chapter 3 marden table formulaton for stability...
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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:
49
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.
51
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.
52
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)
Step 1: Verify the necessary condition for stability by employing the
stability constraints (i) and (ii) listed in Equation (3.9).
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
Order z0 z1 z2 z3 z4 Constraints
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
56
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)
Step 1: Verify the necessary condition for stability by employing the
stability constraints (i) and (ii) listed in Equation (3.9).
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
Order z0 z1 z2 z3 z4 Constraints
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
conditions on coefficients for the characteristic equation to have all its roots
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).
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 3.6.1.2, based on Jury
algorithm is shown in Table 3.8.
Table 3.8 Jury Table for Example 3.6.1.2
Order z0 z1 z2 z3 z4 Constraints
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
Result: It is noticed from the above test, the necessary and sufficient
conditions on coefficients for stability of F(z) are not met. So we conclude
that the system is unstable.
Marden Algorithm (1949)
The formulated stability table for Example 3.6.1.2, based on
Marden algorithm is shown in Table 3.9.
Table 3.9 Marden Table for Example 3.6.1.2
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.
The number of arithmetic operations of each method for the
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)
Step 1: Verify the necessary condition for stability by employing the
stability constraints (i) and (ii) listed in Equation (3.9).
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
Order z0 z1 z2 z3 z4 Constraints
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
conditions on coefficients for the characteristic equation to have all its roots
inside the unit circle is not satisfied. So we conclude that the system is
unstable.
Remark: Result is in accordance with Jury (1967).
The problem is solved using Jury (1964) and Marden (1949)
stability criterion. It is compared with the proposed procedure.
63
Jury Algorithm (1964)
The formulated stability table for Example 3.6.1.3, based on Jury
algorithm is shown in Table 3.12.
Table.3.12 Jury Table for Example 3.6.1.3
Order z0 z1 z2 z3 z4 Constraints
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
Result: It is noticed from the above test, the necessary and sufficient
conditions on coefficients of F(z) are not met. So we conclude that the system
is unstable.
Marden Algorithm (1949)
The formulated stability table for Example 3.6.1.3, based on
Marden algorithm is shown in Table 3.13.
Table 3.13 Marden Table for Example 3.6.1.3
Order z0 z1 z2 z3 z4
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
The number of arithmetic operations of each method for the
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
Example 1 Example 2 Example 3
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
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
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.
Remark: Result is in agreement with Jury (1967) and Raible (1974)
algorithms.
75
Jury Method (1964)
The formulated stability table for Example 3.7.3.1, based on Jury
algorithm is shown in Table 3.20.
Table 3.20 Jury Table for Example 3.7.3.1
Order z4 z3 z2 z1 z0
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)
The formulated stability table for Example 3.7.3.1, based on
Raible algorithm is shown in Table 3.21.
76
Table 3.21 Raible Table for Example 3.7.3.1
Order z4 z3 z2 z1 z0
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.
Table 3.22 Arithmetic Operation for Example 3.7.3.1
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.
Consider a characteristic equation with real coefficients given in
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)
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 stability table for Example 3.7.3.2,
based on the proposed procedure I is shown in Table 3.23.
Table 3.23 Proposed Procedure I Table for Example 3.7.3.2
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
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.3.23, 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.
Remark: Result is in agreement with Jury (1967) and Raible (1974)
algorithms.
Proposed Procedure II
The formulated absolute stability table for Example 3.7.3.2, based
on the proposed procedure II is shown in Table 3.24.
Table 3.24 Proposed Procedure II Table for Example 3.7.3.2
Order z0 z1 z2 z3 z4
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
Table 3.25 Root Distribution with Respect to Unit Circle for Example 3.7.3.2
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)
The formulated stability table for Example 3.7.3.2, based on Jury
algorithm is shown in Table 3.26.
Table 3.26 Jury Table for Example 3.7.3.2
Order z4 z3 z2 z1 z0
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
Result: It is noticed from the Table 3.26, it is found that the given linear time
invariant discrete system is stable and having two roots inside the unit circle
and two roots outside the unit circle.
80
Raible Method (1974)
The formulated stability table for Example 3.7.3.2, based on
Raible algorithm is shown in Table 3.27.
Table 3.27 Raible Table for Example 3.7.3.2
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
Result: It is noticed from the Table 3.27, 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.
The formulated absolute stability table for Example 3.7.3.2, based
on the proposed procedure II is shown in Table 3.28.
Table 3.28 Arithmetic Operation for Example 3.7.3.2
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.
Consider a characteristic equation with real coefficients given in
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)
Note: The same problem is solved using Jury (1964) and Raible (1974)
stability table and compared with the proposed procedure I and II.
Proposed Procedure I
The formulated stability table for Example 3.7.3.3, based on the
proposed procedure I is shown in Table 3.29.
Table 3.29 Proposed Procedure I Table for Example 3.7.3.3
Order z0 z1 z2 z3 z4
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
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.3.29, 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.
Remark: Result is in accordance with Jury (1967) and Raible (1974)
algorithms.
Proposed Procedure II
The formulated stability table for Example 3.7.3.3, based on the
proposed procedure II is shown in Table 3.30.
Table 3.30 Proposed Procedure II Table for Example 3.7.3.3
Order z0 z1 z2 z3 z4
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
Table 3.31 Root Distribution with Respect to Unit Circle for Example 3.7.3.3
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.
Remark: Result is in accordance with Jury (1967) and Raible (1974)
algorithms.
Jury Method (1964)
The formulated stability table for Example 3.7.3.3, based on Jury
algorithm is shown in Table 3.32.
Table 3.32 Jury Table for Example 3.7.3.3
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
Result: It is noticed from the Table 3.32, 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.
Raible Method (1974)
The formulated stability table for Example 3.7.3.3, based on
Raible algorithm is shown in Table 3.33.
Table 3.33 Raible Table for Example 3.7.3.3
Order z4 z3 z2 z1 z0
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
Result: It is noticed from the Table 3.33, 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.
85
The formulated absolute stability table for Example 3.7.3.3, based
on the proposed procedure II is shown in Table 3.34.
Table 3.34 Arithmetic Operation for Example 3.7.3.3
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
Proposed Procedure-II
0 00
5
10
15
20
25
Num
ber
of O
pera
tions
Division Multiplication Subtraction
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
Division Multiplication Subtraction
Jury Table Raible Table Proposed Procedure-I
Proposed Procedure-II
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
Example 1 Example 2 Example 3
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
91
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.
95
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
96
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
97
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.