[PDF] ECE 602 Lecture Notes: Realizations and Minimal - IUPUI

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ECE 602 Lecture Notes: Realizations and Minimal - IUPUI

ECE 602 Lumped Systems Theory April 29, 2019 5 = h 0 1 i 2 6 6 4 s 1 s2 2s 7 4 s2 2s 7 3 7 7 5 (5) 4 s2 2s 7: (6) Now, let’s reverse the process and nd a realization of this transfer function

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ECE 602 Lumped Systems Theory April 29, 20191

ECE 602 Lecture Notes:

Realizations and Minimal Realizations

We have learned several processes for obtaining realizations. The rst, from Section 4.5.1 nds a MIMO realization from a transfer function matrix.

1This MIMO realization is

seldom minimal. To check for minimality, we compute the controllability and observability matrices. By theorem 7.M2, the realization of a proper rational matrix^G(s) is minimal if and only if (i) it is both controllable and observable. Another process, from Section 7.2, Implications of Coprimeness, nds a SISO realization from a single proper rational transfer function. If we start with a transfer function that is a coprime ratio of polynomials, this SISO realization will be minimal. This process has the added benet that if the polynomials are not coprime, we still get a realization { it is just not minimal. As stated in Theorem 7.2, the realization is minimal i the controllability and observability matrices both have full rank. Thus if we nd that the realization is not minimal we can go back and nd the common root(s), then factor them out to obtain a coprime ratio of polynomials. The second equivalent condition, dimA= deg ^g(s) must be used with extreme caution. Thedegree of a transfer functionis dened within the text

2to be the degree of the denominator of the transfer function when its

numerator and denominator are coprime! Thus, the theorem is technically true, but if the numerator and denominator are not coprime, the dimension of the resultingAmatrix may still equal the degree of the denominator of ^g(s), even though the realization is not minimal. We'll see such an example below.

MIMO Realizations from Transfer Matrices

Recall from section 4.5 of the textbook that arealizationof a proper rational transfer matrix^G(s) is a quadruple (A;B;C;D) of nite-dimensional matrices such that

G(s) =C(sIA)1B+D:

We obtained a MIMO realization in block controllable form as follows: Step 0We dene \^G(1)" to be the limit assgoes to innity of^G(s);then decompose the transfer matrix into^G(s) =^G(1) +^Gsp(s):

ThenD=^G(1).1

This method can also be used to nd a realization of a transfer function, but since it might not be minimal, we prefer the second procedure for transfer functions.

2in section 7.2.1, on page 227

ECE 602 Lumped Systems Theory April 29, 20192

Step 1Find the least common denominator of the transfer functions ^gij(s) contained in ^Gsp(s), and call it d(s) =sk+1sk1+k1s+k: Step 2Decompose the strictly proper part of the transfer matrix into

Gsp(s) =1d(s)N(s);

where

N(s) =N1sk1+N2sk2++Nk1s+Nk:

Note that thekhere is the samekas in the common denominatord(s).

Step 3TheA,B;andCmatrices are now

A=2 6

6666666641I2I k1IkI

I 00 0

0 I0 0............

0 00 0

0 0I 03

7

777777775;B=2

6

6666664I

0... 0 03 7

7777775;

and C=hN

1N2Nk1Nki;

where the dimension of the identity and zero matrices inAandBisp. The di- mensionpis the number of inputs to the system described by the transfer matrix, where^G(s) isqp. Step 4Verify that our realization satises the original transfer matrix. The transfer function for the above was veried in the text. The text contains examples of constructing a MIMO realization in this manner.

SISO Realizations from Transfer Functions

In Chapter 7, we were introduced to minimal realizations. We considered only the SISO case. The process of obtaining a minimal realization in the MIMO case is beyond the scope of our course. In the SISO case, we have a single proper rational transfer function ^g(s) rather than a proper rational transfer matrix^G(s). The SISO realization will have the form (A;b;c;d) wherebandcare vectors of appropriate dimensions anddis a scalar. The process of nding a minimal realization of a proper rational transfer function is as follows:

ECE 602 Lumped Systems Theory April 29, 20193

Step 0Decompose the transfer function as

^g(s) = ^g(1) + ^gsp(s) and simplify the transfer function ^gsp(s) so that the numerator and denominator are coprime. The direct term isd:= ^g(1): Step 1Identify the coecients of the numerator and denominator polynomialsN(s) and

D(s) of ^gsp(s) as

^gsp(s) =N(s)D(s)=1sn1+2sn2++n1s+ns n+1sn1+2sn2++n1s+n: Step 2Dene the pseudostatev(t) via its Laplace transform ^v(s) =D1(s)^u(s):

Step 3Dene the state variablex(t) by

x(t) :=2 6

6666664v

(n1)(t) v n2(t)... _v(t) v(t)3 7

7777775;soL(x(t)) =^x(s) =2

6

6666664s

n1 s n2 s 13 7

7777775^v(s):

Step 4Now the state equations are

_x(t) =Ax(t) +bu(t) =2 6

666666412 n1n

1 00 0

0 10 0

0 0 ...0 0

0 01 03

7

7777775x(t) +2

6

66666641

0... 0 03 7

7777775u(t):

Step 5By denition of ^v(s), we have that

^y(s) =N(s)^v(s) so y(t) =cx(t) =h

12n1nix(t):

Step 6Verify that the resulting realization satises ^g(s) =c(sIA)1b+d: The formulas were veried in the text, but we need to always check our work when we calculate a specic realization. Step 7Check minimality. (This was veried for the formulas in the text, but we need to check when we solve problems.)

ECE 602 Lumped Systems Theory April 29, 20194

How did we get from Step 3 to Step 4?

Since _xk(t) =xk1(t)3,

D(s)^v(s) = ^u(s)

becomes sn+1sn1+2sn2++n1s+n^v(s) = ^u(s); or s n^v(s) +1sn1^v(s) +2sn2^v(s) ++n1s^v(s) +n^v(s) = ^u(s); so using that s n^v(s) =s^x1(s); sn1^v= ^x1(s); sn2= ^x2(s); :::; and rearranging we have s^x1(s) =(1^x1(s) +2^x2(s) ++n1^xn1(s) +n^xn(s)) + ^u(s): Inverse Laplace transforming, with the assumption that allvk(0) = 0, we have _x1(t) =h12 n1nix(t) +u(t) so we obtain the realization of Step 4.

Example:

Consider the following system:

_x(t) ="1 2 4 1# x(t) +"1 0# u(t) (1) y(t) =h0 1ix(t):(2) Since ^g(s) =c(sIA)1b; the corresponding transfer function is ^g(s) =h0 1i"s12 4s1# 1"1 0# (3) h0 1i2 6 64s1s

22s72s

22s74s

22s7s1s

22s73
7 75"1
0# (4)3

Note that this is the opposite of our usual convention in which, for example, a positionx1has velocity

x

2= _x1.

ECE 602 Lumped Systems Theory April 29, 20195

h0 1i2 6 64s1s

22s74s

22s73
7 75(5)
4s

22s7:(6)

Now, let's reverse the process and nd a realization of this transfer function. We cannot assume that we will get the sameA,b, andc, matrices back, but we know by Theorem

7.3 on page 229 that the realizations will be equivalent. That means that there will be a

similarity transformPsuch that our new state~x=Px:

Step 0Our transfer function

^g(s) =4s 22s7
is already strictly proper and has coprime numerator and denominator. Step 1We haveD(s) =s22s7 andN(s) = 0s+ 4, thus the coecients are:

1=22=71= 02= 4:

Step 2The pseudostate is

v(t) =L1(^v(s)) =L1(D^u(s)) =L1(s22s7^u(s)) = u(t)2_u(t)7u(t)

Step 3Our new state~x(t) is

~x(t) :="_v(t) v(t)#

Step 4We have the realization

_x(t) =~A~x(t) +~bu(t) "12 1 0# ~x(t) +"1 0# u(t): and

A="2 7

1 0# and ~b="1 0#

ECE 602 Lumped Systems Theory April 29, 20196

Step 5By denition of ^v(s), we have that

^y(s) =N(s)^v(s) soy(t) =h

12i~x(t)

and ~c=h

12i=h0 4i:

Step 6Verication:

~g(s) =~csI~A

1~b(7)

h0 4i"s27 1s# 1"1 0# (8) h0 4i"ss

22s77s

22s71s

22s7s2s

22s7#"

1 0# (9) h0 4i"ss

22s71s

22s7#
(10) 4s

22s7:(11)

We have now veried the accuracy of our realization. Step 7We'll verify below that the realization is both controllable and observable, hence minimal. Let's make sure that our new realization is equivalent to the original one. As part of the proof of Theorem 7.3, the similarity transformation is determined to be

P:=~O1O:

We have

O="c cA# (12) (13) ="0 1 4 1# (14) and O="~c ~c ~A# (15) (16) ="0 4 1 7# (17)

ECE 602 Lumped Systems Theory April 29, 20197

so

P="0 4

1 7# 1"0 1 4 1# ="1 1=4

0 1=4#

:(18) Verifying that this is indeed the similarity transformation we were seeking, Matlab yields >> P = [0 4;1 7]\[0 1;4 1] P =

1.0000 0.2500

0 0.2500

>> P*A/P ans = 2 7 1 0 >> P*b ans = 1 0 >> c/P ans = 0 4 Thus we have veried that, as required by Theorem 7.3, the two realizations are equivalent.

Note that the observability matrix

~Ohas full rank, as it shhould to satisfy Theorem 7.2, which says that if this is a minimal realization of the proper rational transfer function ^g(s), (~A;~c) must be observable and (~A;~b) controllable. Checking, we nd that

C=h~b~A~bi="1 2

0 1# (19) which has full rank so ( ~A;~b) is controllable. Suppose that inStep 0our numerator and denominator were not coprime. In that case, if we computed the controllability and observability matrices, we would nd that either one or both were not full rank.

ECE 602 Lumped Systems Theory April 29, 20198

Example: A Non-Coprime Transfer Function

Consider the transfer function

^g(s) =s+ 2s

3+ 2s24s8:

First Attempt

Step 0The transfer function is already strictly proper so d= 0:

Step 1The denominator polynomial is

D(s) =s3+ 2s24s8;

sok= 3 and

N(s) = 0s2+ 1s+ 2:

We now have coecient vectors

=h

123i=h248i

and =h

123i=h0 1 2i:

Since we are only interested in what happens to the realization when the polynomial fraction is not coprime, we'll skip Steps 2 and 3 for now.

Steps 4 and 5We have the realization

_x(t) =Ax(t) +bu(t) (20) =2 6

42 4 8

1 0 0

0 1 03

7

5x(t) +2

6 41
0 03 7

5u(t) (21)

y(t) =cx(t) (22) =h0 1 2ix(t):(23)

ECE 602 Lumped Systems Theory April 29, 20199

Step 6We compute

^g(s) =h0 1 2i2 6

4s+ 248

1s0 01s3 7 512
6 41
0 03 7 5(24) h0 1 2i2 6 64s

2D(s)4s

248sD(s)sD(s)ss

248D(s)

1D(s)1s

24s

22s+4D(s)3

7 752
6 41
0 03 7 5(25) h0 1 2i2 6 64s

2D(s)sD(s)1D(s)3

7

75(26)

s+ 2s

3+ 2s24s8(27)

where I have substitutedD(s) for its value to reduce space requirements. Thus wequotesdbs_dbs7.pdfusesText_13