev also contains some useful constants, see EqnEnv class for details.
All components in Jaxonomy inherit from the base ComponentBase class. In order to implement a custom component, we can inherit from this base class and implement the functionality of the component. However, each domain provides subclasses of ComponentBase which can be used for creating one port or two port components for that domain with gerater ease.
As an example, the translational domain library includes TranslationalOnePort and TranslationalTwoPort, which can be used as follows to create custom components.
Use TranslationalOnePort to create a component which imposes aerodynamic drag force per: $ F _d = \frac{1}{2}*p*C _d*A*v^2 $.
class AeroDrag(TranslationalOnePort):
"""
Applies force equal to the aerodynamic drag based on the velocity of
its port.
"""
def __init__(self,
ev,
name=None,
p=1.2, # density of air, kg/m**3
A=1.0, # cross sectional area, m**2
Cd=0.1, # drag coefficient
)
self.name = self.__class__.__name__ if name is None else name
super().__init__(ev,self.name)
# Cd has been chosen as parameter available for optmization.
# p and A are static, so the model must be recompiled if these are changed.
self.Cd = self.declare_symbol(
ev,
"Cd",
self.name,
kind=SymKind.param,
val=M,
validator=lambda Cd: Cd > 0.0,
invalid_msg=f"Component {self.__class__.__name__} {self.name} must have Cd>0",
)
# add the equation that defines the component behavior.
self.add_eqs([sp.Eq(self.f.s, 0.5 * p * A * self.Cd * self.v.s**2)])
Use TranslationalTwoPort to create a fixed length bar with non-zero mass.
class RigidBar(TranslationalTwoPort):
"""
A rigid bar with length L, and mass M.
"""
def __init__(self,
ev,
name=None,
L=1.0,
M=1.0,
initial_velocity_A=0.0,
initial_velocity_A_fixed=False,
initial_position_A=0.0,
initial_position_A_fixed=False,
)
self.name = self.__class__.__name__ if name is None else name
super().__init__(
ev,
self.name,
v1_ic=initial_velocity_A,
v1_ic_fixed=initial_velocity_A_fixed,
x1_ic=initial_position_A,
x1_ic_fixed=initial_position_A_fixed,
v2_ic=initial_velocity_A,
v2_ic_fixed=initial_velocity_A_fixed,
x2_ic=initial_position_A + L,
x2_ic_fixed=initial_position_A_fixed,
)
# The component equation 0 = f1 + f1 was inherited from TranslationalTwoPort.
# Note: neither L nor M may be modified in optimization.
self.add_eqs(
[
sp.Eq(self.x1.s + L, self.x2.s),
sp.Eq(self.v1.s, self.v2.s),
sp.Eq(self.a1.s, self.a2.s),
sp.Eq(self.f1.s, M * self.a1.s),
]
)
As noted in the comments, declaring parameters using the declare_symbol method is necessary if (i) automatic differentiation with respect to these parameters is required; or (ii) the parameters need to be updated in the context. If these are not requirements, the parameters can be used in equations directly, or stored as instance attributes for other uses.
ComponentBase implementation¶
In order to implement components with more ports, or ports from multiple domains, we need to inherit from the ComponentBase class. While we refer to the documentation of the ComponentBase class for full details, here we provide an overview of basic functionality followed by a complete component example.
Acausal ports¶
To delcare a port, we use the declare_<domain>_port methods, and pass the appropriate arguments for the port domain. These methods will declare necessary symbols, declare any require derivative relationshsips between them, and return the symbols for potential and flow variables at the port. For example, to add an electrical port to our component we can write port_v, port_i = self.declare_electrical_port(ev, 'plus') in the initializer of the ComponentBase subclass. The port object added is then accessible by writing self.ports['plus'].
Parameters¶
As shown in the component examples above, parameters of components may be either static or dynamic.
To declare a parameter, we use the declare_symbol method, and pass in the appropriate arguments for a parameter. See Sym class for details. For example, to declare a parameter $a$ with a scalar value of 1.0, we can write a = self.declare_symbol(ev, "a", self.name, val=1.0) in the initializer of the ComponentBase subclass. The returned object cannot be directly used in the component equations, but rather the sympy.symbol attribute of the object, e.g. a.s.
Equations¶
AcausalCompiler handles equations which serve many different purposes in the context of the 'system of equations' for the acausal model. Some equations are defined by the components, while others are added by AcausalCompiler to complete the 'system of equations'. The available equation kinds are listed in the EqnKind class.
Component Equations¶
The behavior of components is realized by the 'component equations', identified by the EqnKind.comp kind. Component equations are necessary to define the relationship(s) between the potential and flow variables at the components' ports. Component equations can be declared by writting self.eqs([sp.Eq(Sym1.s, Sym2.s + Sym3.s)]). Note that a list is passed as the argument to this method, any number of component equations can be included in the list. Note that these equations will be classed as 'component equations' with the EqnKind.comp kind, so any equation which should not have this classification should not be included in the list.
Output Equations¶
In order to have AcausalSystem include a variable as a causal output, it is necessary to declare an output equation. This can be done by writting self.declare_equation(sp.Eq(output_symbol.s, some_component_variable.s), kind=EqnKind.outp). Note that it is necessary to first declare an output symbol to act as the referencne symbol for the output. This can be done by writting output_symbol = self.declare_symbol(ev, "output_symbol", self.name, kind=SymKind.outp). Note that the symbol must be classifed as SymKind.outp in order for AcausalCompiler to treat it approriately. Note also that the output equation must be explicitly classified as EqnKind.outp in order for AcausalCompiler to treat it appropriately.
Conditionals¶
It is common in component equations to need an If/Then/Else conditional to switch behavior depending on some condition. Symbols for conditional functions can be declared by writting:
cond = self.declare_conditional(
ev,
<conditional_expression>,
<true_expression>,
<false_expression>,
cond_name="cond",
)
See electrical.Diode for example uses.
Lookup Tables¶
One-dimensional and two-dimensional lookup tables can be used in component equations provided they are declared first. The declare_1D_lookup_table and declare_2D_lookup_table methods will return a symbol which is a symbolic representation of the function. For example, you can create a 2D efficiency map as follows. Note these methods also create the symbols for the breakpoint arrays and table matrix.
eff_lut = self.declare_2D_lookup_table(
ev,
spd.s, # symbol of the x axis of the map
"eff_spd", # name of x axis breakpoints symbol
eff_spd, # values of the x axis breakpoints
trq.s, # symbol of the y axis of the map
"eff_trq", # name of y axis breakpoints symbol
eff_trq, # values of the y axis breakpoints
"eff_eff", # name of the table points symbol
eff_eff, # values of the table points
"eff_lut", # name of the lookup talbe symbol
)
See electrical.IntegratorMotor for example uses.
Output ports¶
See 'Output Equations' above.
Events¶
Events can be delcared using the declare_zc method. These zero crossing events are necessary for the DAEs to be correctly integrated over discontinuties in themselves. This is rarely used on its own, but used when creating conditional symbols.
Putting it all together¶
Electrical Linear Actuator¶
Let's implement an electrical linear actuator. The component will have two eletrical ports and one translational port. It will have some mass associated with its translational port, and equations which convert energy between the electrical and translational domains. The component is ideal, and hence no energy loss for conversion in either direction. The equations are:
$ 0 = I _{pos} + I _{neg} $
$ 0 = f + K * I _{pos} - mass * a $
$ backEMF = K * v $
$ V - backEMF = I * R + \frac{\partial I}{\partial t} * L $
Further, let's include a force transducer in the component, so that the force becomes an output of the AcausalSystem.
This component is essentially the linear version of IdealMotor from the electrical library.
import sympy as sp
from jaxonomy.experimental.acausal.component_library.base import SymKind, EqnKind
from jaxonomy.experimental.acausal.component_library.component_base import (
ComponentBase,
)
class LinearActuator(ComponentBase):
def __init__(
self,
ev,
name=None,
R=1.0, # electrical resistance
K=1.0,
mass=1.0,
L=1.0,
):
self.name = self.__class__.__name__ if name is None else name
super().__init__()
# declare ports
Vp, Ip = self.declare_electrical_port(ev,"pos", I_ic=0.0, I_ic_fixed=True)
Vn, In = self.declare_electrical_port(ev, "neg")
f, x, v, a = self.declare_translational_port(ev,"flange")
# declare parameters
R = self.declare_symbol(
ev,
"R",
self.name,
kind=SymKind.param,
val=R,
validator=lambda R: R > 0.0,
invalid_msg=f"Component {self.__class__.__name__} {self.name} must have R>0",
)
K = self.declare_symbol(
ev,
"K",
self.name,
kind=SymKind.param,
val=K,
validator=lambda K: K > 0.0,
invalid_msg=f"Component {self.__class__.__name__} {self.name} must have K>0",
)
L = self.declare_symbol(
ev,
"L",
self.name,
kind=SymKind.param,
val=L,
validator=lambda L: L > 0.0,
invalid_msg=f"Component {self.__class__.__name__} {self.name} must have L>0",
)
mass = self.declare_symbol(
ev,
"mass",
self.name,
kind=SymKind.param,
val=mass,
validator=lambda mass: mass > 0.0,
invalid_msg=f"Component {self.__class__.__name__} {self.name} must have mass>0",
)
dI = self.declare_symbol(
ev, "dI", self.name, kind=SymKind.var, int_sym=Ip, ic=0.0
)
# declare component equations
self.add_eqs(
[
sp.Eq(0, Ip.s + In.s),
sp.Eq(0, f.s + K.s * Ip.s - mass.s * a.s),
sp.Eq( # 3rd and 4th equations combined
Vp.s - Vn.s - (K.s * v.s), Ip.s * R.s + dI.s * L.s
),
]
)
# create the output for the force
f_out = self.declare_symbol(ev, "f_out", self.name, kind=SymKind.outp)
self.declare_equation(sp.Eq(f_out.s, f.s), kind=EqnKind.outp)
We can now include the component in an AcausalDiagram. We'll make a system that drives the actuator with a constant voltage source. As a complement to our component's force sensor, we'll include a current sensor and translational motion sensor to observe the system.
import jaxonomy
import matplotlib.pyplot as plt
from jaxonomy.experimental import AcausalCompiler, AcausalDiagram, EqnEnv
from jaxonomy.experimental import electrical as elec
from jaxonomy.experimental import translational as trn
ev = EqnEnv()
ad = AcausalDiagram()
v = elec.VoltageSource(ev, name="v1", v=50.0, enable_voltage_port=False)
gnd = elec.Ground(ev, name="gnd")
act = LinearActuator(ev, name="act")
motion = trn.MotionSensor(ev, name="motion", enable_flange_b=False, enable_position_port=True)
curr = elec.CurrentSensor(ev, name="curr")
ad.connect(v, "p", curr, "p")
ad.connect(curr, "n", act, "pos")
ad.connect(v, "n", act, "neg")
ad.connect(v, "n", gnd, "p")
ad.connect(act, "flange", motion, "flange_a")
ac = AcausalCompiler(ev, ad)
acausal_system = ac()
builder = jaxonomy.DiagramBuilder()
acausal_system = builder.add(acausal_system)
diagram = builder.build()
context = diagram.create_context(check_types=True)
# run the simulation
spd_idx = acausal_system.outsym_to_portid[motion.get_sym_by_port_name("v_rel")]
x_idx = acausal_system.outsym_to_portid[motion.get_sym_by_port_name("x_rel")]
I_idx = acausal_system.outsym_to_portid[curr.get_sym_by_port_name("i")]
f_idx = acausal_system.outsym_to_portid[act.get_sym_by_port_name("f_out")]
recorded_signals = {
"spd": acausal_system.output_ports[spd_idx],
"x": acausal_system.output_ports[x_idx],
"I": acausal_system.output_ports[I_idx],
"f": acausal_system.output_ports[f_idx],
}
results = jaxonomy.simulate(
diagram,
context,
(0.0, 10.0),
recorded_signals=recorded_signals,
)
t = results.time
spd = results.outputs["spd"]
x = results.outputs["x"]
I = results.outputs["I"] # noqa
f = results.outputs["f"]
fig, (ax1, ax2, ax3, ax4) = plt.subplots(4, 1, figsize=(8, 3))
ax1.plot(t, spd, label="spd")
ax1.legend()
ax1.grid()
ax2.plot(t, x, label="x")
ax2.legend()
ax2.grid()
ax3.plot(t, I, label="I")
ax3.legend()
ax3.grid()
ax4.plot(t, f, label="f")
ax4.legend()
ax4.grid()
plt.show()
nodes={4: {(MotionSensor_motion, 'flange_a'), (LinearActuator_act, 'flange')}, 2: {(LinearActuator_act, 'neg'), (Ground_gnd, 'p'), (VoltageSource_v1, 'n')}, 1: {(LinearActuator_act, 'pos'), (CurrentSensor_curr, 'n')}, 0: {(CurrentSensor_curr, 'p'), (VoltageSource_v1, 'p')}}
AcausalCompiler.add_node_flow_eqs()
self.nodes={4: {(MotionSensor_motion, 'flange_a'), (LinearActuator_act, 'flange')}, 2: {(LinearActuator_act, 'neg'), (Ground_gnd, 'p'), (VoltageSource_v1, 'n')}, 1: {(LinearActuator_act, 'pos'), (CurrentSensor_curr, 'n')}, 0: {(CurrentSensor_curr, 'p'), (VoltageSource_v1, 'p')}}
AcausalCompiler.add_stream_balance_eqs()
self.nodes={4: {(MotionSensor_motion, 'flange_a'), (LinearActuator_act, 'flange')}, 2: {(LinearActuator_act, 'neg'), (Ground_gnd, 'p'), (VoltageSource_v1, 'n')}, 1: {(LinearActuator_act, 'pos'), (CurrentSensor_curr, 'n')}, 0: {(CurrentSensor_curr, 'p'), (VoltageSource_v1, 'p')}}
AcausalCompiler.add_node_potential_eqs()
self.nodes={4: {(MotionSensor_motion, 'flange_a'), (LinearActuator_act, 'flange')}, 2: {(LinearActuator_act, 'neg'), (Ground_gnd, 'p'), (VoltageSource_v1, 'n')}, 1: {(LinearActuator_act, 'pos'), (CurrentSensor_curr, 'n')}, 0: {(CurrentSensor_curr, 'p'), (VoltageSource_v1, 'p')}}
np_base_name='np4_translational_spd' port_set={(MotionSensor_motion, 'flange_a'), (LinearActuator_act, 'flange')}
cmp=MotionSensor_motion port_id='flange_a'
cmp=LinearActuator_act port_id='flange'
node_alias_map={-1: np4_translational_spd_n1, 0: np4_translational_spd_0, 1: np4_translational_spd_p1}
np_base_name='np2_electrical_volt' port_set={(LinearActuator_act, 'neg'), (Ground_gnd, 'p'), (VoltageSource_v1, 'n')}
cmp=LinearActuator_act port_id='neg'
cmp=Ground_gnd port_id='p'
cmp=VoltageSource_v1 port_id='n'
node_alias_map={0: np2_electrical_volt_0}
np_base_name='np1_electrical_volt' port_set={(LinearActuator_act, 'pos'), (CurrentSensor_curr, 'n')}
cmp=LinearActuator_act port_id='pos'
cmp=CurrentSensor_curr port_id='n'
node_alias_map={0: np1_electrical_volt_0}
np_base_name='np0_electrical_volt' port_set={(CurrentSensor_curr, 'p'), (VoltageSource_v1, 'p')}
cmp=CurrentSensor_curr port_id='p'
cmp=VoltageSource_v1 port_id='p'
node_alias_map={0: np0_electrical_volt_0}
add weak IC for x_dot_el=Derivative(v1_n_I(t), t)
add weak IC for x_dot_el=Derivative(np4_translational_spd_n1(t), t)
add weak IC for x_dot_el=Derivative(np4_translational_spd_0(t), t)
X=[np4_translational_spd_0(t), np4_translational_spd_n1(t), act_dI(t), Derivative(v1_n_I(t), t), Derivative(np4_translational_spd_n1(t), t), Derivative(np4_translational_spd_0(t), t), np4_translational_spd_p1(t), v1_n_I(t)]
Too few strong initial conditions. Determining acceptable weak initial conditions.
The following weaks ICs were converted to strong ICs:
np4_translational_spd_0(t)=0.0
np4_translational_spd_n1(t)=0.0
Strong ICs post tweaking for numerical computation:
v1_n_I(t) = 0.0
np4_translational_spd_0(t) = 0.0
np4_translational_spd_n1(t) = 0.0
Weak ICs post tweaking for numerical computation:
Derivative(np4_translational_spd_n1(t), t) = 0.0
act_dI(t) = 0.0
Derivative(v1_n_I(t), t) = 0.0
np4_translational_spd_p1(t) = 0.0
Derivative(np4_translational_spd_0(t), t) = 0.0
Proceeding with numerical computation of ICs.
##### Initial conditions #####
np4_translational_spd_0(t) = 0.0
np4_translational_spd_n1(t) = 0.0
act_dI(t) = 50.0
Derivative(v1_n_I(t), t) = 50.0
Derivative(np4_translational_spd_n1(t), t) = 0.0
Derivative(np4_translational_spd_0(t), t) = 0.0
np4_translational_spd_p1(t) = 0.0
v1_n_I(t) = 0.0
Proceeding with numerical computation of ICs.
12:47:30.764 - [jaxonomy][INFO]: max_major_steps=200 by default since no discrete period in system 12:47:30.765 - [jaxonomy][INFO]: Simulator ready to start: SimulatorOptions(math_backend=jax, enable_tracing=True, max_major_step_length=None, max_major_steps=200, ode_solver_method=auto, rtol=1e-06, atol=1e-08, min_minor_step_size=None, max_minor_step_size=None, zc_bisection_loop_count=40, save_time_series=True, recorded_signals=4, return_context=True), BDFSolver(system=Diagram(root, 1 nodes), rtol=1e-06, atol=1e-08, max_step_size=None, min_step_size=None, method='auto', enable_autodiff=False, max_checkpoints=16, supports_mass_matrix=True)
##### Initial conditions ##### np4_translational_spd_0(t) = 0.0 np4_translational_spd_n1(t) = 0.0 act_dI(t) = 50.0 Derivative(v1_n_I(t), t) = 50.0 Derivative(np4_translational_spd_n1(t), t) = 0.0 Derivative(np4_translational_spd_0(t), t) = 0.0 np4_translational_spd_p1(t) = 0.0 v1_n_I(t) = 0.0
