LTI Systems (lib601.lti)

Framework for modeling and simulating Linear Time Invariant systems.

class lib601.lti.Cascade(s1, s2)

Subclass of System representing the Cascade composition of two System s. Cascade systems pass input into their first subsystem, pass the output of their first subsystem as the input into the second subsystem, and finally output the second subsystem’s output.

Parameters:
  • s1 – the first System in this cascade
  • s2 – the second System in this cascade
Variables:
  • initial_state – a tuple representing the initial states of self.s1 and self.s2
  • s1 – the first System in this cascade
  • s2 – the second System in this cascade
calculate_step(state, inp)
Parameters:
  • state – a tuple representing current hypothetical states of self.s1 and self.s2
  • inp – input value to the system
Returns:

a tuple containing the cascade system’s new state (a tuple of self.s1’s new state and self.s2’s new state) and output after a hypothetical step through each subsystem.

Note that this is a pure function which does not store or modify any class, instance, or global variables.

dominant_pole()
Returns:the dominant pole of this system.
poles()
Returns:a list containing the poles of this system.
class lib601.lti.FeedbackAdd(s1, s2)

Subclass of System representing the feedback addition of two System s. In FeedbackAdd systems, the second subsystem’s output is added to the input to the FeedbackAdd system. This sum is passed as input into the first subsystem, whose consequent output is both the output of the overall system and the input back into the second subsystem. For this to work, we assume that at least one of the subsystems has output that doesn’t depend directly on its input.

Parameters:
  • s1 – the first System in this feedback addition
  • s2 – the second System in this feedback addition
Variables:
  • initial_state – a tuple representing the initial states of self.s1 and self.s2
  • s1 – the first System in this feedback addition
  • s2 – the second System in this feedback addition
calculate_step(state, inp)
Parameters:
  • state – a tuple representing current hypothetical states of self.s1 and self.s2
  • inp – input value to the system
Returns:

a tuple containing the cascade system’s new state (a tuple of self.s1’s new state and self.s2’s new state) and output after a hypothetical step through each subsystem.

Note that this is a pure function which does not store or modify any class, instance, or global variables.

dominant_pole()
Returns:the dominant pole of this system.
poles()
Returns:a list containing the poles of this system.
class lib601.lti.FeedforwardAdd(s1, s2)

Subclass of System representing the feedforward addition of two System s. FeedforwardAdd systems pass the same input into each of their two subsystems, and output the sum of their subsystems’ outputs.

Parameters:
  • s1 – the first System in this feedforward addition
  • s2 – the second System in this feedforward addition
Variables:
  • initial_state – a tuple representing the initial states of self.s1 and self.s2
  • s1 – the first System in this feedforward addition
  • s2 – the second System in this feedforward addition
calculate_step(state, inp)
Parameters:
  • state – a tuple representing current hypothetical states of self.s1 and self.s2
  • inp – input value to the system
Returns:

a tuple containing the FeedforwardAdd system’s new state (a tuple of self.s1’s new state and self.s2’s new state) and output after a hypothetical step through each subsystem.

Note that this is a pure function which does not store or modify any class, instance, or global variables.

dominant_pole()
Returns:the dominant pole of this system.
poles()
Returns:a list containing the poles of this system.
class lib601.lti.Gain(k)

Subclass of System representing Gain systems. Gain systems output at time \(n\) the input passed into the system at time \(n\), multiplied by a gain k.

Parameters:

k – the value this gain system should multiply inputs by
Variables:
  • initial_state – the initial state of this system
  • k – the value this gain system should multiply inputs by
calculate_step(state, inp)
Parameters:
  • state – object representing a current hypothetical state of the system
  • inp – input value to the system
Returns:

a tuple containing the system’s new state and output after a hypothetical step using the given state and input.

Note that this is a pure function which does not store or modify any class, instance, or global variables.

dominant_pole()
Returns:the dominant pole of this system.
poles()
Returns:a list containing the poles of this system.
class lib601.lti.R(output0=0)

Subclass of System representing Delay systems. Delay systems output at time \(n\) the input passed into the system at time \((n-1)\).

Parameters:output0 – the value this system should output at time zero
Variables:initial_state – the initial state of this system
calculate_step(state, inp)
Parameters:
  • state – object representing a current hypothetical state of the system
  • inp – input value to the system
Returns:

a tuple containing the system’s new state and output after a hypothetical step using the given state and input.

Note that this is a pure function which does not store or modify any class, instance, or global variables.

dominant_pole()
Returns:the dominant pole of this system.
poles()
Returns:a list containing the poles of this system.
class lib601.lti.System(numerator, denominator=None, previous_inputs=None, previous_outputs=None)

Represents Linear Time Invariant systems, and supports step-by-step simulation.

Parameters:
  • numerator – a list of numbers representing the numerator of the system functional, starting with lowest order coefficient
  • denominator – a list of numbers representing the denominator of the system functional, starting with lowest order coefficient
  • previous_inputs – a list of previous input numbers to store in the initial state
  • previous_outputs – a list of previous output numbers to store in the initial state
Variables:
  • numerator – list of numbers representing the numerator of the system functional
  • denominator – list of numbers representing the denominator of the system functional
  • initial_state – tuple of two lists representing the previous inputs and outputs to the system
calculate_step(state, inp)
Parameters:
  • state – object representing a current hypothetical state of the system
  • inp – input value to the system
Returns:

a tuple containing the system’s new state and output after a hypothetical step using the given state and input.

Note that this is a pure function which does not store or modify any class, instance, or global variables.

dominant_pole()
Returns:the dominant pole of this system.
poles()
Returns:a list containing the poles of this system.
class lib601.lti.SystemSimulator(system)

Simulates Linear Time Invariant systems, and supports resetting and stepping for individual or lists of inputs.

Parameters:system – the instance of System to simulate
Variables:system – the instance of System to simulate
get_response(inputs, reset=True)
Parameters:
  • inputs – a list of consecutive input values to this simulator
  • reset – whether to reset this simulator prior to executing with these inputs
Returns:

a list of output values of the simulator after executing with the given inputs.

Note that this function updates the system’s internal state.

reset()

Resets this simulator’s state to self.system’s initial state.

step(inp)
Parameters:inp – input value to this simulator
Returns:the output value of this simulator after executing with the given input

Note that this function updates this simulator’s internal state.