State space transformations
• Let us consider the following linear timeinvariant system:
˙x(t) = A x(t) + B u(t)
y(t) = C x(t) + D u(t) (1)
• A state space transformation can be obtained using a biunivocal linear transformation which links the old state vector x with the new vector x:
x = T x where T is a square nonsingular matrix.
• Applying this transformation to system (1) one obtains “a different but equivalent” mathematical description of the given dynamic system:
˙x(t) = A x(t) + B u(t) y(t) = C x(t) + Du(t)
• The matrices of the two systems are linked together as follows:
A = T^{−}^{1}AT, B = T^{−}^{1}B, C = CT
• Properly choosing matrix T it is possible to obtain mathematical descrip tions of the given system (the canonical forms) characterized by matrices A, B, C and D which have particularly simple structures.
• For each choice of matrice T one obtains a “different but equivalent”
mathematical description of the given system. All these different mathe matical 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.).
Eigenvalues and eigenvectors of a matrix A
• Let A be a square matrix of dimension n. If there exists a nonzero vector v and a scalar λ such that:
Av = λv ↔ (A − λI)v = 0
then
λ is an eigenvalue of matrix A;
v is an eigenvector of matrix A associated to eigenvalue λ.
∆A(λ) = det(λI − A) is the characteristic polynomial of matrix A.^{△}
∆A(λ) = 0 is the characteristic equation of matrix A.
• The eigenvalues λi of matrix A are the solutions of the characteristic equation ∆A(λ) = 0.
• The molteplicity ri of the eigenvalue λi as a solution of the characteristic equation is known as algebraic molteplicity of the eigenvalue λi.
• The set of all the eigenvalues λi of matrix A is the spectrum of matrix A.
• Let λi be an eigenvalue of matrix A. The set of all the solutions vi of the system:
(A − λiI) vi = 0
is a vector space called the autospace Uλ_{i} associated to the eigenvalue λi.
• The dimension mi of the autospace Uλ_{i} is called geometric molteplicity of eigenvalue λi.
• Property. The geometric molteplicity mi is always smaller or equal to the algebraic molteplicity ri: mi ≤ ri.
• Property. “Distinct” eigenvalues λ1, . . . , λ_{h} are always associated to “li near independent” eigenvectors v1, . . . , v_{h}. Two autospaces Uλ_{i} and Uλ_{j}
associated to distinct eigenvalues λi and λj, are disjoined.
same eigenvalue λ = 1. This eigenvalue has algebraic molteplicity r = 2 for both the matrices, but it has different geometric molteplocity m:
A = 1 0 0 1
, (λ − 1)^{2} = 0, r = 2, m = 2
A^{′} = 1 1 0 1
, (λ − 1)^{2} = 0, r = 2, m = 1
• Property. Two similar matrices have the same characteristic polynomial.
Let T be a non singular matrix and let A be the matrix obtained from A applying the similitude transformation:
A = T^{−}^{1}AT The characteristic polynomial of matrix A is:
∆_{A}(λ) = det(λI − A) = det(λI − T^{−}^{1}AT)
= det(λT^{−}^{1}T − T^{−}^{1}AT) = det(T^{−}^{1}(λI − A)T)
= det T^{−}^{1} det(λ − A) det T = det(λ − A) = ∆A(λ)
• If the matrices A and A have the same characteristic polynomial, then they also have the same eigenvalues.
• Matlab example:
Matlab commands  A=[1 4 2;...
9 2 4;...
1 0 5]; % Matrix A
[V, D]=eig(A) % D = Eigenvalues; V = Eigenvectors;
poles=roots(poly(A))’ % Roots of the characteristic polynomial Matlab output
V =
0.6024 0.5121 0.4378 0.7957 0.8440 0.6627 0.0634 0.1593 0.6075 D =
4.4944 0 0
0 8.2151 0
0 0 4.2793
poles =
4.4944 8.2151 4.2793

Example. Let us consider a mechanical dynamic system composed by a mass M , a spring with stiffness K, a viscous friction b and with an external force F acting on mass M:
F
1
¨ x

1 M
1 s
?
?
?
˙x

b
6
6
 
x
2

6 1 s 6
K
6
 0
x(t) b
M F K
1 2
The dynamic behavior of the system is described by the following differential equation:
F = M ¨x+ b ˙x + Kx
The transfer function G(s) of the considered dynamic system is:
G(s) = x(s)
F(s) = 1
M s^{2} + bs + K =
1 M
s^{2} + _{M}^{b} s+ _{M}^{K} The poles s1 and s2 of the transfer function G(s) are:
s_{1} = − b 2M −
s
b 2M
2
− K
M, s_{2} = − b 2M +
s
b 2M
2
− K M
Let x = [x, ˙x]^{T} and y = x. The considered dynamic system can also be described in the state space as follows:
˙x
¨ x
=
0 1
−^{K}
M − ^{b}
M
x
˙x
+ 0
1 M
F
y =
1 0
x The system matrices have the following form:
A =
0 1
−^{K}
M − ^{b}
M
, b = 0
1 M
, c = 1 0 The characteristic polynomial of matrix A is:
det(λI − A) = λ^{2} + λ b
M + K M
The zeros λ1 and λ2 of the characteristic polynomial are equal to the eigenvalues of matrix A:
λ_{1} = − f 2M −
s
f 2M
2
− K
M, λ_{2} = − f
2M + s
f 2M
2
− K M
The eigenvalues λ1 and λ2 of matrix A are equal to the poles s1 and s2 of the transfer function G(s) = c(sI − A)^{−1}b of the considered dynamic system.
˙x = A x,
let v be an eigenvector of matrix A associated to eigenvalue λ:
(λI − A)v = 0
and let x be a state vector proportional to the eigenvector v:
x = α v ∈ Im[v].
• Under these hypotheses, it follows that the velocity vector ˙x is parallel to x:
˙x = A x = λ x.
• Graphical representation:
x
˙x=λx
x_{1} x_{2}
Im[v]
a) λ > 0
˙x=λx x
x_{1} x_{2}
Im[v]
b) λ < 0
• 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 x_{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: x(t) = x_{0}.
x_{0}
x(t)
x_{1} x_{2} a) λ > 0
Im[v]
x_{0} x(t)
x_{1} x_{2} b) λ < 0
Im[v]