Various State Space Description – Similarity Transformation

8.5 Various State Space Description – Similarity Transformation

As can be seen in Sect. 1.3.4, we could construct many state diagrams for a given transfer function and each of them can be represented by a state equation. This implies that there is no unique state space model for a certain system.

8.5 Various State Space Description – Similarity Transformation 377 As a matter of fact, we can use similarity transformation to derive any number

of different state equations, all of which are equivalent in terms of the input-output relationship. Consider the state and output equation of an LTI system as

State equation: x ′ (t ) = Ax(t) + Bu(t)

x[n + 1] = Ax[n] + Bu[n] (8.5.1b) Output equation:

(8.5.1a)

y[n] = Cx[n] + Du[n] (8.5.2b) With a nonsingular transformation matrix P and a new state vector w(t )/w[n], we

y(t) = Cx(t) + Du(t)

(8.5.2a)

substitute x(t) = P w(t)

x[n] = P w[n] (8.5.3b) into Eqs. (8.5.1) and (8.5.2) to write w ′ (t ) = P −1

(8.5.3a)

(t ) + P −1 −1 Bu (t ) w[n + 1] = P A Pw [n] + P Bu [n] y(t) = C Pw(t) + Du(t)

A Pw

y[n] = C Pw[n] + Du[n] This can be written as another set of state and output equations: w ′ (t ) = A p w(t) + B p u(t)

w[n + 1] = A p w[n] + B p u[n] (8.5.4b) y(t) = C p w(t) + D p u(t)

(8.5.4a)

y[n] = C p w[n] + D p u[n] (8.5.5b) where

(8.5.5a)

p B, C = C P, and D p =D (8.5.6) Remark 8.2 Similarity Transformation – Equivalence Transformation

(1) Note that, with different nonsingular matrices P, we could obtain different set of state and output equations as many as we want. (2) Transfer function, characteristic equation, and eigenvalues are not changed under similarity transformation:

A P| = |P −1 sIP−P |s I − A −1 | = |s I − P

p −1

A P| = |P −1 ||s I − A||P| = |s I − A| (8.5.7)

(3) The determinant/trace of A p is the same as that of A, since the determinant/trace of a matrix is equal to the product/sum of the eigenvalues. Especially if a system has distinct characteristic values, we may derive a state equation with a diagonal system matrix having the characteristic values on its diagonal, which is referred to as the Jordan canonical or normal form.

378 8 State Space Analysis of LTI Systems Suppose we have an N × N system matrix A with N distinct eigenvalues λ i ’s for

i = 1 : N and the corresponding eigenvectors m i ’s. Then we can write Am i =λ i m i for

i = 1, 2, · · · , N (8.5.8) or in a more compact form,

A ⎥ [m 1 m 2 ···m N ] = [m 1 m 2 ···m N ] ⎢ ······ ⎥ ⎢ ······ ⎥ ⎢

⎥ ⎣ ······ ⎦ 00···λ N ;

(8.5.9) where M, called a modal matrix, is composed of the eigenvectors as columns, and

AM = MΛ

M −1

A M = Λ is a diagonal matrix with the eigenvalues on its diagonal. The modal matrix M is nonsingular if the eigenvalues are distinct and consequently, the

eigenvectors are linearly independent. This implies that, with the similarity trans- formation P = M, the resulting system matrix A p will appear to be diagonal in the

new state equation. Example 8.6 Diagonal/Jordan Canonical Form

Consider a system described by the following discrete-time state and output equations:

1 [n + 1] 1 [n]

u [n] (E8.6.1) x 2 [n + 1]

We can get its eigenvalues and the corresponding eigenvectors as below: |s I − A| = (s − 1)(s − 2) = 0; s 1 = 1, s 2 =2

;m : arbitrary, m ;m = 0, m 11 : arbitrary

= 3m 22

Thus we have a modal matrix and its inverse as

11 m 12 M=

, M −1 =

(E8.6.3) m 21 m 22 01 0 1

Problems 379 Now, with the similarity transformation P = M, we use Eq. (8.5.6) to obtain another

set of state and output equations with

A p =M −1 AM = 01 02 01 = 02 01 = 02 : diagonal (E8.6.4a)

B p =M −1 B=

(E8.6.4b)

C ( p =CM= '1 0(

01 = '1 3

(E8.6.4c)

We can use the MATLAB function ‘ jordan()’ to get a modal matrix and the corresponding Jordan canonical form of state equation:

>>A=[1 3; 0 2]; [M,Ap]=jordan(A) % [M,L]=eig(A) M= -3

3 Ap = 1 0 0 1 0 2

>>B=[0; 1]; C=[1 0]; Bp=M\B, Cp=C*M

Bp = 1 Cp = -3