ELEC 3035, Lecture 3: Autonomous systems Ivan Markovsky


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1 ELEC 3035, Lecture 3: Autonomous systems Ivan Markovsky Equilibrium points and linearization Eigenvalue decomposition and modal form State transition matrix and matrix exponential Stability ELEC 3035 (Part I, Lecture 3) Autonomous systems 1 / 20
2 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
3 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
4 Example: harmonic oscillator A = [ ] x x 1 ELEC 3035 (Part I, Lecture 3) Autonomous systems 4 / 20
5 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
6 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 = [ [ ] fi a ij = 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 = A x } (Linearlization of a nonlinear system will be covered in part 2.) ELEC 3035 (Part I, Lecture 3) Autonomous systems 6 / 20
7 Initial conditions A trajectory of an autonomous system is uniquely determined by the initial state x(0) or initial conditions: in discretetime (DT) y( l+1),y( l+2), y(0) in continuoustime (CT) ( d dt ) l+1 y(0), ( d dt ) l+2 y(0),... ( d dt ) 0 y(0). In the DT case In the CT case y(t) = CA t x(0), t > 0. the matrix power A t is replaced by the matrix exponential e At. ELEC 3035 (Part I, Lecture 3) Autonomous systems 7 / 20
8 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
9 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 = [ ] 1 v 1 v n λ... }{{}}{{} V V λ n }{{} Λ ELEC 3035 (Part I, Lecture 3) Autonomous systems 9 / 20
10 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)= C x(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
11 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
12 Matrix exponential If the system is in a modal form B(Λ,CV) so that d dt x = Λ x = d dt x i = λ i x i, for i = 1,...,n. x i (t) = e λ t x i i (0) = x(t) = e λ 1t... x(0) e λnt } {{ } 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
13 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
14 State construction Consider a scalar autonomous system B(P), where P(z) = P 0 z 0 + P 1 z 1 + +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
15 Characteristic polynomial of a matrix The polynomial equation det(λi n A) = c 0 λ 0 + c 1 λ 1 + +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 1 + +c n z n are equal to the eigenvalues of A. CayleyHamilton thm: Every matrix satisfies its own char. polynomial c 0 A 0 + c 1 A 1 + +c n A n = 0. ELEC 3035 (Part I, Lecture 3) Autonomous systems 15 / 20
16 Example: harmonic oscillator A = [ ] Characteristic equation det(λi A) = det ([ ]) λ 1 = λ = 0 1 λ Eigenvalues and eigenvectors [ ] 1 λ 1,2 = ±i, v 1,2 =. ±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
17 Sampling a continuoustime system x CT trajectory, x d DT trajectory x : R R n x d : Z R n Let x d (t) := x(ht), h is the sampling time. Then x d (t) = e Aht x(0) = A t d x(0), A d := e Ah. ELEC 3035 (Part I, Lecture 3) Autonomous systems 17 / 20
18 Stability An autonomous system B = {x σx = f(x)} is stable if x B implies x(t) 0 as t. For a linear timeinvariant system, B = {x σx = Ax } the eigenvalues of A determine the stability property of the system. CT LTI system is stable iff all eigenvalues have negative real parts. DT LTI system is stable iff all eigenvalues have absolute value < 1. ELEC 3035 (Part I, Lecture 3) Autonomous systems 18 / 20
19 Qualitative behaviour of the system If the eigenvalues are distinct y i = α i1 e λ1t + +α in e λ nt where α ij depend on the initial condition x(0) real λ j exponentially decaying or growing term complex λ j exponentially decaying or growing sinusoidal terms In CT R(λ j ) > 0 exponentially growing mode R(λ j ) < 0 exponentially decaying mode R(λ j ) = 0 a periodic or constant mode Repeated eigenvalues give rise to polynomial terms in the solution. ELEC 3035 (Part I, Lecture 3) Autonomous systems 19 / 20
20 Qualitative behaviour of the system In DT λ j > 1 exponentially growing mode λ j < 1 exponentially decaying mode λ j = 1 a periodic or constant mode Repeated eigenvalues give rise to polynomial terms in the solution. ELEC 3035 (Part I, Lecture 3) Autonomous systems 20 / 20
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