Control API
lti
- class sympy.physics.control.lti.TransferFunction(num, den, var)[source]
A class for representing LTI (Linear, time-invariant) systems that can be strictly described by ratio of polynomials in the Laplace transform complex variable. The arguments are
num,den, andvar, wherenumanddenare numerator and denominator polynomials of theTransferFunctionrespectively, and the third argument is a complex variable of the Laplace transform used by these polynomials of the transfer function.numanddencan be either polynomials or numbers, whereasvarhas to be a Symbol.- Parameters
num : Expr, Number
The numerator polynomial of the transfer function.
den : Expr, Number
The denominator polynomial of the transfer function.
var : Symbol
Complex variable of the Laplace transform used by the polynomials of the transfer function.
- Raises
TypeError
When
varis not a Symbol or whennumordenis not a number or a polynomial.ValueError
When
denis zero.
Explanation
Generally, a dynamical system representing a physical model can be described in terms of Linear Ordinary Differential Equations like -
\(\small{b_{m}y^{\left(m\right)}+b_{m-1}y^{\left(m-1\right)}+\dots+b_{1}y^{\left(1\right)}+b_{0}y= a_{n}x^{\left(n\right)}+a_{n-1}x^{\left(n-1\right)}+\dots+a_{1}x^{\left(1\right)}+a_{0}x}\)
Here, \(x\) is the input signal and \(y\) is the output signal and superscript on both is the order of derivative (not exponent). Derivative is taken with respect to the independent variable, \(t\). Also, generally \(m\) is greater than \(n\).
It is not feasible to analyse the properties of such systems in their native form therefore, we use mathematical tools like Laplace transform to get a better perspective. Taking the Laplace transform of both the sides in the equation (at zero initial conditions), we get -
\(\small{\mathcal{L}[b_{m}y^{\left(m\right)}+b_{m-1}y^{\left(m-1\right)}+\dots+b_{1}y^{\left(1\right)}+b_{0}y]= \mathcal{L}[a_{n}x^{\left(n\right)}+a_{n-1}x^{\left(n-1\right)}+\dots+a_{1}x^{\left(1\right)}+a_{0}x]}\)
Using the linearity property of Laplace transform and also considering zero initial conditions (i.e. \(\small{y(0^{-}) = 0}\), \(\small{y'(0^{-}) = 0}\) and so on), the equation above gets translated to -
\(\small{b_{m}\mathcal{L}[y^{\left(m\right)}]+\dots+b_{1}\mathcal{L}[y^{\left(1\right)}]+b_{0}\mathcal{L}[y]= a_{n}\mathcal{L}[x^{\left(n\right)}]+\dots+a_{1}\mathcal{L}[x^{\left(1\right)}]+a_{0}\mathcal{L}[x]}\)
Now, applying Derivative property of Laplace transform,
\(\small{b_{m}s^{m}\mathcal{L}[y]+\dots+b_{1}s\mathcal{L}[y]+b_{0}\mathcal{L}[y]= a_{n}s^{n}\mathcal{L}[x]+\dots+a_{1}s\mathcal{L}[x]+a_{0}\mathcal{L}[x]}\)
Here, the superscript on \(s\) is exponent. Note that the zero initial conditions assumption, mentioned above, is very important and cannot be ignored otherwise the dynamical system cannot be considered time-independent and the simplified equation above cannot be reached.
Collecting \(\mathcal{L}[y]\) and \(\mathcal{L}[x]\) terms from both the sides and taking the ratio \(\frac{ \mathcal{L}\left\{y\right\} }{ \mathcal{L}\left\{x\right\} }\), we get the typical rational form of transfer function.
The numerator of the transfer function is, therefore, the Laplace transform of the output signal (The signals are represented as functions of time) and similarly, the denominator of the transfer function is the Laplace transform of the input signal. It is also a convention to denote the input and output signal’s Laplace transform with capital alphabets like shown below.
\(H(s) = \frac{Y(s)}{X(s)} = \frac{ \mathcal{L}\left\{y(t)\right\} }{ \mathcal{L}\left\{x(t)\right\} }\)
\(s\), also known as complex frequency, is a complex variable in the Laplace domain. It corresponds to the equivalent variable \(t\), in the time domain. Transfer functions are sometimes also referred to as the Laplace transform of the system’s impulse response. Transfer function, \(H\), is represented as a rational function in \(s\) like,
\(H(s) =\ \frac{a_{n}s^{n}+a_{n-1}s^{n-1}+\dots+a_{1}s+a_{0}}{b_{m}s^{m}+b_{m-1}s^{m-1}+\dots+b_{1}s+b_{0}}\)
Examples
>>> from sympy.abc import s, p, a >>> from sympy.physics.control.lti import TransferFunction >>> tf1 = TransferFunction(s + a, s**2 + s + 1, s) >>> tf1 TransferFunction(a + s, s**2 + s + 1, s) >>> tf1.num a + s >>> tf1.den s**2 + s + 1 >>> tf1.var s >>> tf1.args (a + s, s**2 + s + 1, s)
Any complex variable can be used for
var.>>> tf2 = TransferFunction(a*p**3 - a*p**2 + s*p, p + a**2, p) >>> tf2 TransferFunction(a*p**3 - a*p**2 + p*s, a**2 + p, p) >>> tf3 = TransferFunction((p + 3)*(p - 1), (p - 1)*(p + 5), p) >>> tf3 TransferFunction((p - 1)*(p + 3), (p - 1)*(p + 5), p)
To negate a transfer function the
-operator can be prepended:>>> tf4 = TransferFunction(-a + s, p**2 + s, p) >>> -tf4 TransferFunction(a - s, p**2 + s, p) >>> tf5 = TransferFunction(s**4 - 2*s**3 + 5*s + 4, s + 4, s) >>> -tf5 TransferFunction(-s**4 + 2*s**3 - 5*s - 4, s + 4, s)
You can use a Float or an Integer (or other constants) as numerator and denominator:
>>> tf6 = TransferFunction(1/2, 4, s) >>> tf6.num 0.500000000000000 >>> tf6.den 4 >>> tf6.var s >>> tf6.args (0.5, 4, s)
You can take the integer power of a transfer function using the
**operator:>>> tf7 = TransferFunction(s + a, s - a, s) >>> tf7**3 TransferFunction((a + s)**3, (-a + s)**3, s) >>> tf7**0 TransferFunction(1, 1, s) >>> tf8 = TransferFunction(p + 4, p - 3, p) >>> tf8**-1 TransferFunction(p - 3, p + 4, p)
Addition, subtraction, and multiplication of transfer functions can form unevaluated
SeriesorParallelobjects.>>> tf9 = TransferFunction(s + 1, s**2 + s + 1, s) >>> tf10 = TransferFunction(s - p, s + 3, s) >>> tf11 = TransferFunction(4*s**2 + 2*s - 4, s - 1, s) >>> tf12 = TransferFunction(1 - s, s**2 + 4, s) >>> tf9 + tf10 Parallel(TransferFunction(s + 1, s**2 + s + 1, s), TransferFunction(-p + s, s + 3, s)) >>> tf10 - tf11 Parallel(TransferFunction(-p + s, s + 3, s), TransferFunction(-4*s**2 - 2*s + 4, s - 1, s)) >>> tf9 * tf10 Series(TransferFunction(s + 1, s**2 + s + 1, s), TransferFunction(-p + s, s + 3, s)) >>> tf10 - (tf9 + tf12) Parallel(TransferFunction(-p + s, s + 3, s), TransferFunction(-s - 1, s**2 + s + 1, s), TransferFunction(s - 1, s**2 + 4, s)) >>> tf10 - (tf9 * tf12) Parallel(TransferFunction(-p + s, s + 3, s), Series(TransferFunction(-1, 1, s), TransferFunction(s + 1, s**2 + s + 1, s), TransferFunction(1 - s, s**2 + 4, s))) >>> tf11 * tf10 * tf9 Series(TransferFunction(4*s**2 + 2*s - 4, s - 1, s), TransferFunction(-p + s, s + 3, s), TransferFunction(s + 1, s**2 + s + 1, s)) >>> tf9 * tf11 + tf10 * tf12 Parallel(Series(TransferFunction(s + 1, s**2 + s + 1, s), TransferFunction(4*s**2 + 2*s - 4, s - 1, s)), Series(TransferFunction(-p + s, s + 3, s), TransferFunction(1 - s, s**2 + 4, s))) >>> (tf9 + tf12) * (tf10 + tf11) Series(Parallel(TransferFunction(s + 1, s**2 + s + 1, s), TransferFunction(1 - s, s**2 + 4, s)), Parallel(TransferFunction(-p + s, s + 3, s), TransferFunction(4*s**2 + 2*s - 4, s - 1, s)))
These unevaluated
SeriesorParallelobjects can convert into the resultant transfer function using.doit()method or by.rewrite(TransferFunction).>>> ((tf9 + tf10) * tf12).doit() TransferFunction((1 - s)*((-p + s)*(s**2 + s + 1) + (s + 1)*(s + 3)), (s + 3)*(s**2 + 4)*(s**2 + s + 1), s) >>> (tf9 * tf10 - tf11 * tf12).rewrite(TransferFunction) TransferFunction(-(1 - s)*(s + 3)*(s**2 + s + 1)*(4*s**2 + 2*s - 4) + (-p + s)*(s - 1)*(s + 1)*(s**2 + 4), (s - 1)*(s + 3)*(s**2 + 4)*(s**2 + s + 1), s)
References
- dc_gain()[source]
Computes the gain of the response as the frequency approaches zero.
The DC gain is infinite for systems with pure integrators.
Examples
>>> from sympy.abc import s, p, a, b >>> from sympy.physics.control.lti import TransferFunction >>> tf1 = TransferFunction(s + 3, s**2 - 9, s) >>> tf1.dc_gain() -1/3 >>> tf2 = TransferFunction(p**2, p - 3 + p**3, p) >>> tf2.dc_gain() 0 >>> tf3 = TransferFunction(a*p**2 - b, s + b, s) >>> tf3.dc_gain() (a*p**2 - b)/b >>> tf4 = TransferFunction(1, s, s) >>> tf4.dc_gain() oo
- property den
Returns the denominator polynomial of the transfer function.
Examples
>>> from sympy.abc import s, p >>> from sympy.physics.control.lti import TransferFunction >>> G1 = TransferFunction(s + 4, p**3 - 2*p + 4, s) >>> G1.den p**3 - 2*p + 4 >>> G2 = TransferFunction(3, 4, s) >>> G2.den 4
- expand()[source]
Returns the transfer function with numerator and denominator in expanded form.
Examples
>>> from sympy.abc import s, p, a, b >>> from sympy.physics.control.lti import TransferFunction >>> G1 = TransferFunction((a - s)**2, (s**2 + a)**2, s) >>> G1.expand() TransferFunction(a**2 - 2*a*s + s**2, a**2 + 2*a*s**2 + s**4, s) >>> G2 = TransferFunction((p + 3*b)*(p - b), (p - b)*(p + 2*b), p) >>> G2.expand() TransferFunction(-3*b**2 + 2*b*p + p**2, -2*b**2 + b*p + p**2, p)
- classmethod from_rational_expression(expr, var=None)[source]
Creates a new
TransferFunctionefficiently from a rational expression.- Parameters
expr : Expr, Number
The rational expression representing the
TransferFunction.var : Symbol, optional
Complex variable of the Laplace transform used by the polynomials of the transfer function.
- Raises
ValueError
When
expris of typeNumberand optional parametervaris not passed.When
exprhas more than one variables and an optional parametervaris not passed.ZeroDivisionError
When denominator of
expris zero or it hasComplexInfinityin its numerator.
Examples
>>> from sympy.abc import s, p, a >>> from sympy.physics.control.lti import TransferFunction >>> expr1 = (s + 5)/(3*s**2 + 2*s + 1) >>> tf1 = TransferFunction.from_rational_expression(expr1) >>> tf1 TransferFunction(s + 5, 3*s**2 + 2*s + 1, s) >>> expr2 = (a*p**3 - a*p**2 + s*p)/(p + a**2) # Expr with more than one variables >>> tf2 = TransferFunction.from_rational_expression(expr2, p) >>> tf2 TransferFunction(a*p**3 - a*p**2 + p*s, a**2 + p, p)
In case of conflict between two or more variables in a expression, SymPy will raise a
ValueError, ifvaris not passed by the user.>>> tf = TransferFunction.from_rational_expression((a + a*s)/(s**2 + s + 1)) Traceback (most recent call last): ... ValueError: Conflicting values found for positional argument `var` ({a, s}). Specify it manually.
This can be corrected by specifying the
varparameter manually.>>> tf = TransferFunction.from_rational_expression((a + a*s)/(s**2 + s + 1), s) >>> tf TransferFunction(a*s + a, s**2 + s + 1, s)
varalso need to be specified whenexpris aNumber>>> tf3 = TransferFunction.from_rational_expression(10, s) >>> tf3 TransferFunction(10, 1, s)
- property is_biproper
Returns True if degree of the numerator polynomial is equal to degree of the denominator polynomial, else False.
Examples
>>> from sympy.abc import s, p, a, b >>> from sympy.physics.control.lti import TransferFunction >>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s) >>> tf1.is_biproper True >>> tf2 = TransferFunction(p**2, p + a, p) >>> tf2.is_biproper False
- property is_proper
Returns True if degree of the numerator polynomial is less than or equal to degree of the denominator polynomial, else False.
Examples
>>> from sympy.abc import s, p, a, b >>> from sympy.physics.control.lti import TransferFunction >>> tf1 = TransferFunction(b*s**2 + p**2 - a*p + s, b - p**2, s) >>> tf1.is_proper False >>> tf2 = TransferFunction(p**2 - 4*p, p**3 + 3*p + 2, p) >>> tf2.is_proper True
- is_stable()[source]
Returns True if the transfer function is asymptotically stable; else False.
This would not check the marginal or conditional stability of the system.
Examples
>>> from sympy.abc import s, p, a >>> from sympy import symbols >>> from sympy.physics.control.lti import TransferFunction >>> q, r = symbols('q, r', negative=True) >>> tf1 = TransferFunction((1 - s)**2, (s + 1)**2, s) >>> tf1.is_stable() True >>> tf2 = TransferFunction((1 - p)**2, (s**2 + 1)**2, s) >>> tf2.is_stable() False >>> tf3 = TransferFunction(4, q*s - r, s) >>> tf3.is_stable() False >>> tf4 = TransferFunction(p + 1, a*p - s**2, p) >>> tf4.is_stable() is None # Not enough info about the symbols to determine stability True
- property is_strictly_proper
Returns True if degree of the numerator polynomial is strictly less than degree of the denominator polynomial, else False.
Examples
>>> from sympy.abc import s, p, a, b >>> from sympy.physics.control.lti import TransferFunction >>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s) >>> tf1.is_strictly_proper False >>> tf2 = TransferFunction(s**3 - 2, s**4 + 5*s + 6, s) >>> tf2.is_strictly_proper True
- property num
Returns the numerator polynomial of the transfer function.
Examples
>>> from sympy.abc import s, p >>> from sympy.physics.control.lti import TransferFunction >>> G1 = TransferFunction(s**2 + p*s + 3, s - 4, s) >>> G1.num p*s + s**2 + 3 >>> G2 = TransferFunction((p + 5)*(p - 3), (p - 3)*(p + 1), p) >>> G2.num (p - 3)*(p + 5)
- poles()[source]
Returns the poles of a transfer function.
Examples
>>> from sympy.abc import s, p, a >>> from sympy.physics.control.lti import TransferFunction >>> tf1 = TransferFunction((p + 3)*(p - 1), (p - 1)*(p + 5), p) >>> tf1.poles() [-5, 1] >>> tf2 = TransferFunction((1 - s)**2, (s**2 + 1)**2, s) >>> tf2.poles() [I, I, -I, -I] >>> tf3 = TransferFunction(s**2, a*s + p, s) >>> tf3.poles() [-p/a]
- to_expr()[source]
Converts a
TransferFunctionobject to SymPy Expr.Examples
>>> from sympy.abc import s, p, a, b >>> from sympy.physics.control.lti import TransferFunction >>> from sympy import Expr >>> tf1 = TransferFunction(s, a*s**2 + 1, s) >>> tf1.to_expr() s/(a*s**2 + 1) >>> isinstance(_, Expr) True >>> tf2 = TransferFunction(1, (p + 3*b)*(b - p), p) >>> tf2.to_expr() 1/((b - p)*(3*b + p)) >>> tf3 = TransferFunction((s - 2)*(s - 3), (s - 1)*(s - 2)*(s - 3), s) >>> tf3.to_expr() ((s - 3)*(s - 2))/(((s - 3)*(s - 2)*(s - 1)))
- property var
Returns the complex variable of the Laplace transform used by the polynomials of the transfer function.
Examples
>>> from sympy.abc import s, p >>> from sympy.physics.control.lti import TransferFunction >>> G1 = TransferFunction(p**2 + 2*p + 4, p - 6, p) >>> G1.var p >>> G2 = TransferFunction(0, s - 5, s) >>> G2.var s
- zeros()[source]
Returns the zeros of a transfer function.
Examples
>>> from sympy.abc import s, p, a >>> from sympy.physics.control.lti import TransferFunction >>> tf1 = TransferFunction((p + 3)*(p - 1), (p - 1)*(p + 5), p) >>> tf1.zeros() [-3, 1] >>> tf2 = TransferFunction((1 - s)**2, (s**2 + 1)**2, s) >>> tf2.zeros() [1, 1] >>> tf3 = TransferFunction(s**2, a*s + p, s) >>> tf3.zeros() [0, 0]
- class sympy.physics.control.lti.Series(*args, evaluate=False)[source]
A class for representing a series configuration of SISO systems.
- Parameters
args : SISOLinearTimeInvariant
SISO systems in a series configuration.
evaluate : Boolean, Keyword
When passed
True, returns the equivalentSeries(*args).doit(). Set toFalseby default.- Raises
ValueError
When no argument is passed.
varattribute is not same for every system.TypeError
Any of the passed
*argshas unsupported typeA combination of SISO and MIMO systems is passed. There should be homogeneity in the type of systems passed, SISO in this case.
Examples
>>> from sympy.abc import s, p, a, b >>> from sympy.physics.control.lti import TransferFunction, Series, Parallel >>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s) >>> tf2 = TransferFunction(s**3 - 2, s**4 + 5*s + 6, s) >>> tf3 = TransferFunction(p**2, p + s, s) >>> S1 = Series(tf1, tf2) >>> S1 Series(TransferFunction(a*p**2 + b*s, -p + s, s), TransferFunction(s**3 - 2, s**4 + 5*s + 6, s)) >>> S1.var s >>> S2 = Series(tf2, Parallel(tf3, -tf1)) >>> S2 Series(TransferFunction(s**3 - 2, s**4 + 5*s + 6, s), Parallel(TransferFunction(p**2, p + s, s), TransferFunction(-a*p**2 - b*s, -p + s, s))) >>> S2.var s >>> S3 = Series(Parallel(tf1, tf2), Parallel(tf2, tf3)) >>> S3 Series(Parallel(TransferFunction(a*p**2 + b*s, -p + s, s), TransferFunction(s**3 - 2, s**4 + 5*s + 6, s)), Parallel(TransferFunction(s**3 - 2, s**4 + 5*s + 6, s), TransferFunction(p**2, p + s, s))) >>> S3.var s
You can get the resultant transfer function by using
.doit()method:>>> S3 = Series(tf1, tf2, -tf3) >>> S3.doit() TransferFunction(-p**2*(s**3 - 2)*(a*p**2 + b*s), (-p + s)*(p + s)*(s**4 + 5*s + 6), s) >>> S4 = Series(tf2, Parallel(tf1, -tf3)) >>> S4.doit() TransferFunction((s**3 - 2)*(-p**2*(-p + s) + (p + s)*(a*p**2 + b*s)), (-p + s)*(p + s)*(s**4 + 5*s + 6), s)
Notes
All the transfer functions should use the same complex variable
varof the Laplace transform.See also
- doit(**kwargs)[source]
Returns the resultant transfer function obtained after evaluating the transfer functions in series configuration.
Examples
>>> from sympy.abc import s, p, a, b >>> from sympy.physics.control.lti import TransferFunction, Series >>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s) >>> tf2 = TransferFunction(s**3 - 2, s**4 + 5*s + 6, s) >>> Series(tf2, tf1).doit() TransferFunction((s**3 - 2)*(a*p**2 + b*s), (-p + s)*(s**4 + 5*s + 6), s) >>> Series(-tf1, -tf2).doit() TransferFunction((2 - s**3)*(-a*p**2 - b*s), (-p + s)*(s**4 + 5*s + 6), s)
- property is_biproper
Returns True if degree of the numerator polynomial of the resultant transfer function is equal to degree of the denominator polynomial of the same, else False.
Examples
>>> from sympy.abc import s, p, a, b >>> from sympy.physics.control.lti import TransferFunction, Series >>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s) >>> tf2 = TransferFunction(p, s**2, s) >>> tf3 = TransferFunction(s**2, 1, s) >>> S1 = Series(tf1, -tf2) >>> S1.is_biproper False >>> S2 = Series(tf2, tf3) >>> S2.is_biproper True
- property is_proper
Returns True if degree of the numerator polynomial of the resultant transfer function is less than or equal to degree of the denominator polynomial of the same, else False.
Examples
>>> from sympy.abc import s, p, a, b >>> from sympy.physics.control.lti import TransferFunction, Series >>> tf1 = TransferFunction(b*s**2 + p**2 - a*p + s, b - p**2, s) >>> tf2 = TransferFunction(p**2 - 4*p, p**3 + 3*s + 2, s) >>> tf3 = TransferFunction(s, s**2 + s + 1, s) >>> S1 = Series(-tf2, tf1) >>> S1.is_proper False >>> S2 = Series(tf1, tf2, tf3) >>> S2.is_proper True
- property is_strictly_proper
Returns True if degree of the numerator polynomial of the resultant transfer function is strictly less than degree of the denominator polynomial of the same, else False.
Examples
>>> from sympy.abc import s, p, a, b >>> from sympy.physics.control.lti import TransferFunction, Series >>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s) >>> tf2 = TransferFunction(s**3 - 2, s**2 + 5*s + 6, s) >>> tf3 = TransferFunction(1, s**2 + s + 1, s) >>> S1 = Series(tf1, tf2) >>> S1.is_strictly_proper False >>> S2 = Series(tf1, tf2, tf3) >>> S2.is_strictly_proper True
- property var
Returns the complex variable used by all the transfer functions.
Examples
>>> from sympy.abc import p >>> from sympy.physics.control.lti import TransferFunction, Series, Parallel >>> G1 = TransferFunction(p**2 + 2*p + 4, p - 6, p) >>> G2 = TransferFunction(p, 4 - p, p) >>> G3 = TransferFunction(0, p**4 - 1, p) >>> Series(G1, G2).var p >>> Series(-G3, Parallel(G1, G2)).var p
- class sympy.physics.control.lti.Parallel(*args, evaluate=False)[source]
A class for representing a parallel configuration of SISO systems.
- Parameters
args : SISOLinearTimeInvariant
SISO systems in a parallel arrangement.
evaluate : Boolean, Keyword
When passed
True, returns the equivalentParallel(*args).doit(). Set toFalseby default.- Raises
ValueError
When no argument is passed.
varattribute is not same for every system.TypeError
Any of the passed
*argshas unsupported typeA combination of SISO and MIMO systems is passed. There should be homogeneity in the type of systems passed.
Examples
>>> from sympy.abc import s, p, a, b >>> from sympy.physics.control.lti import TransferFunction, Parallel, Series >>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s) >>> tf2 = TransferFunction(s**3 - 2, s**4 + 5*s + 6, s) >>> tf3 = TransferFunction(p**2, p + s, s) >>> P1 = Parallel(tf1, tf2) >>> P1 Parallel(TransferFunction(a*p**2 + b*s, -p + s, s), TransferFunction(s**3 - 2, s**4 + 5*s + 6, s)) >>> P1.var s >>> P2 = Parallel(tf2, Series(tf3, -tf1)) >>> P2 Parallel(TransferFunction(s**3 - 2, s**4 + 5*s + 6, s), Series(TransferFunction(p**2, p + s, s), TransferFunction(-a*p**2 - b*s, -p + s, s))) >>> P2.var s >>> P3 = Parallel(Series(tf1, tf2), Series(tf2, tf3)) >>> P3 Parallel(Series(TransferFunction(a*p**2 + b*s, -p + s, s), TransferFunction(s**3 - 2, s**4 + 5*s + 6, s)), Series(TransferFunction(s**3 - 2, s**4 + 5*s + 6, s), TransferFunction(p**2, p + s, s))) >>> P3.var s
You can get the resultant transfer function by using
.doit()method:>>> Parallel(tf1, tf2, -tf3).doit() TransferFunction(-p**2*(-p + s)*(s**4 + 5*s + 6) + (-p + s)*(p + s)*(s**3 - 2) + (p + s)*(a*p**2 + b*s)*(s**4 + 5*s + 6), (-p + s)*(p + s)*(s**4 + 5*s + 6), s) >>> Parallel(tf2, Series(tf1, -tf3)).doit() TransferFunction(-p**2*(a*p**2 + b*s)*(s**4 + 5*s + 6) + (-p + s)*(p + s)*(s**3 - 2), (-p + s)*(p + s)*(s**4 + 5*s + 6), s)
Notes
All the transfer functions should use the same complex variable
varof the Laplace transform.See also
- doit(**kwargs)[source]
Returns the resultant transfer function obtained after evaluating the transfer functions in parallel configuration.
Examples
>>> from sympy.abc import s, p, a, b >>> from sympy.physics.control.lti import TransferFunction, Parallel >>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s) >>> tf2 = TransferFunction(s**3 - 2, s**4 + 5*s + 6, s) >>> Parallel(tf2, tf1).doit() TransferFunction((-p + s)*(s**3 - 2) + (a*p**2 + b*s)*(s**4 + 5*s + 6), (-p + s)*(s**4 + 5*s + 6), s) >>> Parallel(-tf1, -tf2).doit() TransferFunction((2 - s**3)*(-p + s) + (-a*p**2 - b*s)*(s**4 + 5*s + 6), (-p + s)*(s**4 + 5*s + 6), s)
- property is_biproper
Returns True if degree of the numerator polynomial of the resultant transfer function is equal to degree of the denominator polynomial of the same, else False.
Examples
>>> from sympy.abc import s, p, a, b >>> from sympy.physics.control.lti import TransferFunction, Parallel >>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s) >>> tf2 = TransferFunction(p**2, p + s, s) >>> tf3 = TransferFunction(s, s**2 + s + 1, s) >>> P1 = Parallel(tf1, -tf2) >>> P1.is_biproper True >>> P2 = Parallel(tf2, tf3) >>> P2.is_biproper False
- property is_proper
Returns True if degree of the numerator polynomial of the resultant transfer function is less than or equal to degree of the denominator polynomial of the same, else False.
Examples
>>> from sympy.abc import s, p, a, b >>> from sympy.physics.control.lti import TransferFunction, Parallel >>> tf1 = TransferFunction(b*s**2 + p**2 - a*p + s, b - p**2, s) >>> tf2 = TransferFunction(p**2 - 4*p, p**3 + 3*s + 2, s) >>> tf3 = TransferFunction(s, s**2 + s + 1, s) >>> P1 = Parallel(-tf2, tf1) >>> P1.is_proper False >>> P2 = Parallel(tf2, tf3) >>> P2.is_proper True
- property is_strictly_proper
Returns True if degree of the numerator polynomial of the resultant transfer function is strictly less than degree of the denominator polynomial of the same, else False.
Examples
>>> from sympy.abc import s, p, a, b >>> from sympy.physics.control.lti import TransferFunction, Parallel >>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s) >>> tf2 = TransferFunction(s**3 - 2, s**4 + 5*s + 6, s) >>> tf3 = TransferFunction(s, s**2 + s + 1, s) >>> P1 = Parallel(tf1, tf2) >>> P1.is_strictly_proper False >>> P2 = Parallel(tf2, tf3) >>> P2.is_strictly_proper True
- property var
Returns the complex variable used by all the transfer functions.
Examples
>>> from sympy.abc import p >>> from sympy.physics.control.lti import TransferFunction, Parallel, Series >>> G1 = TransferFunction(p**2 + 2*p + 4, p - 6, p) >>> G2 = TransferFunction(p, 4 - p, p) >>> G3 = TransferFunction(0, p**4 - 1, p) >>> Parallel(G1, G2).var p >>> Parallel(-G3, Series(G1, G2)).var p
- class sympy.physics.control.lti.Feedback(sys1, sys2=None, sign=- 1)[source]
A class for representing closed-loop feedback interconnection between two SISO input/output systems.
The first argument,
sys1, is the feedforward part of the closed-loop system or in simple words, the dynamical model representing the process to be controlled. The second argument,sys2, is the feedback system and controls the fed back signal tosys1. Bothsys1andsys2can either beSeriesorTransferFunctionobjects.- Parameters
sys1 : Series, TransferFunction
The feedforward path system.
sys2 : Series, TransferFunction, optional
The feedback path system (often a feedback controller). It is the model sitting on the feedback path.
If not specified explicitly, the sys2 is assumed to be unit (1.0) transfer function.
sign : int, optional
The sign of feedback. Can either be
1(for positive feedback) or-1(for negative feedback). Default value is \(-1\).- Raises
ValueError
When
sys1andsys2are not using the same complex variable of the Laplace transform.When a combination of
sys1andsys2yields zero denominator.TypeError
When either
sys1orsys2is not aSeriesor aTransferFunctionobject.
Examples
>>> from sympy.abc import s >>> from sympy.physics.control.lti import TransferFunction, Feedback >>> plant = TransferFunction(3*s**2 + 7*s - 3, s**2 - 4*s + 2, s) >>> controller = TransferFunction(5*s - 10, s + 7, s) >>> F1 = Feedback(plant, controller) >>> F1 Feedback(TransferFunction(3*s**2 + 7*s - 3, s**2 - 4*s + 2, s), TransferFunction(5*s - 10, s + 7, s), -1) >>> F1.var s >>> F1.args (TransferFunction(3*s**2 + 7*s - 3, s**2 - 4*s + 2, s), TransferFunction(5*s - 10, s + 7, s), -1)
You can get the feedforward and feedback path systems by using
.sys1and.sys2respectively.>>> F1.sys1 TransferFunction(3*s**2 + 7*s - 3, s**2 - 4*s + 2, s) >>> F1.sys2 TransferFunction(5*s - 10, s + 7, s)
You can get the resultant closed loop transfer function obtained by negative feedback interconnection using
.doit()method.>>> F1.doit() TransferFunction((s + 7)*(s**2 - 4*s + 2)*(3*s**2 + 7*s - 3), ((s + 7)*(s**2 - 4*s + 2) + (5*s - 10)*(3*s**2 + 7*s - 3))*(s**2 - 4*s + 2), s) >>> G = TransferFunction(2*s**2 + 5*s + 1, s**2 + 2*s + 3, s) >>> C = TransferFunction(5*s + 10, s + 10, s) >>> F2 = Feedback(G*C, TransferFunction(1, 1, s)) >>> F2.doit() TransferFunction((s + 10)*(5*s + 10)*(s**2 + 2*s + 3)*(2*s**2 + 5*s + 1), (s + 10)*((s + 10)*(s**2 + 2*s + 3) + (5*s + 10)*(2*s**2 + 5*s + 1))*(s**2 + 2*s + 3), s)
To negate a
Feedbackobject, the-operator can be prepended:>>> -F1 Feedback(TransferFunction(-3*s**2 - 7*s + 3, s**2 - 4*s + 2, s), TransferFunction(10 - 5*s, s + 7, s), -1) >>> -F2 Feedback(Series(TransferFunction(-1, 1, s), TransferFunction(2*s**2 + 5*s + 1, s**2 + 2*s + 3, s), TransferFunction(5*s + 10, s + 10, s)), TransferFunction(-1, 1, s), -1)
See also
- doit(cancel=False, expand=False, **kwargs)[source]
Returns the resultant transfer function obtained by the feedback interconnection.
Examples
>>> from sympy.abc import s >>> from sympy.physics.control.lti import TransferFunction, Feedback >>> plant = TransferFunction(3*s**2 + 7*s - 3, s**2 - 4*s + 2, s) >>> controller = TransferFunction(5*s - 10, s + 7, s) >>> F1 = Feedback(plant, controller) >>> F1.doit() TransferFunction((s + 7)*(s**2 - 4*s + 2)*(3*s**2 + 7*s - 3), ((s + 7)*(s**2 - 4*s + 2) + (5*s - 10)*(3*s**2 + 7*s - 3))*(s**2 - 4*s + 2), s) >>> G = TransferFunction(2*s**2 + 5*s + 1, s**2 + 2*s + 3, s) >>> F2 = Feedback(G, TransferFunction(1, 1, s)) >>> F2.doit() TransferFunction((s**2 + 2*s + 3)*(2*s**2 + 5*s + 1), (s**2 + 2*s + 3)*(3*s**2 + 7*s + 4), s)
Use kwarg
expand=Trueto expand the resultant transfer function. Usecancel=Trueto cancel out the common terms in numerator and denominator.>>> F2.doit(cancel=True, expand=True) TransferFunction(2*s**2 + 5*s + 1, 3*s**2 + 7*s + 4, s) >>> F2.doit(expand=True) TransferFunction(2*s**4 + 9*s**3 + 17*s**2 + 17*s + 3, 3*s**4 + 13*s**3 + 27*s**2 + 29*s + 12, s)
- property sensitivity
Returns the sensitivity function of the feedback loop.
Sensitivity of a Feedback system is the ratio of change in the open loop gain to the change in the closed loop gain.
Note
This method would not return the complementary sensitivity function.
Examples
>>> from sympy.abc import p >>> from sympy.physics.control.lti import TransferFunction, Feedback >>> C = TransferFunction(5*p + 10, p + 10, p) >>> P = TransferFunction(1 - p, p + 2, p) >>> F_1 = Feedback(P, C) >>> F_1.sensitivity 1/((1 - p)*(5*p + 10)/((p + 2)*(p + 10)) + 1)
- property sign
Returns the type of MIMO Feedback model.
1for Positive and-1for Negative.
- property sys1
Returns the feedforward system of the feedback interconnection.
Examples
>>> from sympy.abc import s, p >>> from sympy.physics.control.lti import TransferFunction, Feedback >>> plant = TransferFunction(3*s**2 + 7*s - 3, s**2 - 4*s + 2, s) >>> controller = TransferFunction(5*s - 10, s + 7, s) >>> F1 = Feedback(plant, controller) >>> F1.sys1 TransferFunction(3*s**2 + 7*s - 3, s**2 - 4*s + 2, s) >>> G = TransferFunction(2*s**2 + 5*s + 1, p**2 + 2*p + 3, p) >>> C = TransferFunction(5*p + 10, p + 10, p) >>> P = TransferFunction(1 - s, p + 2, p) >>> F2 = Feedback(TransferFunction(1, 1, p), G*C*P) >>> F2.sys1 TransferFunction(1, 1, p)
- property sys2
Returns the feedback controller of the feedback interconnection.
Examples
>>> from sympy.abc import s, p >>> from sympy.physics.control.lti import TransferFunction, Feedback >>> plant = TransferFunction(3*s**2 + 7*s - 3, s**2 - 4*s + 2, s) >>> controller = TransferFunction(5*s - 10, s + 7, s) >>> F1 = Feedback(plant, controller) >>> F1.sys2 TransferFunction(5*s - 10, s + 7, s) >>> G = TransferFunction(2*s**2 + 5*s + 1, p**2 + 2*p + 3, p) >>> C = TransferFunction(5*p + 10, p + 10, p) >>> P = TransferFunction(1 - s, p + 2, p) >>> F2 = Feedback(TransferFunction(1, 1, p), G*C*P) >>> F2.sys2 Series(TransferFunction(2*s**2 + 5*s + 1, p**2 + 2*p + 3, p), TransferFunction(5*p + 10, p + 10, p), TransferFunction(1 - s, p + 2, p))
- property var
Returns the complex variable of the Laplace transform used by all the transfer functions involved in the feedback interconnection.
Examples
>>> from sympy.abc import s, p >>> from sympy.physics.control.lti import TransferFunction, Feedback >>> plant = TransferFunction(3*s**2 + 7*s - 3, s**2 - 4*s + 2, s) >>> controller = TransferFunction(5*s - 10, s + 7, s) >>> F1 = Feedback(plant, controller) >>> F1.var s >>> G = TransferFunction(2*s**2 + 5*s + 1, p**2 + 2*p + 3, p) >>> C = TransferFunction(5*p + 10, p + 10, p) >>> P = TransferFunction(1 - s, p + 2, p) >>> F2 = Feedback(TransferFunction(1, 1, p), G*C*P) >>> F2.var p
- class sympy.physics.control.lti.TransferFunctionMatrix(arg)[source]
A class for representing the MIMO (multiple-input and multiple-output) generalization of the SISO (single-input and single-output) transfer function.
It is a matrix of transfer functions (
TransferFunction, SISO-Seriesor SISO-Parallel). There is only one argument,argwhich is also the compulsory argument.argis expected to be strictly of the type list of lists which holds the transfer functions or reducible to transfer functions.- Parameters
arg : Nested
List(strictly).Users are expected to input a nested list of
TransferFunction,Seriesand/orParallelobjects.
Examples
Note
pprint()can be used for better visualization ofTransferFunctionMatrixobjects.>>> from sympy.abc import s, p, a >>> from sympy import pprint >>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix, Series, Parallel >>> tf_1 = TransferFunction(s + a, s**2 + s + 1, s) >>> tf_2 = TransferFunction(p**4 - 3*p + 2, s + p, s) >>> tf_3 = TransferFunction(3, s + 2, s) >>> tf_4 = TransferFunction(-a + p, 9*s - 9, s) >>> tfm_1 = TransferFunctionMatrix([[tf_1], [tf_2], [tf_3]]) >>> tfm_1 TransferFunctionMatrix(((TransferFunction(a + s, s**2 + s + 1, s),), (TransferFunction(p**4 - 3*p + 2, p + s, s),), (TransferFunction(3, s + 2, s),))) >>> tfm_1.var s >>> tfm_1.num_inputs 1 >>> tfm_1.num_outputs 3 >>> tfm_1.shape (3, 1) >>> tfm_1.args (((TransferFunction(a + s, s**2 + s + 1, s),), (TransferFunction(p**4 - 3*p + 2, p + s, s),), (TransferFunction(3, s + 2, s),)),) >>> tfm_2 = TransferFunctionMatrix([[tf_1, -tf_3], [tf_2, -tf_1], [tf_3, -tf_2]]) >>> tfm_2 TransferFunctionMatrix(((TransferFunction(a + s, s**2 + s + 1, s), TransferFunction(-3, s + 2, s)), (TransferFunction(p**4 - 3*p + 2, p + s, s), TransferFunction(-a - s, s**2 + s + 1, s)), (TransferFunction(3, s + 2, s), TransferFunction(-p**4 + 3*p - 2, p + s, s)))) >>> pprint(tfm_2, use_unicode=False) # pretty-printing for better visualization [ a + s -3 ] [ ---------- ----- ] [ 2 s + 2 ] [ s + s + 1 ] [ ] [ 4 ] [p - 3*p + 2 -a - s ] [------------ ---------- ] [ p + s 2 ] [ s + s + 1 ] [ ] [ 4 ] [ 3 - p + 3*p - 2] [ ----- --------------] [ s + 2 p + s ]{t}
TransferFunctionMatrix can be transposed, if user wants to switch the input and output transfer functions
>>> tfm_2.transpose() TransferFunctionMatrix(((TransferFunction(a + s, s**2 + s + 1, s), TransferFunction(p**4 - 3*p + 2, p + s, s), TransferFunction(3, s + 2, s)), (TransferFunction(-3, s + 2, s), TransferFunction(-a - s, s**2 + s + 1, s), TransferFunction(-p**4 + 3*p - 2, p + s, s)))) >>> pprint(_, use_unicode=False) [ 4 ] [ a + s p - 3*p + 2 3 ] [---------- ------------ ----- ] [ 2 p + s s + 2 ] [s + s + 1 ] [ ] [ 4 ] [ -3 -a - s - p + 3*p - 2] [ ----- ---------- --------------] [ s + 2 2 p + s ] [ s + s + 1 ]{t}
>>> tf_5 = TransferFunction(5, s, s) >>> tf_6 = TransferFunction(5*s, (2 + s**2), s) >>> tf_7 = TransferFunction(5, (s*(2 + s**2)), s) >>> tf_8 = TransferFunction(5, 1, s) >>> tfm_3 = TransferFunctionMatrix([[tf_5, tf_6], [tf_7, tf_8]]) >>> tfm_3 TransferFunctionMatrix(((TransferFunction(5, s, s), TransferFunction(5*s, s**2 + 2, s)), (TransferFunction(5, s*(s**2 + 2), s), TransferFunction(5, 1, s)))) >>> pprint(tfm_3, use_unicode=False) [ 5 5*s ] [ - ------] [ s 2 ] [ s + 2] [ ] [ 5 5 ] [---------- - ] [ / 2 \ 1 ] [s*\s + 2/ ]{t} >>> tfm_3.var s >>> tfm_3.shape (2, 2) >>> tfm_3.num_outputs 2 >>> tfm_3.num_inputs 2 >>> tfm_3.args (((TransferFunction(5, s, s), TransferFunction(5*s, s**2 + 2, s)), (TransferFunction(5, s*(s**2 + 2), s), TransferFunction(5, 1, s))),)
To access the
TransferFunctionat any index in theTransferFunctionMatrix, use the index notation.>>> tfm_3[1, 0] # gives the TransferFunction present at 2nd Row and 1st Col. Similar to that in Matrix classes TransferFunction(5, s*(s**2 + 2), s) >>> tfm_3[0, 0] # gives the TransferFunction present at 1st Row and 1st Col. TransferFunction(5, s, s) >>> tfm_3[:, 0] # gives the first column TransferFunctionMatrix(((TransferFunction(5, s, s),), (TransferFunction(5, s*(s**2 + 2), s),))) >>> pprint(_, use_unicode=False) [ 5 ] [ - ] [ s ] [ ] [ 5 ] [----------] [ / 2 \] [s*\s + 2/]{t} >>> tfm_3[0, :] # gives the first row TransferFunctionMatrix(((TransferFunction(5, s, s), TransferFunction(5*s, s**2 + 2, s)),)) >>> pprint(_, use_unicode=False) [5 5*s ] [- ------] [s 2 ] [ s + 2]{t}
To negate a transfer function matrix,
-operator can be prepended:>>> tfm_4 = TransferFunctionMatrix([[tf_2], [-tf_1], [tf_3]]) >>> -tfm_4 TransferFunctionMatrix(((TransferFunction(-p**4 + 3*p - 2, p + s, s),), (TransferFunction(a + s, s**2 + s + 1, s),), (TransferFunction(-3, s + 2, s),))) >>> tfm_5 = TransferFunctionMatrix([[tf_1, tf_2], [tf_3, -tf_1]]) >>> -tfm_5 TransferFunctionMatrix(((TransferFunction(-a - s, s**2 + s + 1, s), TransferFunction(-p**4 + 3*p - 2, p + s, s)), (TransferFunction(-3, s + 2, s), TransferFunction(a + s, s**2 + s + 1, s))))
subs()returns theTransferFunctionMatrixobject with the value substituted in the expression. This will not mutate your originalTransferFunctionMatrix.>>> tfm_2.subs(p, 2) # substituting p everywhere in tfm_2 with 2. TransferFunctionMatrix(((TransferFunction(a + s, s**2 + s + 1, s), TransferFunction(-3, s + 2, s)), (TransferFunction(12, s + 2, s), TransferFunction(-a - s, s**2 + s + 1, s)), (TransferFunction(3, s + 2, s), TransferFunction(-12, s + 2, s)))) >>> pprint(_, use_unicode=False) [ a + s -3 ] [---------- ----- ] [ 2 s + 2 ] [s + s + 1 ] [ ] [ 12 -a - s ] [ ----- ----------] [ s + 2 2 ] [ s + s + 1] [ ] [ 3 -12 ] [ ----- ----- ] [ s + 2 s + 2 ]{t} >>> pprint(tfm_2, use_unicode=False) # State of tfm_2 is unchanged after substitution [ a + s -3 ] [ ---------- ----- ] [ 2 s + 2 ] [ s + s + 1 ] [ ] [ 4 ] [p - 3*p + 2 -a - s ] [------------ ---------- ] [ p + s 2 ] [ s + s + 1 ] [ ] [ 4 ] [ 3 - p + 3*p - 2] [ ----- --------------] [ s + 2 p + s ]{t}
subs()also supports multiple substitutions.>>> tfm_2.subs({p: 2, a: 1}) # substituting p with 2 and a with 1 TransferFunctionMatrix(((TransferFunction(s + 1, s**2 + s + 1, s), TransferFunction(-3, s + 2, s)), (TransferFunction(12, s + 2, s), TransferFunction(-s - 1, s**2 + s + 1, s)), (TransferFunction(3, s + 2, s), TransferFunction(-12, s + 2, s)))) >>> pprint(_, use_unicode=False) [ s + 1 -3 ] [---------- ----- ] [ 2 s + 2 ] [s + s + 1 ] [ ] [ 12 -s - 1 ] [ ----- ----------] [ s + 2 2 ] [ s + s + 1] [ ] [ 3 -12 ] [ ----- ----- ] [ s + 2 s + 2 ]{t}
Users can reduce the
SeriesandParallelelements of the matrix toTransferFunctionby usingdoit().>>> tfm_6 = TransferFunctionMatrix([[Series(tf_3, tf_4), Parallel(tf_3, tf_4)]]) >>> tfm_6 TransferFunctionMatrix(((Series(TransferFunction(3, s + 2, s), TransferFunction(-a + p, 9*s - 9, s)), Parallel(TransferFunction(3, s + 2, s), TransferFunction(-a + p, 9*s - 9, s))),)) >>> pprint(tfm_6, use_unicode=False) [ -a + p 3 -a + p 3 ] [-------*----- ------- + -----] [9*s - 9 s + 2 9*s - 9 s + 2]{t} >>> tfm_6.doit() TransferFunctionMatrix(((TransferFunction(-3*a + 3*p, (s + 2)*(9*s - 9), s), TransferFunction(27*s + (-a + p)*(s + 2) - 27, (s + 2)*(9*s - 9), s)),)) >>> pprint(_, use_unicode=False) [ -3*a + 3*p 27*s + (-a + p)*(s + 2) - 27] [----------------- ----------------------------] [(s + 2)*(9*s - 9) (s + 2)*(9*s - 9) ]{t} >>> tf_9 = TransferFunction(1, s, s) >>> tf_10 = TransferFunction(1, s**2, s) >>> tfm_7 = TransferFunctionMatrix([[Series(tf_9, tf_10), tf_9], [tf_10, Parallel(tf_9, tf_10)]]) >>> tfm_7 TransferFunctionMatrix(((Series(TransferFunction(1, s, s), TransferFunction(1, s**2, s)), TransferFunction(1, s, s)), (TransferFunction(1, s**2, s), Parallel(TransferFunction(1, s, s), TransferFunction(1, s**2, s))))) >>> pprint(tfm_7, use_unicode=False) [ 1 1 ] [---- - ] [ 2 s ] [s*s ] [ ] [ 1 1 1] [ -- -- + -] [ 2 2 s] [ s s ]{t} >>> tfm_7.doit() TransferFunctionMatrix(((TransferFunction(1, s**3, s), TransferFunction(1, s, s)), (TransferFunction(1, s**2, s), TransferFunction(s**2 + s, s**3, s)))) >>> pprint(_, use_unicode=False) [1 1 ] [-- - ] [ 3 s ] [s ] [ ] [ 2 ] [1 s + s] [-- ------] [ 2 3 ] [s s ]{t}
Addition, subtraction, and multiplication of transfer function matrices can form unevaluated
SeriesorParallelobjects.For addition and subtraction: All the transfer function matrices must have the same shape.
For multiplication (C = A * B): The number of inputs of the first transfer function matrix (A) must be equal to the number of outputs of the second transfer function matrix (B).
Also, use pretty-printing (
pprint) to analyse better.>>> tfm_8 = TransferFunctionMatrix([[tf_3], [tf_2], [-tf_1]]) >>> tfm_9 = TransferFunctionMatrix([[-tf_3]]) >>> tfm_10 = TransferFunctionMatrix([[tf_1], [tf_2], [tf_4]]) >>> tfm_11 = TransferFunctionMatrix([[tf_4], [-tf_1]]) >>> tfm_12 = TransferFunctionMatrix([[tf_4, -tf_1, tf_3], [-tf_2, -tf_4, -tf_3]]) >>> tfm_8 + tfm_10 MIMOParallel(TransferFunctionMatrix(((TransferFunction(3, s + 2, s),), (TransferFunction(p**4 - 3*p + 2, p + s, s),), (TransferFunction(-a - s, s**2 + s + 1, s),))), TransferFunctionMatrix(((TransferFunction(a + s, s**2 + s + 1, s),), (TransferFunction(p**4 - 3*p + 2, p + s, s),), (TransferFunction(-a + p, 9*s - 9, s),)))) >>> pprint(_, use_unicode=False) [ 3 ] [ a + s ] [ ----- ] [ ---------- ] [ s + 2 ] [ 2 ] [ ] [ s + s + 1 ] [ 4 ] [ ] [p - 3*p + 2] [ 4 ] [------------] + [p - 3*p + 2] [ p + s ] [------------] [ ] [ p + s ] [ -a - s ] [ ] [ ---------- ] [ -a + p ] [ 2 ] [ ------- ] [ s + s + 1 ]{t} [ 9*s - 9 ]{t} >>> -tfm_10 - tfm_8 MIMOParallel(TransferFunctionMatrix(((TransferFunction(-a - s, s**2 + s + 1, s),), (TransferFunction(-p**4 + 3*p - 2, p + s, s),), (TransferFunction(a - p, 9*s - 9, s),))), TransferFunctionMatrix(((TransferFunction(-3, s + 2, s),), (TransferFunction(-p**4 + 3*p - 2, p + s, s),), (TransferFunction(a + s, s**2 + s + 1, s),)))) >>> pprint(_, use_unicode=False) [ -a - s ] [ -3 ] [ ---------- ] [ ----- ] [ 2 ] [ s + 2 ] [ s + s + 1 ] [ ] [ ] [ 4 ] [ 4 ] [- p + 3*p - 2] [- p + 3*p - 2] + [--------------] [--------------] [ p + s ] [ p + s ] [ ] [ ] [ a + s ] [ a - p ] [ ---------- ] [ ------- ] [ 2 ] [ 9*s - 9 ]{t} [ s + s + 1 ]{t} >>> tfm_12 * tfm_8 MIMOSeries(TransferFunctionMatrix(((TransferFunction(3, s + 2, s),), (TransferFunction(p**4 - 3*p + 2, p + s, s),), (TransferFunction(-a - s, s**2 + s + 1, s),))), TransferFunctionMatrix(((TransferFunction(-a + p, 9*s - 9, s), TransferFunction(-a - s, s**2 + s + 1, s), TransferFunction(3, s + 2, s)), (TransferFunction(-p**4 + 3*p - 2, p + s, s), TransferFunction(a - p, 9*s - 9, s), TransferFunction(-3, s + 2, s))))) >>> pprint(_, use_unicode=False) [ 3 ] [ ----- ] [ -a + p -a - s 3 ] [ s + 2 ] [ ------- ---------- -----] [ ] [ 9*s - 9 2 s + 2] [ 4 ] [ s + s + 1 ] [p - 3*p + 2] [ ] *[------------] [ 4 ] [ p + s ] [- p + 3*p - 2 a - p -3 ] [ ] [-------------- ------- -----] [ -a - s ] [ p + s 9*s - 9 s + 2]{t} [ ---------- ] [ 2 ] [ s + s + 1 ]{t} >>> tfm_12 * tfm_8 * tfm_9 MIMOSeries(TransferFunctionMatrix(((TransferFunction(-3, s + 2, s),),)), TransferFunctionMatrix(((TransferFunction(3, s + 2, s),), (TransferFunction(p**4 - 3*p + 2, p + s, s),), (TransferFunction(-a - s, s**2 + s + 1, s),))), TransferFunctionMatrix(((TransferFunction(-a + p, 9*s - 9, s), TransferFunction(-a - s, s**2 + s + 1, s), TransferFunction(3, s + 2, s)), (TransferFunction(-p**4 + 3*p - 2, p + s, s), TransferFunction(a - p, 9*s - 9, s), TransferFunction(-3, s + 2, s))))) >>> pprint(_, use_unicode=False) [ 3 ] [ ----- ] [ -a + p -a - s 3 ] [ s + 2 ] [ ------- ---------- -----] [ ] [ 9*s - 9 2 s + 2] [ 4 ] [ s + s + 1 ] [p - 3*p + 2] [ -3 ] [ ] *[------------] *[-----] [ 4 ] [ p + s ] [s + 2]{t} [- p + 3*p - 2 a - p -3 ] [ ] [-------------- ------- -----] [ -a - s ] [ p + s 9*s - 9 s + 2]{t} [ ---------- ] [ 2 ] [ s + s + 1 ]{t} >>> tfm_10 + tfm_8*tfm_9 MIMOParallel(TransferFunctionMatrix(((TransferFunction(a + s, s**2 + s + 1, s),), (TransferFunction(p**4 - 3*p + 2, p + s, s),), (TransferFunction(-a + p, 9*s - 9, s),))), MIMOSeries(TransferFunctionMatrix(((TransferFunction(-3, s + 2, s),),)), TransferFunctionMatrix(((TransferFunction(3, s + 2, s),), (TransferFunction(p**4 - 3*p + 2, p + s, s),), (TransferFunction(-a - s, s**2 + s + 1, s),))))) >>> pprint(_, use_unicode=False) [ a + s ] [ 3 ] [ ---------- ] [ ----- ] [ 2 ] [ s + 2 ] [ s + s + 1 ] [ ] [ ] [ 4 ] [ 4 ] [p - 3*p + 2] [ -3 ] [p - 3*p + 2] + [------------] *[-----] [------------] [ p + s ] [s + 2]{t} [ p + s ] [ ] [ ] [ -a - s ] [ -a + p ] [ ---------- ] [ ------- ] [ 2 ] [ 9*s - 9 ]{t} [ s + s + 1 ]{t}
These unevaluated
SeriesorParallelobjects can convert into the resultant transfer function matrix using.doit()method or by.rewrite(TransferFunctionMatrix).>>> (-tfm_8 + tfm_10 + tfm_8*tfm_9).doit() TransferFunctionMatrix(((TransferFunction((a + s)*(s + 2)**3 - 3*(s + 2)**2*(s**2 + s + 1) - 9*(s + 2)*(s**2 + s + 1), (s + 2)**3*(s**2 + s + 1), s),), (TransferFunction((p + s)*(-3*p**4 + 9*p - 6), (p + s)**2*(s + 2), s),), (TransferFunction((-a + p)*(s + 2)*(s**2 + s + 1)**2 + (a + s)*(s + 2)*(9*s - 9)*(s**2 + s + 1) + (3*a + 3*s)*(9*s - 9)*(s**2 + s + 1), (s + 2)*(9*s - 9)*(s**2 + s + 1)**2, s),))) >>> (-tfm_12 * -tfm_8 * -tfm_9).rewrite(TransferFunctionMatrix) TransferFunctionMatrix(((TransferFunction(3*(-3*a + 3*p)*(p + s)*(s + 2)*(s**2 + s + 1)**2 + 3*(-3*a - 3*s)*(p + s)*(s + 2)*(9*s - 9)*(s**2 + s + 1) + 3*(a + s)*(s + 2)**2*(9*s - 9)*(-p**4 + 3*p - 2)*(s**2 + s + 1), (p + s)*(s + 2)**3*(9*s - 9)*(s**2 + s + 1)**2, s),), (TransferFunction(3*(-a + p)*(p + s)*(s + 2)**2*(-p**4 + 3*p - 2)*(s**2 + s + 1) + 3*(3*a + 3*s)*(p + s)**2*(s + 2)*(9*s - 9) + 3*(p + s)*(s + 2)*(9*s - 9)*(-3*p**4 + 9*p - 6)*(s**2 + s + 1), (p + s)**2*(s + 2)**3*(9*s - 9)*(s**2 + s + 1), s),)))
See also
- elem_poles()[source]
Returns the poles of each element of the
TransferFunctionMatrix.Note
Actual poles of a MIMO system are NOT the poles of individual elements.
Examples
>>> from sympy.abc import s >>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix >>> tf_1 = TransferFunction(3, (s + 1), s) >>> tf_2 = TransferFunction(s + 6, (s + 1)*(s + 2), s) >>> tf_3 = TransferFunction(s + 3, s**2 + 3*s + 2, s) >>> tf_4 = TransferFunction(s + 2, s**2 + 5*s - 10, s) >>> tfm_1 = TransferFunctionMatrix([[tf_1, tf_2], [tf_3, tf_4]]) >>> tfm_1 TransferFunctionMatrix(((TransferFunction(3, s + 1, s), TransferFunction(s + 6, (s + 1)*(s + 2), s)), (TransferFunction(s + 3, s**2 + 3*s + 2, s), TransferFunction(s + 2, s**2 + 5*s - 10, s)))) >>> tfm_1.elem_poles() [[[-1], [-2, -1]], [[-2, -1], [-5/2 + sqrt(65)/2, -sqrt(65)/2 - 5/2]]]
See also
- elem_zeros()[source]
Returns the zeros of each element of the
TransferFunctionMatrix.Note
Actual zeros of a MIMO system are NOT the zeros of individual elements.
Examples
>>> from sympy.abc import s >>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix >>> tf_1 = TransferFunction(3, (s + 1), s) >>> tf_2 = TransferFunction(s + 6, (s + 1)*(s + 2), s) >>> tf_3 = TransferFunction(s + 3, s**2 + 3*s + 2, s) >>> tf_4 = TransferFunction(s**2 - 9*s + 20, s**2 + 5*s - 10, s) >>> tfm_1 = TransferFunctionMatrix([[tf_1, tf_2], [tf_3, tf_4]]) >>> tfm_1 TransferFunctionMatrix(((TransferFunction(3, s + 1, s), TransferFunction(s + 6, (s + 1)*(s + 2), s)), (TransferFunction(s + 3, s**2 + 3*s + 2, s), TransferFunction(s**2 - 9*s + 20, s**2 + 5*s - 10, s)))) >>> tfm_1.elem_zeros() [[[], [-6]], [[-3], [4, 5]]]
See also
- classmethod from_Matrix(matrix, var)[source]
Creates a new
TransferFunctionMatrixefficiently from a SymPy Matrix ofExprobjects.- Parameters
matrix :
ImmutableMatrixhavingExpr/Numberelements.var : Symbol
Complex variable of the Laplace transform which will be used by the all the
TransferFunctionobjects in theTransferFunctionMatrix.
Examples
>>> from sympy.abc import s >>> from sympy.physics.control.lti import TransferFunctionMatrix >>> from sympy import Matrix, pprint >>> M = Matrix([[s, 1/s], [1/(s+1), s]]) >>> M_tf = TransferFunctionMatrix.from_Matrix(M, s) >>> pprint(M_tf, use_unicode=False) [ s 1] [ - -] [ 1 s] [ ] [ 1 s] [----- -] [s + 1 1]{t} >>> M_tf.elem_poles() [[[], [0]], [[-1], []]] >>> M_tf.elem_zeros() [[[0], []], [[], [0]]]
- property num_inputs
Returns the number of inputs of the system.
Examples
>>> from sympy.abc import s, p >>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix >>> G1 = TransferFunction(s + 3, s**2 - 3, s) >>> G2 = TransferFunction(4, s**2, s) >>> G3 = TransferFunction(p**2 + s**2, p - 3, s) >>> tfm_1 = TransferFunctionMatrix([[G2, -G1, G3], [-G2, -G1, -G3]]) >>> tfm_1.num_inputs 3
See also
- property num_outputs
Returns the number of outputs of the system.
Examples
>>> from sympy.abc import s >>> from sympy.physics.control.lti import TransferFunctionMatrix >>> from sympy import Matrix >>> M_1 = Matrix([[s], [1/s]]) >>> TFM = TransferFunctionMatrix.from_Matrix(M_1, s) >>> print(TFM) TransferFunctionMatrix(((TransferFunction(s, 1, s),), (TransferFunction(1, s, s),))) >>> TFM.num_outputs 2
See also
- property shape
Returns the shape of the transfer function matrix, that is,
(# of outputs, # of inputs).Examples
>>> from sympy.abc import s, p >>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix >>> tf1 = TransferFunction(p**2 - 1, s**4 + s**3 - p, p) >>> tf2 = TransferFunction(1 - p, p**2 - 3*p + 7, p) >>> tf3 = TransferFunction(3, 4, p) >>> tfm1 = TransferFunctionMatrix([[tf1, -tf2]]) >>> tfm1.shape (1, 2) >>> tfm2 = TransferFunctionMatrix([[-tf2, tf3], [tf1, -tf1]]) >>> tfm2.shape (2, 2)
- transpose()[source]
Returns the transpose of the
TransferFunctionMatrix(switched input and output layers).
- property var
Returns the complex variable used by all the transfer functions or
Series/Parallelobjects in a transfer function matrix.Examples
>>> from sympy.abc import p, s >>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix, Series, Parallel >>> G1 = TransferFunction(p**2 + 2*p + 4, p - 6, p) >>> G2 = TransferFunction(p, 4 - p, p) >>> G3 = TransferFunction(0, p**4 - 1, p) >>> G4 = TransferFunction(s + 1, s**2 + s + 1, s) >>> S1 = Series(G1, G2) >>> S2 = Series(-G3, Parallel(G2, -G1)) >>> tfm1 = TransferFunctionMatrix([[G1], [G2], [G3]]) >>> tfm1.var p >>> tfm2 = TransferFunctionMatrix([[-S1, -S2], [S1, S2]]) >>> tfm2.var p >>> tfm3 = TransferFunctionMatrix([[G4]]) >>> tfm3.var s
- class sympy.physics.control.lti.MIMOSeries(*args, evaluate=False)[source]
A class for representing a series configuration of MIMO systems.
- Parameters
args : MIMOLinearTimeInvariant
MIMO systems in a series configuration.
evaluate : Boolean, Keyword
When passed
True, returns the equivalentMIMOSeries(*args).doit(). Set toFalseby default.- Raises
ValueError
When no argument is passed.
varattribute is not same for every system.num_outputsof the MIMO system is not equal to thenum_inputsof its adjacent MIMO system. (Matrix multiplication constraint, basically)TypeError
Any of the passed
*argshas unsupported typeA combination of SISO and MIMO systems is passed. There should be homogeneity in the type of systems passed, MIMO in this case.
Examples
>>> from sympy.abc import s >>> from sympy.physics.control.lti import MIMOSeries, TransferFunctionMatrix >>> from sympy import Matrix, pprint >>> mat_a = Matrix([[5*s], [5]]) # 2 Outputs 1 Input >>> mat_b = Matrix([[5, 1/(6*s**2)]]) # 1 Output 2 Inputs >>> mat_c = Matrix([[1, s], [5/s, 1]]) # 2 Outputs 2 Inputs >>> tfm_a = TransferFunctionMatrix.from_Matrix(mat_a, s) >>> tfm_b = TransferFunctionMatrix.from_Matrix(mat_b, s) >>> tfm_c = TransferFunctionMatrix.from_Matrix(mat_c, s) >>> MIMOSeries(tfm_c, tfm_b, tfm_a) MIMOSeries(TransferFunctionMatrix(((TransferFunction(1, 1, s), TransferFunction(s, 1, s)), (TransferFunction(5, s, s), TransferFunction(1, 1, s)))), TransferFunctionMatrix(((TransferFunction(5, 1, s), TransferFunction(1, 6*s**2, s)),)), TransferFunctionMatrix(((TransferFunction(5*s, 1, s),), (TransferFunction(5, 1, s),)))) >>> pprint(_, use_unicode=False) # For Better Visualization [5*s] [1 s] [---] [5 1 ] [- -] [ 1 ] [- ----] [1 1] [ ] *[1 2] *[ ] [ 5 ] [ 6*s ]{t} [5 1] [ - ] [- -] [ 1 ]{t} [s 1]{t} >>> MIMOSeries(tfm_c, tfm_b, tfm_a).doit() TransferFunctionMatrix(((TransferFunction(150*s**4 + 25*s, 6*s**3, s), TransferFunction(150*s**4 + 5*s, 6*s**2, s)), (TransferFunction(150*s**3 + 25, 6*s**3, s), TransferFunction(150*s**3 + 5, 6*s**2, s)))) >>> pprint(_, use_unicode=False) # (2 Inputs -A-> 2 Outputs) -> (2 Inputs -B-> 1 Output) -> (1 Input -C-> 2 Outputs) is equivalent to (2 Inputs -Series Equivalent-> 2 Outputs). [ 4 4 ] [150*s + 25*s 150*s + 5*s] [------------- ------------] [ 3 2 ] [ 6*s 6*s ] [ ] [ 3 3 ] [ 150*s + 25 150*s + 5 ] [ ----------- ---------- ] [ 3 2 ] [ 6*s 6*s ]{t}
Notes
All the transfer function matrices should use the same complex variable
varof the Laplace transform.MIMOSeries(A, B)is not equivalent toA*B. It is always in the reverse order, that isB*A.See also
- doit(cancel=False, **kwargs)[source]
Returns the resultant transfer function matrix obtained after evaluating the MIMO systems arranged in a series configuration.
Examples
>>> from sympy.abc import s, p, a, b >>> from sympy.physics.control.lti import TransferFunction, MIMOSeries, TransferFunctionMatrix >>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s) >>> tf2 = TransferFunction(s**3 - 2, s**4 + 5*s + 6, s) >>> tfm1 = TransferFunctionMatrix([[tf1, tf2], [tf2, tf2]]) >>> tfm2 = TransferFunctionMatrix([[tf2, tf1], [tf1, tf1]]) >>> MIMOSeries(tfm2, tfm1).doit() TransferFunctionMatrix(((TransferFunction(2*(-p + s)*(s**3 - 2)*(a*p**2 + b*s)*(s**4 + 5*s + 6), (-p + s)**2*(s**4 + 5*s + 6)**2, s), TransferFunction((-p + s)**2*(s**3 - 2)*(a*p**2 + b*s) + (-p + s)*(a*p**2 + b*s)**2*(s**4 + 5*s + 6), (-p + s)**3*(s**4 + 5*s + 6), s)), (TransferFunction((-p + s)*(s**3 - 2)**2*(s**4 + 5*s + 6) + (s**3 - 2)*(a*p**2 + b*s)*(s**4 + 5*s + 6)**2, (-p + s)*(s**4 + 5*s + 6)**3, s), TransferFunction(2*(s**3 - 2)*(a*p**2 + b*s), (-p + s)*(s**4 + 5*s + 6), s))))
- property num_inputs
Returns the number of input signals of the series system.
- property num_outputs
Returns the number of output signals of the series system.
- property shape
Returns the shape of the equivalent MIMO system.
- property var
Returns the complex variable used by all the transfer functions.
Examples
>>> from sympy.abc import p >>> from sympy.physics.control.lti import TransferFunction, MIMOSeries, TransferFunctionMatrix >>> G1 = TransferFunction(p**2 + 2*p + 4, p - 6, p) >>> G2 = TransferFunction(p, 4 - p, p) >>> G3 = TransferFunction(0, p**4 - 1, p) >>> tfm_1 = TransferFunctionMatrix([[G1, G2, G3]]) >>> tfm_2 = TransferFunctionMatrix([[G1], [G2], [G3]]) >>> MIMOSeries(tfm_2, tfm_1).var p
- class sympy.physics.control.lti.MIMOParallel(*args, evaluate=False)[source]
A class for representing a parallel configuration of MIMO systems.
- Parameters
args : MIMOLinearTimeInvariant
MIMO Systems in a parallel arrangement.
evaluate : Boolean, Keyword
When passed
True, returns the equivalentMIMOParallel(*args).doit(). Set toFalseby default.- Raises
ValueError
When no argument is passed.
varattribute is not same for every system.All MIMO systems passed do not have same shape.
TypeError
Any of the passed
*argshas unsupported typeA combination of SISO and MIMO systems is passed. There should be homogeneity in the type of systems passed, MIMO in this case.
Examples
>>> from sympy.abc import s >>> from sympy.physics.control.lti import TransferFunctionMatrix, MIMOParallel >>> from sympy import Matrix, pprint >>> expr_1 = 1/s >>> expr_2 = s/(s**2-1) >>> expr_3 = (2 + s)/(s**2 - 1) >>> expr_4 = 5 >>> tfm_a = TransferFunctionMatrix.from_Matrix(Matrix([[expr_1, expr_2], [expr_3, expr_4]]), s) >>> tfm_b = TransferFunctionMatrix.from_Matrix(Matrix([[expr_2, expr_1], [expr_4, expr_3]]), s) >>> tfm_c = TransferFunctionMatrix.from_Matrix(Matrix([[expr_3, expr_4], [expr_1, expr_2]]), s) >>> MIMOParallel(tfm_a, tfm_b, tfm_c) MIMOParallel(TransferFunctionMatrix(((TransferFunction(1, s, s), TransferFunction(s, s**2 - 1, s)), (TransferFunction(s + 2, s**2 - 1, s), TransferFunction(5, 1, s)))), TransferFunctionMatrix(((TransferFunction(s, s**2 - 1, s), TransferFunction(1, s, s)), (TransferFunction(5, 1, s), TransferFunction(s + 2, s**2 - 1, s)))), TransferFunctionMatrix(((TransferFunction(s + 2, s**2 - 1, s), TransferFunction(5, 1, s)), (TransferFunction(1, s, s), TransferFunction(s, s**2 - 1, s))))) >>> pprint(_, use_unicode=False) # For Better Visualization [ 1 s ] [ s 1 ] [s + 2 5 ] [ - ------] [------ - ] [------ - ] [ s 2 ] [ 2 s ] [ 2 1 ] [ s - 1] [s - 1 ] [s - 1 ] [ ] + [ ] + [ ] [s + 2 5 ] [ 5 s + 2 ] [ 1 s ] [------ - ] [ - ------] [ - ------] [ 2 1 ] [ 1 2 ] [ s 2 ] [s - 1 ]{t} [ s - 1]{t} [ s - 1]{t} >>> MIMOParallel(tfm_a, tfm_b, tfm_c).doit() TransferFunctionMatrix(((TransferFunction(s**2 + s*(2*s + 2) - 1, s*(s**2 - 1), s), TransferFunction(2*s**2 + 5*s*(s**2 - 1) - 1, s*(s**2 - 1), s)), (TransferFunction(s**2 + s*(s + 2) + 5*s*(s**2 - 1) - 1, s*(s**2 - 1), s), TransferFunction(5*s**2 + 2*s - 3, s**2 - 1, s)))) >>> pprint(_, use_unicode=False) [ 2 2 / 2 \ ] [ s + s*(2*s + 2) - 1 2*s + 5*s*\s - 1/ - 1] [ -------------------- -----------------------] [ / 2 \ / 2 \ ] [ s*\s - 1/ s*\s - 1/ ] [ ] [ 2 / 2 \ 2 ] [s + s*(s + 2) + 5*s*\s - 1/ - 1 5*s + 2*s - 3 ] [--------------------------------- -------------- ] [ / 2 \ 2 ] [ s*\s - 1/ s - 1 ]{t}
Notes
All the transfer function matrices should use the same complex variable
varof the Laplace transform.See also
- doit(**kwargs)[source]
Returns the resultant transfer function matrix obtained after evaluating the MIMO systems arranged in a parallel configuration.
Examples
>>> from sympy.abc import s, p, a, b >>> from sympy.physics.control.lti import TransferFunction, MIMOParallel, TransferFunctionMatrix >>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s) >>> tf2 = TransferFunction(s**3 - 2, s**4 + 5*s + 6, s) >>> tfm_1 = TransferFunctionMatrix([[tf1, tf2], [tf2, tf1]]) >>> tfm_2 = TransferFunctionMatrix([[tf2, tf1], [tf1, tf2]]) >>> MIMOParallel(tfm_1, tfm_2).doit() TransferFunctionMatrix(((TransferFunction((-p + s)*(s**3 - 2) + (a*p**2 + b*s)*(s**4 + 5*s + 6), (-p + s)*(s**4 + 5*s + 6), s), TransferFunction((-p + s)*(s**3 - 2) + (a*p**2 + b*s)*(s**4 + 5*s + 6), (-p + s)*(s**4 + 5*s + 6), s)), (TransferFunction((-p + s)*(s**3 - 2) + (a*p**2 + b*s)*(s**4 + 5*s + 6), (-p + s)*(s**4 + 5*s + 6), s), TransferFunction((-p + s)*(s**3 - 2) + (a*p**2 + b*s)*(s**4 + 5*s + 6), (-p + s)*(s**4 + 5*s + 6), s))))
- property num_inputs
Returns the number of input signals of the parallel system.
- property num_outputs
Returns the number of output signals of the parallel system.
- property shape
Returns the shape of the equivalent MIMO system.
- property var
Returns the complex variable used by all the systems.
Examples
>>> from sympy.abc import p >>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix, MIMOParallel >>> G1 = TransferFunction(p**2 + 2*p + 4, p - 6, p) >>> G2 = TransferFunction(p, 4 - p, p) >>> G3 = TransferFunction(0, p**4 - 1, p) >>> G4 = TransferFunction(p**2, p**2 - 1, p) >>> tfm_a = TransferFunctionMatrix([[G1, G2], [G3, G4]]) >>> tfm_b = TransferFunctionMatrix([[G2, G1], [G4, G3]]) >>> MIMOParallel(tfm_a, tfm_b).var p
- class sympy.physics.control.lti.MIMOFeedback(sys1, sys2, sign=- 1)[source]
A class for representing closed-loop feedback interconnection between two MIMO input/output systems.
- Parameters
sys1 : MIMOSeries, TransferFunctionMatrix
The MIMO system placed on the feedforward path.
sys2 : MIMOSeries, TransferFunctionMatrix
The system placed on the feedback path (often a feedback controller).
sign : int, optional
The sign of feedback. Can either be
1(for positive feedback) or-1(for negative feedback). Default value is \(-1\).- Raises
ValueError
When
sys1andsys2are not using the same complex variable of the Laplace transform.Forward path model should have an equal number of inputs/outputs to the feedback path outputs/inputs.
When product of
sys1andsys2is not a square matrix.When the equivalent MIMO system is not invertible.
TypeError
When either
sys1orsys2is not aMIMOSeriesor aTransferFunctionMatrixobject.
Examples
>>> from sympy import Matrix, pprint >>> from sympy.abc import s >>> from sympy.physics.control.lti import TransferFunctionMatrix, MIMOFeedback >>> plant_mat = Matrix([[1, 1/s], [0, 1]]) >>> controller_mat = Matrix([[10, 0], [0, 10]]) # Constant Gain >>> plant = TransferFunctionMatrix.from_Matrix(plant_mat, s) >>> controller = TransferFunctionMatrix.from_Matrix(controller_mat, s) >>> feedback = MIMOFeedback(plant, controller) # Negative Feedback (default) >>> pprint(feedback, use_unicode=False) / [1 1] [10 0 ] \-1 [1 1] | [- -] [-- - ] | [- -] | [1 s] [1 1 ] | [1 s] |I + [ ] *[ ] | * [ ] | [0 1] [0 10] | [0 1] | [- -] [- --] | [- -] \ [1 1]{t} [1 1 ]{t}/ [1 1]{t}
To get the equivalent system matrix, use either
doitorrewritemethod.>>> pprint(feedback.doit(), use_unicode=False) [1 1 ] [-- -----] [11 121*s] [ ] [0 1 ] [- -- ] [1 11 ]{t}
To negate the
MIMOFeedbackobject, use-operator.>>> neg_feedback = -feedback >>> pprint(neg_feedback.doit(), use_unicode=False) [-1 -1 ] [--- -----] [ 11 121*s] [ ] [ 0 -1 ] [ - --- ] [ 1 11 ]{t}
See also
- doit(cancel=True, expand=False, **kwargs)[source]
Returns the resultant transfer function matrix obtained by the feedback interconnection.
Examples
>>> from sympy import pprint >>> from sympy.abc import s >>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix, MIMOFeedback >>> tf1 = TransferFunction(s, 1 - s, s) >>> tf2 = TransferFunction(1, s, s) >>> tf3 = TransferFunction(5, 1, s) >>> tf4 = TransferFunction(s - 1, s, s) >>> tf5 = TransferFunction(0, 1, s) >>> sys1 = TransferFunctionMatrix([[tf1, tf2], [tf3, tf4]]) >>> sys2 = TransferFunctionMatrix([[tf3, tf5], [tf5, tf5]]) >>> F_1 = MIMOFeedback(sys1, sys2, 1) >>> pprint(F_1, use_unicode=False) / [ s 1 ] [5 0] \-1 [ s 1 ] | [----- - ] [- -] | [----- - ] | [1 - s s ] [1 1] | [1 - s s ] |I - [ ] *[ ] | * [ ] | [ 5 s - 1] [0 0] | [ 5 s - 1] | [ - -----] [- -] | [ - -----] \ [ 1 s ]{t} [1 1]{t}/ [ 1 s ]{t} >>> pprint(F_1.doit(), use_unicode=False) [ -s s - 1 ] [------- ----------- ] [6*s - 1 s*(6*s - 1) ] [ ] [5*s - 5 (s - 1)*(6*s + 24)] [------- ------------------] [6*s - 1 s*(6*s - 1) ]{t}
If the user wants the resultant
TransferFunctionMatrixobject without canceling the common factors then thecancelkwarg should be passedFalse.>>> pprint(F_1.doit(cancel=False), use_unicode=False) [ 25*s*(1 - s) 25 - 25*s ] [ -------------------- -------------- ] [ 25*(1 - 6*s)*(1 - s) 25*s*(1 - 6*s) ] [ ] [s*(25*s - 25) + 5*(1 - s)*(6*s - 1) s*(s - 1)*(6*s - 1) + s*(25*s - 25)] [----------------------------------- -----------------------------------] [ (1 - s)*(6*s - 1) 2 ] [ s *(6*s - 1) ]{t}
If the user wants the expanded form of the resultant transfer function matrix, the
expandkwarg should be passed asTrue.>>> pprint(F_1.doit(expand=True), use_unicode=False) [ -s s - 1 ] [------- -------- ] [6*s - 1 2 ] [ 6*s - s ] [ ] [ 2 ] [5*s - 5 6*s + 18*s - 24] [------- ----------------] [6*s - 1 2 ] [ 6*s - s ]{t}
- property sensitivity
Returns the sensitivity function matrix of the feedback loop.
Sensitivity of a closed-loop system is the ratio of change in the open loop gain to the change in the closed loop gain.
Note
This method would not return the complementary sensitivity function.
Examples
>>> from sympy import pprint >>> from sympy.abc import p >>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix, MIMOFeedback >>> tf1 = TransferFunction(p, 1 - p, p) >>> tf2 = TransferFunction(1, p, p) >>> tf3 = TransferFunction(1, 1, p) >>> sys1 = TransferFunctionMatrix([[tf1, tf2], [tf2, tf1]]) >>> sys2 = TransferFunctionMatrix([[tf1, tf3], [tf3, tf2]]) >>> F_1 = MIMOFeedback(sys1, sys2, 1) # Positive feedback >>> F_2 = MIMOFeedback(sys1, sys2) # Negative feedback >>> pprint(F_1.sensitivity, use_unicode=False) [ 4 3 2 5 4 2 ] [- p + 3*p - 4*p + 3*p - 1 p - 2*p + 3*p - 3*p + 1 ] [---------------------------- -----------------------------] [ 4 3 2 5 4 3 2 ] [ p + 3*p - 8*p + 8*p - 3 p + 3*p - 8*p + 8*p - 3*p] [ ] [ 4 3 2 3 2 ] [ p - p - p + p 3*p - 6*p + 4*p - 1 ] [ -------------------------- -------------------------- ] [ 4 3 2 4 3 2 ] [ p + 3*p - 8*p + 8*p - 3 p + 3*p - 8*p + 8*p - 3 ] >>> pprint(F_2.sensitivity, use_unicode=False) [ 4 3 2 5 4 2 ] [p - 3*p + 2*p + p - 1 p - 2*p + 3*p - 3*p + 1] [------------------------ --------------------------] [ 4 3 5 4 2 ] [ p - 3*p + 2*p - 1 p - 3*p + 2*p - p ] [ ] [ 4 3 2 4 3 ] [ p - p - p + p 2*p - 3*p + 2*p - 1 ] [ ------------------- --------------------- ] [ 4 3 4 3 ] [ p - 3*p + 2*p - 1 p - 3*p + 2*p - 1 ]
- property sign
Returns the type of feedback interconnection of two models.
1for Positive and-1for Negative.
- property sys1
Returns the system placed on the feedforward path of the MIMO feedback interconnection.
Examples
>>> from sympy import pprint >>> from sympy.abc import s >>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix, MIMOFeedback >>> tf1 = TransferFunction(s**2 + s + 1, s**2 - s + 1, s) >>> tf2 = TransferFunction(1, s, s) >>> tf3 = TransferFunction(1, 1, s) >>> sys1 = TransferFunctionMatrix([[tf1, tf2], [tf2, tf1]]) >>> sys2 = TransferFunctionMatrix([[tf3, tf3], [tf3, tf2]]) >>> F_1 = MIMOFeedback(sys1, sys2, 1) >>> F_1.sys1 TransferFunctionMatrix(((TransferFunction(s**2 + s + 1, s**2 - s + 1, s), TransferFunction(1, s, s)), (TransferFunction(1, s, s), TransferFunction(s**2 + s + 1, s**2 - s + 1, s)))) >>> pprint(_, use_unicode=False) [ 2 ] [s + s + 1 1 ] [---------- - ] [ 2 s ] [s - s + 1 ] [ ] [ 2 ] [ 1 s + s + 1] [ - ----------] [ s 2 ] [ s - s + 1]{t}
- property sys2
Returns the feedback controller of the MIMO feedback interconnection.
Examples
>>> from sympy import pprint >>> from sympy.abc import s >>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix, MIMOFeedback >>> tf1 = TransferFunction(s**2, s**3 - s + 1, s) >>> tf2 = TransferFunction(1, s, s) >>> tf3 = TransferFunction(1, 1, s) >>> sys1 = TransferFunctionMatrix([[tf1, tf2], [tf2, tf1]]) >>> sys2 = TransferFunctionMatrix([[tf1, tf3], [tf3, tf2]]) >>> F_1 = MIMOFeedback(sys1, sys2) >>> F_1.sys2 TransferFunctionMatrix(((TransferFunction(s**2, s**3 - s + 1, s), TransferFunction(1, 1, s)), (TransferFunction(1, 1, s), TransferFunction(1, s, s)))) >>> pprint(_, use_unicode=False) [ 2 ] [ s 1] [---------- -] [ 3 1] [s - s + 1 ] [ ] [ 1 1] [ - -] [ 1 s]{t}
- property var
Returns the complex variable of the Laplace transform used by all the transfer functions involved in the MIMO feedback loop.
Examples
>>> from sympy.abc import p >>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix, MIMOFeedback >>> tf1 = TransferFunction(p, 1 - p, p) >>> tf2 = TransferFunction(1, p, p) >>> tf3 = TransferFunction(1, 1, p) >>> sys1 = TransferFunctionMatrix([[tf1, tf2], [tf2, tf1]]) >>> sys2 = TransferFunctionMatrix([[tf1, tf3], [tf3, tf2]]) >>> F_1 = MIMOFeedback(sys1, sys2, 1) # Positive feedback >>> F_1.var p