State space transformations

Transcription

1 Capitolo 0. INTRODUCTION. State space transformations Let us consider the following linear time-invariant system: { ẋ(t) = A(t)+Bu(t) y(t) = C(t)+Du(t) () A state space transformation can be obtained using a biunivocal linear transformation which links the old state vector with the new vector : = T where T is a square nonsingular matri. Applying this transformation to system () one obtains a different but equivalent mathematical description of the given dynamic system: { ẋ(t) = A(t)+Bu(t) y(t) = C(t)+Du(t) The matrices of the two systems are linked together as follows: A = T AT, B = T B, C = CT Properly choosing matri T it is possible to obtain mathematical descriptions of the given system (the canonical forms) characterized by matrices A, B, C and D which have particularly simple structures. or each choice of matrice T one obtains a different but equivalent mathematical description of the given system. All these different mathematical models maintain the basic physical properties of the given dynamic system: stability, controllability and observability. The transformed matrices A, B and C maintain the same geometric and mathematical internal properties of matrices A, B and C of the given system: matrices A and A have the same eigenvalues, the reachability and observability subspaces do not change, etc.). Zanasi Roberto - System Theory. A.A. 205/206

2 Capitolo. SYSTE THEORY.2 Eigenvalues and eigenvectors of a matri A Let A be a square matri of dimension n. If there eists a nonzero vector v and a scalar λ such that: then Av = λv (A λi)v = 0 λ is an eigenvalue of matri A; v is an eigenvector of matri A associated to eigenvalue λ. A (λ) = det(λi A) is the characteristic polynomial of matri A. A (λ) = 0 is the characteristic equation of matri A. The eigenvalues λ i of matri A are the solutions of the characteristic equation A (λ) = 0. The molteplicity r i of the eigenvalue λ i as a solution of the characteristic equation is known as algebraic molteplicity of the eigenvalue λ i. The set of all the eigenvalues λ i of matri A is the spectrum of matri A. Let λ i be an eigenvalue of matri A. The set of all the solutions v i of the system: (A λ i I)v i = 0 is a vector space called the autospace U λi associated to the eigenvalue λ i. The dimension m i of the autospace U λi is called geometric molteplicity of eigenvalue λ i. Property. The geometric molteplicity m i is always smaller or equal to the algebraic molteplicity r i : m i r i. Property. Distinct eigenvalues λ,...,λ h are always associated to linear independent eigenvectors v,...,v h. Two autospaces U λi and U λj associated to distinct eigenvalues λ i and λ j, are disjoined. Zanasi Roberto - System Theory. A.A. 205/206

3 Capitolo. SYSTE THEORY.3 Eample. The following two matrices A and A are characterized by the same eigenvalue λ =. This eigenvalue has algebraic molteplicity r = 2 for both the matrices, but it has different geometric molteplocity m: [ ] 0 A =, (λ ) 2 = 0, r = 2, m = 2 0 [ ] A =, (λ ) 2 = 0, r = 2, m = 0 Property. Two similar matrices have the same characteristic polynomial. Let T be a non singular matri and let A be the matri obtained from A applying the similitude transformation: A = T AT The characteristic polynomial of matri A is: A (λ) = det(λi A) = det(λi T AT) = det(λt T T AT) = det(t (λi A)T) = dett det(λ A) dett = det(λ A) = A (λ) If the matrices A and A have the same characteristic polynomial, then they also have the same eigenvalues. atlab eample: ---atlab commands A=[ 4 2; ; ]; % atri A [V, D]=eig(A) % D = Eigenvalues; V = Eigenvectors; poles=roots(poly(a)) % Roots of the characteristic polynomial ---atlab output V = D = poles = Zanasi Roberto - System Theory. A.A. 205/206

4 Capitolo. SYSTE THEORY.4 Eample. Let us consider a mechanical dynamic system composed by a mass, a spring with stiffness K, a viscous friction b and with an eternal force acting on mass : s ẋ ẍ b 2 K s 0 The dynamic behavior of the system is described by the following differential equation: = ẍ+bẋ+k K (t) The transfer function G(s) of the considered dynamic system is: G(s) = (s) (s) = s 2 +bs+k = s 2 + b s+ K The poles s and s 2 of the transfer function G(s) are: s = b 2 ( b 2 ) 2 K, s 2 = b ( b 2 b ) 2 K Let = [, ẋ] T and y =. The considered dynamic system can also be described in the state space as follows: [ẋ ] ẍ = [ 0 K b ] [ ] ẋ + [ 0 ] y = [ 0 ] The system matrices have the following form: [ ] [ 0 0 A = K b, b = The characteristic polynomial of matri A is: ], c = [ 0 ] det(λi A) = λ 2 +λ b + K The zeros λ and λ 2 of the characteristic polynomial are equal to the eigenvalues of matri A: ( λ = f ) 2 ( f 2 K 2, λ 2 = f ) 2 f 2 + K 2 The eigenvalues λ and λ 2 of matri A are equal to the poles s and s 2 of the transfer function G(s) = c(si A) b of the considered dynamic system. Zanasi Roberto - System Theory. A.A. 205/206

5 Capitolo. SYSTE THEORY.5 Let us consider the following autonomous system: ẋ = A, let v be an eigenvector of matri A associated to eigenvalue λ: (λi A)v = 0 and let be a state vector proportional to the eigenvector v: = αv. Under these hypotheses, it follows that the velocity vector ẋ is parallel to : ẋ = A = λ. Graphical representation: a) λ > 0 ẋ=λ b) λ < 0 ẋ=λ The straight lines associated to real eigenvector v are, within the state space, the only straight line trajectories which can be found in the given autonomous dynamic system. If the initial state 0 of a linear dynamic system belongs to the straight line associated to eigenvector v, then the corresponding free evolution of the system belongs to the same straight line. If λ > 0 the trajectory moves away from the origin, if λ < 0 trajectory moves toward the origin, and finally if λ = 0 the trajectory coincides with the initial condition: (t) = 0. a) λ > 0 b) λ < 0 (t) 0 (t) 0 Zanasi Roberto - System Theory. A.A. 205/206