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//! # Continuous time linear system //! //! The time evolution of the system is performed through ODE (ordinary //! differential equation) [solvers](../solver/index.html). use nalgebra::{ComplexField, DVector, RealField}; use num_traits::Float; use crate::{ linear_system::{ solver::{Order, Radau, Rk, Rkf45}, Equilibrium, SsGen, }, units::Seconds, Continuous, }; /// State-space representation of continuous time linear system pub type Ss<T> = SsGen<T, Continuous>; /// Implementation of the methods for the state-space impl<T: ComplexField> Ss<T> { /// Calculate the equilibrium point for continuous time systems, /// given the input condition /// ```text /// x = - A^-1 * B * u /// y = - (C * A^-1 * B + D) * u /// ``` /// /// # Arguments /// /// * `u` - Input vector /// /// # Example /// /// ``` /// use automatica::Ss; /// let a = [-1., 1., -1., 0.25]; /// let b = [1., 0.25]; /// let c = [0., 1., -1., 1.]; /// let d = [0., 1.]; /// /// let sys = Ss::new_from_slice(2, 1, 2, &a, &b, &c, &d); /// let u = 0.0; /// let eq = sys.equilibrium(&[u]).unwrap(); /// assert_eq!((0., 0.), (eq.x()[0], eq.y()[0])); /// ``` pub fn equilibrium(&self, u: &[T]) -> Option<Equilibrium<T>> { assert_eq!(u.len(), self.b.ncols(), "Wrong number of inputs."); let u = DVector::from_row_slice(u); // 0 = A*x + B*u let bu = -&self.b * &u; let lu = &self.a.clone().lu(); // A*x = -B*u let x = lu.solve(&bu)?; // y = C*x + D*u let y = &self.c * &x + &self.d * u; Some(Equilibrium::new(x, y)) } } /// Implementation of the methods for the state-space impl<T: ComplexField + Float + RealField> Ss<T> { /// System stability. Checks if all A matrix eigenvalues (poles) are negative. /// /// # Example /// /// ``` /// use automatica::Ss; /// let sys = Ss::new_from_slice(2, 1, 1, &[-2., 0., 3., -7.], &[1., 3.], &[-1., 0.5], &[0.1]); /// assert!(sys.is_stable()); /// ``` #[must_use] pub fn is_stable(&self) -> bool { self.poles().iter().all(|p| p.re.is_negative()) } } /// Implementation of the methods for the state-space impl Ss<f64> { /// Time evolution for the given input, using Runge-Kutta second order method /// /// # Arguments /// /// * `u` - input function returning a vector (column mayor) /// * `x0` - initial state (column mayor) /// * `h` - integration time interval /// * `n` - integration steps pub fn rk2<F>(&self, u: F, x0: &[f64], h: Seconds<f64>, n: usize) -> Rk<F, f64> where F: Fn(Seconds<f64>) -> Vec<f64>, { Rk::new(self, u, x0, h, n, Order::Rk2) } /// Time evolution for the given input, using Runge-Kutta fourth order method /// /// # Arguments /// /// * `u` - input function returning a vector (column mayor) /// * `x0` - initial state (column mayor) /// * `h` - integration time interval /// * `n` - integration steps pub fn rk4<F>(&self, u: F, x0: &[f64], h: Seconds<f64>, n: usize) -> Rk<F, f64> where F: Fn(Seconds<f64>) -> Vec<f64>, { Rk::new(self, u, x0, h, n, Order::Rk4) } /// Runge-Kutta-Fehlberg 45 with adaptive step for time evolution. /// /// # Arguments /// /// * `u` - input function returning a vector (column vector) /// * `x0` - initial state (column vector) /// * `h` - integration time interval /// * `limit` - time evaluation limit /// * `tol` - error tolerance pub fn rkf45<F>( &self, u: F, x0: &[f64], h: Seconds<f64>, limit: Seconds<f64>, tol: f64, ) -> Rkf45<F, f64> where F: Fn(Seconds<f64>) -> Vec<f64>, { Rkf45::new(self, u, x0, h, limit, tol) } /// Radau of order 3 with 2 steps method for time evolution. /// /// # Arguments /// /// * `u` - input function returning a vector (column vector) /// * `x0` - initial state (column vector) /// * `h` - integration time interval /// * `n` - integration steps /// * `tol` - error tolerance pub fn radau<F>(&self, u: F, x0: &[f64], h: Seconds<f64>, n: usize, tol: f64) -> Radau<F, f64> where F: Fn(Seconds<f64>) -> Vec<f64>, { Radau::new(self, u, x0, h, n, tol) } } #[cfg(test)] mod tests { use super::*; #[test] fn equilibrium() { let a = [-1., 1., -1., 0.25]; let b = [1., 0.25]; let c = [0., 1., -1., 1.]; let d = [0., 1.]; let sys = Ss::new_from_slice(2, 1, 2, &a, &b, &c, &d); let u = 0.0; let eq = sys.equilibrium(&[u]).unwrap(); assert_eq!((0., 0.), (eq.x()[0], eq.y()[0])); assert!(!format!("{}", eq).is_empty()); } #[test] fn stability() { let eig1 = -2.; let eig2 = -7.; let sys = Ss::new_from_slice( 2, 1, 1, &[eig1, 0., 3., eig2], &[1., 3.], &[-1., 0.5], &[0.1], ); assert!(sys.is_stable()) } #[test] fn new_rk2() { let a = [-1., 1., -1., 0.25]; let b = [1., 0.25]; let c = [0., 1.]; let d = [0.]; let sys = Ss::new_from_slice(2, 1, 1, &a, &b, &c, &d); let iter = sys.rk2(|_| vec![1.], &[0., 0.], Seconds(0.1), 30); assert_eq!(31, iter.count()); } #[test] fn new_rk4() { let a = [-1., 1., -1., 0.25]; let b = [1., 0.25]; let c = [0., 1.]; let d = [0.]; let sys = Ss::new_from_slice(2, 1, 1, &a, &b, &c, &d); let iter = sys.rk4(|_| vec![1.], &[0., 0.], Seconds(0.1), 30); assert_eq!(31, iter.count()); } #[test] fn new_rkf45() { let a = [-1., 1., -1., 0.25]; let b = [1., 0.25]; let c = [0., 1.]; let d = [0.]; let sys = Ss::new_from_slice(2, 1, 1, &a, &b, &c, &d); let iter = sys.rkf45(|_| vec![1.], &[0., 0.], Seconds(0.1), Seconds(2.), 1e-5); assert_relative_eq!(2., iter.last().unwrap().time().0, max_relative = 0.01); } #[test] fn new_radau() { let a = [-1., 1., -1., 0.25]; let b = [1., 0.25]; let c = [0., 1.]; let d = [0.]; let sys = Ss::new_from_slice(2, 1, 1, &a, &b, &c, &d); let iter = sys.radau(|_| vec![1.], &[0., 0.], Seconds(0.1), 30, 1e-5); assert_eq!(31, iter.count()); } }