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//! # Transfer functions for continuous time systems.
//!
//! Specialized struct and methods for continuous time transfer functions
//! * time delay
//! * initial value and initial derivative value
//! * sensitivity function
//! * complementary sensitivity function
//! * control sensitivity function
//! * root locus plot
//! * bode plot
//! * polar plot
//! * static gain
use std::{
cmp::Ordering,
marker::PhantomData,
ops::{Add, Div, Mul, Neg, Sub},
};
use polynomen::Poly;
use crate::{
eigen::EigenConst,
enums::Continuous,
plots::{root_locus::RootLocus, Plotter},
rational_function::Rf,
transfer_function::TfGen,
units::Seconds,
wrappers::PWrapper,
Abs, Complex, Cos, Epsilon, Exp, Hypot, Infinity, Inv, Max, One, Pow, Sign, Sin, Sqrt, Zero,
};
/// Continuous transfer function
pub type Tf<T> = TfGen<T, Continuous>;
impl<T> Tf<T>
where
T: Clone + Cos + Exp + Mul<Output = T> + Neg<Output = T> + PartialEq + Sin,
{
/// Time delay for continuous time transfer function.
/// `y(t) = u(t - tau)`
/// `G(s) = e^(-tau * s)`
///
/// # Arguments
///
/// * `tau` - Time delay
///
/// # Example
/// ```
/// use automatica::{Complex, Seconds, Tf};
/// let d = Tf::delay(Seconds(2.));
/// assert_eq!(1., d(Complex::new(0., 10.)).norm());
/// ```
pub fn delay(tau: Seconds<T>) -> impl Fn(Complex<T>) -> Complex<T> {
move |s| (-s * tau.0.clone()).exp()
}
}
impl<T> Tf<T>
where
T: Add<Output = T>
+ Clone
+ Div<Output = T>
+ Infinity
+ Mul<Output = T>
+ One
+ PartialEq
+ Sub<Output = T>
+ Zero,
{
/// System inital value response to step input.
/// `y(0) = G(s->infinity)`
///
/// # Example
/// ```
/// use automatica::Tf;
/// let tf = Tf::new([4.], [1., 5.]);
/// assert_eq!(0., tf.init_value());
/// ```
#[must_use]
pub fn init_value(&self) -> T {
let n = self.num_int().0.degree();
let d = self.den_int().0.degree();
match n.cmp(&d) {
Ordering::Less => T::zero(),
Ordering::Equal => {
self.num_int().0.leading_coeff().0 / self.den_int().0.leading_coeff().0
}
Ordering::Greater => T::infinity(),
}
}
/// System derivative inital value response to step input.
/// `y'(0) = s * G(s->infinity)`
///
/// # Example
/// ```
/// use automatica::Tf;
/// let tf = Tf::new([1., -3.], [1., 3., 2.]);
/// assert_eq!(-1.5, tf.init_value_der());
/// ```
#[must_use]
pub fn init_value_der(&self) -> T {
let n = self.num_int().0.degree();
let d = self.den_int().0.degree().map(|d| d - 1);
match n.cmp(&d) {
Ordering::Less => T::zero(),
Ordering::Equal => {
self.num_int().0.leading_coeff().0 / self.den_int().0.leading_coeff().0
}
Ordering::Greater => T::infinity(),
}
}
/// Sensitivity function for the given controller `r`.
/// ```text
/// 1
/// S(s) = -------------
/// 1 + G(s)*R(s)
/// ```
///
/// # Arguments
///
/// * `r` - Controller
///
/// # Example
/// ```
/// use automatica::Tf;
/// let g = Tf::new([1.], [0., 1.]);
/// let r = Tf::new([4.], [1., 1.]);
/// let s = g.sensitivity(&r);
/// assert_eq!(Tf::new([0., 1., 1.], [4., 1., 1.]), s);
/// ```
#[must_use]
pub fn sensitivity(&self, r: &Self) -> Self {
let n = self.num_int().0.clone() * r.num_int().0.clone();
let d = self.den_int().0.clone() * r.den_int().0.clone();
Self {
rf: Rf::new_from_poly(d.clone(), n + d),
time: PhantomData,
}
}
/// Complementary sensitivity function for the given controller `r`.
/// ```text
/// G(s)*R(s)
/// F(s) = -------------
/// 1 + G(s)*R(s)
/// ```
///
/// # Arguments
///
/// * `r` - Controller
///
/// # Example
/// ```
/// use automatica::Tf;
/// let g = Tf::new([1.], [0., 1.]);
/// let r = Tf::new([4.], [1., 1.]);
/// let f = g.compl_sensitivity(&r);
/// assert_eq!(Tf::new([4.], [4., 1., 1.]), f);
/// ```
#[must_use]
pub fn compl_sensitivity(&self, r: &Self) -> Self {
let l = self * r;
l.feedback_n()
}
/// Sensitivity to control function for the given controller `r`.
/// ```text
/// R(s)
/// Q(s) = -------------
/// 1 + G(s)*R(s)
/// ```
///
/// # Arguments
///
/// * `r` - Controller
///
/// # Example
/// ```
/// use automatica::Tf;
/// let g = Tf::new([1.], [0., 1.]);
/// let r = Tf::new([4.], [1., 1.]);
/// let q = g.control_sensitivity(&r);
/// assert_eq!(Tf::new([0., 4.], [4., 1., 1.]), q);
/// ```
#[must_use]
pub fn control_sensitivity(&self, r: &Self) -> Self {
Self {
rf: Rf::new_from_poly(
r.num_int().0.clone() * self.den_int().0.clone(),
r.num_int().0.clone() * self.num_int().0.clone()
+ r.den_int().0.clone() * self.den_int().0.clone(),
),
time: PhantomData,
}
}
}
impl<T> Tf<T>
where
T: Abs
+ Add<Output = T>
+ Clone
+ Copy
+ Div<Output = T>
+ EigenConst
+ Epsilon
+ Hypot
+ Inv<Output = T>
+ Max
+ Mul<Output = T>
+ Neg<Output = T>
+ One
+ PartialOrd
+ Pow<T>
+ Sign
+ Sqrt
+ Sub<Output = T>
+ Zero,
{
/// System stability. Checks if all poles are negative.
///
/// # Example
/// ```
/// use polynomen::Poly;
/// use automatica::Tf;
/// let tf = Tf::new([1.0], Poly::new_from_roots(&[-1., -2.]));
/// assert!(tf.is_stable());
/// ```
#[must_use]
pub fn is_stable(&self) -> bool {
self.complex_poles().iter().all(|p| p.re.is_sign_negative())
}
/// Root locus for the given coefficient `k`
///
/// # Arguments
///
/// * `k` - Transfer function constant
///
/// # Example
/// ```
/// use polynomen::Poly;
/// use automatica::{Complex, Tf};
/// let l = Tf::new([1.0], Poly::new_from_roots(&[-1., -2.]));
/// let locus = l.root_locus(0.25);
/// assert_eq!(Complex::new(-1.5, 0.), locus[0]);
/// ```
#[must_use]
pub fn root_locus(&self, k: T) -> Vec<Complex<T>> {
let p = (self.num_int().0.clone() * PWrapper(k)) + self.den_int().0.clone();
let p2 = Poly::new_from_coeffs_iter(p.as_slice().iter().copied());
p2.complex_roots()
.iter()
.map(|(re, im)| Complex::new(re.0, im.0))
.collect()
}
}
impl<T> Tf<T>
where
T: Clone + PartialOrd + Zero,
{
/// Create a `RootLocus` plot
///
/// # Arguments
///
/// * `min_k` - Minimum transfer constant of the plot
/// * `max_k` - Maximum transfer constant of the plot
/// * `step` - Step between each transfer constant
///
/// `step` is linear.
///
/// # Panics
///
/// Panics if the step is not strictly positive of the minimum transfer constant
/// is not lower than the maximum transfer constant.
///
/// # Example
/// ```
/// use polynomen::Poly;
/// use automatica::{Complex, Tf};
/// let l = Tf::new([1.0_f32], Poly::new_from_roots(&[-1., -2.]));
/// let locus = l.root_locus_plot(0.1, 1.0, 0.05).into_iter();
/// assert_eq!(19, locus.count());
/// ```
pub fn root_locus_plot(self, min_k: T, max_k: T, step: T) -> RootLocus<T> {
RootLocus::new(self, min_k, max_k, step)
}
}
impl<T: Clone + PartialEq> Tf<T> {
/// Static gain `G(0)`.
/// Ratio between constant output and constant input.
/// Static gain is defined only for transfer functions of 0 type.
///
/// Example
///
/// ```
/// use automatica::Tf;
/// let tf = Tf::new([4., -3.],[2., 5., -0.5]);
/// assert_eq!(2., tf.static_gain());
/// ```
#[must_use]
pub fn static_gain<'a>(&'a self) -> T
where
T: Zero,
&'a T: 'a + Div<&'a T, Output = T>,
{
&self.num_int().0[0].0 / &self.den_int().0[0].0
}
}
impl<T> Plotter<T> for Tf<T>
where
T: Abs
+ Add<Output = T>
+ Clone
+ Div<Output = T>
+ Inv<Output = T>
+ Mul<Output = T>
+ Neg<Output = T>
+ PartialEq
+ PartialOrd
+ Sub<Output = T>
+ Zero,
{
/// Evaluate the transfer function at the given value.
///
/// # Arguments
///
/// * `s` - angular frequency at which the function is evaluated
fn eval_point(&self, s: T) -> Complex<T> {
self.eval(&Complex::new(T::zero(), s))
}
}
#[cfg(test)]
mod tests {
use proptest::prelude::*;
use polynomen::Poly;
use crate::{
plots::{bode::Bode, polar::Polar},
units::RadiansPerSecond,
};
use super::*;
#[test]
fn delay() {
let d = Tf::delay(Seconds(2.));
assert_relative_eq!(1., d(Complex::new(0., 10.)).norm());
assert_relative_eq!(-1., d(Complex::new(0., 0.5)).arg());
}
proptest! {
#[test]
fn qc_static_gain(g: f32) {
let tf = Tf::new([g, -3.], [1., 5., -0.5]);
assert_relative_eq!(g, tf.static_gain());
}
}
#[test]
fn stability() {
let stable_den = Poly::new_from_roots(&[-1.0_f64, -2.]);
let stable_tf = Tf::new([1., 2.], stable_den);
assert!(stable_tf.is_stable());
let unstable_den = Poly::new_from_roots(&[0.0_f64, -2.]);
let unstable_tf = Tf::new([1., 2.], unstable_den);
assert!(!unstable_tf.is_stable());
}
#[test]
fn bode() {
let tf = Tf::new(Poly::one(), Poly::new_from_roots(&[-1.]));
let b = Bode::new(tf, RadiansPerSecond(0.1), RadiansPerSecond(100.0), 0.1);
for g in b.into_iter().into_db_deg() {
assert!(g.magnitude() < 0.);
assert!(g.phase() < 0.);
}
}
#[test]
fn polar() {
let tf = Tf::new([5.], Poly::new_from_roots(&[-1., -10.]));
let p = Polar::new(tf, RadiansPerSecond(0.1), RadiansPerSecond(10.0), 0.1);
for g in p {
assert!(g.magnitude() < 1.);
assert!(g.phase() < 0.);
}
}
#[test]
fn initial_value() {
let tf = Tf::new([4.], [1., 5.]);
assert_relative_eq!(0., tf.init_value());
let tf = Tf::new([4., -12.], [1., 5.]);
assert_relative_eq!(-2.4, tf.init_value());
let tf = Tf::new([-3., 4.], [5.]);
assert_relative_eq!(f32::INFINITY, tf.init_value());
}
#[test]
fn derivative_initial_value() {
let tf = Tf::new([1., -3.], [1., 3., 2.]);
assert_relative_eq!(-1.5, tf.init_value_der());
let tf = Tf::new([1.], [1., 3., 2.]);
assert_relative_eq!(0., tf.init_value_der());
let tf = Tf::new([1., 0.5, -3.], [1., 3., 2.]);
assert_relative_eq!(f32::INFINITY, tf.init_value_der());
}
#[test]
fn complementary_sensitivity() {
let g = Tf::new([1.], [0., 1.]);
let r = Tf::new([4.], [1., 1.]);
let f = g.compl_sensitivity(&r);
assert_eq!(Tf::new([4.], [4., 1., 1.]), f);
}
#[test]
fn sensitivity() {
let g = Tf::new([1.], [0., 1.]);
let r = Tf::new([4.], [1., 1.]);
let s = g.sensitivity(&r);
assert_eq!(Tf::new([0., 1., 1.], [4., 1., 1.]), s);
}
#[test]
fn control_sensitivity() {
let g = Tf::new([1.], [0., 1.]);
let r = Tf::new([4.], [1., 1.]);
let q = g.control_sensitivity(&r);
assert_eq!(Tf::new([0., 4.], [4., 1., 1.]), q);
}
#[test]
fn root_locus() {
let l = Tf::new([1.0_f64], Poly::new_from_roots(&[-1., -2.]));
let locus1 = l.root_locus(0.25);
assert_eq!(Complex::new(-1.5, 0.), locus1[0]);
let locus2 = l.root_locus(-2.);
assert_eq!(Complex::new(-3., 0.), locus2[0]);
assert_eq!(Complex::new(0., 0.), locus2[1]);
}
#[test]
fn root_locus_iterations() {
let l = Tf::new([1.0_f32], Poly::new_from_roots(&[0., -3., -5.]));
let loci = l.root_locus_plot(1., 130., 1.).into_iter();
let last = loci.last().unwrap();
assert_relative_eq!(130., last.k());
assert_eq!(3, last.output().len());
assert!(last.output().iter().any(|r| r.re > 0.));
}
}