CHAPTER 3 MARDEN TABLE FORMULATON FOR STABILITY...

40

CHAPTER 3

MARDEN TABLE FORMULATON

FOR STABILITY ANALYSIS

3.1 INTRODUCTION

Stability tests within the unit circle are very important in the field

of linear discrete systems. The stability of systems can be checked by finding

the roots of characteristic polynomial. In certain applications, the designer

may simply need to know whether a system is stable or unstable and the

values of the poles of the transfer function may not be required. In such

applications, the stability of the system can be checked quickly through the

use of one of several available stability tests (Bishop 1974). A logical and

useful approach to the problem of stability is to be able to obtain a stability

criterion directly in the z-plane. One of the first direct methods devised for

testing the location of roots of a polynomial in z-plane with respect to the unit

circle in the z-plane is the Schur & Cohn (1922) criterion. The criterion gives

necessary and sufficient conditions for the roots to lie inside the unit circle in

terms of the signs of the Schur & Cohn determinants. The Jury (1964) test is

based on the same mathematical relationships as the Schur & Cohn.

A simplified version of the Schur & Cohn stability criterion is the Jury

criterion.

The Schur & Cohn criterion involves the evaluation of the

determinants of N matrices of dimensions ranging from 2x2 to 2Nx2N which

would require large amount of computation. A more efficient stability

41

criterion was developed by Fujiwara (1924). This is actually a modified

version of the Schur & Cohn criterion. Marden (1949) represented the Schur-

Cohn determinants in terms of second order determinants. Marden has given

an algorithm to test the stability of linear discrete system of any order whose

characteristic polynomial has real coefficients (Porter 1967). Another form of

the stability test for the unit circle is table form. A stability table based on

Marden algorithm (1967) used for testing the absolute stability of linear

discrete time systems is called Marden table. The limitation of the Marden

table is that it cannot provide information about root distribution in the case of

unstable system.

Jury (1964) developed a table to test the absolute stability of linear

discrete system by using the work of Rouche, Cohn and the relationships due

to Marden for Schur-Cohn determinants. There are several variations of the

Jury table, such as Raible table (1974) in which the information about root

distribution is obtained in addition to the absolute stability determination.

This research analyses the absolute stability, root distribution determination

and design of linear time invariant (LTIDS) discrete regular systems

described by real polynomials. The absolute stability is determined by

combining a novel implementation procedure and proposed stability

constraint with Marden (1949) table. If the system is unstable, the information

on the number of roots that lie inside and outside the unit circle is obtained by

using certain new inferences in the Marden (1949) table. The necessary

conditions are applied to the given characteristic polynomial for extracting the

approximate range of values of the design parameters. The values are further

sharpened using bisection principle along with Marden table (1949) to obtain

the exact range of parameters.

42

3.2 REPRESENTATION OF LINEAR TIME INVARIANT

DISCRETE SYSTEM

A discrete system is characterized by a rule of correspondence that

describes the relationship of the output signal produced with respect to the

signal applied at the input of the system. Depending on the rule of

correspondence, a discrete system can be linear or nonlinear, time invariant or

time dependent and causal or non causal (Antoniou 1990).

Discrete systems can be characterized in terms of difference

equations, state equations and transfer functions. The transfer function of a

discrete system is defined as the ratio of the z-transform of the response to the

z-transform of the excitation. The transfer function is the z-transform of the

impulse response of the system. For a causal linear time invariant discrete

system (LTIDS) ,the transfer function assumes the general form as,

knzn

0k kb

kmzm

0k kaH(z)

(3.1)

Where ak and bk are the real coefficients of the numerator and

denominator polynomials respectively.

Transfer function of a casual linear time invariant discrete system

can be expressed in polynomial form as,

nb....1nz1bnz0bma.....1mz1amz0a

F(z)A(z)H(z)

(3.2)

i.e. the ratio of two polynomial in z. The order of F (z) should be

equal to or greater than the order of A (z).

43

Through one of the z-transforms, a discrete system can be

characterized in terms of discrete transfer function which is a complete

representation of the system in z domain. The transfer function can be used to

find response of a given system to an arbitrary time domain excitation to find

its frequency response and to ascertain whether the system is stable or

unstable. Also the transfer function serves as the stepping stone between

desired specifications and system design.

3.3 ASPECT OF STABILITY AND INSTABILITY IN LTI

DISCRETE SYSTEM

For the LTIDS described by the Equation (3.2), is said to be

Bounded Input Bounded Output (BIBO) stable, when the output is bounded

words, the impulse response g (k) of the system satisfies the condition in

Equation (3.3),

0k|g(k)| (3.3)

Where, g (k) is the impulse response sequence. The Equation (3.3)

is a necessary and sufficient condition for stability.

Although Equation (3.3) is both correct and fundamental, it is not

particularly useful. If it is to be used as a stability test, an infinite sum must be

evaluated. Nearly truncating the sum is unsatisfactory because a truncated

sequence will always be finite. Furthermore Equation (3.3) requires that the

impulse response be available. Linear time invariant discrete system design

algorithms usually provide the transfer function. Hence the stability of the

system can be determined from its transfer function.

44

General Rule

Let the n characteristic equation roots of the system described by

Equation (3.2) be represented by zi

in Equation (3.3) the magnitude of zi must be less than one. In other words the

roots of the characteristic equation must all be inside the unit circle in the

z -plane. The stability and instability conditions are as follows:

(i) If all the roots of the characteristic polynomial lie inside the

unit circle (|zi|<1) in z-plane, then the impulse response is

bounded and decays to zero. That means the system is stable.

(ii) If the condition |zi|<1 is not satisfied then the system is said to

be unstable and has at least one root of the characteristic

polynomial lie outside or on the unit circle in z plane.

Marden (1949) proposed an algorithm in table form to determine

the stability condition of linear time invariant discrete system, which is

similar to Routh Table (Nagrath et al 1981), used for stability analysis of

linear time invariant continuous systems.

3.4 MARDEN ALGORITHM

Marden (1949) had proposed an algorithm which analyses the roots

of the equation Fn(z) where n has any value and the characteristic equation

has real coefficients.

n

given in the general form (Porter 1967),

Fn (z) = b0zn+ b1zn-1+ b2zn-2n-1z+ bn = 0 (3.4)

45

The product of all n roots of the Equation (3.4) is (-1)n * bn/b0. If all

those roots lie within the unit circle, their product will be less than unity so

that |bn/b0| < 1. This is equivalent to the condition

1 0 2 bn 2 > 0 (3.5)

The algorithm proceeds by forming a sequence of polynomials

Fn-1(z), Fn-2(z), Fn-3(z), Fn-4(z), F0(z) whose roots all lie within the unit circle if

those of Fn(z) have the same property. In this way (n-1) conditions similar to

Equation (3.5) can be obtained.

The first step in the formation of Marden table is to reverse the

coefficients of the given characteristic polynomial in Equation (3.4), and then

the formed new polynomial is

Fn*(z) =bnzn+ bn-1zn-1+ bn-2zn-2

n-1z1+ b0 (3.6)

Using Equations (3.4) and (3.6), the Marden table is formulated as

in Table 3.1. The first two rows consist of the coefficients of F*n(z) (Reversed

Polynomial) and Fn(z) (Given Polynomial) arranged in descending order of

powers of z.

Table 3.1 Marden Table for Fn(z) in Equation (3.2)

Order zn zn-1 zn-2 z2 z1 z0 Fn

*(z) bn bn-1 bn-2 b2 b1 b0 Fn(z) b0 b1 b2 bn-2 bn-1 bn

F*n-1(z) cn-1 cn-2 cn-3 c1 c0 1 Fn-1(z) c0 c1 c2 cn-2 cn-1

F*n-2 (z) dn-2 dn-3 dn-4 d0 2 . . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. F0

*(z) z0 n

46

In Table 3.1 the elements in the third row Fn-1*(z) are calculated as

given below:

bn bk

b0 bn-k For k = n-1,n-2,n-

ck = (b0*bk) (bn*bn-k)

and these elements form the reduced polynomial of order (n-1) as given

below,

F*n-1(z) = cn-1zn-1+ cn-2zn-21z1+c0 (3.7)

The coefficients of the polynomial in Equation (3.7) reversed to

form the fourth row Fn-1(z) in the Marden Table.

The above computations are repeated to formulate the polynomials

F*n-2(z), Fn-2 *0(z), F0(z), and this completes the formulation of entire

-1) Marden polynomials

can be formulated up to constant term.

The constant terms in F*n-1(z), F*n-2 *0 1 2

n all greater than zero , if all the roots of the equation satisfy |z| < 1, then the

system is said to be stable.

3.4.1 Observations from Marden Algorithm

From the Marden algorithm, the following observations are made,

i.

depends on the Leading co-efficient, and the constant term of

ck =

47

the -

n/b0| < 1 is the necessary condition

given by Bishop (1975) for all the roots of a polynomial to be

less than one in magnitude (i.e., to lie within the unit circle).

ii. k = [(Leading co-efficient)2 (Constant Term)2] of the

previous order polynomial. For k= 1 to n as mentioned in

Table 3.1.

n/b0| < 1 constraint. If the condition

iii The constant term in the reduced order polynomial is equal to

+ = b0 2 bn 2 > 0 if |an/a0 - = b0 2 bn

2 < 0.

+ - iv. In other words, Marden algorithm is mainly the verification of

the following inequality |b0|>|bn| |Leading Coefficient of

F(z)|>|Constant term of F(z)|.

3.5 JURY ALGORITHM

The application of Jury algorithm (1964) for testing the stability of

a system requires the formation of a table. Jury table is formulated using a

sequence of reduced order polynomials for testing instability of the given

system. In the table the first two rows consisting of the co-efficient in Fn(z)

arranged in ascending order of powers of z in row 1 and the reverse order in

row 2. All even number rows are simply the reverse of the immediately

preceding odd number row. The elements for row 3 through 2n-3 are

calculated from the following determinants.

48

bn bn-1-k ck = k -1, b0 bk+1

cn-1 cn-2-k

dk = c0 ck+1 k -2,

The process continues until the (2n-3) rd row is formed

p3 p2-k qk = k = 0,1,2. p0 pk+1

This will contain exactly three elements.

Table 3.2 Jury Table for Fn (z) in Equation (3.4)

Row/Column z0 z1 z2 zn-2 zn-1 zn

1 bn bn-1 bn-2 b2 b1 b0

2 b0 b1 b2 bn-2 bn-1 bn

3 cn-1 cn-2 cn-3 c1 c0

4 c0 c1 c2 cn-2 cn-1

5 dn-2 dn-3 dn-4 d0

.

.

.

.

.

.

.

.

.

.

.

.

2n-5 p3 p2 p1 p0

2n-4 p0 p1 p2 p3

2n-3 q2 q1 q0

By Jury Stability Criterion (1964), a system is stable if all of the

following conditions hold:

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i. F(z=1) > 0 ;

ii. (-1)n F(z=-1) > 0

iii. The coefficients in the Jury table meet the following n-1

constraints:

|bn|<|b0|, |cn-1|>|c0|, |dn-1|>|d0|, . . . . |q2|>|q0|. (3.8)

3.5.1 Observations from Jury Algorithm

From the Jury algorithm, the following observations are made,

i. In Jury algorithm the conditions (i), (ii) and (iii) given in

Equation (3.8) are tested for stability of the system. Instead of

proceeding to the next step to get the reduced order

polynomial as in the case of Marden algorithm, it compares

whether the magnitude of leading coefficient is greater than

the constant term of the present order polynomial. And if the

condition is not satisfied, the table is terminated and the

system is declared as unstable.

ii. -efficient,

and the constant term of the polynomial results in verification

of the necessary condition for stability,

Co- n/b0| < 1 only. But the testing

terminates in Jury test one step ahead of Marden algorithm.

This leads to further reduction of Computational Complexity.

iii. Before proceeding to the next reduced order polynomial, jury

verifies the same constraint (|bn/b0| < 1) in a different

approach.

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iv. In other words Jury algorithm is simply the verification of the

following inequality |b0|>|bn|, which is equivalent to testing k

in Marden algorithm.

|bn|<|b0|

|Constant Term of Fn(z) | <

|Leading coefficient of Fn(z)|

|cn-1|>|c0|

|Constant Term of Fn-1(z) | >

|Leading coefficient of Fn-1(z)|

v. Further in jury Algorithm the constraints involving the

constant term and the Leading Coefficient are compared in all

rows from 1 to 2n-3.

vi. In both cases of Marden table and Jury table the reduced order

polynomial is derived by reducing the nth order polynomial

arrays into 2x2 matrixes. The coefficients of the reduced order

polynomial in the case of Jury is found out using determinant

values, which is negative of the corresponding value found in

Marden algorithm.

3.6 PROPOSED PROCEDURE FOR STABILITY ANALYSIS OF

LTIDS

Comparing Marden algorithm (1949) and Jury algorithm (1964) for

testing stability, we can say both the methods use the same stability constraint

and the result is interpreted in a different way. But in Jury test, reductions in

computations are observed, and the condition for instability of a system is

observed one step ahead of Marden table test. In the case of Marden test,

construction of the table is continued till to zero order because it requires n

constant terms to determine th

at (2n-3)th order polynomial itself.

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The basis for Marden, Jury tests and related, current efficient

stability tests are order reduction by iteration. The proposed procedure is

based on novel implementation of Marden algorithm and employing a

proposed stability constraint, i.e. |b1/b0| < n, equally |b1/nb0

order of the polynomial) instead of |bn/b0| < 1, in addition to the necessary

stability constraints.

In the proposed procedure the necessary and sufficient condition for

the roots of F(z) to be inside the unit circle are,

(i) F(z= 1) > 0

(ii) (-1)n F(z= -1) > 0 (3.9)

(iii) |b1/b0| < n i.e.|Succeeding coefficients /leading coefficients|< n

The steps involved in the proposed procedure for testing stability,

Step 1: Verify the necessary condition for stability by employing the

stability constraints (i) and (ii) listed in Equation (3.9).

IF the necessary condition is TRUE, then proceed to Step 2 to

check sufficient conditions for stability.

ELSE declare the system is unstable.

Step 2: Determine the leading coefficient and succeeding coefficient of

next reduced order polynomial, then check the stability constraint

(iii) in Equation (3.9).

IF the constraint is satisfied, then proceed to Step 3.

ELSE Stop the computation and declare the system is unstable.

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Step 3: Determine the remaining coefficients in the reduced order

polynomial by using the following formula, then go to Step 1.

bn bk ck = b0 bn-k -1.

Where ck = (b0*bk) (bn*bn-k) (3.10)

Step 4: Formulate the next reduced order polynomials by repeating Step1to

3 using the above procedure until 2nd order polynomial is reached

or terminate when any one of the constraints in Equation (3.9) is

not satisfied.

3.6.1 Illustrations

Example: 3. 6.1.1. Consider a characteristic equation with real

coefficients given in Bistritz (1983), Jury (1964) and check for its stability

using the proposed procedure.

F (z) =z4 - 1.368z3 +0.4126z2 +0.08z +0.0025 (3.11)



F (z=1) = 1-1.368+0.4126+0.0800+0.0025=0.1271>0,

and,

F (z=-1) =1+1.368+0.4126-0.0800+0.0025=2.7031>0,

Since this is an even order polynomial, it meets the first two

necessary conditions of the proposed procedure.

53

Step 2: Determine the leading coefficient and succeeding coefficient of

next reduced order polynomial using Equation (3.10), then check

the stability constraint (iii) in Equation (3.9).

0.0025 1.0000 c0 = 1.0000 0.0025 For, k=0

Where c0 = (1*1) (0.0025*0.0025) =1.0000

0.0025 -1.3680 c1 = 1.0000 0.0800 For, k=1

Where c1 = (1*-1.3680) (0.0025*0.0800) =-1.3682

Applying the stability constraint (iii) in Equation (3.10), we

obtained

|c1|=-1.3682, |c0|=1.0000 therefore |c1/c0|=1.3682< 3, the

constraint (iii) is satisfied.

Step 3: Determine the remaining coefficients in the reduced order

polynomial

0.0025 0.4126 C2 = 1.0000 0.4126 for k=2

Where c2 = (1*0.4126) (0.0025*0.4126) =0.4116

0.0025 0.0800 C3 = 1.0000 -1.3680 for k=3

54

Where c3 = (1*0.0800) (0.0025*-1.3680) =0.0834

Order z0 z1 z2 z3 z4 F*4(z) 0.0025 0.0800 0.4126 -1.3680 1.0000 F4(z) 1.0000 -1.3680 0.4126 0.0800 0.0025

F*3(z) 0.0834 0.4116 -1.3682 1.0000

Verify the necessary conditions for the reduced order polynomial F*3(Z),

F (1) = 1.0000-1.3682+0.4116+0.0834=0.1268>0 and F (-1) = -1.0000-1.3682-0.4116+0.0834=-2.6963<0

From the above conditions, it does not show instability. Hence the above process is repeated till it violates any one of the stability constraint in Equation (3.9) or up to a 2nd order polynomial is obtained.

Step 4: The formulated stability table for Example 3. 6.1.1. based on the proposed algorithm is shown in Table 3.3.

Table 3.3 Proposed Procedure Based Marden Table for Example 3.6.1.1

Order z0 z1 z2 z3 z4 Constraints

F*4(z) 0.0025 0.0800 0.4126 -1.3680 1.0000

|b1/b0|=1.3680< 4, F(1)=0.1271, F(-1)=2.7031 True

F4(z) 1.0000 -1.3680 0.4126 0.0800 0.0025

F*3(z) 0.0834 0.4116 -1.3682 1.0000

|c1/c0|=1.3682< 3, F(1)=0.1268, F(-1)=-2.6963 True

F3(z) 1.0000 -1.3682 0.4116 0.0834

F*2(z) 0.5257 -1.4025 0.9930

|d1/d0|=1.4124< 2, F(1)=0.1162, F(-1)=2.9213 True

55

Result: It is noticed from the Table 3.3 that the necessary and sufficient

conditions on coefficients for the characteristic equation to have all its roots

inside the unit circle, starting from the first row F4(z) to F2(z). So we can

conclude that the system is stable.

Remark: Result is in accordance with Jury (1964) and Bistritz (1983).

The problem is solved using Jury (1964) and Marden (1949)

stability criterion. It is compared with the proposed procedure.

Jury Algorithm (1964)

The formulated stability table for Example 3.6.1.1, based on Jury

(1964) algorithm is shown in Table 3.4.

Table 3.4 Jury Table for Example 3.6.1.1


F*4(z) 0.0025 0.0800 0.4126 -1.3680 1.0000

|1|>|0.0025| , F(1)=0.1271,

F(-1)=2.7031

True

F4(z) 1.0000 -1.3680 0.4126 0.0800 0.0025

F*3(z) -1.0000 1.3682 -0.4116 -0.0834 |1.0000|>|0.0834| True

F3(z) -0.0834 -0.4116 1.3682 -1.0000

F*2(z) 0.9930 -1.4025 0.5257 |0.9930|>|0.5257| True

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Stability Test

F (z=1) = 1-1.368+0.4126+0.0800+0.0025=0.1271>0,

And,

F (z=-1) =1+1.368+0.4126-0.0800+0.0025=2.7031>0,

|b0| < |bn| = |0.0025| < |1|

|c0| > |cn-1| = |1.0000| > |0.0834,

|d0| > |dn-2| = |0.9930| > |0.5257|

Result: It is noticed from the above test, the necessary and sufficient

conditions on coefficients of F(z) are met. So we can conclude that the system

is stable.

Marden Algorithm (1949)

The formulated stability table for Example 3.6.1.1, based on

Marden algorithm is shown in Table 3.5.

Table 3.5 Marden Table for Example 3.6.1.1

Order z0 z1 z2 z3 z4

Constant Term

(Reverse polynomial)

F*4(z) 0.0025 0.08 0.4146 -1.368 1.0 F4(z) 1.0 -1.368 0.4146 0.08 0.0025 F*3(z) 0.0834 0.4136 -1.3682 Positive F3(z) 1.0 -1.3682 0.4136 0.0834 F*2(z) 0.5277 -1.4027 Positive F2(z) 0.993 -1.4027 0.5277 F*1(z) -0.6527 Positive F1(z) 0.7076 -0.6527 F*0(z) Positive

57

Result: It is noticed from the Table 3.5, all the 1 2 4 are

greater than zero satisfy the complete set of necessary and sufficient

conditions for the stability of the system (i.e. all the roots of the equation

satisfy |z| < 1). Hence the system is said to be stable.

3.6.2 Comparison of Computational Efficiency

The following are the three methods used for comparing for their

computational efficiencies.

(i) Jury (1964) table

(ii) Marden (1949) table

(iii) Proposed procedure based Marden table

The number of arithmetic operations of each method for the

Example: 3.6.1.1 is given in Table 3.6.

Table 3.6 Arithmetic Operations for Example 3.6.1.1

Operations Marden Table (1949)

Jury Table (1964)

Proposed procedure based Marden Table

Multiplication 20 18 18 Subtraction 10 9 9 Total 30 27 27

Comment: From the above Table 3.6, it is found that the proposed procedure

based Marden table involves less number of multiplications and subtractions

when compared with Marden (1949) table. The proposed algorithm based

Marden table is having equal number of multiplications and subtractions as

that of Jury (1964) table.

Example: 3.6.1.2. Consider a characteristic equation with real coefficients

given in Jury (1967) and check for its stability using the proposed procedure.

58

F(z) = 2z4 4z3 +5z2 -2z +1 (3.12)



F (1) =2-4+5-2+1=2>0,

and

F (-1) =2+4+5+2+1=14>0

Since this is an even polynomial, it meets the necessary condition

of the proposed procedure.

Step 2: Formulate the Marden table by using the proposed algorithm.

Step 3: Construct the table as per the step 2.

Step 4: The formulated stability table for Example 3.6.1.2, based on the

proposed procedure is shown in Table 3.7.

Table 3.7 Proposed Procedure based Marden Table for Example 3.6.1.2


F*4(z) 1.0000 -2.0000 5.0000 -4.0000 2.0000

|b1/b0|=2.0000< 4,F(1)=2.0000, F(-1)=14.0000 True

F4(z) 2.0000 -4.0000 5.0000 -2.0000 1.0000

F*3(z) 0.0000 5.0000 -6.0000 3.0000

|c1/c0|=2.0000< 3,F(1)=2.0000, F(-1)=-14.0000 True

F3(z) 3.0000 -6.0000 5.0000 0.0000

F*2(z) -18.0000 9.0000

|d1/d0|=2.0000< 2,F(1)=6.0000, F(-1)=42.0000 False

59

Result: It is noticed from the Table 3.7 that the necessary and sufficient


not inside the unit circle, starting from the first row F4(z) to F2(z). So we

conclude that the system is unstable.

Remark: Result is in accordance with Jury (1967).





algorithm is shown in Table 3.8.



F*4(z) 1.0000 -2.0000 5.0000 -4.0000 2.0000

|2|>|1| , F(1)=2.0000,

F(-1)=14.0000

True

F4(z) 2.0000 -4.0000 5.0000 -2.0000 1.0000

F*3(z) -3.0000 6.0000 -5.0000 0.0000 |3.0000|>|0.0000|

True

F3(z) 0.0000 -5.0000 6.0000 -3.0000

F*2(z) 9.0000 -18.0000 15.0000

|9.0000|>|15.0000|

False

60

Stability Test:

F (1) =2-4+5-2+1=2>0,

And

F (-1) =2+4+5+2+1=14>0,

|b0| < |bn|, |1| < |2|, True

|c0| > |cn-1|, |3| > |0|, True

|d0| > |dn-2|, |9| > |15|, False


conditions on coefficients for stability of F(z) are not met. So we conclude

that the system is unstable.





Order z0 z1 z2 z3 z4 Constant Term

(Reverse polynomial) F*4(z) 1.0 -2.0 5.0 -4.0 2.0 F4(z) 2.0 -4.0 5.0 -2.0 1.0 F*3(z) 0.0 5.0 -6.0 Positive F3(z) 3.0 -6.0 5.0 0.0 F*2(z) 15.0 -18.0 Positive F2(z) 9.0 -18.0 15.0 F*1(z) 108.0 - Negative F1(z) -144.0 108.0 F0(z) Positive

61

Result: 1 2 are

greater 3 is less than zero, the required necessary and sufficient

conditions for the stability of the system are not satisfied. i.e. at least one root

is outside the unit circle, hence the system is said to be unstable.


Example: 3.6.1.2 is given in Table.3.10.

Table 3.10 Arithmetic Operation for Example 3.6.1.2

Operations Marden Table

(1949) Jury Table

(1964) Proposed procedure based Marden Table

Multiplication 20 14 12

Subtraction 10 7 6

Total 30 21 18

Comment: From the above Table.3.10, it is found that the proposed

procedure based Marden table involves less number of multiplications and

subtractions when compared with Marden (1949) table and Jury (1964) table.

Example: 3.6.1.3. Consider a characteristic equation with real coefficients

given in Jury (1967) and check for its stability using the proposed procedure.

F (z) =2z4 +7z3 +10z2 +4z +1 (3.13)



F (z=1) = 2+7+10+4+1=24>0,

and,

F (z=-1) =2-7+10-4+1=2>0,

62

Since this is an even order polynomial, it meets the necessary conditions of

the proposed algorithm.

Step 2: Formulate the Marden table using the proposed procedure.

Step 3: Construct the table as per the step 2.

Step 4: The formulated stability table for Example 3.6.1.3, based on the

proposed algorithm is shown in Table 3.11.

Table.3.11 Proposed Procedure based Marden Table for Example 3.6.1.3


F*4(z) 1.0000 4.0000 10.0000 7.0000 2.0000

|b1/b0|=3.5000< 4,

F(1)=24.0000,

F(-1)=2.0000

True

F4(z) 2.0000 7.0000 10.0000 4.0000 1.0000

F*3(z) 10.0000 3.0000 |b1/b0|=3.333< 3,

False

Result: It is noticed from the Table 3.11, that the necessary and sufficient


inside the unit circle is not satisfied. So we conclude that the system is

unstable.

Remark: Result is in accordance with Jury (1967).



63




Table.3.12 Jury Table for Example 3.6.1.3


F*4(z) 1.0000 4.0000 10.0000 7.0000 2.0000

F4(z) 2.0000 7.0000 10.0000 4.0000 1.0000

F(1)=24.0000,

F(-1)=2.0000,

True

F*3(z) -3.0000 -10.0000 -10.0000 -1.0000 |3.0000|>|1.0000|

True

F3(z) -1.0000 -10.0000 -10.0000 -3.0000

F*2(z) 8.0000 20.0000 20.0000 |8.0000|>|20.0000|

False

Stability Test:

F (1) =8+20+20=48>0,

And

F (-1) =8-20+20=8>0,

|b0| < |bn|, |1| < |2|, True

|c0| > |cn-1|, |3|> |1|, True

|d0| > |dn-2|, |8|> |20|, False

64


conditions on coefficients of F(z) are not met. So we conclude that the system

is unstable.






Constant Term

(Reverse polynomial)

F*4(z) 1.0000 4.0000 10.0000 7.0000 2.0000

F4(z) 2.0000 7.0000 10.0000 4.0000 1.0000

F*3(z) 1.0000 10.0000 10.0000 Positive

F3(z) 3.0000 10.0000 10.0000 1.0000

F*2(z) 20.0000 20.0000 Positive

F2(z) 8.0000 20.0000 20.0000

F*1(z) -240.0000 - Negative

F1(z) -336.0000 -240.0000

F0(z) 55296.0000= 4 Positive

Result: : 1 2

4 3 is less than zero, the required necessary and

sufficient condition for the stability of the system is not satisfied. Hence the

system is said to be unstable.

65


Example: 3.6.1.3 is given in Table.3.14.

Table.3.14 Arithmetic Operations for Example 3.6.1.3

Operations Marden Table

(1949) Jury Table

(1964) Proposed procedure based Marden Table

Multiplication 20 14 4

Subtraction 10 7 2

Total 30 21 6

Comment: From the above Table.3.6, it is found that the proposed procedure

based Marden table involves less number of multiplications and subtractions

when compared with Marden (1949) table and Jury (1964) table.

3.6.3 Discussion

The proposed procedure based Marden table, original Marden

(1949) table and Jury (1964) table are applied on three types of linear time

invariant discrete systems represented by real characteristic polynomials in

Equations (3.11, 3.12 and 3.13). The results bring out the following salient

points.

i. In the case of first example the stability was determined using

proposed procedure with 18 multiplications and 9

subtractions only whereas Marden algorithm uses 20

multiplications and 10 subtractions, and Jury algorithm uses

18 multiplications and 9 subtractions to solve the same

problem employing that the new method involves less

computations.

66

ii. In the second example, the instability was found out by

proposed procedure using 12 multiplications and 6 subtractions;

whereas it is 20 multiplications and 10 subtractions in the case of

Marden and 18 multiplications and 9 subtractions in the case

of Jury. The number of multiplications has been brought down

from 18 to 12 using the proposed procedure.

iii. In the case of third example, the instability was found out by

proposed procedure using 4 multiplications and 2 subtractions;

whereas it is 20 multiplications and 10 subtractions in the case

of Marden and 14 multiplications and 7 subtractions in the

case of Jury. The number of multiplications has been brought

down from 20(Marden) to 4 using the proposed procedure.

Summary of the results given in Table 3.15 and the bar graph

representations given in Figures 3.1 and 3.2 clearly indicates that with the

proposed procedure, the stability can be determined with less arithmetic

operations compared to Marden and Jury algorithms for stability

determination of LTID systems.

Table 3.15 Results Summary of Examples

Polynomials Marden Table

(1949) Jury Table (1964) Proposed procedure based Marden Table

M S Total M S Total M S Total Example: 3.6.1.1 Bistritz

20 10 30 18 9 27 18 9 27

Example: 3.6.1.2 E.I. Jury

20 10 30 14 7 21 12 6 18

Example: 3.6.1.3 Nagrath and Gopal

20 10 30 14 7 21 4 2 6

M Multiplication S Subtraction

67

A similar set of steps may be used to establish the fact that a

continuous time LTI system can be analyzed for stability by the proposed

method by suitably combining the bilinear transformation along with the

proposed procedure for LTID systems.

Figure 3.1 Comparison of Arithmetic operations in Marden (1949), Jury (1964) and Proposed Algorithm based Marden Table

Figure 3.2 Comparison of Total Number of Operations in Marden (1949), Jury (1964) and Proposed Algorithm based Marden Table

0

5

10

15

20

25

Num

ber

of O

pera

tions

Arithmetic Operations

Marden Table-MultiplicationMarden Table - SubtractionJury Table - MultiplicationJury Table - SubtractionProposed Algorithm Based Marden Table -MultiplicationProposed Algorithm Based Marden Table - Subtraction

Example 1 Example 2 Example 3

05

101520253035

Num

ber

of O

pera

tions

Result Summary

Marden Table

Jury Table

Proposed Algorithm Based Marden Table


68

3.7 PROPOSED SCHEME FOR ROOT DISTRIBUTION OF

LINEAR TIME INVARIANT DISCRETE SYSTEM USING

MARDEN TABLE

The stability of a linear time invariant discrete system was studied

by Schur (1917), Cohn (1922) and Marden (1949) among others Jury gave a

procedure to ascertain the number of roots of a polynomial that lie inside and

outside the unit circle. Raible reported a simplification of Jury table in 1974.

It is already proved that the Marden algorithm is a very efficient model

reduction algorithm. Tabular methods of determining root distribution of

polynomials with respect to the unit circle in the complex plane, typically

utilize a sequence of polynomial that are of descending order.

The table proposed by Marden (1949) reveals only the asymptotic

stability which is further investigated for distribution of roots by applying a

novel procedure and certain new inferences.The new procedure for root

distribution analysis of linear time invariant discrete systems consist two

methods. In the first method, the Marden table is formulated with an

additional checking of sign of the leading coefficients of successive

polynomials in the table. In the second method, the Marden table (1949) used

for the absolute stability testing is formulated, from the same table the

information about the root distribution obtained by using certain new

inferences. In both the procedures, because of the proposed scheme the table

proposed by Marden (1949) reveals not only absolute stability but also

information about the root distribution.

3.7.1 Proposed Procedure I

The procedure uses Marden table with an additional testing of sign

of leading coefficient of successive reduced order polynomials. The

coefficients of the next reduced order polynomials are determined based on

69

the sign of the leading coefficients of previous order polynomials. If the sign

is positive, the coefficients are derived based on Marden (1949) algorithm, if

sign is negative then the rows involved in the generation of next order

coefficients are interchanged and same Marden algorithm used. This is

continued till to the formulation zero order polynomial. In the derived table,

the number of positive signed leading coefficients 1 2 3 n) of the

reverse polynomial are equal to the number of roots inside the unit circle and

the number of negative signed leading coefficients 1 2 3 n) are equal

to the number roots outside the unit circle. The root distribution information

of the given linear discrete time system is determined by using the sign of

leading coefficient and the table proposed by Marden (1949). More details are

given in Appendix 1.

Algorithm for Procedure I

Step 1: Fill the first two rows of the table by polynomial coefficients

according to the increasing, and decreasing, powers in z

respectively.

Step 2: Check sign of leading coefficient. Determine the coefficients of the

next reduced order polynomials based on the sign of the leading

coefficients of previous order polynomial.

Step 3: If the sign of previous order leading coefficient ( ) is positive,

then calculate the coefficients of next reduced order polynomial

based on Marden algorithm.

Step 4: Else calculate the coefficients of next reduced order polynomial by

interchanging the rows involved in the determination coefficients

and use the Marden algorithm.

Step 5: Continue step 3 and step 4 till zero order polynomial is reached.

70

Step 6: Location of roots with respect to the unit circle is determined from

the sign of leading coefficients ( ) of the odd rows in the table.

Step 7: If the leading coefficient is positive, then a root is inside the unit

circle.

Else, root is outside of the unit circle.

Step 8: Ascertain the root distribution (Number of positive leading

coefficients = Number of roots inside the unit circle; Number of

Negative leading coefficients = Number of roots outside the unit

circle).

3.7.2 Proposed Procedure II

In this procedure, a new expression is formulated by using the

relationship between the sign of leading coefficients of the successive

polynomials in the Marden table (1949) based on the proof given by

Bhattacharya et al (1988) for the Jury table (1964). The expression alone is

used to ascertain the information on root distribution along with the table

proposed by Marden (1949) without modifying it.

In this method the Marden table is formulated by using the

procedure based on Marden algorithm (Porter 1967). By using this table it is

not possible to know the information on root distribution, because Marden

(1949) reveals only absolute stability. A simple relationship due to

Bhattacharya (1989) provides a means by which the root distribution

information is obtained from the Marden table. The sign relationship between

the successive leading coefficients of polynomials in the Marden table is

modified based on the relationship due to Bhattacharya (1989). More details

are given in Appendix 1.

71

i.e. Sign of k = [Sign of k ] * [Sign of 'k-1] (3.14)

Where k=1, 2 ...n.

Sign 0 =sign [b0]

0=b0=leading coefficient of the given polynomial

Sign [ k

Sign [ k

1, 2, 3, n are the leading coefficients of the polynomials in

the Marden table.

n = Order of the given polynomial

By using the Equation (3.14) the root distribution table formulated

as shown in Table 3.16.

Table 3.16 Root Distribution with Respect to Unit Circle

Leading coefficient ( 0 1 n-1 n

Sign

Sign of k k-1 k (+) (+) (-)

Location of root IUC IUC OUC

The last row in the Table 3.16 provides information on root

distribution. i.e. Number of roots inside the unit circle= Number of positive

sign k and Number of roots outside the unit circle = Number of Negative

sign k. This method is simple and direct and comparable to Jury (1964) and

Raible (1974) methods.

72

Algorithm for Procedure II

Step 1: Fill the first two rows of the table by polynomial coefficients

according to the increasing, and decreasing, powers in z

respectively.

Step 2: Complete the formulation of Marden table by using Marden

algorithm for determination of absolute stability.

Step 3: Determine the sign of k from the sign of 0, 1, n-1

(leading coefficients in the Marden table) by using the following

expression, Sign of k = [Sign of k ] * [Sign of 'k-1].

Step 4: Ascertain the root distribution of the given real polynomial

(Number of positive k = Number of roots inside the unit circle;

Number of Negative k = Number of roots outside the unit circle).

3.7.3 Illustration

Example: 3.7.3.1

Consider a characteristic equation with real coefficients given in

Bistritz (1983), Jury (1964) and determine the root distribution using the

procedure I and II of proposed scheme.

F (z) = z4 - 1.368z3 +0.4126z2 +0.08z +0.0025 (3.15)

Note: The same problem is solved using Jury (1964) and Raible (1974)

stability table and compared with the proposed scheme.

Proposed Procedure I

The formulated root distribution table for Example 3.7.3.1, based

on the proposed procedure I is shown in Table 3.17.

73

Table 3.17 Proposed Procedure I Table for Example 3.7.3.1


F*4(z) 0.0025 0.0800 0.4126 -1.3680 1.0000

F4(z) 1.0000 -1.3680 0.4126 0.0800 0.0025

F*3(z) 0.0834 0.4116 -1.3682 1

F3(z) 1.0000 -1.3682 0.4116 0.0834

F*2(z) 0.5257 -1.4025 2

F2(z) 0.9930 -1.4025 0.5257

F*1(z) -0.6554 3

F1(z) 0.7097 -0.6554

F0(z) 4

Result: Number of positive leading coefficients = Number of roots inside the

unit circle (IUC); Number of Negative leading coefficients = Number of roots

outside the unit circle (OUC). From the Table 3.17, it is found that the given

linear time invariant discrete system is stable and having all the roots inside

the unit circle because the leading coefficients of all odd numbered rows of

the formulated table based on the proposed scheme procedure I are all

positive.

Remark: Result is in agreement with Jury (1967) and Raible (1974)

algorithms.

Comment: The proposed scheme procedure I uses original Marden (1949)

table with an additional checking of leading coefficients in all reduced order

polynomials. Because of this novel implementation of Marden algorithm, the

root distribution of the linear time invariant discrete system represented in

Equation (3.15) with respect to the unit circle is obtained from the Marden

table itself.

74

Proposed Procedure II

The formulated absolute stability table for Example 3.7.3.1, based

on the proposed procedure II is shown in Table 3.18.

Table 3.18 Proposed Procedure II Table for Example 3.7.3.1

Order z0 z1 z2 z3 z4 F*4(z) 0.0025 0.0800 0.4126 -1.3680 0 F4(z) 1.0000 -1.3680 0.4126 0.0800 0.0025 F*3(z) 0.0834 0.4116 -1.3682 1 F3(z) 1.0000 -1.3682 0.4116 0.0834 F*2(z) 0.5257 -1.4025 2 F2(z) 0.9930 -1.4025 0.5257 F*1(z) -0.6554 3 F1(z) 0.7097 -0.6554 F0(z) 4

Table 3.19 Root Distribution with Respect to Unit Circle for Example 3.7.3.1

0 1 2 3 4 Sign (+)1 (+)1 (+)0.9930 (+)0.7097 (+)0.0741 Sign of k k-1 k (+) (+) (+) (+) Location of root IUC IUC IUC IUC

Result: The sign of k is determined from the sign of 0 1 2 and 3 (leading coefficients in the Table 3.18 ) by using the following expression,

Sign of k = [Sign of k] * [Sign of 'k-1]. It is noticed from the Table 3.19 it

is found that the given linear time invariant discrete system is stable and

having all the roots inside the unit circle.


algorithms.

75

Jury Method (1964)





F4(z) 1.0000 -1.3680 0.4126 0.0800 0.0025

F*4(z) 0.0000 0.0002 0.0010 -0.0034 0.0025

F*3(z) 1.0000 -1.3682 0.4116 0.0834

F3(z) 0.0070 0.0343 -0.114 0.0834

F*2(z) 0.9930 -1.4025 0.5257

F2(z) 0.2783 -0.7425 0.5257

F1(z) 0.7147 -0.660

F*1(z) 0.6095 -0.660

F0(z) 0.1052

Result: It is noticed from the Table 3.20 it is found that the given linear time

invariant discrete system is stable and having all the roots inside the unit

circle.

Raible Method (1974)


Raible algorithm is shown in Table 3.21.

76

Table 3.21 Raible Table for Example 3.7.3.1


F4(z) 1.0000 -1.3680 0.4126 0.0800 0.0025 k=0.0025

F*4(z) 0.0000 0.0002 0.0010 -0.0034 0.0025

F*3(z) 1.0000 -1.368 0.4116 0.0834 k=0.0834

F3(z) 0.0070 0.0343 -0.1141 0.0834

F*2(z) 0.9930 -1.4025 0.5257 k=0.5294

F2(z) 0.2783 -0.7425 0.5257

F1(z) 0.7147 -0.6600 k=-0.9235

F*1(z) 0.6095 -0.660

F0(z) 0.1052

Result: It is noticed from the Table 3.21, it is found that the given linear time

invariant discrete system is stable and having all the roots inside the unit

circle.

The number of arithmetic operation of each method for the example

3.7.3.1 is given in Table 3.22.


Operations Jury Table

(1964) Raible (1974)

Proposed

procedure I Proposed

procedure II

Division 4 4 0 0

Multiplication 14 14 20 20

Subtraction 14 14 10 10

Total 32 32 30 30

77

Example: 3.7.3.2.


Jury (1967) and determine the root distribution using the procedure I and II of

proposed scheme.

F (z) =2z4 4z3 +5z2 -2z +1 (3.16)


stability table and compared with the proposed scheme.


The formulated root distribution stability table for Example 3.7.3.2,

based on the proposed procedure I is shown in Table 3.23.


Order z0 z1 z2 z3 z4 F*4(z) 1.0000 -2.0000 5.0000 -4.0000 2.0000

F4(z) 2.0000 -4.0000 5.0000 -2.0000 1.0000 F*3(z) 0.0000 5.0000 -6.0000 1 F3(z) 3.0000 -6.0000 5.0000 0.0000 F*2(z) 15.0000 -18.0000 2 F2(z) 9.0000 -18.0000 15.0000 F*1(z) 108.0000 - 3 F1(z) -144.0000 108.0000 F0(z) - 4



outside the unit circle (OUC). From the Table.3.23, it is found that the given

linear time invariant discrete system is unstable and having two roots inside

78

the unit circle and two roots outside the unit circle because in the odd

numbered rows of the constructed table based on the proposed scheme-

procedure I, two are positive and two are negative.


algorithms.






F*4(z) 1.0000 -2.0000 5.0000 -4.0000 0

F4(z) 2.0000 -4.0000 5.0000 -2.0000 1.0000

F*3(z) 0.0000 5.0000 -6.0000 1

F3(z) 3.0000 -6.0000 5.0000 0.0000

F*2(z) 15.0000 -18.0000 2

F2(z) 9.0000 -18.0000 15.0000

F*1(z) 108.0000 - 3

F1(z) -144.0000 108.0000

F0(z) 4


Leading coefficient ( 0 1 2 3 4

Sign (+)2 (+)3 (+)9 (-)144 (+)9072

Sign of k k-1 k (+) (+) (-) (-)

Location of root IUC IUC OUC OUC

79

Result: The sign of k 0 1 2 and 3

(leading coefficients in the Table 3.24) by using the following expression,

Sign of k = [Sign of k] * [Sign of k-1]. It is noticed from the Table 3.25, it

is found that the given linear time invariant discrete system is unstable and

two roots are inside the unit circle and two roots are outside the unit circle.

Remark: Result is in accordance with Jury (1967) and Raible (1974)

algorithms.

Jury Method (1964)





F4(z) 2.0000 -4.0000 5.0000 -2.0000 1.0000

F*4(z) 0.5000 -1.0000 2.5000 -2.0000 1.0000

F*3(z) 1.5000 -3.0000 2.5000 0.0000

F3(z) 0.0000 0.0000 -0.0000 0.0000

F*2(z) 1.5000 -3.0000 2.5000

F2(z) 4.1667 -5.0000 2.5000

F1(z) -2.6667 2.0000

F*1(z) -1.5000 2.0000

F0(z) -1.1667


invariant discrete system is stable and having two roots inside the unit circle

and two roots outside the unit circle.

80





Order z4 z3 z2 z1 z0 F4(z) 2.0000 -4.0000 5.0000 -2.0000 1.0000 k=0.5000

F*4(z) 0.5000 -1.0000 2.5000 -2.0000 1.0000 F*3(z) 1.5000 -3.0000 2.5000 0.0000 k=0.0000 F3(z) 0.0000 0.0000 -0.0000 0.0000

F*2(z) 1.5000 -3.0000 2.5000 k=1.6667 F2(z) 4.1667 -5.0000 2.5000 F1(z) -2.6667 2.0000 k=-0.7500

F*1(z) -1.5000 2.0000 F0(z) -1.1667


invariant discrete system is unstable and having two roots inside the unit

circle and two roots outside the unit circle.




Operations

Jury Table (1964)

Raible Table (1974)

Proposed Procedure-I

Proposed Procedure-II

Division 4 4 0 0 Multiplication 14 14 20 20 Subtraction 14 14 10 10 Total 32 32 30 30

81

Example: 3.7.3.3.


Jury (1967) and determine the root distribution using the procedure I and II of

proposed scheme.

F (z) =2z4 +7z3 +10z2 +4z +1 (3.17)


stability table and compared with the proposed procedure I and II.


The formulated stability table for Example 3.7.3.3, based on the

proposed procedure I is shown in Table 3.29.



F*4(z) 1.0000 4.0000 10.0000 7.0000 2.0000

F4(z) 2.0000 7.0000 10.0000 4.0000 1.0000

F*3(z) 1.0000 10.0000 10.0000 1

F3(z) 3.0000 10.0000 10.0000 1.0000

F*2(z) 20.0000 20.0000 2

F2(z) 8.0000 20.0000 20.0000

F*1(z) -240.0000 - 3

F1(z) -336.0000 -240.0000

F0(z) - 4

82



outside the unit circle (OUC). From the Table.3.29, it is found that the given

linear time invariant discrete system is unstable and having two roots inside

the unit circle and two roots outside the unit circle because in the odd

numbered rows of the constructed table based on the proposed procedure I,

two are positive and two are negative.


algorithms.


The formulated stability table for Example 3.7.3.3, based on the

proposed procedure II is shown in Table 3.30.



F*4(z) 1.0000 4.0000 10.0000 7.0000 0

F4(z) 2.0000 7.0000 10.0000 4.0000 1.0000

F*3(z) 1.0000 10.0000 10.0000 1

F3(z) 3.0000 10.0000 10.0000 1.0000

F*2(z) 20.0000 20.0000 2

F2(z) 8.0000 20.0000 20.0000

F*1(z) -240.0000 - 3

F1(z) -336.0000 -240.0000

F0(z) 4

83


Leading coefficient ( 0 1 2 3 4 Sign (+)1 (+)3 (+)8 (-)336 (+)55296 Sign of k k-1 k (+) (+) (-) (-) Location of root IUC IUC OUC OUC

Result: The sign of k 0 1 2, 3 and 4

(leading coefficients in the Table 3.30 ) by using the following expression,

Sign of k = [Sign of k] * [Sign of k-1]. It is noticed from the Table 3.31, it

is found that the given linear time invariant discrete system is unstable and

two roots are inside the unit circle and two roots are outside the unit circle.


algorithms.

Jury Method (1964)




Order z4 z3 z2 z1 z0 F4(z) 2.0000 7.0000 10.0000 4.0000 1.0000 F*4(z) 0.5000 2.0000 5.0000 3.5000 1.0000 F*3(z) 1.5000 5.0000 5.0000 0.5000 F3(z) 0.1667 1.6667 1.6667 0.5000 F*2(z) 1.3333 3.3333 3.3333 F2(z) 8.3333 8.3333 3.3333 F1(z) -7.0000 -5.0000 F*1(z) -3.5714 -5.0000 F0(z) -3.4286

84









F4(z) 2.0000 7.0000 10.0000 4.0000 1.0000 k=0.5000

F*4(z) 0.5000 2.0000 5.0000 3.5000 1.0000

F*3(z) 1.5000 5.0000 5.0000 0.5000 k=0.3333

F3(z) 0.1667 1.6667 1.6667 0.5000

F*2(z) 1.3333 3.3333 3.3333 k=2.5000

F2(z) 8.3333 8.3333 3.3333

F1(z) -7.0000 -5.0000 k=0.7143

F*1(z) -3.5714 -5.0000

F0(z) -3.4286




85




Operations Jury Table

(1964) Raible (1974)

Proposed procedure I

Proposed

Procedure II

Division 4 4 0 0

Multiplication 14 14 20 20

Subtraction 14 14 10 10

Total 32 32 30 30

3.7.4 Discussion

The linear time invariant discrete systems represented by real

polynomials in Equations (3.15, 3.16, and 3.17) have been tested for root

distribution using procedure I and procedure II with Marden algorithm, Jury

(1964) algorithm and Raible (1974) algorithm.

From the above three examples it is inferred that Jury and Raible

methods possess the same computational effort while the suggested procedure

do not contain division operation.

In all the three examples, the information on root distribution was

determined using proposed procedure I and II with 20 multiplications and 10

subtractions only whereas Jury algorithm uses 14 multiplications, 14

and 14 subtractions and 4 divisions to solve the same problem employing that

the proposed procedures involves less computations, simple and direct.

Results summary of Examples (3.6.1.1, 3.6.1.2 and 3.6.1.3 are given in

Table 3.35

86

Table 3.35 Results Summary of Examples

Polynomials

Jury Table

(1964)

Raible Table

(1974)

Proposed

procedure I

Proposed

procedure II

D M S T D M S T D M S T D M S T

Example:3.6.1.1

(Bistritz) 4 14 14 32 4 14 14 32 0 20 10 30 0 20 10 30

Example:3.6.1.2

(E.I. Jury) 4 14 14 32 4 14 14 32 0 20 10 30 0 20 10 30

Example:3.6.1.3

(Nagrath and

Gopal)

4 14 14 32 4 14 14 32 0 20 10 30 0 20 10 30

D: Division, M: Multiplication, S: Subtraction, T: Total.

3.7.5 Comparison of Computational Efficiency

The construction of the Jury table and Raible table requires the

calculation of same number of entries. The involved numbers of elementary

multiplicative, subtraction and division operations are exactly equal. The

number of subtraction operations is less and nil division operation is present

in the proposed algorithm based Marden table. The number of multiplicative

operations in the new table is higher by one operation for each entry in the

table. The Figures 3.3, 3.4, 3.5 and 3.6 illustrate the number of each type of

operation and the number of total arithmetic operations involved in the root

distribution analysis of different types LTI discrete systems by the various

methods used in this research.

87

Figure 3.3 Comparison of Arithmetic Operation for Example: 3.6.1.1

Figure 3.4 Comparison of Arithmetic Operation for Example: 3.6.1.2

0 00

5

10

15

20

25N

umbe

r of

Ope

ratio

ns

Division Multiplication Subtraction

Jury Table Raible Table Proposed Procedure-I


0 00

5

10

15

20

25

Num

ber

of O

pera

tions


Jury Table Raible Table

Proposed Procedure-I

Proposed Procedure -II

88

Figure.3.5 Comparison of Arithmetic Operation for Example: 3.6.1.3

Figure 3.6 Comparison of Total Number of Operations in Jury, Raible and Proposed Procedure-I and II

0 00

5

10

15

20

25

Num

ber

of O

pera

tions


Jury Table Raible Table Proposed Procedure-I


29

29.5

30

30.5

31

31.5

32

32.5

Num

ber

of O

pera

tions

Result Summary

Jury Table Raible Table

Proposed Procedure I Proposed Procedure II


89

3.8 PROPOSED PROCEDURE FOR LTIDS DESIGN USING

MARDEN TABLE

The design of automatic control system is perhaps the most

important function that the control engineer carries out. Every control system

designed for a specific application has to meet certain performance

specifications. Merely by gain adjustment it may be possible to meet the

given specifications on performance of simple control systems. In such cases

the gain adjustment seems to be the most direct and simple method of design.

In general, the design of single and multi parameters existing as

coefficients in the characteristic polynomial of a linear time invariant discrete

system can be performed using the methods proposed by Anderson et al

(1973), Bandopadhyay et al (1988), Bistritz (1984), De La Sen et al (2003),

Engelborghs et al (2001), Franklin et al (2006), Fuller (1955), Jury et al

(1961,1974), Kuo et al (2003), Marden (1940, 1966), Nagrath et al (2007),

Park & Ikeda (2004), Raible (1974), Tantaris et al (2003), Wu et al (2007),

et al (1974) is applied but the exact value of the interested parameter in the

system cannot be predicted. When designing control systems, it is often

desirable to know the range of an adjustable parameter that results in a stable

system. Marden stability criterion is of limited usefulness in linear time

invariant discrete system analysis mainly because it does not suggest the way

to improve relative stability or how to stabilize an unstable system. It is

possible, however to determine the effect of changing one or two parameters

of a system by examining the values that cause instability. Also, it should be

noted that in the case of the higher order characteristic polynomial with

unknown design parameters, the application of Jury table and Marden

algorithm become tedious.

90

Read the characteristic polynomial

Evaluate F (z=1) and F (z=-1), and obtain the approximate range of K (Kmin,Kmax)

Bisect the values of Kmin, Kmax

Stop

Start

Apply the bisection principle and Marden table to get the Critical value of K

NO

YES

IF Lowest and highest real values of K are obtained?

To circumvent this situation, the necessary conditions are applied to

the given characteristic polynomial for extracting the approximate range of

values of the design parameters. Then the approximate ranges of parameters

are further tuned using bisection principle along with the Marden table

(Marden 1966) to obtain the exact range of parameters.

The necessary conditions given in Equation (3.9) are applied

successively for extracting the lower and upper limiting values of design

parameters. The procedure is depicted in Figure 3.7 as flowchart and an

algorithmic form in section 3.8.1.

Figure 3.7 Proposed Scheme for Discrete System Design

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3.8.1 Algorithm for Proposed Procedure

The various steps involved in the algorithm are as follows:

Step 1: Read the given characteristic polynomial F (z).

(Containing a parameter, say, K to be designed for stability)

Step 2: Evaluate F (z=1) and F (z=-1), and obtain the approximate range of

K (Kmin,Kmax).

Step 3: Bisect the values of Kmin, Kmax,

2

KKK maxmin

b (3.18)

Substitute the value of Kb in the characteristic polynomial F (z),

Then compute F (z=1) and F (z=-1).

Step 4: Repeat step 3, until the lowest and highest real values of K are

obtained.

(Based on the magnitude variations of F(1) or F(-1) , i.e. increasing

decreasing-increasing or decreasing increasing decreasing form of

variation).

i.e., Lowest real value of K = K1 and Highest real value of K = K2.

Thus K1 < K < K2 (3.19)

Step 5: Form Marden table for F(z) with K = K1 and K = K2. Apply

bisection principle (midpoint between any two given values),

sharpen this range by checking if the coefficient of )z(F0 of Marden

table tends to zero, to get the critical value of K.

92

-

+ R(s) C(s)

se1 sT

G(s)

Step 6: Thus the sharpened value of K is obtained.

Step 7: Stop

The above proposed procedure is applied to the following

illustration.

3.8.2 Illustrations

The proposed procedure is applied to the sampled data system

design in this section.

Assume a sampled data system, shown in Figure.3.8 with open loop

transfer function as (Jury et al. 1974),

1)s(sKsT'

es

Tse1G(s) (3.20)

Where sampling period T =1 and = 1.25.

Figure 3.8 Sampled Data Feedback System

The z-transform of Equation (3.20) is,

0.368)1)(z(z2z1.755)0.03)(z(z0.2223KG(z) (3.21)

The characteristic equation of equation (3.21) is, 1+ G (z) = 0

93

F(z) = z4-1.368z3+(0.368+0.2223K)z2+(0.3974K)z+0.0123K = 0

(3.22)

Now, applying the proposed algorithm to the characteristic equation

in Equation (3.22),

Step 1: Read the characteristic equation given in Equation (3.22)

Step 2: Evaluating F (z) in Equation (3.22) at z = 1 and -1,

i) F(1) = 0.6327K > 0

K > 0 (3.23)

ii) F(-1) = 2.736 0.1621K > 0

K < 2.736 / 0.1621

K < 16.8785 (3.24)

From equations (3.23), (3.24) the approximate range of K is,

0 < K < 16.8785 (3.25)

Step 3: Bisecting the value of K,

Kb = 8.439252

16.87850 (3.26)

Substituting the value of K in Equation (3.26) to Equation (3.22),

we get,

F(z) = z4-1.368z3+2.25z2+3.3538z+0.1038 (3.27)

Compute F(z) at z = 1 and -1,

1.3681)F(5.3396F(1) (3.28)

94

Step 4: Repeat step 3 until lowest and highest real values of K are obtained.

Table.3.35 shows the operation involved in step 3 for further

bisections of K value. i.e., it provides F(z) at z = 1 and -1 for different values

of K.

Table 3.36 Operation Involved in Step 3 for Various Values of K

S. No K F(1) F(-1)

1 8.43925 5.3396 1.368

2 4.2196 2.6698 2.052

3 2.1098 1.3349 4.0709

4 1.0549 0.6674 2.5650

5 0.52745 0.3337 3.0697

6 0.2637 0.1668 2.9028

By observing the calculated values of F(1) and F(-1) for various

values obtained by the bisection rule, the stopping point for the process can be

determined. The locus of the roots of the characteristic polynomial is directly

dependent on the value of K. Whenever a root locus intersects the unit circle

in the complex plane the sign of some of the coefficients of the characteristic

polynomial will change, this in turn observed as a direction change

(increasing decreasing-increasing or decreasing increasing decreasing

form of variation) in the magnitude of the calculated value at z=1 and z=-1.

In Table 3.36, it can be observed that for K = 1.0549, F(-1) has

decreased and for K = 0.52745, it has again increased, as a result, these two K

values can be chosen as approximate highest and lowest real values, and can

be further tuned to get the critical value.

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i.e., Lowest real value of K = K1 = 0.52745

Highest real value of K = K2 = 1.0549

Thus, 0.52745< K < 1.0549 (3.29)

Step 5: Formulate Marden table for F (z) with K = K1 =0.52745 and

K =K2 = 1.0549 as in Table3.37 and Table.3.38 respectively.

Table 3.37 Marden Table for F (z) in Equation (3.22) with K = 0.52745

Order z0 z1 z2 z3 z4 Necessary and

Sufficient Conditions

F4* (z) 0.0065 0.2096 0.4856 -1.368 1

F4(z) 1 -1.368 0.4856 0.2096 0.0065

|b1/b0|=1.368< 4, F(1)=0.3337, F(-1)=2.6505, True

F3* (z) 0.2185 0.4824 -1.3694 1.0000

F3(z) 1.0000 -1.3694 0.4824 0.2185

|b1/b0|=1.3694< 3, F(1)=0.3315, F(-1)=-2.6333, True

F2* (z) 0.7816 -1.4747 0.9522

F2(z) 0.9522 -1.4747 0.7816

|b1/b0|=1.5488< 2, F(1)=0.2591, F(-1)=3.2085, True

F1* (z) -0.2515 0.2957

F1(z) 0.2957 -0.2515

|b1/b0|=0.8506< 1, F(1)=0.0442, F(-1)=-0.5472, True

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Table 3.38 Marden Table for F(z) in Equation (3.22) with K = 1.0549

Order z0 z1 z2 z3 z4 Conditions

F4* (z) 0.013 0.4192 0.6032 -1.368 1

F4(z) 1 -1.368 0.6032 0.4192 0.013

|b1/b0|=1.3680< 4, F(1)=0.6674,

F(-1)=2.5650,

True

F3* (z) 0.4370 0.5954 -1.3734 0.9998

F3(z) 0.9998 -1.3734 0.5954 0.4370 |b1/b0|=1.3737< 3, F(1)=0.6587,

F(-1)=-2.5317, True

F2* (z) 1.1955 -1.6333 0.8086

F2(z) 0.8086 -1.6333 1.1955 |b1/b0|=2.0197<2

False

From Table.3.37, it can be noted that F (z) at K = 0.52745 is stable.

From Table 3.38, it is observed that F2 (z) at K = 1.0549 is not satisfying the

conditions for stability. Since F2 (z) does not satisfy one of the necessary

conditions, the system is unstable for the choice of K = 1.0549.

For different choices of K this process is carried out with the help

of proposed procedure and the range of values of K for stability is shown in

Table 3.39

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Table 3.39 Achieved Results during Design for Critical Value of K

Approximate range of K Condition in Marden Table of

F(z) i.e., value of )z(F0

0.52745 Stable

1.0549 Unstable

0.7912 Unstable

0.6593 Stable

0.7252 Unstable

0.69225 Marginally stable

From Table 3.39, it can be noted that for K = 0.69225, the

computed element )z(F0 in Marden table becomes zero indicating marginal

stability condition.

Step 6: Thus the sharpened value of K is,

K = 0.69225

Step 7: Stop

Thus the critical value of K = 0.69225 obtained using the proposed

procedure is in agreement with that given in Jury et al (1974).

3.8.3 Discussion

The proposed procedure employs a improved concept of evaluating

the characteristic equations with the help of necessary conditions. The

necessary conditions are utilized to evaluate the approximate range of design

parameters in a given system and are sharpened using bisection principle. The

suggested approach is applied to the sampled data control system. The

illustration depicts single parameter design of the given characteristic

98

polynomial. This suggested procedure reduces the computational complexity

compared to the direct application of Jury methods (Jury et al. 1974) and

Bistritz (1985) method to the original higher order characteristic equation. By

adding linear transformation techniques with procedure proposed for

designing discrete linear systems can also be used to handle the design of

linear time invariant continuous systems.

3.9 SUMMARY

In this chapter the proposed procedures for testing the absolute

stability as well as designing of single parameter in a control system were

carried out. In the case of unstable systems the information on root

distribution was obtained by using two different procedures both are

computationally efficient. It is observed that the suggested procedures are

direct and straight forward in its application having lesser amount of

computations compared to that of original Mardens Table as well as that of

Jury and Raible method.

CHAPTER 3 MARDEN TABLE FORMULATON FOR STABILITY...

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