# Autonomous system = system without inputs

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1 Autonomous system = system without inputs State space representation B(A,C) = {y there is x, such that σx = Ax, y = Cx } x is the state, n := dim(x) is the state dimension, y is the output Polynomial representation where P R p p [z] and det(p) = 0. B(P) = {y P(σ)y = 0} ELEC 3035 (Part I, Lecture 3) Autonomous systems 2 / 20

2 Phase plane In σx = Ax, Ax is a velocity vector it shows how x changes in time. x 2 Ax(t) x(t) x 1 For n = 2, the plot of Ax over x R n is called phase plane. ELEC 3035 (Part I, Lecture 3) Autonomous systems 3 / 20

3 Example: harmonic oscillator A = x ELEC 3035 (Part I, Lecture 3) Autonomous systems 4 / 20 x 1

4 Equilibrium point of a dynamical system Consider a nonlinear autonomous system B = {x σx = f (x)} where f : R n R n and suppose that f (x e ) = x e, for some x e R n. x e is called an equilibrium point of B If x(t 1 ) = x e for some t 1, x(t) = x e, for all t > t 1. The set of equilibrium points of and LTI autonomous system B = {x σx = Ax } is ker(a I) the nullspace of A I. ELEC 3035 (Part I, Lecture 3) Autonomous systems 5 / 20

5 Linearization around an equilibrium point Suppose that x(t) is near an equilibrium point x e. Then σx = f (x) f (x e ) + A(x x e ), where A = a ij = fi x j. xe,j The dynamics of the deviation from x e x = x x e is described approximately be a linear system B = { x σ x = Ax } (Linearlization of a nonlinear system will be covered in part 2.) ELEC 3035 (Part I, Lecture 3) Autonomous systems 6 / 20

6 Initial conditions A trajectory of an autonomous system is uniquely determined by the initial state x(0) or initial conditions: in discrete-time (DT) y( l + 1),y( l + 2), y(0) in continuous-time (CT) d dt l+1 y(0), d dt l+2 y(0),... d dt 0 y(0). In the DT case y(t) = CA t x(0), t > 0. In the CT case the matrix power A t is replaced by the matrix exponential e At. ELEC 3035 (Part I, Lecture 3) Autonomous systems 7 / 20

7 Modal form Assume that there is a nonsingular matrix V, such that 1 V 1 AV = λ... λ n =: Λ. λ 1,..., λ n are the eigenvalues of A the columns of V are the corresponding eigenvectors. Then B(A,C) = B(Λ, C), where C := CV. The state equation of σx = Λx is a set of n decoupled equations. λ i pole of the system e λ i t (in CT) or λ t i (in DT) mode of the system ELEC 3035 (Part I, Lecture 3) Autonomous systems 8 / 20

8 Eigenvalues and eigenvectors of a matrix Consider a square matrix A R n n. v C n is an eigenvectors of A if Av = λv, for some λ C λ is called an eigenvalue of A, corresponding to v. Computing λ and v for given A involves solving a nonlinear equation. Suppose that A has n linearly independent eigenvectors v 1,...,v n, then Av i = λ i v i, i = 1,...,n = A v 1 v n V = 1 v 1 v n λ... V λ n Λ ELEC 3035 (Part I, Lecture 3) Autonomous systems 9 / 20

9 Let x be the state vector of B(Λ, C). In the DT case, x(t) = Λ t x(0) = λ t 1... x(0) λn t so that and therefore x i (t) = λ t i x i (0) y = Cx(t) = Cx(t) = c 1 x 1 (t)+ c n x n (t) = α 1 λ t 1 + α n λ t n, α i = c i x i (0) B(A,C) = B(Λ, C) is a linear combination of its modes λ 1,...,λ n. ELEC 3035 (Part I, Lecture 3) Autonomous systems 10 / 20

10 Complex poles The complex eigenvalues of A R n n can always be grouped in complex conjugate pairs λ i = a + bi = αe iω, λ j = a bi = αe iω (i := 1) so the sum of the two complex modes λ t i and λ t j gives one real mode λ t i + λ t j = α t e iωt + α t e iωt = 2α t cos(ωt) α damping factor ω frequency A real mode is of the form λ t i exponential ELEC 3035 (Part I, Lecture 3) Autonomous systems 11 / 20

11 Matrix exponential If the system is in a modal form B(Λ,CV ) d dt x = Λx = d dt x i = λ i x i, for i = 1,...,n. so that x i (t) = e λ i t x i (0) = x(t) = e λ 1t... e λnt x(0) e Λt Going back to the original basis we have x(t) = V e Λt V 1 x(0). e At ELEC 3035 (Part I, Lecture 3) Autonomous systems 12 / 20

12 State transition matrix The dynamics of the sate vector x is given by the equation x(t) = Φ(t)x(0) where Φ(t) = A t in DT and Φ(t) = e At in CT. The matrix Φ(t) is called state transition matrix. Φ(t) shows how the initial state x(0) is propagated in t time steps Note: if t < 0, Φ(t) propagates backwards in time. ELEC 3035 (Part I, Lecture 3) Autonomous systems 13 / 20

13 State construction Consider a scalar autonomous system B(P), where P(z) = P 0 z 0 + P 1 z P n 1 z n 1 + Iz n. How can we represent this system in a state space form B(A,C)? Choose x(t) = col y(t 1),...,y(t n). Then P n 1 P n 2 P 1 P 0 I A =. 0 I I 0 C = P n 1 P n 2 P 1 P 0 companion matrix of P ELEC 3035 (Part I, Lecture 3) Autonomous systems 14 / 20

14 Characteristic polynomial of a matrix The polynomial equation det(λi n A) = c 0 λ 0 + c 1 λ c n λ n = 0 is called the characteristic equation of the matrix A R n n. The roots of the characteristic polynomial c(z) = c 0 z 0 + c 1 z c n z n are equal to the eigenvalues of A. Cayley-Hamilton thm: Every matrix satisfies its own char. polynomial c 0 A 0 + c 1 A c n A n = 0. ELEC 3035 (Part I, Lecture 3) Autonomous systems 15 / 20

15 Example: harmonic oscillator A = Characteristic equation λ 1 det(λi A) = det 1 λ = λ = 0 Eigenvalues and eigenvectors λ 1,2 = ±i, v 1,2 = 1. ±i Matrix exponential e At = 1 1 e i i i e i = i i cos(t) sin(t) sin(t). cos(t) ELEC 3035 (Part I, Lecture 3) Autonomous systems 16 / 20

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