Continuous-time systems¶

Linear time invariant filters¶

Linear time invariant filters are created with the LTIFilter class:

>>> fil = LTIFilter(b, a)

where b is a list or array of the transfer function numerator

coefficients and a is a list or array of the transfer function denominator coefficients. For example,

>>> fil = LTIFilter(('b0', ), ('a0', 'a1'))

The coefficients can be found from the a and b attributes:

>>> fil.a
(a₀, a₁)

>>> fil.b
(b₀,)

The filter’s transfer function is found with the transfer_function() method:

>>> fil.transfer_function()

    b₀
─────────
a₀⋅s + a₁

its frequency response is found with the frequency_response() method:

>>> fil.frequency_response()
       b₀
───────────────
2⋅ⅉ⋅π⋅a₀⋅f + a₁

its angular frequency response is found with the angular_frequency_response() method:

>>> fil.angular_frequency_response()
      b₀
  ──────────
  ⅉ⋅a₀⋅ω + a₁

its group delay is found with the group_delay() method:

>>> fil.group_delay()
        a₀⋅a₁
 ─────────────────
    2   2  2     2
 4⋅π ⋅a₀ ⋅f  + a₁

its impulse response is found with the impulse_response() method:

>>> fil.impulse_response()
    -a₁⋅t
    ──────
      a₀
b₀⋅ℯ      ⋅u(t)
───────────────
       a₀

and its step response is found with the step_response() method:

>>> fil.step_response()
   ⎛      -a₁⋅t ⎞
   ⎜      ──────⎟
   ⎜        a₀  ⎟
   ⎜1    ℯ      ⎟
b₀⋅⎜── - ───────⎟⋅u(t)
   ⎝a₁      a₁  ⎠

The filter’s differential equation is found with the differential_equation() method:

>>> fil.differential_equation()
               d
a₀⋅y(t) = - a₁⋅──(y(t)) + b₀⋅x(t)
               dt

The input and output symbols can be changed with the inputsym and outputsym arguments.

The response due to intial conditions is found with the initial_response() method:

>>> fil.initial_response(('y0', ))
       ⎛           -a₁⋅t      ⎞
       ⎜           ──────     ⎟
       ⎜             a₀       ⎟
       ⎜δ(t)   a₁⋅ℯ      ⋅u(t)⎟
-a₁⋅y₀⋅⎜──── - ───────────────⎟
       ⎜ a₀            2      ⎟
       ⎝             a₀       ⎠

An LTI filter can also be constructed from its transfer function in zero-pole-gain representation with the LTIFilter.from_ZPK class method. For example,

>>> fil = LTIFilter.from_ZPK(('z1',),('p1','p2'), 'K')
>>> fil.a
(1, -p₁ - p₂, p₁⋅p₂)
>>> fil.b
(K, -K⋅z₁)

A continuous-time LTI filter can be approximated by discrete-time LTI filter using the discretize() method:

Continuous-time linear time invariant filter attributes¶

a denominator coefficients as a list
b numerator coefficients as a list
is_marginally_stable True if impulse response marginally stable
is_stable True if impulse response stable

Continuous-time linear time invariant filter methods¶

differential_equation() creates continuous-time differential equation
impulse_response() creates continuous-time domain impulse response
step_response() returns continuous time-domain step response
initial_response() returns continuous time-domain response due to initial conditions
response() returns continuous time-domain response due to input signal and initial conditions
transfer_fnction() creates s-domain transfer function
sdomain_initial_response() returns s-domain response due to initial conditions
group_delay() returns the group delay as function of frequency, f

Butterworth filters¶

Butterworth filters are created with the Butterworth class method. For example:

>>> B = Butterworth(order=2, Wn=omega0, btype='lowpass')
>>> B.transfer_function()
          2
        ω₀
 ──────────────────
   2              2
 ω₀  + √2⋅ω₀⋅s + s


 >>> abs(B.frequency_response())
           2
         ω₀
 ───────────────────
    ________________
   ╱     4  4     4
 ╲╱  16⋅π ⋅f  + ω₀


 >>> B.group_delay()
       2  2           3
 4⋅√2⋅π ⋅f ⋅ω₀ + √2⋅ω₀
 ──────────────────────
         4  4     4
     16⋅π ⋅f  + ω₀

Bessel filters¶

Bessel filters are created with the Bessel class method. For example:

>>> B = Bessel(order=2, Wn=omega0, btype='lowpass')
>>> B.transfer_function()
            2
        3⋅ω₀
 ───────────────────
     2             2
 3⋅ω₀  + 3⋅ω₀⋅s + s

 >>> abs(B.frequency_response())
                 2
             3⋅ω₀
 ──────────────────────────────
      2  2                    2
 - 4⋅π ⋅f  + 6⋅ⅉ⋅π⋅f⋅ω₀ + 3⋅ω₀

 >>> B.group_delay()
           2  2          3
       12⋅π ⋅f ⋅ω₀ + 9⋅ω₀
 ───────────────────────────────
     4  4       2  2   2       4
 16⋅π ⋅f  + 12⋅π ⋅f ⋅ω₀  + 9⋅ω₀

Differential equations¶

Differential equations are represented by the DifferentialEquation class. They are usually created by the differential_equation() method of the LTIFilter class or from transfer functions. For example:

>>> fil = LTIFilter(('b0', ), ('a0', 'a1'))
>>> de = fil.differential_equation()
>>> de
               d
a₀⋅y(t) = - a₁⋅──(y(t)) + b₀⋅x(t)
               dt

There are two attributes: lhs for the left-hand-side and rhs for the right-hand-side,

>>> de.lhs
a₀⋅y(t)
>>> de.rhs
     d
- a₁⋅──(y(t)) + b₀⋅x(t)
     dt

A transfer function is created with the transfer_function() method:

>>> de.transfer_function()
    b₀
 ─────────
 a₀ + a₁⋅s

An LTFilter object is created with the lti_filter() method:

>>> fil = de.lti_filter()

Differential equation attributes¶

lhs left-hand-side of the equation
rhs right-hand-side of the equation
inputsym input symbol, usually ‘x’
outputsym input symbol, usually ‘y’

Differential equation methods¶

dlti_filter() creates continuous-time linear time invariant filter (LTIFilter) object
separate() separates the input expressions from the output expressions.
transfer_function() creates s-domain transfer function

Continuous-time state-space representation¶

Lcapy has two state-space representations: StateSpace for continuous-time linear time-invariant systems and DTStateSpace for discrete-time linear time-invariant systems. Both representations share many methods and attributes, see State-space analysis.

A state-space object is created from the state matrix, A, input matrix, B, output matrix C, and feed-through matrix D:

>>> ss = StateSpace(A, B, C, D)

A state-space object can also be created from lists of the numerator and denominator coefficients b and a:

>>> ss = StateSpace.from_ba(b, a)

By default, the controllable canonical form CCF is created. The observable canonical form OCF is created with:

>>> ss = StateSpace.from_ba(b, a, form='OCF')

Similarly, the diagonal canonical form DCF is created with:

>>> ss = StateSpace.from_ba(b, a, form='DCF')

For the DCF, the poles of the transfer function must be unique.

State-space from transfer function¶

Transfer functions (and impedances and admittances) can be converted to a state-space representation. Here’s an example:

>>> Z = (s**2 + a) / (s**3 + b * s + c)
>>> ss = Z.state_space('CCF')

State-space representation are not unique; Lcapy uses the controllable canonical form (CCF), the observable canonical form (OCF), and the diagonal canonical form (DCF). The CCF form of the state-space matrices are:

>>> ss.A
⎡0   1   0⎤
⎢         ⎥
⎢0   0   1⎥
⎢         ⎥
⎣-c  -b  0⎦

>>> ss.B
⎡0⎤
⎢ ⎥
⎢0⎥
⎢ ⎥
⎣1⎦

>>> ss.C
[a  0  1]

>>> ss.D
[0]

Transfer function from state-space¶

For a single-input single-output (SISO) system the transfer function is obtained with the transfer_function() method, for example:

>>> ss = StateSpace(A, B, C, D)
>>> G = ss.transfer_function

State-space operations¶

Model balancing¶

This returns a new StateSpace object that has the controllability and observability gramians equal to the diagonal matrix with the Hankel singular values on the diagonal. For example:

>>> ss2 = ss.balance()

Note, this requires numerical A, B, C, D matrices.

Model reduction¶

A balanced reduction can be performed using:

>>> ss2 = ss.balance_reduce(threshold=0.1)

where states are removed with a Hankel singular value below the threshold. Note, this requires numerical A, B, C, D matrices.

Alternatively, specific states can be removed. For example:

>>> ss2 = ss.reduce(elim_states=[1, 3])

Discrete-time systems¶

Difference equations¶

Difference equations can be generated from transfer functions and impulse responses. Both FIR and IIR (direct form I) can be generated. For example:

>>> H = (z + 2) / z**2
>>> H.difference_equation('x', 'y', 'fir')
y(n) = 2⋅x(n - 2) + x(n - 1)

Difference equations can be created explicitly, for example:

>>> de = difference_equation('y(n)', '2 * x(n - 2) + x(n - 1)')

The separate() method separates the input expressions from the output expressions. For example:

>>> de = difference_equation('y(n)', '2 * y(n - 1) + x(n)')
>>> de.separate()
y(n) - 2⋅y(n - 1) = x(n)

Difference equation attributes¶

lhs left-hand-side of the equation
rhs right-hand-side of the equation
inputsym input symbol, usually ‘x’
outputsym input symbol, usually ‘y’

Difference equation methods¶

dlti_filter() creates discrete-time linear time invariant filter (DLTIFilter) object
separate() separates the input expressions from the output expressions.
transfer_function() creates z-domain transfer function

Discrete-time transfer functions¶

A discrete-time transfer functions can be determined from a difference equation or a DLTI filter. For example:

>>> de = difference_equation('y(n)', '2 * x(n - 2) + x(n - 1)')
>>> H = de.transfer_function()
>>> H
z + 2
─────
  2
 z

Discrete-time transfer function methods¶

dlti_filter() creates discrete-time linear time invariant filter (DLTIFilter) object
difference_equation() creates discrete-time difference equation

Discrete-time linear time invariant filters¶

A discrete-time linear time invariant filter can be specified by its numerator and denominator coefficients. For example, a first-order, discrete-time, recursive low-pass filter can be created with:

>>> a = symbol('a')
>>> lpf = DLTIFilter((1 - a, ), (1, -a))

The difference equation can be printed using:

>>> lpf.difference_equation()
y(n) = a⋅y(n - 1) + (1 - a)⋅x(n)

The transfer function can be printed using:

>>> lpf.transfer_function()
z⋅(a - 1)
─────────
  a - z

The impulse response can be printed using:

>>> lpf.impulse_response()
 n
a ⋅(1 - a)⋅u[n]

The general response to an input x(n) can be printed using:

>>> lpf.response(x, ni=(0, 5))

For a recursive filter, the initial conditions can also be specified:

>>> lpf.response(x, ic=[1], ni=(0, 5))

The input to the filter can be a DiscreteTimeDomainExpression or a sequence. The output is a sequence.

A discrete-time LTI filter can be created from difference equations and transfer functions. For example:

>>> de = DifferenceEquation('2 * y(n)', '4 * y(n + 1) - 3 * y(n-3) -2 * x(n) - 5 * x(n-3)')
>>> fil = de.dlti_filter()
>>> fil.a
[4, -2, 0, 0, -3]
>>> fil.b
[0, 2, 0, 0, 5]
>>> fil.difference_equation()

Discrete-time linear time invariant filter attributes¶

a denominator coefficients as a list
b numerator coefficients as a list
is_marginally_stable True if impulse response marginally stable
is_stable True if impulse response stable

Discrete-time linear time invariant filter methods¶

difference_equation() creates discrete-time difference equation
impulse_response() creates discrete-time domain impulse response
initial_response() returns discrete time-domain response due to initial conditions
inverse() creates an inverse filter by switching numerator and denominator coefficients
response() returns discrete time-domain response due to input signal and initial conditions
transfer_function() creates z-domain transfer function
zdomain_initial_response() returns z-domain response due to initial conditions

Discrete-time state-space representation¶

Discrete-time state-space objects are defined in a similar manner to continuous-time state-space objects and share many methods and attributes. A discrete-time state-space object is created from the state matrix, A, input matrix, B, output matrix C, and feed-through matrix D:

>>> ss = DTStateSpace(A, B, C, D)