Design of second order linear time invariant systems for deadbeat response

1. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14)
30-31, December, 2014, Ernakulam, India
51
DESIGN OF SECOND ORDER LINEAR TIME INVARIANT
SYSTEMS FOR DEADBEAT RESPONSE
Dr.K.SREEKALA1
, Dr.S.N.SIVANADAM2
1
EEE, Sngce, Kadayiruppu, India
2
CSE, Karpagam Institute of Technology, Coimbatore, India
ABSTRACT
In this paper employing Routh’s table, an algebraic criteria for the deadbeat stability analysis of linear time-
invariant systems having real coefficients are formulated. The proposed algebraic stability criterion check whether the
given linear system response is deadbeat or not. The additional significance of the criterion is , it can be used to count the
number of complex roots of a system having real coefficients which is not possible by the use of original Routh’s Table.
This procedure can also be used for the design of linear time invariant systems. In the proposed method, the characteristic
equations having real coefficients are first converted to complex coefficient equations using Romonov’s transformation.
These complex coefficients are used to form the Modified Routh’s tables for the proposed scheme named as Sign Pair
Criterion.
Keywords: Stability Analysis, Deadbeat Response, Complex Roots of a Polynomial, Modified Routh’s table, Sign pair
criterion
I. INTRODUCTION
The stability analysis can be carried out by the knowledge of root distribution of it’s characteristic equation in
the case of a linear invariant continuous system. To analyze the stability of complex polynomials the generalized Routh-
Hurwitz method was investigated in [1] - [6]. Frank [1] and Agashe [2] developed a new Routh like algorithm to
determine the number of RHP roots in the complex case. Benidir and Picinbono [3] proposed an extended Routh table
which considers singular cases of vanishing leading array element. By adding intermediate rows in the Routh array,
Shyan and Jason [4] developed a tabular column, which is also a complicated one. Adel [5] has done the stability analysis
of complex polynomials using the J-fraction expansion, Hurwitz Matrix determinant and also generalized Routh’s Array.
Formation of Routh’s Table by retaining the ‘j’ terms of the complex coefficients and the stability analysis using Sign
Pair Criterion were done in [6].
The deadbeat responses represent a response without any oscillations and overshoot. Information about the
deadbeat response of a control system is of paramount importance for any design problem. This is generally used for the
design of instrumentation systems, network analysis and automatic controls. The existences of real and distinct roots in
the negative real axis determine the deadbeat behavior of a linear system. The presence of any complex roots shows that
the system is unstable. To analyze the deadbeat stability, a generalized method was investigated in [7] by Fuller.
Romonov [8] developed a new transformation to determine the deadbeat behavior of the linear system which results in
complex coefficient polynomials. A popular but cumbersome method for determining the number of real roots in
polynomial with real coefficients is by Sturm’s theorem [9]. Itzhack presented [10] a three-steps transformation
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2. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14)
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procedure which develops a polynomial whose number of right hand plane poles equals the number of complex roots
present in the original polynomial.
In the proposed method, the characteristic equations having real coefficients are first converted to complex
coefficient equations using Romonov’s transformation. These complex coefficients are used to form the Modified
Routh’s table for the proposed scheme named as Sign Pair Criterion The beauty of the Routh’s algorithm lies in finding
the deadbeat stability of the system without determining the roots of the system. In this approach, formation of Routh’s
Table is done by retaining the ‘j’ terms of the complex coefficients and the stability analysis is done using Sign Pair
Criterion .The proof of the method is given in [6]. Then by the use of the proposed algorithm, the deadbeat behavior of
linear systems, and also the number of complex roots of the characteristic equations can be determined.
II. DEADBEAT STABILITY ANALYSIS
A linear time invariant control system represented by the characteristic equation F(s) =0, with real coefficients is
a periodically stable (gives deadbeat response) only when all roots are distinct real and lie on the negative real axis of‘s’
plane. To analyze this situation, Romonov [8] suggested a transformed polynomial of F(s) into a complex polynomial
defined as given in equation (1).
‫)ݏ(ܨ‬ = [‫ | )ݏ(ܥ‬௦ୀ௝௦ + ݆ (݀‫]))ݏ݆(݀/ )ݏ݆(ܥ‬ (1)
This transformation results in a polynomial possessing complex coefficients and the newly proposed method can
be extended to test for deadbeat condition. The number of sign pairs obtained from the first column of the Routh-like
table, which do not obey the criterion gives the number of complex roots of the characteristic equation. The number of
complex roots is twice as that of the number of pairs which violate the criterion, since complex roots exist as conjugate
pairs.
III. PROPOSED METHOD
For stability, the real parts of all the roots of the characteristic equation must lie on the left half of ‘s’ plane. If
all the coefficients are real, Routh-Hurwitz stability test is directly applicable as explained in (Porter 1967). In the
Routh’s table, all the elements in the first column computed using Routh’s algorithm should have same sign for stability.
If the coefficients are complex, ‘Sign Pair Criterion is formulated as given below. In this approach, the elements of
Routh-like table are used to formulate the stability criterion.
The characteristic equation with complex coefficients can be written as shown below.
‫)ݏ(ܥ‬ = ‫ݏ‬௡
+ (ܽଵ + ݆ ܾଵ)‫ݏ‬௡ିଵ
+ (ܽଶ + ݆ ܾଶ)‫ݏ‬௡ିଶ
+ ⋯ + (ܽ௞ + ݆ ܾ௞) = 0
(2)
The first two rows of Routh-like table are written as shown below.
3.1Certain Guidelines for Completing the Routh Table
i) When the first element in any row of the Routh table is zero, it can be replaced by a small positive value or
+0.01.
ii) When any one of the element in the first column, starting from the third row is zero, it shows the presence of
complex or imaginary roots.
iii) If all the elements in arrow become zero, then the auxiliary polynomial is formed using the previous row elements
and differentiated once; the coefficients of this modified polynomial are entered instead of zeros and the table is
completed by applying the Routh multiplication rule.
iv) A row with full of zeros represents the existence of imaginary roots and the order of the auxiliary equation formed
by the elements of the previous row, gives the number of roots on imaginary axis.
v) All the elements of any row can be divided by a positive constant during the table formulation.
1 ݆ܾଵ ܽଶ ݆ܾଷ ܽସ . . .
ܽଵ ݆ܾଶ ܽଷ ݆ܾସ ܽହ . . .

3. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14)
30-31, December, 2014, Ernakulam, India
53
3.2 Modified Routh Table
Applying the standard Routh multiplication rule, the subsequent elements of Routh-like table are computed and
the table is as shown.
After the formation of Routh-like table, the Sign pair criterion can be applied to find the stability of the system. The
criterion is stated as shown below.
3.3 Sign Pair Criterion
For an nth
order characteristic equation there are 2n pairs in the Routh –like table and each pair carries pure real
or pure imaginary terms with same sign when the system is a periodically stable. Any violation for this condition
indicates a response with oscillations or overshoot.
If the two elements in any of the pair do not possess same sign, the system is not a periodically stable. The
number of pairs which does not obey the criterion gives the number of complex root pairs.
Using the first column elements, sign pairs are formed as
It is ascertained that each element of the pair has to maintain the same sign for the roots of characteristic
equation to lie on the left of s-plane. Thus the necessary and sufficient conditions for SPC are established.
IV. DESIGN PROBLEM
Consider the system given by Itzhack & Calise [10]
Let
‫)ݏ(ܨ‬ = ‫ݏ‬ଶ
+ ‫ݏܭ‬ + 2 = 0
Design the value of ‘K’ for the system to be of deadbeat response.
‫ܨ‬′(‫)ݏ‬ = (݆‫)ݏ‬ଶ
+ ‫)ݏ݆(ܭ‬ + 2 + ݆[2(݆‫)ݏ‬ + ‫]ܭ‬
‫ܨ‬′(‫)ݏ‬ = ‫ݏ‬ଶ
+ (2 − ݆‫ݏ)ܭ‬ + (−2 − ݆‫)ܭ‬
1 ݆ܾଵ ܽଶ ݆ܾଷ ܽସ . . .
ܽଵ ݆ܾଶ ܽଷ ݆ܾସ ܽହ . . .
‫ݎ‬ଷଵ ‫ݎ‬ଷଶ ‫ݎ‬ଷସ ‫ݎ‬ଷହ . . .
‫ݎ‬ସଵ ‫ݎ‬ସଶ ‫ݎ‬ସଷ ‫ݎ‬ସସ . . .
‫ݎ‬ହଵ ‫ݎ‬ହଶ ‫ݎ‬ହଷ . . .
‫ݎ‬଺ଵ ‫ݎ‬଺ଶ ‫ݎ‬଺ଷ . . .
‫ݎ‬଻ଵ ‫ݎ‬଻ଶ . . .
‫ݎ‬଼ଵ . . .
. . . .
. . . .

4. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14)
30-31, December, 2014, Ernakulam, India
54
Modified Routh like table is as shown below.
+1 −݆‫ܭ‬ −2
+2 −݆‫ܭ‬
−݆(0.5)‫ܭ‬ −2
−݆(
8
‫ܭ‬
− ‫)ܭ‬
For the system to have deadbeat response, the conditions are given by
i) 0.5 K > 0
ii) 8 – K2
>0
Hence the conditions for stability are
i) ‫ܭ‬ > 0
ii) K2
>8 (K > 2.83)
The design is verified with the result of Itzhack & Calise [10]. Let us verify the result for two different values of K.
4.1 Stability Analysis for K= 2 (K < 2.83)
‫)ݏ(ܨ‬ = ‫ݏ‬ଶ
+ 2‫ݏ‬ + 2 = 0 (3)
‫ܨ‬′(‫)ݏ‬ = (݆‫)ݏ‬ଶ
+ 2 + ݆ሾ2(݆‫)ݏ‬ + 2ሿ = 0‫ܨ‬′(‫)ݏ‬ = ‫ݏ‬ଶ
+ (2 − ݆2)‫ݏ‬ + (−2 − ݆2) = 0 (4)
Modified Routh like table for K=2 is formed as
+1 −݆2 −2
+2 −݆2
−݆1 −2
+݆2
Pairs are formed as
The elements in the second pair do not possess same sign hence this pair does not obey SPC. So the system does
not possess deadbeat response and there exists two numbers of complex roots for the given equation. The roots of the
equation are -1 + j1 and -1 – j 1and, which verifies the result.
Verification using Step Response
The step response is obtained using MATLAB simulation as shown below.
Figure 1 Step Response of the system for K=2

5. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14)
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From the response, it is clear that the system is stable since it finally settles to the desired value, but it is not
having deadbeat response due to the presence of overshoot.
4.2 Stability Analysis for K= 3 (K >2.83)
The Modified Routh like table for K=3 is formed as
+1 −݆3 −2
+2 −݆3
−݆1.5 −2
−݆0.33
Pairs are formed as
It is noted that the two elements in each pair have the same sign and obey SPC. Hence the system has deadbeat
response. The roots of are found as – 1 and - 2 which are real values, this verifies the result.
Verification using Step Response
The step response is obtained using MATLAB simulation as shown below.
Figure 2 Step Response of the system for K=3
From the response, it is clear that the system is stable and has deadbeat response due to the absence of
oscillations and overshoot.

6. Proceedings of the International Conference on Emerging Trends in Engineering and Management (ICETEM14)
30-31, December, 2014, Ernakulam, India
56
V. CONCLUSION
In this paper, extending the Routh criterion, a stability criterion is formulated which is directly applicable to
handle the characteristic equation having complex coefficients. For inferring the deadbeat stability, the computations
remain the same as that of original Routh array. Further Sign Pair Criterion (SPC) is used to find the distributions of
roots. Thus the proposed approach shows the unique extension of the results deduced from the Routh Table for stability
investigation of Linear Time Invariant Systems for deadbeat response.
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