A Constraint Propagation Algorithm for Determining the Stability Margin. The paper addresses the stability margin assessment for linear systems

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1 A Contraint Propagation Algorithm for Determining the Stability Margin of Linear Parameter Circuit and Sytem Lubomir Kolev and Simona Filipova-Petrakieva Abtract The paper addree the tability margin aement for linear ytem under interval parameter uncertaintie. The original robut tability problem i initially tranformed into an equivalent problem of etimating the eigenvalue range of matrice whoe element are non-linear function of independent interval parameter. A new algorithm for finding the exact value of tability margin (within error bound) i uggeted. It i baed on the ue of the inner and outer bound on the right end of the eigenvalue conidered in order to determine a a narrow initial uncertainty region a poible. Then the contraint propagation approach i applied. It conit of two tep. Firt, one weep of contraint propagation, relative to the interval component of the eigenvalue eigenvector pair, i carried out, keeping the parameter interval fixed. Next, a econd weep of contraint propagation, relative to the component of the interval parameter, i applied, keeping the reduced interval of the eigenvalue eigenvector pair fixed.

2 A numerical example, illutrating the applicability of the algorithm uggeted, i olved at the end.. INTRODUCTION It i well known that tability analyi of linear circuit and ytem under parameter uncertaintie can be formulated a the problem of etimating the range of the eigenvalue of interval matrice (ee e.g. [3] [9]) becaue the tability margin of the curcuit tudied i equal of the right end of the eigenvalue which i mot to the right from the imaginary axi of the complex plane. Let A be a real n x n matrix, A - an interval matrix containing A, and + A, A, A and R A the left end, the right end, the center and the radiu of A, repectively. We conider the following perturbed eigenvalue problem: _ + = λ = = + A A Ax x, A [ A, A ] A [ R, R ] A. () Ordinary letter will denote real quantitie while bold face letter will tand for their interval counterpart. Each matrix A Ai non-ingular. For implicity, the element a ij of interval matrix A can be conidered a independent interval, but in general, they are non-linear function of m parameter, which take their value within precribed interval, i.e. aij ( p) = aij ( p,..., p m ), i, j =,..., n. (2) p p, l =,..., m l l then the eigenvalue problem conidered tranform a follow:

3 A( p) x = λx, p p. (3) Each matrix A( p), p p i non-ingular. Some of the known method for aeing the tability margin in the above formulation are baed on the olution of the following ub-problem: (i) find an inner bound on the right end of the range for the eigenvalue conidered; (ii) find an outer bound on the right end of the range for the eigenvalue conidered a well a outer bound on the right end of the component of the aociated eigenvector; (iii) the final olution of the problem i then found by determining the exact right end of that eigenvalue range that i mot to the right. Each of thee ub-problem i olved for independent interval element of matrix A (problem ()) in [3], [4] and [6], a well a in the cae of dependent interval element of matrix A (problem (3)) [5]. Uing the above inner and outer bound it i uggeted a new algorithm for obtainding the exact tability margin a a narrow initial uncertainty region a poible. Thi algorithm i baed on an approach called contraint propagation. The preent paper dicue the problem of determination the correponding exact right end of the eigenvalue of matrice whoe element are nonlinear function of independent interval parameter. The problem tatement i defined in Section 2. A new contrain propagation algorithm for obtaining the exact value of the right end of the conidered interval i uggeted in Section 3. It i generalization of the contraint propagation approach

4 uggeted in [7]. The algorithm procedure include two tep. Firt, it i applied the contraint propagation technique relative to the interval component of the eigenvalue-eigenvector pair, keeping the parameter interval fixed. Second, the ame technique, relative to the component of the interval parameter, i carried out, keeping the reduced interval of the eigenvalue-eigenvector pair fixed. A numerical example, illutrating the implicity and applicability of the algorithm uggeted, i conidered in Section 4. The paper end with concluion remark in Section PROBLEM STATEMENT The olution of initial baic eigenvalue problem (3) will be found making the following tranformation. Let the ytem (3) i written for the central parameter vector p A( p ) x λx =. (4) We will etimate only the interval of the maximum eigenvalue of matrix A( p ). In general, the new algorithm uggeted later can be applied for any other real eigenvalue. Let ( ( p )), k = n * λ ( p ) = max λk,..., * i a maximum eigenvalue while x ( x x x ) T = i the correponding,,..., n 2 eigenvector. We make the following aumption (enuring tructural tability of the problem).

5 * p Aumption A: Let λ ( ) and x * ( p ), correponding to all p p, remain real. On account of Aumption A, the range * { λ( p) : p } λ = p (6) i a real interval. Without any lo of generality we need a econd aumption. If the pair ( x, λ ) i the olution of (4) then Aumption A2: We aume that the abolute value of the th component x of vector x i the larget component of the other component, i.e. i, x x i (7) Now x i normalized through x =. (8) Further, we require that (8) be alo valid for x ( p) =, p p. (8a) We introduce the n-dimenional real vector ( ) T y = y, y2,..., yn (9) with y = x ( p), i =,..., n, n y i i = λ ( p) () Uing (), the eigenvalue problem (3) i

6 n i= i n i= i a ( p) y y y + a ( p) =, i =,..., n, i ij j i i a ( p) y y + a ( p) = j j. () Then we obtain the exact value of the tability margin conidered applying the contraint propagation approach to the non-linear ytem (). (The eence of thi approach i decribed in [] and [7].) In the framework (), the contraint propagation technique conit in ucceive atifaction of ome contraint, given a equalitie, and reduction of the initial region (box) of uncertain interval parameter. In the context of the problem conidered, the contraint are given by the component of the interval eigenvalue problem under conideration. 3. A NEW CONSTRAIN-PROPAGATION ALGORITHM Thi algorithm find the exact value of the tability margin within error bound. It baed on the fact that tability margin i equal of the right end of the eigenvalue range that i mot to the right. Let the element of matrix A( p ) are non-linear function of independent interval parameter p, l,..., m l = following (2). We write them in the following linear form with repect to the element of vector p [2]: a ( p) = α p + b, p p (2) ij ij ij and ubtitude in ytem ()

7 b + α p y y y + b + α p =, i =,..., n, i n m m i= ij l= ijl l j i i l= il l i b + α p y y + b + α p = n m m i= j l= jl l j l= l l i. (3) Then we implify the non-linear ytem (3) with repect to the component of parameter vector p and get the following ytem m gil pl = fi, pl p l, i =,..., n, (4) l= where. g = α + α y, i =,..., n il il ijl j j= j n n f = b b y + y y, i =,..., n, i i i ij j i j= j f = b b y + y n j j j= j (4a) The ytem (4) i linear with repect to the component of the parameter vector p. The known method olved the non-linear ytem with interval coeffcian () with repect to the component of interval vector y [5, 7]. Now we will ugget the algorithm, baed on contraint propagation approach, to obtain the ame component of y uing the linear parameter ytem (4) yet. It conit of two iterative procedure which call each other recurively. It can be decribed with the following way. The main procedure include 3 tep.

8 Procedure Step: We tart with the initial uncertainty region interval vector (ee Step and 2 in [7]) y y = out = () y, (5) out where y i an interval vector whoe component are the outer bound of y( p), p p, calculated by the method uggeted in [5]. We ubtitute the th component of y with interval y = z = in + out + ( y ), ( y ), (6) in where ( y ) + out and ( ) + y are the right end on the inner and outer bound * on the exact range y, calculated by method uggeted in [5]. Step 2: We apply the contraint propagation procedure (Procedure 2 from [5]) to obtain the component of the interval vector y conit of the eigenvalue conidered and the component of the repective eigenvector, keeping the parameter interval fixed. The top criteria i two erial value of the interval component of vector y to be cloe enough. Step 3: We conider the component of vector y a an independent interval and apply the contrain propagation to the linear ytem (4) with repect to

9 the independent component of parameter vector p. Thu we calculate the new component of the interval vector p a follow: For i = to n do p ( i) q p = p m fi il l g p l= l q =, q =..., n g p iq ( i) q q q (7) End The top criteria i two erial value of the interval component of vector p to be cloe enough. Go to Step 2. In the firt iteration of the procedure we tart with initial vector () y, calculated by (5) and (6), but in the other iteration we ue the component of interval vector y obtained a a reult in the end of the Step 3. The Procedure i valid for all eigenvalue of interval matrix A( p), p p but to implify the preentation we decribed it only for the maximum one. The right end of the eigenvalue (the th component y ) of the interval vector y, which i mot to the right i equal to the tability margin of the analyzed ytem. 4. NUMERICAL EXAMPLE

10 The circuit tudied i hown in Fig. 4.. Auming that the vector of parameter i p = ( R, R2, R3, L, C) with = [ ] T Ri Ri, L L, C C, R, 97, 3 R = [ ], R = [ ], = [ , 5.] mh C 2 98, 22 = [ 238, 267 ] µf electric circuit ,. L,. Such ytem arie in tolerance analyi of linear AC + - i (t) e(t) R L i 2 (t) R 2 i 3 (t) R3 C Figure 4. We apply the Procedure to obtain the tability margin of the circuit analyzed. Firt, in Step we find the initial interval vector y according to (5) and (6) in the following way. It i een from the example that the expreion (2) are non-linear function of 5 parameter R + R2 L. k L. k A =, k = + C. k R3C. k R R 2 3 M = L. (8) The componenet of the initial centre and radii parameter vector are: ( 2 5* 25* ) T p =, (9a) -3-6 Start Start -3-6 ( 3 2..* 2* ) T Rp =. (9b)

11 The linearization of aij ( p ) i made by the method uggeted in [2] uing (9). So we get the a ( p ) a the affine function: ij [ ] a = p p.334 p p a = p.4446 p p [ ] a = p p 5.34 * 6 p * 3 + [ ] a =.446 p p * 6 p [ ] (2) The number of the maximum eigenvalue and the maximum component of it eigenvector are k = 2 and = 2, repectively. Hence, we are intereted in the econd component of interval vector y. According to (9) the vector y i: y x y = = 2 y 2 λ rep. (2) y x y = = = y λ. (22) Applying the method uggeted in [5] we get the following outer bound of the eigenvector conidered component [ ] [ ] out y =, (23) the right end of the inner bound on the eigenvalue conidered ( in ) ( y in ) ( in y ) λ = = 2 = (24) and the right end of the exact range exact + ( ) ( y exact + λ ) = = (25)

12 Then baed on (5) and (6) and uing (23) and (24) we get the initial interval vector [ ] [ ] () y =. (26) Second, we repeat the Step 2 and 3 from the Procedure until we obtain two erial value of the interval component of vector p to be cloe enough with the accurancy ε 9 =. The final componenet of the centre and radii parameter vector are reached on the 6th iteration and they are following: ( 2 5* 25* ) T p =, (27a) -3-6 Final Rp = Final -3-6 ( * * ) T Thu the right end of the th component of the vector y i. (27b) ( y ) + = (28) But out out ( y ) ( y ) ( λ ) then = = (29) out out ( y ) + ( λ ) = = (3) It i obviou from (25) and (3) that the right end of outer bound and of the exact range conidered, calculated by the new algorithm and the method from [5], are equal. Therefore, the new method uggeted determine the right end of the tability margin of the circuit tudied. The calculation time for thi example are hown in Table.

13 Table Step in Procedure (method from [5]) Whole the Procedure CONCLUSION The new algorithm uggeted obtain the exact value of the tability margin within error bound. Some of the known method for aeing it are baed on determining the exact range and the repective inner and outer bound of the conidered eigenvalue-eigenvector pair olving the non-linear ytem () with repect to the component of the interval vector y. Unlike thi approach, a new contraint propagation algorithm which applie to the linear parameter ytem (4) i propoed with repect to the component of the parameter vector p. The algorithm uggeted obtain directly the right end of the exact range of the eigenvalue analyzed and thu determine the exact value of the tability margin. 6. REFERENCES [] Jaulin L., M. Kieffer, O. Didrit and E. Walter: Applied Interval Analyi with Example in Parameter and State Etimation. Robut Control and Robotic, Springer-Verlag, 2. [2] Kolev L.: Automatic Computation of a Linear Interval Encloure, Reliable Computing 7, 2, pp

14 [3] Kolev L., S. Filipova-Petrakieva, N. Vrabchev: Outer bound on the real eigenvalue of interval matrice. Proceeding of XI International Sympoium on Theoretical electrical engineering ISTET', Linz, Autria, 9-22 Augut 2, pp [4] Kolev L., S. Filipova-Petrakieva: Inner bound and exact olution for the range of the eigenvalue of interval matrice. Technical idea, BAS, vol. 39, No. -2, 22, pp [5] Kolev L., S. Filipova-Petrakieva: Stability analyi of linear interval parametric ytem via aeing the eigenvalue range. Proceeding of XII International Sympoium on Theoretical electrical engineering ISTET'3, July 6-9, 23, Waraw, Poland, pp [6] Kolev L., S. Petrakieva: Aeing the tability of linear time-invariant interval dynamic ytem. IEEE Tran. on Automatic Control, vol. 5, Iue 3, March 25, pp [7] Kolev L., S. Petrakieva: Two-ided bound on the tability margin of linear interval parameter. Proceeding of XIII International Sympoium on Theoretical electrical engineering ISTET'5, July 4-7, 25, Lviv, Ukraine, pp [8] Juang Y., Ch. Shao: Stability analyi of dynamic interval ytem, International Journal of Control, vol. 49, No. 4, 989, pp [9] Rachid A.: Robutne of dicrete ytem under tructured uncertaintie. International Journal of Control, vol. 5, No. 4, 989, pp

15 [] Wang S., W. Lin: A new approach to the tability analyi of interval ytem. Control - Theory and Advanced Technology, vol. 7, No. 2, June 99, pp Prof. DSc Lubomir Kolev, Ait. Prof. Phd Simona Filipova-Petrakieva Department of Theoretical Electrical Engineering Faculty of Automatica Technical Univerity of Sofia 8, Kliment Ohridki Str., Sofia, Bulgaria lkolev@tu-ofia.bg, petrakieva-te@tu-ofia.bg