Constant gain stabilization with desired damping

Computers and Chemical Engineering 30 (2006) 1072–1075

Constant gain stabilization with desired damping S. Narayanan a,∗ , P. Kanagasabapathy b , J. Prakash c a

Department of Electrical Engineering, Thiagarajar College of Engineering, Madurai, Tamil Nadu, India b Madras Institute of Technology, Anna University, Chennai 600044, Tamil Nadu, India c Department of Instrumentation Engineering, Madras Institute of Technology, Anna University, Chennai 600044, Tamil Nadu, India Received 29 September 2004; received in revised form 31 October 2005; accepted 9 February 2006 Available online 17 April 2006

Abstract In this short note, we have proposed an approach to find the solution for constant gain stabilization problem with desired damping. This short note illustrates that it is easy to find the bound on the constant gain as compared to Aniruddha Datta’s proposed method. © 2006 Elsevier Ltd. All rights reserved. Keywords: Constant gain stabilization; Hermite–Biehler theorem

1. Introduction Let the system transfer function is G(s) = (N(s)/D(s)) and the controller transfer function is Gc (s) = k. The characteristic polynomial is δ(s,k) = D(s) + kN(s). The Hermite–Biehler theorem gives necessary and sufficient conditions for the Hurwitz stability of a polynomial in terms of certain interlacing conditions. These results have been successfully used in Datta, Ho, and Bhattacharyya (2000) to obtain the analytical results on P, PI and PID stabilization of single input and single output system. However, in some cases, there will be some difficulties in the implementation of the above said algorithm. The difficulty is that both real and imaginary parts (p(ω,k) and q(ω,k), respectively) depend on controller parameter k which makes this quite formidable problem to solve, even when p(ω,k) and q(ω,k) is of lower order. Datta has suggested some modifications in the existing approach to overcome the above said difficulty. The procedure followed by Datta to make either p(ω,k) or q(ω,k) independent of k is outlined in Datta et al. (2000). This difficulty arises only when the system has some open loop zeros. If the system has only open loop poles, then this modification is not necessary. Even though, the above said approach is useful and simple to obtain the unknown controller parameters, it has some limitations. The limitation is that it provides the results for absolute stability condition only, i.e., it is

∗

Corresponding author. Tel.: +91 4422230850; fax: +91 44 22232403. E-mail address: rvsn [email protected] (S. Narayanan).

0098-1354/$ – see front matter © 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compchemeng.2006.02.004

not possible to make p(ω,k) or q(ω,k) independent of k, when relative stability condition is to be satisfied. But, Hermite–Biehler theorem applicable to real polynomial provides absolutely no information about its root distribution when the polynomial is not Hurwitz. Further, We need the information regarding the number of right half plane roots of δ(s), i.e., we have to determine the difference between the number of roots of the polynomial in the left half and right half of s-plane. Due to this reason, generalized versions of Hermite–Biehler theorem (GHBT) has been developed by S.P. Bhattacharyya. The advantage of the generalized version is that the given polynomial need not be Hurwitz. By investigating the characteristic polynomial of the system using a generalized Hermite–Biehler theorem, Ho, Datta and Bhattacharyya obtained all stabilizing gains for SISO system. Though there are computational difficulties in solving stabilization problem using generalized Hermite–Biehler theorem, the statement of Datta et al. (2000) and Ho, Datta, and Bhattacharyya (1999) about the features of generalized Hermite’s theorem are valid. Further, the analysis procedure using GHBT can be extended to complex polynomials. Indeed, a special case of these results has been successfully used in Datta et al. (2000) to obtain the analytical results on constant gain stabilization with desired damping. It should, however, be pointed out that a graphical solution to the same problem can be obtained using standard root locus technique. By selecting a suitable controller structure, it is possible to make the closed loop polynomial Hurwitz. As long as the closed loop system is stable, there will not be any necessity to go for the generalized version of Hermite theorem. Moreover, the computational effort gets reduced

S. Narayanan et al. / Computers and Chemical Engineering 30 (2006) 1072–1075

if interlacing condition is used for stability analysis. Further, Tan, Kaya, and Atherton (2003) cited that the computation time for Datta’s approach increases in an exponential manner with the order of the system being considered. The approach suggested by Nusret Tan et al. is a graphical one and also it requires range of frequencies and test point among the frequencies. In our approach, this information is not required. Our approach is based on Hermite–Biehler theorem. The theorem implies that all the roots of δr (ω) and δi (ω) are real and distinct. In addition, the roots of δr (ω) and δi (ω) should satisfy the interlacing condition. The equivalent representation is that δ(jω) = 0 for all real ω. Hence, we utilize the benefits of Hermite–Biehler theorem for solving the above said problem. 2. Preliminary results

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Uk+1 (ξ) − 2ξUk (ξ) + Uk−1 (ξ) = 0 with T0 (ξ) = 1, T1 (ξ) = ξ, U0 (ξ) = 0, U1 (ξ) = 1. For pertinent of ξ, the chebyshev functions may be computed. By substituting Eq. (2.5) in Eq. (2.4), one can obtain the function δ(s) and it can be expressed as: δ(ωn , ξ) = δr (ωn , ξ) + jδi (ωn , ξ) (2.6) n k k where δr (ωn , ξ) = k=0 (−1) αk ωn Tk (ξ) and δi (ωn , ξ) =  n k+1 2 1−ξ αk ωnk Uk (ξ). k=0 (−1) 2.2. Relative stability analysis based on pole assignment  Let Xk = (−1)k ωnk Tk (ξ), Yk = (−1)k+1 ωnk Uk (ξ) 1 − ξ 2 The real and imaginary part can be rewritten as:

In this section, we briefly review some results reported by Stojic and Siljak (1965) and Siljak (1966). The results, which are relevant to this work only are highlighted.

δr (σ, ω) =

2.1. Relative stability analysis based on damping factor specification

δi (σ, ω) =

n


ak X k

k=0 n


a k Yk

k=0

δ(σ, ω) = δr (σ, ω) + jδi (σ, ω)

The characteristic equation is: δ(s) = α0 + α1 s + α2 s2 + · · · + αn sn = 0

(2.1)

X0 = 1;

X1 = σ;

Y0 = 0;

(2.7) Y1 = ω;

The Eq. (2.1) can be represented as: δ(s) =

n


αk s

Xk+1 = 2X1 Xk − [X12 + Y12 ]Xk−1

k

(2.2)

k=0

where the coefficients αk are real. If the root of the Eq. (2.2) is located in desired dominant place, it can be stated as:   s = −ωn ξ + jωn 1 − ξ 2 = ωn (−ξ + j 1 − ξ 2 ) = ωn (cos θ + j sin θ) s = ωn ejθ

(2.3)

Substituting (2.3) in Eq. (2.2): δ(s) =

n


αk e

can be converted to the form: (2.5)

where bk = αk Tk (−ξ) = (−1)k αk Tk (ξ) 

G(s) =

N(s) a 0 + a 1 s + a 2 s 2 + · · · + an s n = D(s) b0 + b 1 s + b 2 s 2 + · · · + b n s n

3.1. Relative stability based on pole assignment ejkθ

= bk + jck

ck = αk

A simplified procedure to compute the set of K of all S Hurwitz stabilizing gain values for the G(s) is as follows: the key idea is to use the results of Siljak (1966) and Stojic and Siljak (1965) in conjunction with Hermite–Biehler theorem. Consider the system transfer function as:

(2.4)

k=0

jkθ

3. Main results

where all the coefficients of system transfer function are real.

k

αk (ejθ ωn )

The coefficients αk

Yk+1 = 2X1 Yk − [X12 + Y12 ]Yk−1

1 − ξ 2 Uk (−ξ) = (−1)k+1 αk



1 − ξ 2 Uk (ξ)

Using Eq. (2.7), the characteristic equation of the closed loop system can be rewritten as:   Ne (σ, ω) + jNo (σ, ω) 1 + kc =0 (3.1) De (σ, ω) + jDo (σ, ω) Ne (σ, ω) = a0 X0 + a1 X1 + a2 X2 + · · · + an Xn ; No (σ, ω) = a0 Y0 + a1 Y1 + a2 Y2 + · · · + an Yn ;

these functions are obtained from the following recurrence formulae:

De (σ, ω) = b0 X0 + b1 X1 + b2 X2 + · · · + bn Xn ;

Tk+1 (ξ) − 2ξTk (ξ) + Tk−1 (ξ) = 0

Do (σ, ω) = b0 Y0 + b1 Y1 + b2 Y2 + · · · + bn Yn

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S. Narayanan et al. / Computers and Chemical Engineering 30 (2006) 1072–1075

The characteristic Eq. (3.1) can be rewritten as:

The characteristic Eq. (3.4) can be rewritten as:

δ(σ, ω, kc ) = [De (σ, ω) + jDo (σ, ω)]

δ(ξ, ωn , kc ) = [De (ξ, ωn ) + jDo (ξ, ωn )] + kc [Ne (ξ, ωn )

+ kc [Ne (σ, ω) + jNo (σ, ω)] = 0

(3.2)

+ jNo (ξ, ωn )] = 0

(3.5)

The Eq. (3.2) can be rewritten as:

The Eq. (3.5) can be rewritten as:

[De (σ, ω) + kc Ne (σ, ω)] + j[Do (σ, ω) + kc No (σ, ω)] = 0

[De (ξ, ωn ) + kc Ne (ξ, ωn )] + j[Do (ξ, ωn ) + kc No (ξ, ωn )] = 0

p(σ, ω, kc ) + jq(σ, ω, kc ) = 0

p(ξ, ωn , kc ) + jq(ξ, ωn , kc ) = 0

where p(σ, ω, kc ) = De (σ, ω) + kc Ne (σ, ω); q(σ, ω, kc ) = Do (σ, ω) + kc No (σ, ω) Our objective is to analytically determine kc , if any, for which δ(σ,ω,kc ) is Hurwitz. First, we have to determine the frequencies ω at which q(σ,ω,kc ) = 0. Here, both p(σ,ω,kc ) and q(σ,ω,kc ) depend on kc makes this quite a formidable problem to solve, even, when p(σ,ω,kc ) and q(σ,ω,kc ) are of low order. This difficulty can be overcome by recasting the problem with p(σ,ω,kc ) or q(σ,ω,kc ) independent of kc . This can be done as follows: δ(σ, ω, kc ) × [Ne (σ, ω) − jNo (σ, ω)] = 0

δ(ξ, ωn , kc ) × [Ne (ξ, ωn ) − jNo (ξ, ωn )] = 0

[(De (σ, ω) + jDo (σ, ω)) + kc (Ne (σ, ω) + jNo (σ, ω))] ×[Ne (σ, ω) − jNo (σ, ω)] = 0

[(De (ξ, ωn ) + jDo (ξ, ωn )) + kc (Ne (ξ, ωn ) + jNo (ξ, ωn ))] ×[Ne (ξ, ωn ) − jNo (ξ, ωn )] = 0

[De (σ, ω)Ne (σ, ω) + kc Ne2 (σ, ω) + Do (σ, ω)No (σ, ω) + kc No2 (σ, ω)] + j[Do (σ, ω)Ne (σ, ω) − De (σ, ω)No (σ, ω)]=0 p∗ (σ, ω, kc ) + jq∗ (σ, ω) = 0

where p(ξ, ωn , kc ) = De (ξ, ωn ) + kc Ne (ξ, ωn ); q(ξ, ωn , kc ) = Do (ξ, ωn ) + kc No (ξ, ωn ).Our objective is to analytically determine kc , if any, for which δ(ξ,ωn ,kc ) is Hurwitz. First, we have to determine the frequencies ω at which q(ξ,ωn ,kc ) = 0. Here, both p(ξ,ωn ,kc ) and q(ξ,ωn ,kc ) depend on kc which makes this quite a formidable problem to solve, even, when p(ξ,ωn ,kc ) and q(ξ,ωn ,kc ) are of low order. This difficulty can be overcome by recasting the problem with p(ξ,ωn ,kc ) or q(ξ,ωn ,kc ) independent of kc . This can be done as follows:

(3.3)

[De (ξ, ωn )Ne (ξ, ωn ) + kc Ne2 (ξ, ωn ) + Do (ξ, ωn )No (ξ, ωn ) + kc No2 (ξ, ωn )] + j[Do (ξ, ωn )Ne (ξ, ωn ) − De (ξ, ωn )No (ξ, ωn )] = 0

where p∗ (σ, ω, kc ) = De (σ, ω)Ne (σ, ω) + kc Ne2 (σ, ω) + Do (σ, ω)No (σ, ω) + kc No2 (σ, ω); ∗

q (σ, ω) = [Do (σ, ω)Ne (σ, ω) − De (σ, ω)No (σ, ω)] It is clear that is independent of kc . Let ω0 , ω1 , ω2 , . . . denote the non-negative real zeros of q* (σ,ω). Using interlacing theorem, we can determine the range of kc values for the abovementioned frequencies. q* (σ,ω)

p∗ (ξ, ωn , kc ) + jq∗ (ξ, ωn ) = 0

(3.6)

p∗ (ξ, ωn , kc ) = De (ξ, ωn )Ne (ξ, ωn ) + kc Ne2 (ξ, ωn ) + Do (ξ, ωn )No (ξ, ωn ) + kc No2 (ξ, ωn , ); q∗ (ξ, ωn ) = [Do (ξ, ωn )Ne (ξ, ωn ) − De (ξ, ωn )No (ξ, ωn )] It is clear that q* (σ,ω) is independent of kc . Let ω0 , ω1 , ω2 , . . . denote the non-negative real zeros of q* (σ,ω). Using interlacing

3.2. Relative stability based on damping coefficient Using Eq. (2.6), the characteristic equation of the closed loop system can be rewritten as,   Ne (ξ, ωn ) + jNo (ξ, ωn ) 1 + kc =0 (3.4) De (ξ, ωn ) + jDo (ξ, ωn ) Ne (ξ, ωn ) = a0 ωn0 T0 (ξ)−a1 ωn T1 (ξ) + a2 ωn2 T2 (ξ) − a3 ωn3 T3 (ξ),  No (ξ, ωn ) = 1 − ξ 2 (−a0 ωn0 U0 (ξ)+a1 ωn1 U1 (ξ)−a2 ωn2 U2 (ξ)), De (ξ, ωn ) = b0 ωn0 T0 (ξ)−b1 ωn T1 (ξ) + b2 ωn2 T2 (ξ)−b3 ωn3 T3 (ξ),  Do (ξ, ωn ) = 1 − ξ 2 (−b0 ωn0 U0 (ξ)+b1 ωn1 U1 (ξ)−b2 ωn2 U2 (ξ))

Fig. 1. The region S with a specified damping ratio and damped natural frequency.

S. Narayanan et al. / Computers and Chemical Engineering 30 (2006) 1072–1075

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Fig. 2. (a) The −α shifted Hurwitz stability region S−α , (b) the ␸ rotated Hurwitz stability region Sϕ and (c) the−␸ rotated Hurwitz stability region S−ϕ .

theorem, we can determine the range of kc values for the abovementioned frequencies.

To illustrate the benefits of our proposal, consider the open loop transfer function given as: G(s) = ((s2 + 2s − 2)/ (s3 + 3s2 + 4s)) and we have to determine the values of K for which all the roots of the corresponding characteristic equation have negative real parts σ ≤ −0.5 and damping coefficient ξ ≥ 0.5 Solution : −b0 = 0; b1 = 4; b2 = 3; b3 = 1; a0 = −2; a1 = 2; a2 = 1 Step 1. First, we have to determine the values of K for which all the roots of the corresponding characteristic equation have negative real parts σ ≤ −0.5. From Eq. (3.1), we obtain: No (−0.5, ω) = ω;

De (−0.5, ω) = −1.375 − 1.5ω2 ;

Do (−0.5, ω) = 1.75ω−ω3

p∗ (σ, ω, kc ) = (3.78125 + 7.25ω2 + 0.5ω4 ) + kc (7.5625 + 6.5ω + ω ); 2

4

q (σ, ω) = ω + 2.5ω3 − 3.4375ω 5

ω = 0 and 0.9930 are the real, non-negative zeros of q* (σ,ω). The values of kc are (−0.7639, −0.5). For this range of kc only, the roots of p* (σ,ω,kc ) and q* (σ,ω) are real and distinct as well as they satisfy the interlacing property. Step 2. We have to determine the values of K for which all the roots of the corresponding characteristic equation have damping coefficient ξ ≥ 0.5. From Eq. (3.4), we obtain: Ne (0.5, ωn ) = −2 − ωn − 0.5ωn2 ; No (0.5, ωn ) = (2ωn − ωn2 )0.866; De (0.5, ωn ) = −2ωn − 1.5ωn2 + ωn3 ; Do (0.5, ωn ) = (4ωn − 3ωn2 )0.866

ω = 0 and 1.12087 are the real, non-negative zeros of q* (σ,ω). The values of kc are (−0.725,0). For this range of kc only, the roots of p* (σ,ω,kc ) and q* (σ,ω) are real and distinct as well as they satisfy the interlacing property. We have to determine the constant gain such that closed loop poles will lie in the region S as shown in Fig. 1. The region S (Fig. 2) has been conveniently characterized by using a parallel shift and two rotations of the imaginary axis as shown in Fig. 2 and, then Datta has determined the set K−α consisting of those values of k, for which the closed loop system, is −α Hurwitz stable and the sets Kϕ and K−ϕ consisting of those values of k, for which the closed loop system is ϕ and −ϕ Hurwitz stable. The set K of all values of k for which the closed loop characteristic polynomial is S Hurwitz stable is given by, K = K−α ∩K−φ ∩Kφ . The set of K of all S Hurwitz stabilizing gain values for the G(s) is given by K = (−0.725, −0.5). 5. Conclusions

From Eq. (3.3), we obtain:

∗

q∗ (0.5, ωn ) = [Do (ξ, ωn )Ne (ξ, ωn ) − De (ξ, ωn )No (ξ, ωn )]; q∗ (0.5, ωn ) = 0.866ωn (−8 + 6ωn + 2ωn2 − 2ωn3 + ωn4 )

4. Simulation results

Ne (−0.5, ω) = −2.75 − ω2 ;

From Eq. (3.6), we obtain:

In this paper, we have obtained the constant gain values, which satisfies desired performance measure for linear time invariant (LTIV) system. Also, a procedure is presented to determine the set of stabilizing controller parameters for a given LTIV system References Datta, A., Ho, M. T., & Bhattacharyya, S. P. (2000). Structure and synthesis of PID controllers. London: Springer-Verlag. Ho, M. T., Datta, A., & Bhattacharyya, S. P. (1999). Generalization of the Hermite–Biehler theorem. Linear Algebra and its Applications, 302–303, 135–153. Siljak, D. D. (1966). A note on the generalized Nyquist criterion. IEEE Transactions on Automatic Control, 250–254. Stojic, M. R., & Siljak, D. D. (1965). Generalization of Hurwitz, Nyquist and Mikhailov stability criteria. IEEE Transactions Automatic Control, 250–254. Tan, N., Kaya, I., & Atherton, D. P. (2003). Computation of stabilizing PI and PID controllers. In IEEE Conference Proceedings (pp. 876–881).