Breaking Algebraic Loops
It this tutorial, we will simulate model consisting a closed loop feedback system. The model has an algebraic loop.
Algebraic Loops
An algebraic loop is a closed-loop consisting of one or more components whose outputs are directly dependent on their inputs. If algebraic loops exist in a model, the simulation gets stuck because none of the components in the loop can generate output to break the loop. Such a problem can be broken by rearranging the model without algebraic loops, solving the feed-forward algebraic equation of the loop, or inserting a memory component with a certain initial condition anywhere in the loop. Causal provides all these loop-breaking solutions. During the inspection stage, in case they are detected, all the loops are broken. Otherwise, a report is printed to notify the user to insert memory components to break the loops.
Breaking Algebraic Loops Automatically
Before initializing and running the simulation, Causal inspects the model first. See Simulation Stages for more information of simulation stages. In case the they exist in the model, all the algebraic loops are tried to be broken automatically without requiring a user intervention. Consider the following model
where
Note that there exist an algebraic loop consisting of adder and gain. Solving this algebraic loop, we have
The following script constructs and simulates the model.
using Causal
# Describe the model
@defmodel model begin
@nodes begin
gen = RampGenerator()
adder = Adder(signs=(+,-))
gain = Gain()
writerout = Writer()
writerin = Writer()
end
@branches begin
gen[1] => adder[1]
adder[1] => gain[1]
gain[1] => adder[2]
gen[1] => writerin[1]
gain[1] => writerout[1]
end
end
# Simulate the model
ti, dt, tf = 0., 1. / 64., 1.
sim = simulate!(model, ti, dt, tf, withbar=false)
# Read the simulation data and plot
using Plots
t, y = read(getnode(model, :writerout).component)
t, r = read(getnode(model, :writerin).component)
plot(t, r, label="r(t)", marker=(:circle, 3))
plot!(t, y, label="y(t)", marker=(:circle, 3))[ Info: 2020-08-27T08:56:40.331 Started simulation... [ Info: 2020-08-27T08:56:40.4 Inspecting model... ┌ Info: The model has algrebraic loops:[[2, 3]] └ Trying to break these loops... [ Info: Loop [2, 3] is broken [ Info: 2020-08-27T08:56:41.606 Done. [ Info: 2020-08-27T08:56:41.606 Initializing the model... [ Info: 2020-08-27T08:56:42.044 Done... [ Info: 2020-08-27T08:56:42.392 Running the simulation... [ Info: 2020-08-27T08:56:49.873 Done... [ Info: 2020-08-27T08:56:49.873 Terminating the simulation... [ Info: 2020-08-27T08:56:50.034 Done.
Breaking Algebraic Loops With a Memory
It is also possible to break algebraic loops by inserting a Memory component at some point the loop. For example, consider the model consider following the model which is the model in which a memory component is inserted in the feedback path.
Note that the input to adder is not $y(t)$, but instead is $\hat{y}(t)$ which is one sample delayed form of $y(t)$. That is, we have, $\hat{y}(t) = y(t - dt)$ where $dt$ is the step size of the simulation. If $dt$ is small enough, $\hat{y}(t) \approx y(t)$.
The script given below simulates this case.
using Causal
# Simulation time settings.
ti, dt, tf = 0., 1. / 64., 1.
# Describe the model
@defmodel model begin
@nodes begin
gen = RampGenerator()
adder = Adder(signs=(+,-))
gain = Gain()
writerout = Writer()
writerin = Writer()
mem = Memory(delay=dt, initial=zeros(1))
end
@branches begin
gen[1] => adder[1]
adder[1] => gain[1]
gain[1] => mem[1]
mem[1] => adder[2]
gen[1] => writerin[1]
gain[1] => writerout[1]
end
end
# Simulate the model
sim = simulate!(model, ti, dt, tf, withbar=false)
# Plot the simulation data
using Plots
t, r = read(getnode(model, :writerin).component)
t, y = read(getnode(model, :writerout).component)
plot(t, r, label="r(t)", marker=(:circle, 3))
plot!(t, y, label="y(t)", marker=(:circle, 3))[ Info: 2020-08-27T08:57:11.575 Started simulation... [ Info: 2020-08-27T08:57:11.575 Inspecting model... ┌ Info: The model has algrebraic loops:[[2, 3, 6]] └ Trying to break these loops... [ Info: Loop [2, 3, 6] has a Memory component. The loops is broken [ Info: 2020-08-27T08:57:11.595 Done. [ Info: 2020-08-27T08:57:11.595 Initializing the model... [ Info: 2020-08-27T08:57:11.635 Done... [ Info: 2020-08-27T08:57:11.749 Running the simulation... [ Info: 2020-08-27T08:57:11.755 Done... [ Info: 2020-08-27T08:57:11.755 Terminating the simulation... [ Info: 2020-08-27T08:57:11.76 Done.
The fluctuation in $y(t)$ because of one-sample-time delay introduced by the mem component is apparent. The smaller the step size is, the smaller the amplitude of the fluctuation introduced by the mem component.
One other important issue with using the memory component is that the initial value of mem directly affects the accuracy of the simulation. By solving the loop equation, we know that
That is the memory should be initialized with an initial value of zero, which is the case in the script above. To observe that how incorrect initialization of a memory to break an algebraic loop, consider the following example in which memory is initialized randomly.
using Causal
using Plots
# Simulation time settings.
ti, dt, tf = 0., 1. / 64., 1.
# Describe the model
@defmodel model begin
@nodes begin
gen = RampGenerator()
adder = Adder(signs=(+,-))
gain = Gain()
writerout = Writer()
writerin = Writer()
mem = Memory(delay=dt, initial=rand(1))
end
@branches begin
gen[1] => adder[1]
adder[1] => gain[1]
gain[1] => mem[1]
mem[1] => adder[2]
gen[1] => writerin[1]
gain[1] => writerout[1]
end
end
# Simulate the model
sim = simulate!(model, ti, dt, tf, withbar=false)
# Plot the results
using Plots
t, r = read(getnode(model, :writerin).component)
t, y = read(getnode(model, :writerout).component)
plot(t, r, label="r(t)", marker=(:circle, 3))
plot!(t, y, label="y(t)", marker=(:circle, 3))[ Info: 2020-08-27T08:57:12.107 Started simulation... [ Info: 2020-08-27T08:57:12.107 Inspecting model... ┌ Info: The model has algrebraic loops:[[2, 3, 6]] └ Trying to break these loops... [ Info: Loop [2, 3, 6] has a Memory component. The loops is broken [ Info: 2020-08-27T08:57:12.107 Done. [ Info: 2020-08-27T08:57:12.107 Initializing the model... [ Info: 2020-08-27T08:57:12.107 Done... [ Info: 2020-08-27T08:57:12.107 Running the simulation... [ Info: 2020-08-27T08:57:12.114 Done... [ Info: 2020-08-27T08:57:12.114 Terminating the simulation... [ Info: 2020-08-27T08:57:12.118 Done.
