MSU LIBRARIES —.‘—_ BETURNINQWMAIERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. FREQUENCY DOMAIN METHODS FOR SYSTEMS WITH SLOW AND FAST DYNAMICS By Douglas William Luse A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree ‘of DOCTOR OF PHILOSOPHY Department of Electrical Engineering and System Science 1983 ABSTRACT FREQUENCY DOMAIN METHODS FOR SYSTEMS WITH SLOW AND FAST DYNAMICS By Douglas William Luse The mathematical treatment of systems with slow and fast dynamics has traditionally involved singular perturbation theory for differential equationsl This thesis suggests a set of conditions to be placed on a frequency domain description of a system to guarantee two time scale behavior. Some basic stability and approxima- tion results are presented. To my parents, Herman and Catherine Luse ii ACKNOWLEDGEMENTS I wish to thank Dr. Hassan Khalil for his patience and guidance; and my committee members, Dr. R. Schlueter, Dr. R. 0. Barr and Dr. J. C. Kurtz, for their help and valuable suggestions. iii TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . . . . . . . vii LIST OF FIGURES . . . . . . . . . . . . . . . . . . vii I. INTRODUCTION . . . . . . . . . . . . . . . . . . 1 II. SYSTEM MATRIX THEORY FOR AN ARBITRARY FIELD . . 8 Definition 2.0.1 . . . . . . . . . . . . . 9 II.1. Basic Theorems from System Matrix Theory . . . . . . . . . . . . . . . . . 10 Definition 2.1.1 10 Definition 2.1.2 11 Definition 2.1.3 11 Definition 2.1.4 11 Definition 2.1.5 . . . . . . . . . . . . . 11 Theorem 2.2.1 . . . . . . . . . . . . . . . 12 Definition 2.1.6 . . . . . . . . . . . . . 13 Definition 2.1.7 . . . . . . . . . . . . . 13 Theorem 2.1.2 . . . . . . . . . . . . . . . 13 Theorem 2.1.3 . . . . . . . . . . . . . . . 14 Theorem 2.1.4 . . . . . . . . . . . . . . . 15 Theorem 2.1 5 . . . . . . . . . . . . . . . l6 II.2. Appendix: Extensions of Rosenbrock's Results . . . . . . . . . . . . . . . . . 16 Theorem R2.4.l . . . . . . . . . . . . . . 17 Theorem R2.5.l . . . . . . . . . . . . . . 19 Theorem R2.6.1 . . . . . . . . . . . . . . 22 Lemma 2.2.1 . . . . . . . . . . . . . . . . 25 Lemma 2.2.2 . . . . . . . . . . . . . . . . 27 Theorem R2.6.2 . . . . . . . . . . . . . . 27 Lemma 2.2.3 . . . . . . . . . . . . . . . . 30 III. TWO FREQUENCY SCALE RATIONAL MATRICES . . . . . 33 Theorem 3.0.1 . . . . . . . . . . . . . . . 33 Definition 3.0.1 . . . . . . . . . . . . . 35 111.1. Algebraic Form of a Two Frequency Scale Rational Matrix . . . . . . . . . 38 IV. VI. Lemma 3.1.1 . Theorem 3.1.1 . Corollary 3.1.1 III.2. Exact Frequency Scale Decomposition Lemma 3. .1 . Lemma 2 . Lemma .3 . 4 Lemma wwww NNNNN .5 . heorem 3.2.1 . Lemma III.3. System Matrices for Two Frequency Scale Transfer Matrices Lemma 3.3.1 . Theorem 3.3.1 . Definition 3.3.1 APPROXIMATION OF TWO FREQUENCY SCALE RATIONAL MATRICES . . . . . . Lemma 4.1.1 . Lemma 4.1.2 . Theorem 4.1.1 . IV.2. Appendix--Bounds on Rational Functions Theorem 4.2.1 . Theorem 4.2.2 . Theorem 4.2.3 . Theorem 4.2.4 . Corollary 4.2.1 . Corollary 4.2.2 . CLOSED LOOP SYSTEMS Theorem 5.1 . Theorem 5.2 . Corollary 5.1 . Corollary 5.2 . APPLICATIONS . VI.1. Steady State LQG Controller for a Singularly Perturbed System . VI.2. Feedback Design Strategies 38 41 45 45 45 49 50 50 51 55 56 57 59 6O 68 69 71 72 73 73 74 74 75 76 78 81 83 85 85 88 88 93 Design Strategy #1 . . . . . . . . . . . . . . 94 Design Strategy #2 . . . . . . . . . . . . . . 96 V1.3. Numerical Examples . . . . . . . . . . . . . 98 VII. CONCLUSION . . . . . . . . . . . . . . . . . . . .107 Table 6.3.1 . Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure C‘O‘O‘O‘O‘O‘O‘O‘GU‘UIUINN NNNNNNHHHWNI—‘NH O‘kflbWNf-‘WNH LIST OF TABLES LIST OF FIGURES vii I. INTRODUCTION The automatic control literature contains a wide° variety of analysis and design methods for linear multi- variable systems. Most of these, however, suffer from the so-called "curse of dimensionality." That is, the amount of computation required increases dramatically with the dimension of the system under consideration. This situa‘ tion creates a need for efficient model reduction and decomposition methods. Some model simplification schemes [e.g., l3] assume no particular structural properties for the system being treated. Others rely on natural subsystem decomposition caused by spatial [e.g., 14] or temporal separation [l]. The subject of this thesis is the latter. There is an extensive literature on time scale separa- tion through the use of singular perturbation methods for differential equations. This thesis considers the same problem from a frequency domain approach. A discussion of some basic time domain results for linear systems is included here for later comparison. An autonomous linear singularly perturbed system is shown in (1.1). The blocks A11’ A12, A21, and A22 are ana- lytic at e = 0. It can be shown [2] that the state matrix A(e) defined in (1.2) can be brought to block diagonal form (1.3) by a similarity transformation T(e) where the following statements hold: 1. T, As’ and Af are analytic at e = 0. -1 A 2. “AS(O) = All(0) - A12(O) A22 (0) A21(O) = A 3. Af(0) = A22(0) 4. T(0) = [I 1 0] A22 A21 I x1 = All(e) x1 + A12(e) x2 (1.1) 6X2 = A21(e) x1 + A22(e) x2, det A22(O) # O Me) A land) A12t -'¢(t,s)H = 0(8) as e-*O (1 6) :20 . II‘H is any matrix norm. The decomposition (1.4) and approximation (1.5) have been generalized for certain e-dependent systems which do not have the form (1.1) [ 6]. There are methods of computing higher order approximations of the matrices T, AS, and Af [ 8]. In [S9], multiple (more than two) time scales are allowed and A(e) is generalized to be a linear mapping on a Banach space. The approximate time scale decomposition above has a number of other applications. As a representative example, its application to the pole assignment problem with state feedback is included here. The system (1.7) is split into slow and fast subsystems (1.8) and (1.9) respectively. The original problem is stated: Assign, by state feedback, the poles A , A , . . ., A , A , A , . . ., A . It 1 2 nl nl+l nl+2 nl+n2 E E E is required that An +1, . . ., An +n be non-zero. The slow 1 l 2 subproblem is: assign the poles Al, . . ., Anl to the system (1.8) by state feedback Gs' Similarly, the fast sub- problem is: assign the poles An +1, . . ., An +n to the l l 2 system (1.9) with state feedback Gf. x1 = All x1 + A12 x2 + Bl u (1.7) 5x2 = A21 x1 + A22 x2 + B2 u, det A22 # 0. All is n1 x n1, A22 is n2 x n2 x8 = A0 XS + Bo uS (1.8) xf = A22 xf + B2 uf (1.9) _ -l _ -1 Bo ‘ Bl ‘ A12 A22 32 Assuming that the slow and fast subproblems have been solved, their solutions are combined in (1.10) and (1.11) to form the composite control law (1.12) for the system (1.7). _ -l -1 G1 - (II+ of A22 32) GS + of A22 A21 (1.10) G2 = Gf (1.11) u = Cl x1 + C2 x2 (1.12) If the feedback (1.12) is applied to (1.7), then the follow— ing result holds: The closed loop poles can be written (as functions ofE). v (e) v , (e) nl+l n1+n2 e , O O O ’ 8 Vl(€), . - -, Vn(€). wherelvj - Ajl + 0 ass + 0 for 15.j5 n1-+ n2 At this time, there are very few applications of fre- quency domain methods for slow-fast subsystem decomposition in the literature. In CID, the "asymptotic forms" (1.14) for the transfer matrix of the system (1.13) are derived. However, no precise theoretical meaning is given to the terminology "asymptotic forms." x 3 A1 A2 x + B1 u (1.13) z A3/e A4/e z 32/8 , y = C1 x + C2 2 where Al is n.x n, A4 is m x m Gl(s) = CO(sIn - AO)‘ BO - c2 A4 32 (1.14) Gh(s) = C2(esIn - A4)-1 32 where A0 = A1 - A2 A4-1 A3 B0 = Bl ' A2 A4_1 B2 C0 = cl - c2 A4’1 A3 The purpose of this thesis is to show frequency domain analogs for the fundamental time domain results. One of the achievements of this work is to give theoretical meaning to the frequency scale decomposition (1.14), in much the same way in which time domain results give meaning to the approx- imation (1.5). Throughout this work, however, the basic system description used is the transfer matrix, with internal descriptions used only for proofs and examples. There are a number of reasons for investigating multiple time scale systems from a frequency domain viewpoint. Fre- quency domain methods have been revived in recent years with generalizations of classical methods to multivariable systems. This work should open the way for investigation of regularities which may occur in the generalized nyquist and root locus plots of multiple time scale systems. It may lead to convenient application of multivariable robustness [e.g.,15] and sensitivity [e.g.,15] methods, which could lead to design schemes which are fundamentally different from time domain methods. Through this approach, multiple time scale methods may be extended to systems described by convolution operators, such as those involving time delays. For theoretical pur- poses, the approach is useful for clarifying some time domain results. This is because no particular internal structure for systems is assumed--only input-output rela- tionships. In summary, this thesis should be regarded as the theoretical basis for future application of frequency response methods to multiple time scale systems. II. SYSTEM MATRIX THEORY FOR AN ARBITRARY FIELD Rosenbrock [3] develops the theory of time invariant linear system through the use of system matrices. Two classes of system matrices are considered: rational and polynomial. Rational system matrices have elements which are rational over some field and may be regarded as a generalization of the concept of transfer matrix. The class of polynomial system matrices, that is, the class of system matrices with polynomial elements only, includes both the class of state space descriptions and the class of matrix fraction descriptions as special cases. Poly- nomial system matrices are useful because they provide very general internal descriptions of linear systems. Operations which preserve the relation of strict system equivalence are analogous to similarity and unimodular transformations for state space and matrix fraction descrip- tions respectively. Strict system equivalence preserves all external descriptions of a system as well as maintain- ing the values of all (possibly internal) poles. In [3], the underlying number field is the field of complex numbers. An extension is needed, for the present work, to allow the parameter c to appear in coefficients of polynomials and rational fractions of the frequency vari- ables. Whatever manner a is allowed to appear, the field 8 properties of the coefficients must be preserved; otherwise most of the theory of system matrices would be lost. Also, all concern will be with small values of the parameter a. With these observations in mind, we make the following definition: Definition 2.0.1: f(e) is analytic at e = 0; r is an integer This variation of coefficients with e is general enough for most physical purposes. A smaller field, such as the field of functions rational in.e could have been chosen, but a: is needed for a polynomial factorization later on. As mentioned before, most of the theory of system matrices carries over when the field 52 is used. All pro- cesses involving only field Operations, such as block Gaussian elimination, are performed in an identical manner. The important Euclid's algorithm for computation of the greatest common factor of two polynomials is still avail- able. There is one major difference, however: there exist nonlinear prime polynomials. Stated differently, there are polynomials over 9; whose roots are not in 5'8. An example is $2 - e = 0. The roots are s = :vE'which are not analytic at e = 0. Note that as a consequence, the Jordan form of a matrix cannot be generated without leaving the underlying field. 10 A number of Rosenbrock's proofs assume that any polynomial can be factorized into linear factors, but all essential results can be derived without using this property. The remainder of this chapter consists of two parts. First, a statement of the key results from [3] which are needed for this work. Second, the modifications of proofs necessary to alleviate the difficulty discussed above. II.l. Basic Theorems from System Matrix Theory This section contains results from system matrix theory which are needed later. All theorems in this section assume that the underlying field is arbitrary, and the field will be denoted by F. The development roughly follows [3]. A reader familiar with the system matrix approach can follow this seCtion if he assumes that the mechanics of "extraction of decoupling zeros" has been extended to system matrices over a field F. Definition 2.1.1: A rational matrix P(s) (over a field F) is a system matrix with m outputs and z inputs if T(s) U(s) P(s) = (2.1.1) —V(s) W(s) where U is r x 2, V is m xfr, and det T(s) # 0. The associated transfer matrix H(s) is given by H= V'r'lu+w (2.1.2) Also, P is called a system matrix representation of H. Relations between the matrices in (2.1.1) will be indicated 11 by subscripts. For instance, Pl will consist of the blocks T1, U1, V1 and W1. Definition 2.1.2: P(s) is a polynomial system matrix of order n if: 1. P(s) is a system matrix with only polynomial elements; 2. deg det T(s) = n; 3. r 2 max (n,£,m). Definition 2.1.3: A polynomial system matrix P(s) of order n with associated transfer matrix H(s) has least order if every other polynomial system matrix representation of H has order n or greater. Definition 2.1.4: Polynomial system matrices P1 and P2 are strictly system equivalent if there exist polynomial matrices X and Y, and unimodular matrices M and N such that M o N Y P = - P - (2.1.3) 2 x I l o I Definition 2.1.5: System matrices P1 and P2 are system equivalent if P2 can be obtained from P1 by one or more of the following two types of transformations: 1. Transformations of the form of (2.1.3) with X,Y, M,N all rational and M,N non-singular; 2. Trivial expansions and contractions as shown in (2.1.4). I 0 E’<——.-[ ] (2.1.4) It can be shown that two equivalent system matrices have the same associated transfer matrix. Theorem 2.2.1: Let H(s) be a rational matrix over a field P. Then there exist polynomial matrices T(s) and V(s) over F with T(s) non-singular such that H(s) has a least order polynomial system matrix representation of the form (2.1.5). (The form of system matrix (2 1.5) will be referred to as Matrix Fraction Description or MFD form). PI 0 : 01 o T(s) j I (2.1.5) Lo -V(s) E o Proof: Let d(s) be a least common denominator for the ele- ments of H(s) so that H(s) = N(s)/d(s) with N(s) polynomial. Then (2.1.6) is a system matrix representation for H(s). d(s)I I (2.1.6) -N(s) 0 All non-unimodular common right divisors of d(s)I and N(s) can be extracted using system equivalence operations followed by an expansion to meet condition 3 of Definition (2.1.2). (2.1.5) is least order by an extension of Theorem 3.2 of [3]. D 13 Definition 2.1.6: The characteristic polynomial of a rational matrix H(s) over a field F is the least common multiple of the denominators of all non-zero minors of all orders of H(s). The characteristic polynomial is assumed to be nor- malized in the case F = J: so that e can be non-trivially 861': to zero. Definition 2.1.7: Let P(s) be a polynomial system matrix over a field F. Then the input decoupling polynomial of P is the product of the diagonal elements of the Smith form of [T(s) U(s)]. The output decoupling polynomial of P is the product of the diagonal elements of the Smith form of [T(s)T -V(s)T]T. Let P1(s) be a system matrix obtained by extracting the input decoupling polynomial from P(s), i.e., by extracting all non-unimodular common left divisions from [T(s) U(s)]in.P(s). Then the input-output decoupling poly— nomial of P(s) is the quotient y(s)/yl(s) where y(s) is the output decoupling polynomial of P(s) and yl(s) is the output decoupling polynomial of P1(s). Figure 2.1.1 —:’O H1 I .7 .+ ‘L " H2 d‘—‘ V Theorem 2.1.2: Let H(s) be rational over a field F. Let P be a least order polynomial system matrix realization of l4 H(s). Denote the blocks of P by T,U,V, and W. Then CP[H(s)] = f - det T (2.1.7) where feF Proof: Follows almost exactly as in [3]. 0 Theorem 2.1.3: Suppose two systems with transfer matrices H1 and H2 over a field F are connected in series, and put in the unity feedback configuration of Figure 2.1.1. Let H1 and H2 be described by system matrices P1 and P2, respec- tively. Then the closed 100p system matrix can be repre- sented by the system matrix PCL as shown in (2.1.8) if det(I + HlHZ) # O. P ' - T1 U1 0 0 o o i o 0 0 0 T2 U2 0 0 ; 0 o I -v w 0 0 —I o ' o o 1 1 i PCL = o 0 -v2 w2 0 -I : o 0 (2.1.8) .0 I 0 o o I 5-1 0 I 0 0 o I -I o : o -I ............................ l------- -v w o o o o ' 0 o 1 1 : 0 -v2 W2 0 o i 0 0_ Furthermore det TCL = :det Tl det T2 det(I + H1H2)' (The sign depends upon the sizes of the blocks.) Proof: Follows from a trivial extension of the derivation in Chapter 5, Section 1, of [3]. D 15 Theorem 2.1 4: Let H1, H2, P1’ P2, etc. be as in Theorem 2.1.3. Suppose, furthermore, that PI and P2 are polynomial system matrices. Let B, y, and 6 be generic input decoupl- ing polynomial, output decoupling polynomial, and input- output decoupling polynomial, respectively. They will be subscripted according to the system matrices to which they refer (1,2,CL). Then BCL = 81 B2 YCL = Y1 Y2 6CL = 51 52 Proof: It is evident from (2.1.8) that BCL = 81 82. (2.1.8) can be transformed by operations of strict system equivalence to (2.1.9). T1 0 U1 0 0 o E 0 0 -v1 0 W1 0 -I o E 0 o . 0 T2 0 U2 0 o E o o 0 -v2 0 W2 0 -I 5 o o o o I o o I E -I 0 (2.1.9) o o o I -I o E o -I . ................................. T"""“' o o o o I o ; o o _ o o o o o I E o o‘ It can now be seen that YCL = Y1 Y2' To show that 6 first extract the input decoupling polynomial CL = 51 52' from (2.1.8) and perform the same operations of strict systan equivalence that were used to arrive at (2.1.9). This new 16 system matrix is identical to (2.1.9) except that the pairs (TlUl) and (T2U2) are replaced by reduced versions. D The next theorem involves a commonly used regularity condition which guarantees that both Open loop and closed loop systems have the same order. Theorem 2.1.5: Suppose two systems with proper transfer matrices H1 and H2 over a field F are connected as in Figure 2.1.1. If det(I + Hl(m) H2(w)) ¢ 0, then the matrix (2.1.8) is a system matrix. Furthermore, the number of closed loop poles of Figure 2.1.1 is equal to the sum of the number of poles of the two open loop systems. Proof: Clearly, det(I + Hl(s)H2(s)) i 0 if det (I + Hl(m) H2(m))# 0. Hence, (2.1 8) is a system matrix. Referring to (2.1.8), :det T CL = det I + H H 2.1.10 det T1 det T2 ( l 2) ( ) If s is allowed to go to w, it is apparent that det TCL and det Tl ° det T2 must have the same degree. D 11.2. Appendix: Extensions of Rosenbrock's Results This section is not self-contained: it assumes the reader is familiar with the reference [3] and the theory of polynomial matrices [21]. In this section, an R preceeding a theorem number means that it corresponds to that theorem in [3]. 17 The concept of ”extracting a decoupling zero" no longer makes sense, since this zero may not be in the field F. The next theorem has been reworded and reproved to reflect this. Theorem R2.4.l: Let P be a polynomial system matrix over F. Let the blocks of P be labeled: T U p = (2.2.1) -V W T If either [T u] or [IT -vT] is not Smith equivalent to [I 0] then there is a polynomial system matrix P1 of lower order (i.e., deg det Tl < deg det T) giving rise to the same transfer matrix. Proof: Suppose [T U] ” [S 0] where [S 0] is in Smith form .with S # I. Then there exist unimodular R and Q such that R [T U] Q = [s 0] or, R [T U] [s 0] Q’1 (2.2.2) The rows of (2.2.2) are divisible by the corresponding diagonal elements of S. The system matrix (2 2.1) can be transformed by strict system equivalence: -l R o T u s 0 Q o I -v w —VQW o I This matrix has the first r rows divisible by the diagonal elements of S. S has no zero diagonal elements because T is nonsingular. Thus 8'1 exists. Define Pl as s‘1 0 R o T U p A 1 o I o I -v w 18 Then P1 is a polynomial system matrix with deg det T1 = deg det T - deg det S < deg det T. As mentioned in the last section, the concept of set of decoupling zeros" must be replaced by ”decoupling poly— nomial.” We now introduce (and repeat some of) the follow- ing terminology for a polynomial system matrix P with associated transfer matrix C. a = pole polynomial of G B = input decoupling polynomial Y = output decoupling polynomial 6 = input-output decoupling polynomial n = pole polynomial of P The definitions of the above are: B Q det S where [T U] ‘ [S 0] where [S 0] is in Smith form. "D Y det S where [TT - VT] ” [S 0] where.[S 0] is in Smith form. 9 A the output decoupling polynomial of a system matrix obtained from P by "removing" the input decoupling polynomial as in Theorem R2.4.l. 6 A g— (shown later to be a polynomial) y g %$ (shown later to be a polynomial) g det T. 19 Theorem R2.5.1: Let a,8,y,5,n, and o be defined as above for a system matrix P over F. Then the polynomial division relations hold: 0 l v, 6 I B, and BY I n5. Also, 8 | n and y | n. Proof: Let Pl be the system matrix resulting from removal of B from the input of P. If [S 0] is the Smith form of [T U] then det S = the greatest common factor of all r x r minors of [T U]. Then det S l det T or B I n By similar reasoning, y I n. Let R and Q be unimodular matrices which transfer [T U] to its Smith form: R [T U] Q = [S 0] Then P can be transformed through strict system equivalence: R 0 T U [S 0] Q-1 0 I -V W [-V W] “D F (2.2.3) Thhstransformation preserves the Smith form of the output pair: 20 We can left multiply (2.2.3) by the matrix (2.2.4). The result is still a polynomial system matrix and the transfer matrix is unchanged. However, the Smith form of the output pair may change. S 0 (2.2.4) Consider the effect of left multiplying P in (2.2.3) by (2.2.4) on an r x r minor of [TT, -VT]T. The minors will be divided by the (possibly non-unity) diagonal ele- ments of S which correspond to the positions of their rows from T. Pl can be written explicitly: 3'11 s'lu p = 1 -V w A T T Thus, if M is an r x r minor of [T , -V ] and N is the cor- responding r x r minor of [T1T, -VT], then N I M. A T r x r minors of : PH’ M2, ..., Mp y = GCF Mi -V ].Si.$p s‘1 I r x r minors of N1, N2, ., N -V P e = GCF Ni (2.2.5) 15 isjp Since Nil Mi for l S i.s p, then the minors of . T T [T , -V ] can be written glfil, 52N2’ ..., gpr; and y = GCF giNl (2.2 o) 21 where each gi is a polynomial. It is now clear that o IY- Since 8 = det S, gi I B for l S i S p. Stated other- wise, B is a common multiple of the gi's. We then have LCM gi I B M 1 i.Sp since the least common multiple divides all other common multiples. Let h be a prime factor of 6 of multiplicity K. R+1I 5. That is, hKI 6 but h Viewing the list (2.2.6) as a modification of the list (2.2.5), it is seen that hKI gj for some j. The same argument can be repeated for each prime factor of 6 to show that 6 I LCM gi 1515p Thus, 6 I B. T T T Since det T1 is an r x r minor of [T1 -V ] i A O I det T . Also, det T det S = det T. Therefore, 1 1 ° 0 - B I (det T1) - B o - B | det T1 - det S 9 ° 8 I det T But det T a det T and n = det T. This gives 0 ° 8 I n substituting o = 7/6 yields YB I n or VB I n6. D 7? 22 Three standard conditions for two polynomial matrices to be c0prime are generalized in the next theorem. Part (i) of this theorem in [3] is no longer applicable. That is, quantities may be involved which are not in 3:. Theorem R2.6 1: Let T and U be polynomial matrices over a field F where [T U] has normal rank r. Then each of the following conditions are equivalent to T and U being left c0prime. (ii) [T U] is Smith equivalent to [I 0]. (iii) There exist polynomial matrices V and W respectively. 2 x r and 2 x 2 such that T U -V W (iv) There exist right c0prime X and Y such that TX + UY = I. (ii) (+) Suppose T and U are left c0prime. Suppose, to the contrary, that [T U] ~ [S 0] where [S 0] is in Smith form with det S # 1. Then there exist unimodular R and Q such that I [T U] =R[S 0]Q = RS [I 0] Q RS is a left. divisor of T and U, but det RS = det R det S which depends on s. This is a contradiction. Therefore, [T U] ~ [I 0]. (iii) (iV) 23 (+) Suppose that [T U] ” [I 0]. Let R be any common left divisor of T and U. Then there exist polynomial matrices T1 and U1 such that [T U] = R [Tl U1]. Let M be an r x r minor of [T U], and let N be the corresponding r x r minor of [T1 U1]. Then M = (detR)-N follows from the Cauchy-Binet formula. Thus, det R divides every r x r minor of [T U]. Det R must divide the greatest common factor of all r x r minors of [T U]. But the latter is equal to unity, because of.a standard theorem on the Smith form. Since det R Il, det R is independent of s and thus R is unimodular. Coprimeness of T and U follows because R was an arbitrary common left. divisor of T and U. It is also clear that the normal rank of [T U] is r. The proof in [3] holds as written. (+) Suppose T and U are left c0prime. Then there exist M and N such that M [I 0] . N [I 0] [h. d] - N o I [I 0] Q'1 where Q.1 = M 0 - N O I [T U] 0 24 Clearly, Q is unimodular. Let blocks of Q be written as Q ij' The last equation can be rewritten: [T u] = [I o] Q11 Q12 Q21 Q22 Multiplying out yields: T Qll + U Q21 = I We now set X = Q11 and Y = Q21. To prove that X and Y are T right c0prime, let [XT YT] have Smith form [S 0]T. Taking determinants of the above equation, det [T u][x] = det I = 1 (2.2.7) Y . The left hand side can be expanded using the Cauchy-Binet formula. Let the r x r minors of [T U] be listed: M1, ..., M ; let the corresponding (column for row) minors of [XT YT] be listed: N1, ..., Np' Then (2.2.7) becomes P I M. N. = 1 (2.2.8) T We know that det S divides each minor Ni of [XT YT] . Write N1 = Ci - det S where each Gi is a polynomial. Then (2.2.8) becomes P det S - 2 Mi G. = 1 i=1 25 Thus, det 8 I1, and S = I. A transposed version of part (ii) of this theorem shows that X and Y are right c0prime. (+) Suppose there exist X and Y such that T X + U Y = I. Both sides can be transposed: XT TT + YT UT = I This can be written in block form [XT YT] TT = I UT The argument starting just before (2.2.7) above can now be T and UT are right c0prime and there- repeated to show that T fore that T and U are left c0prime. Note that the hypo- thesis of X and Y being right c0prime was not needed. D Theorem 2.6.2 of [3] gives several equivalent conditions for the matrices sI — A and B to be left c0prime. Only parts (iii), (iv), and (vi) are necessary for the essential theorems in the remainder of the book [3], so only these parts will be extended. Lemma 2.2.1: The rank defects of the matrices (2.2.9) and (2.2.10), whose elements are in F, are equal. 26 I I o 0 o o o o B- -A I 0 o 0 o B o o -A 0 o o B o 0 (2.2.9) 0 o I o B o o 0 0 o -A B o o o 0 [B AB A2B ... Am'1 B] (2.2.10) where A is n x n, B is n x I, and (2.2.9) has m block rows. (Thus, (2.2.9) is mn x [(m-l) n + m2] and (2.2.10) is n x m2. Proof: Starting with the first block row of (2.2.9), left multiply each block row by A and add to the next row down. Repeating this m-l times gives I 0 o 0 o o o B 0 I o o 0 o B AB 0 o o o o B AB AZB (2.2.11) 0 o I o B Am’4 Am'3B Am'ZB o 0 0 B AB ..Am'3B E’ZB Am'lB The first n(m-l) rows of (2.2.11) are linearly independent from each other and from the last n rows. The last row obviously has the same rank as (2.2.10). Therefore, the rank defects are the same. 0 27 Lemma 2.2.2: Let A and B be matrices whose elements are in F. Let q be the degree of the minimal polynomial of A. Then rank [B AB ... Am’1 B] = rank [B AB ... Aq'l B] (2.2.12) where m 2 q Prggfz Note--while some results such as the Jordan form are lost in the extension to an arbitrary field, others such as the Hamilton—Cayley theorem still hold. Thus, it is still true that q s n. This lemma is proved by expanding the powers of A on the left hand side of (2.2.12) which are higher than q-l in terms of lower powers of A. This is followed by zeroing out these terms by applying appropriate column operations. D Theorem R2.6.2: Let A and B be matrices with elements in F. Then each of the following conditions is equivalent to SI - A and B being left c0prime. A is n x n, B is n x 2, and the degree of the minimal polynomial of A is q. (iii) The matrix (2.2.10) with m set equal to q has rank n. (iv) The matrix (2.2.9) with m set equal to q has rank nq. (vi) There exist X and Y such that (sI - A) X + BY = I (2.2.13) where deg X_<.q - 2 and deg YSq - 1. Proof: Lemma 2.2.1 shows that (iii) and (iv) are equiva- lent to each other. Figure 2.2.1 shows the circle of implications for the proof. 28 sI - A and B III (...) d (, ) are left c0prime t? 111 an IV E Figure 2.2.1 _ Implication I: This follows immediately from R2.6.1 (iv) (recall the last sentence of the proof of R2.6.1). Implication II: Suppose that (iv) is true. To show that polynomial matrices with the specified properties exist, a system of equations, which the coefficients of X and Y must satisfy, is written. Define Hq as Hq 2 Matrix (2.2.9) with m set equal to q. Also define X and E (nq x n): I- q X = X E = [0- q-2 X 0 Oq- 0 xo 5 YO I I O .Yq_1. .I. where X = XO + X13 + + Xq 2 s - and Y=Y+Ys+...+Y s’ 29 Coefficients of s in (2.2.13) can be equated to yield This has a solution if rank Hq = rank [Hq: E]. By assump- tion, Hq has rank equal to its number of rows, so that add- ing more columns cannot change its rank. [This proves Implication II. Implication III: Suppose sI - A and B are left c0prime. Then R2.6.1 (iv) shows that there exist X and Y such that (31 - A) X + BY = I (2.2.14) Since X and Y are polynomial matrices, they can be expanded in terms of their coefficients: X X + X 0 1 s + ... + Xm- Y Y + Y 0 1 s + ... + Y Proceeding in the same manner as above, define X = Xm-Z E = 0 Hm.= matrix (2.2.9) I 0 X0 Y0 . I 0 Ym-l I Coefficients of s in (2.2.14) can be equated: H X = E (2.2.15) 30 The row operations employed in the proof of Lemma 2.2.1 can be applied to (2.2.15). Let Hm be matrix (2.2.11). Note that E does not change when these row operations are applied to it. HmX = E (2.2.16) The last block row of (2.2.16) is [o o ... o B A B ... Am’lB] x = I This shows that rank [B AB ... Am-lB] z n, so that [B AB ... Am-lB] has rank equal to n, its number of rows. There are now three possible cases: 1. If m = q, (iii) is true 2. If m > q, Lemma 2.2.2 implies (iii) 3. If m < q, addition of the columns AmB, ..., Aq-lB leaves the rank equal to n. Again, (iii) is true. This proves Implication III. D There is one point remaining which needs clarification. Theorem 3.2.2 of [3] requires the next lemma for extension. The notation is preserved from the proof in [3]. Lemma 2.2.3: Let M = qu t where Mq_1(s) is a polynomial matrix over F, with q-l-rows, such that M§_l ~ISI 0], M has normal rank q-l, and t is a polynomial row vector. Then there exists a polynomial row vector w such that 31 Proof: There exist unimodular matrices R and Q such that R Mq-l Q = [I 0] We now apply a unimodular transformation to M R O Mq-l Q = IiMq_l(2 _= I 0 (2.2.17) 0 1 t tQ tll t12 where C Q = [tll t12] The row t12 is zero in (2.2.17) since M has normal rank q-l. Then t Q = [tll O] = tll [I O] = t1l R Mq_l Q, Therefore, t = (tll R) Mq-l' The lemma is proved by setting The following list of theorems from [3] consists of those which hold over a general field F. It is not exhaustive: others may have generalizations, especially when F = 3:. In this the theory of analytic functions of several complex variables may be of use. An asterisk indi- cates that the theorem needs superficial restatement. Chapter 1: 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.10 The above theorems are developed for the general case in [21]. 4. 9. must be added Chapter 2: Chapter 3: Chapter 5: WNGMDWNH H 32 1 1*, 9.2* Some uniformity conditions in e to hypotheses here. .1 .l .l, 1.2, 1.3, 1.4, 1.5 .1, 3.2, 3.3, 3.4, 3.5 .1* .1*, 5.2* .1* (except (1)), 6.2* (except (i) and (ii)) .1* 1 (Theory is developed in [21]) 1, 2.2 .l, 3.2 l, 4.2, 4.3 1, 6.2, 6.3, 6.4 .1, 1.2 III. TWO FREQUENCY SCALE RATIONAL MATRICES The method of introducing a into rational matrices which was presented in the previous chapter is very general. We now narrow this down in such a way that "two frequency scale" behavior, in analogy to "two time scale" behavior, is guaranteed. Before this is done, however, the defini- tion is motivated by examining the variation of the roots of a polynomial in 3 whose coefficients are in the field 57. 8 Theorem 3.0.1: Suppose that in Equation (3.0.1), aj(e)e32 for 05 js n. Then each of the n roots of (3 0.1) can be expanded about e = 0 as in Equation (3.0.2) where the Hg's are complex constants, N is an integer, and el/q is a branch of zq = e. an(e)sn + ... + a1(e)s + ao(e) = 0 (3.0.1) b . j/q lsks 3.0.2 N kJe 9 n ( ) "MB S (e) = k 3' Furthermore, if one root has an expansion (3.0.2) with q >1” then there will be q - 1 other roots having expansions with the same coefficients but using different branches of zq = a. Proof: First, (3.0.1) is multiplied by a suitable power of a so that the result is a polynomial in s with coefficients 33 34 analytic at s = 0. Furthermore, setting 5 to zero will not leave the left hand side identically zero. To do this, express the coefficients: aj(e) = er] Cj(e). 0535“ where each r. is an integer and each cj is analytic and non- J zero at e = 0. Define When (3.0.1) is divided by erCn(e), the resulting poly- nomial has the above properties. (3.0.1) becomes tsn + e ( ) sn‘l + + e ( ) = 0 (3 0 3) E n‘]. E .... O E . . a.(e) where e.(e) = —;l———— and 3 e Cn(e) t is a non—negative integer Up to this point, the roots are not changed. Now the sub- stitution p = ets is made in (3.0.3) and the result is multi- plied byefin-1)t: 1 2 pH + en_l(e) pn- + Jen_2(e) p“. + ... + én’lfie0(.) = 0 (3.0.4) Theorem 4.12 ofI2dlshows that the roots of (3.0.4) can be expanded: Pk(€) = E aki ° 6 , 1 S k.$ n 35 The roots of (3 0.1) can now be written co co _ 1 Z i/q _ z a (i-tq)/q Sk(€) “ Et: i=0 “R1 8 ‘ i=0 ki 8 Changing the index of summation gives (3.0.2). The last remark in the theorem is merely a statement that all branches of the qth root appear. 0 The case when the coefficients in (3.0.1) are rational is treated in most standard texts on complex variables under the topic of algebraic functions [e.g.,l9, or the treatise 22]. (3.0.1) as stated defines s as an algebroidal function of s, which behaves locally as an algebraic function. It is assumed that the slow (or low frequency) and the fast (or high frequency) behaviors are each described by transfer matrices which are independent of a. Following the usual time domain treatment, a scaling ratio of p = as is assumed. The next definition is made in view of these observations. Definition 3.0.1: A matrix H(s,e) rational in s over the field 52 is two frequency scale if: 1. H(s,e) is proper in s; 2. H(s,0) is defined and proper; (3.0.5) 3. H(]%” e) 6:: 0 is defined and proper; (3.0.6) 4. The expansions (3.0.1) of each of the poles of H(s,e) about 6 = 0 take one of the following special forms: (3.0.7) II II M 8 0‘ m (.1. \ .0 A. Sp(e) 2 b. ej/q, b0 I 0. (3.0.8) B. Sp(e) 0 J E 3 Note that each term of H(s,e) or H(—E—, s) can be expressed as the ratio of two polynomials in 3 whose coef- ficients are analytic at e = 0 by multiplying numerator and denominator by a suitable power of B. When expressed in this way, the numerator and denominator are defined at e = 0. Now, only indeterminate forms of the 0/0 type can occur when the evaluations (3.0.5) and (3 0.6) are made. These can always be resolved, however, by dividing both numerator and denominator by a suitable power of 6 while maintaining analytic coefficients. Although part 4 of Definition 3.0.1 may seem compli- cated, it is quite easy to verify once the characteristic polynomial q(s,e) of H(s,e) is known. By definition,s:can be set to zero in q(s,e) so that q(s,0) is defined and q(s,0) f 0. Let r be the smallest integer for which all coefficients of the polynomial (in p) erq(—E—,e) are analy- tic at e = 0. Then part 4 of Definition 3.1 is equivalent to (3.0.9) where L is the number of non-zero roots of €rq(-E-. e) E = 0 deg q(s,€) = deg q(s,0) + L (3.0.9) The notation Hs(s) = H(s,0) and HF(p) = H(-E—, e) E = 0 is introduced for convenience. HS(s) and HF(p) can be 37 interpreted as descriptions of the system at low and high frequencies respectively. The following simple examples illustrate these concepts. 3 + l (B + 2)(es +‘17' Example 3.0.1: Let h(s,6) + l, d Then hs(s) = h(s,0) - 3—1—2 an hF(p) = h(-E—, e) ‘ -é.’-+1 ”0 <-§—+2> 5:: O _ E_:_T 1 (s + 1)(es + l)’ Example 3 0.2: Let h(s,€) = _ l - Then [18(8) - m, and = 0 hp‘?’ = (p+€Y(p + 15 e = 0 Example 3.0.3: Suppose a transfer matrix has charac- teristic polynomial £82 + s + 1 = 0 (3.0.9) Letting e = 0, we get 3 + 1 = 0 ' (3.0.10) Substituting p = as and clearing, p2 + p + e = O. 38 Again, setting a = 0, p

> ( - -(0)) j=1 3 al "=75: H i This polynomial has degree K and it is evident that L CK(O) = f(O) jll (-bj(0)) t O. K 1 .: Ci(e)s Define dl(S,€) = 1 0 40 Using (3.1.1) n: W H equ(—§—,e> = f (p - eai>. i L W (P - b-(€)) (3.1.3) j=1 3 From (3 l 2), EKq1(‘E"€) = "PIN O K-l ° . e Ck(e)pl + l K+L C.(e) z . p j=K+l EJ-K j (3.1.4) Evaluting (3.1.3) and (3.1.4) at e = 0 gives K L K f(0)p I (p - b-(0)) = C (0)13 + , j=1 J K L C. (e) . pK 2 1+? p1 i=1 a e = O Ci+K(€) This shows that i is finite for 1:5 i S L, e e = O C (e) and —E:%——— ¢ 0 e €==O Then Ci+k(e) = eiei(e) for 1 S i S L with ei(e) analytic at e = 0, and eL(0) # 0. K i L+K . qi(s.e) = z Ci(e)s + I c.(e)sJ i=0 j=1+K 3 d ( ,e) + 2 .(e)(eS)jSK 1 3 3:1 eJ dl(s,e) + SKd2(eS,e) 41 On the other hand, if ql(s,e) = dl(s,e) + SKd2(eS,€) then q(s,0) = dl(s,0) has K roots, so that K of the expansions will have no negative powers of 6. Therefore, K of the expansions will be of form A of Definition 3 0.1 part 4. If p = as is substituted into q, K K K e - q1(—E—,e) = e dl(—E+,e) + p d2(p,e) (3 1.5) Evaluating at e = 0, K(—P——) =[c+d<0)]K (316) que’€e=0 K 21"p '° The K zero roots of (3.1.6) correspond to the finite roots of d1(s,0). The L non-zero roots of (3.1.6) must have expansions of type A of Definition 3.0.1 part 4, with the additional constraint that they approach non-zero limits as E a 0. But the roots of (3.1.5) are scaled by a, so the roots of ql(s,e) have expansions of type B of Definition 3.0.1 part 4. In the following discussions. it will be assumed that in each term hij(S,€) of H(s,e), the numerator and denomina- tor are c0prime in 3. Theorem 3.1.1: H(s,e) is two frequency scale if and only if each term can be expanded as K n11j(s'e) + s “211)83’5) h..(s e) = ij ' K dlij(8.€) + s d21j(88.€) 42 where 1. nlij’ n2ij’ dlij’ and d2ij are polynomial with coefficients analytic in e at e = 0; deg dlij(s,e) deg dl j(3,0) = K; 1 deg d2ij(p:€) = deg d (p20); Zij deg nlij s K; deg n2ij 5 deg dZij; @Lflwa The constant terms of n2.. and d .. are both zero. 13 213 nggfz Let each term hij(s,e) of H(s,e) be expressed as nij(s,e)/dij(s,e). Then each denominator dij divides the. characteristic polynomial. Thus, if the characteristic polynomial of H(s,€) satisfies the conditions of Lemma 3.1.1, then so must each dij‘ Conversely, suppose that each dij satisfies the conditions of Lemma 3.1.1. Let M(s,e) = nM(s,€)/dM(s,€) be a minor of H(s,€) computed in the follow- ing manner: all products are performed without cancellation and sums are computed by cross-multiplication. Thus, dM(s,e) is the product of some of the dij(S,€). Clearly, dM(s,€) satisfies the conditions of Lemma 3.1.1. Let M(s,€) = nM'(s,€)/dM'(s,€) be the result of performing all possible numerator and denominator cancellations. It can be shown from the last statement of Theorem 3.0.1 that dM'(s,8) can be chosen to have coefficients analytic intaat t = 0. This shows that dM'(s,e) satisfies the conditions of Lemma 3.1.1. [Since the characteristic polynomial is found by taking the least common multiple of all such poly- nomials, it follows (by similar arguments) that the 43 characteristic polynomial also satisfies the conditions of Lemma 3.1.1. Thus, H(s,e) is two frequency scale if and only if each hij(s,e) is so, and it is enough to prove the theorem for the scalar case. n(s,e) 8,8 (->) Let hij(s,e) B“ A EMELELEL “ g Eaa(s,€): H(SJO) O I! d(s,0) imam») d(s,e) B - a 2 0 since hij(S’O) is defined. _ m(5,e) Therefore, hij(S,€) ‘ dl(s’€) + SKd2(€S,€) where m(s,e) has coefficients analytic in t. Properness of H(s,e) implies K+L 1 m(s,e) = X l.(e)s i=0 ’- K+L . . z 1i(€)eK-lpl _P_ = i=0 hij( e .e) K ‘K e d1(—E—,e) + p d2(p,e) Setting 6 = 0, and using the notation Ci(e) from the previous lemma, K+L . . Z li(e)eK--l pl h..(-E— e) = i=0 e =0 13 E e=o pK[CK(0) +d2(p.0>1 44 K+L 1.1(8) i= -K+1 K81 p K[cK + d21 ° K p1 + 1K(0)p 1 ej cK + 02(p.0> 1. (e ) . 4+ K I p3 + 1K(0) e=0 For this to be defined, Ifixe>=e3t¢o.ls.j5L with fj analytic at t = 0. The proof is completed by setting 1 _ 1i(e)S , dlij - dl IIMV: nlij (S,e) i 0 "Mt" H fj(e)(es)j, d .. = d2 n21j(€S’€) = 213 J' (+-) Inspection shows that hij(S,€) is proper. nllj(s,0) h1j(s’0) = d11j(5:07 (3.1.7) This is defined and proper. K _p_ + K h .(Le) :8 “1 1j< ’5) p“213‘ J 5 ' e:=0 E Kd1:j(i' E ,6) + p KdZij(p,€) 5:0 (p.6) X + n21j(p O) (3..].8) = 0K (0) + d21j’(p 0) where x is the coefficient of degree K in n11j(s 0). This is also defined and proper. 45 Examination of (3.1.7) and (3.1.8) shows the following corollary. Corollary.31 1: If H(S,€) is two frequency scale, then HS(m) = HF(0) and III.2. Exact Frequency Scale Decomposition This section presents a frequency scale decomposition analogous to (1.4). It can be viewed as a partial Laplace expansion of a transfer matrix H(s,e). Most of the compli- cations associated with the complete Laplace expansion are avoided here because the denominators in the decomposition are c0prime. The first step is to show that the denomina- tors of the terms of a two frequency scale transfer matrix can be factorized into slow and fast parts, with each of the parts having coefficients in 32. Lemma 3.2.1: Let d(s,e) be a polynomial whose coefficients are analytic in e at e = 0. Suppose that the roots of d(s,t) obey property 4 of Definition 3.0.1. Then d(s,e) has a unique factorization. d(S.€) = f(€) - dS(S.€) - dF(€S.€) where 1. deg d(s,e) = deg dS(s,0) + deg dF(p,0) dF(0,O) # 0 MN all of the following are analytic atez= 0: 46 f, the coefficients of ds(s,€), and the coefficients of dF(p,e) 4. (1S and dF are monic Proof: d(s,e) can be factorized as in the proof of Lemma 3.1.1: ":7: |._A ":3?" H d(s.e>==f> 1 p2 = z B> 0> 1. j?‘ - '8 0(W.(€)) {MW-(ED 904(6)) ?3 i,j,k# 1 J k 3 q pq = 1:1 0(wi(e>> We firSt claim that 48 Z . 27T(ilKl + + inKn) I when q T Kl + K2 + ... + Kq The summation of (3.2.4) is taken over all possible combina- tions of the indices ij’ 1 s j s n with l S ij 5 q for which no indices are repeated. Thus, (3.2.4) has (3) terms. Proof of this claim is by induction on n with q fixed. q j ZniK n = 1 case: 2 e q is seen to be the discrete i=1 Fourier transform of the constant function 1, evaluated at the frequency variable K. It is well known that this is zero if q I K. Suppose now that the claim is true for n = l, 2, ..., n-l. Assume that q I Kl + K2 + ... + Kn' The "missing" terms can be added to (3.2.4): q q . 2n(i K + ... + i K ) z z eJ 1 l q I?“ (3.2.5) 11=1 1n=1 The difference between (3.2.5) and (3.2.4) consists of the terms: (3) summations with 2 of 11, ..., in set equal (3) summations with 3 of i1, ..., in set equal (2) summations with all of il’ ..., in set equal 49 All of these summations are zero by the previous cases. For instance, the last sum is zero by the n = 1 case. (3.2.5) is the n-dimenstional discrete Fourier transform of the con- stant function 1 evaluated at the frequency variables Kl’ ., Kn' As in the n = 1 case, this is zero. This proves the claim (3.2.4). The coefficient pn can be written 3 M Pn(€) in# 9 (W1 (6)) -.- 0 (Wi (8)) il’ ’ 1 n m m K + K + ... K = Z Z bK bK g l 2 qn Kl=0 Kn=0 1 n q :E: ej 217(11Kl + + ann) l 1 D s in # q (3.2.4) shows that all fractional powers of e have zero coefficients. Therefore, each pH is analytic in e. a The next series of lemmas provides a way of finding numerators for the previously mentioned partial Laplace expansion, once the denominator has been factorized. All polynomials are over F in Lemmas 3.2.2, 3.2.3, and 3.2.4. Lemma 3.2.2: Suppose a, b, x and y are non-zero polynomials such that ax + by = 0 (3.2.6) with deg x < deg b and deg y < deg a. Then a and b are not c0prime. Proof: If (3.2.6) is written ax = -by, then the lemma easily follows by cancellation of prime factors on both sides. D 50 _ n . . Lemma 3.2.3: Let h — 3173; be str1ctly proper w1th d1, d 2.9 and n polynomials, and d1 and d2 are c0prime. Suppose h has a strictly proper decomposition: That is, deg a < deg dl and deg b < deg d2. Then a and b are unique. . b - . Proof: Suppose h = 7?— + 71— where a # a or b # b. . . l 2 a-a b-b _ . , . . _ Then 7r” + 7E; — 0; and th1s g1ves (a-a)d2 + (b-b)dl — 0 1 If a = a, then b = B and the lemma follows; and similarly in A the case that b = b. -If a # a and b # b, Lemma 3.2.2 shows that dl and d2 are not c0prime. The lemma follows from this contradiction. a Lemma 3.2.4: Let h = did- be strictly proper with d1, d2, 1 2 and n polynomials, and <11 and d2 c0prime. Then there exist and r2 such that rl r r l 2 h = + HI 3; with deg r1 < deg d1 and deg r2 < deg d2. Proof: By applying Euclid's algorithm, or as a special case of Theorem R2.6.l(iv) in Section 2.2, there exist x and y such that dlx + dzy = l 51 Then ndlx + ndzy = n nd x + nd y h _ 133 2 l 2 _ nx n“ ‘B*% - qld1 + rl + qzd2 + r2 _ d d l 2 where my = qldl + rl and nx = qzd2 + r2 with deg r1 < deg (11 and deg r2 < deg d2. h can now be written since h is Lemma 3.2.5: §+¥+h+h 1 2 ;—.— 1 2 strictly prOper ql must equal -q2. G Let h(s,e) be a two frequency scale scalar. Then h(s,e) can be expressed h(s,e) = h1(S,e) + h2(es,e) + A(e) where l. h1(s,e) and h2(es,e) are both strictly proper and 2. The two frequency scale; poles of h1(s,e) and h2(p,e) approach finite limits as e + 0 and all poles of h2(p,e) approach non-zero limits; 3. A(e) is analytic at e = 0. 52 Proof: By division, h(s,e) can be written h(S.€) = 8(S.€) + A(€) where g is strictly proper. That A(e) is analytic follows from Theorem 3.1.1. Clearly, g(s,e) is two frequency scale. Let g be expressed n(s,e) We) =EIIs—,e:7 where 0 i d(s,0) f.” Let 0i(€), l s i =EK be the slow poles of g and let Bj(€)/€, 13 j SL be the fast poles of g. . K Then d(s,€) = f(€) ° H (S - 01(3)) i=1 L H (as - B.(e)) (3.2.7) j=1 3 where f is analytic at e = 0 and f(0) # 0. Lemma 3.2.1 shows that the two products in (3.2.7) are poly- nomials with analytic coefficients. K Define: dl(s,e) = f(e) - H (S - 0.(6)) i=1 1 ( ) L ( ( )) d ,e = H - B. e 2 P i=1 p J Properties of d1 and d2 include: dl(s,0) t 0, d2(0,0) # 0; d1(s,e) and d2(€s,8) are c0prime; and deg dl(s,0) = deg dl(s,€). These observations show property 2 of the lemma. d1, d2, and n satisfy the hypothesis of Lemma 3.2.4 so that 53 rl(s,e) r2(s,e) g(s,e) = 3123,55 + 323:3,55 The next step is to show that the coefficients of rl and r2 are analytic--we know only that they are in a: at this point. Expand rl and r2 in powers of t: i | "MS rl(s,e) - i ai(S)€ o i r2(s,e) - l ||M8 bj (S>€j 33,, The ai's and bj's are polynomials in s with deg ai(s) < deg dl(s,e), i = i i + 1, deg bj(s) < deg d2(p,e), j = jo, jO + 1, Let Ko = min (10, jo) so that K0 is the smallest integer for which 3K (3) 1 0 or bK (s) 1 0. g can now be expressed o o _ m ai(s) bi(s) i g(s,e) = .2 31:3,ES + 32(es,e) E 1=Ko The quantity in braces is analytic at e = 0 and has a power series expansion. Thus, g(s,e) has a Laurent series expan- sion aKo(S) bKo(S) K °° i 8(S.€) = 'HIIETUY + 357676] E O + 2 81(3)8 1=K0+1 Suppose that Ko< 0. Since g(s,0) is defined, 54 3K (S) bK (S) O O .. m1 8, + m2 , : 0 By letting s + w and observing that aK /dl is strictly o proper, we must have bKo(s) s 0 It then follows that 8K (s) E 0. But this is a contradic- . 0 tion since either aK or bK was assumed to be non-zero. o 0 Therefore, KO 2 0 and it has been shown that r1 and r2 have coefficients which are analytic at e = 0. Only the dependence of h2(p,€) on 5 remains to be shown. Substituting p = as, rl f2(p,e) D hZCP,e) - W ID Theorem 3.2.1: Let H(s,e) be a two frequency scale transfer matrix. Then H(S,€) can be expressed H(S’E) = I'll-(3,5) + H2(ES:€) + A(€) where l. H1(s,€) and H2(€s,€) are both strictly preper and two frequency scale; 2. The poles of Hl(s,€) and H2(p,8) approach finite limits as e:+0 and all poles of H2(p,0) approach non-zero limits; 3. A(e) is analytic at e = 0. 2529;: Apply Lemma 3.2.5 term by term. We have used the fact a pole of a rational matrix must be a pole of at least one of its terms. D Two simple examples are now given to illustrate this theorem. They demonstrate that the major step is factoriza- tion of the denominator. The computation of the numerators in Lemma 3.2.4 is mainly for theoretical purposes. In practice it is more efficient to set up equations for the coefficients of r1 and r2 directly. 56 Example 3.2.1: l (s+1)(es+1) 1 e ITE+ErI Ell eS+1 h(S,e) Example 3.2.2: l esz+s+1 1_ ... '1'48 V1-4e 1- V 1-46 1+ rf-Zm 9—28— es+——Z— h(S,e) = III.3. System Matrices for Two Frequency Scale Transfer Matrices This section specializes the system matrix approach of Chapter II to the case of two frequency scale transfer matrices. This will aid in the study of closed loop systems. It will also be seen that system matrices can be used as a convenient tool for evaluating the slow and fast descriptions HS(s) and HF(p). Theorem 2.1 shows that a rational matrix H(s,e) over 32 can be represented by a least order polynomial system matrix P(s,e) in MFD form. If any coefficients of s in P(s,e) have a pole at e = 0, the columns can be multiplied by suitable powers of a so that all coefficients are analytic at e = 0. Let the system matrix with cleared columns be Pl(s,€). Even if H(s,0) is defined, the matrix P1(s,0) may 57 not be a system matrix because the upper left block of Pl(s,0) may be singular. It will be convenient later to work with system matrices over 3: for which e can be set to zero. The next theorem shows how this difficulty can be alleviated. Lemma 3.3.1: Let H(s,t) be a rational matrix over 3: for which H(s,0) is defined. Let P(s,e) be a polynomial system matrix representation for H(s,e) in MFD form with coeffici- ents analytic at e = 0. Then P(s,e) is strictly system equivalent to a system matrix Pl(s,c) in MFD form where P1(s,0) is a polynomial matrix over c. "I 0 0] Let P(s,e) = 0 T(s,€) I (3.3.1) [0 -V(s,€) 0_ If det T(s,0) = 0, then det T(s,€) = erq(s,€) where q(s,€) is a polynomial in s with q(s,0) 1 0 and r 2 1. Also, if det T(s,0) = 0, then the Smith form of T(s,0) must have some zero diagonal element. Let R(s) and Q(s) be unimodular matrices over c which transform T(s,0) to its Smith form as in (3.3.2). 8(3) 0 R(s)T(s,0)Q(s) = ‘ (3.3.2) 0 Otxt 58 Since premultiplication by R-](s) can be interpreted as a sequence Of row Operations, 8(5) 0 * 0 ’1(s) = (3.3.3) T(s,0)Q(s) = R The *‘s represent possibly non-zero polynomial entries. Con- sider the effect Of performing the column Operations of post- multiplication by Q(s) on the system matrix (3.3.1). -1 0 0] ”I 0 0] 0 T(s,e)Q(s) I = 0 T(s,e) I = P(s,t) [0 -V(s,s)Q(s) 0 L0 41“,.) 0_ (3.3.4) All coefficients in the last t columns of T(s,e) are zero when t = 0. The rule for computing the transfer matrix yields (3.3.5). This is expressed column by column in (3.3.6). H(s,e) = v(s,e)[I(s,e)]'1 (3.3.5) H(s,6:) - T.j(s,e) = {7.j(s,e) (3.3.6) Suppose that T j(3,2) above is one of the last t columns Of T(s,e). Then (3.3.7) holds. v.j(s,0) = H(s,0) - 0 = 0 (3.3.7) This shows that the entire jth column Of P(s,t) is zero when 8 = 0. This column can now be divided by 6 while leaving all Of its coefficients analytic at e = 0. The Operation of 59 dividing this column by 8 produces a new system matrix for which the exponent r in det T(s,e) = Erq(S,€) is reduced by one. In summary, if P(S,E) is a polynomial system matrix representation for H(S,e) in MFD form with det T(s,e) = Erq(S,€), a new system matrix P2(s,e) strictly system equi— valent to P(s,e) can be found for which det T2(s,e) = K Er-1q(S,€) where K is a complex constant. This process can be repeated until a system matrix P1(s,e) is formed for which det Tl(s,0) f 0. If P(S,€) is a system matrix for a two frequency scale transfer matrix H(s,e), then Lemma 3.3.1 can be applied to the system matrix P(—E—,e) after all negative powers of are eliminated from the coefficients. The various results can now be combined. Theorem 3.3.1: Let H(s,e) be a two frequency scale transfer matrix. Then H(s,e) has a least order polynomial system matrix representation P(S,e) in MFD form for which P(s,0) is a polynomial system matrix over c. Furthermore H(—E—,E) also has a least order polynomial system matrix representa- tion P(p,e) in MFD form for which P(p,0) is a polynomial system matrix over ¢. D In Theorem 3.3.1, P(s,0) and P(p,0) may not be of least order even though P(S,€) and P(p,t) are. It is evident that some poles Of H(s,e) may be undergoing numerator-denominator or other types Of cancellations as e-+0. These will be 60 called "lost poles" Of H(s,€) in accordance with the next definition. This behavior is discussed in [4] for singularly perturbed systems. Definition 3 3.1: Let P(s,s) and P(p,e) be least order poly- nomial system matrices for a two frequency scale rational matrix H(s,e) and its scaled version H(—E—,e) respectively. Suppose that P(s,0) and P(p,0) are polynomial system matricai Let P1(s) and Pl(p) be least order polynomial system matrices equivalent to P(s,0) and P(p,0) respectively. The "lost slow poles" of H(s,e) are the roots Of the "lost slow polynomial" (3.3.8). det T(s,0) qLS(S) a det T1(Sjfi (3.3.8) The ”lost fast poles" Of H(s,e) are the non-zero roots Of (3.3.9). M det T(P.0) p qLF(P) a Jet Ti(p) (3.3.9) The "lost fast polynomial” is a polynomial having the lost fast poles as roots. qLS and qLF are chosen to be monic. Since the polynomial det T(p,e) is simply a 'scaled" version of det T(s,e) and since the zeros Of det T(s,e) = 0 Obey property 4 Of Definition 3.0.1, det T(p,0) = pK - ¢(p) (3.3.10) 61 where ¢(0) # 0 and K = deg det T(s,0). It can be seen from Section 2 Of this chapter that HF(p) has no poles at the origin. Thus, det Tl(0) # 0. This shows that M = K in (3.3.11). Alternatively, the lost slow and fast poles are the roots Of (3.3.11) and the non-zero roots Of (3.3.12) respec- tively. CP[H(S,B)]I€=O 3, (3.3.11) CP[H<—§—. 2)]I€=0 pKCP[H(—E—,E)I (3.3.12) e=0J For system matrices in MFD form, the lost slow poles are rOOts Of the output decoupling polynomial of P(s,0) and, similarly, the-lost fast poles are non-zero roots Of the output decoupling polynomial of P(p,0). The next three examples demonstrate the decomposition method on non-trivial systems. Although the use Of system matrices is not necessary here, their use makes the computa— tions simple and Obvious. Example 3.3.1: (Singularly Perturbed System) The system Of equations (3.3.13) determine a transfer matrix for the output y in terms Of the input u, so that Y(s) = H(s,e)u(s). The matrices on the right hand side Of (3.3.13) are analytic at e = 0, and det A22(0) # 0. x1 = All(€)x1 + A12(€)x2 + B1(€)u 8X2 Y A system matrix representation is SI - All(e) -A21(e) E “Cl(€) 62 A21(e)xl + A22(e)x2 + B2(e)u C1(e)xl + C2(e)x2 + D(e)u -A(e) sI - E 'C2(€) A22(e) 31(5) B2(E) 8 D(e) (3.3.13) The second row can be multiplied by c to give a system matrix for which 6 can be set to zero. This yields a system matrix for the slow subsystem. BI - All(0) -A12(0) B1(o) -A21(0) -A22(O) B2(0) . -c1 -c2<0) D<0)- This is easily shown to give the transfer matrix (3.3.l4a) with A0, B Co After substitut- o, and Do defined by (3.3.15). ing p = as, clearing e's from denominators, and setting a = 0, a system matrix for the fast subsystem is Obtained. "pl 0 ”A21 PI ' A22 L-Cl -C2 _ 63 This gives HF(p) as in (3.3.l4b). Since A22(0) is non-singular, all poles are either 0(1) or 0(—%—) as e+ 0. Thus, H(s,e) is two frequency scale. The expressions (3.3.14) are similar to those in.[1fl. The matrices (3.3.15) for the slow sybsystem are the same as those for the reduced subsystem in [2]. HS(s) = CO(sI - AO)’1BO + 00 (3.3.l4a) HF(p) = C2(O)[pI - A22(0)]'1B2(0) + D(0) (3.3.l4b) A0 = A11 ' A12A22.1A21Ie=0 (3 3 15) B6 = B1 ' A12A22.132 I€=0 Co = Cl ’ C2A22-1A21Ie=0 D0 = D ‘ C2A22-132 Ie=0 Example 3.3.2: (Implicit Singularly Perturbed System) Chow'[ 5] considers the nxn system (3.3.16) where A1(e) remains bounded at e = 0. In [ 5] it is shown that if AO satisfies condition (3.3.17), then (3.3.16) can be brought to explicit singularly perturbed form (3.3.13) by a simi- larity transformation. Herezflzwill be shown that condition (3.3.17) is suffi- cient for the transfer matrix defined by (3.3.16) (with appropriate input and output matrices added) to satisfy con- dition 4 Of Definition 3.0.1. 64 ex = (A0 + eAl(e))x (3.3.16) R(AO) e N(AO) = Rn (3.3.17) where R(o) and N(-) indicate range and null spaces respec- tively, and 6 indicates the direct sum of subspaces. It is easily shown that (3.3.17) is equivalent to: There exists a non-singular matrix T such that (3.3.19) holds, where Jl is an rxr, non-singular Jordan matrix. For simplicity, the input and output matrices will be suppressed in the discussion that follows. IAOI‘l = J = diag(Jl, 0) (3.3.18) The system matrix in the unsealed frequency variables is given by (3.3.19). It is assumed here that Al(€) is analytic in E at 8 = 0. P(s,e) = sI - —%— AO - Al(e) (3.3 19) Now the above similarity transformation is applied and the denominators are cleared of E. Pl(S,E) = TP(S,€)T-l diag(e,e, ...,t, 0, 0, ..., 0) r 8's = 681 - J1 - €B11(€) - 8312(8) ' 321(8) SI ‘ 322(6) - I 311(6) 312(5) _1 where B(e) = = TA1(5)T [321(8) 322(8) 65 The matrix Pl(s,0) is a system matrix, and det P(s,0) = det(-Jl). det(sI - B22(0)). Clearly, det P(s,0) has degree n - r. The system matrix for the scaled frequency variable is P(p,e) = pI - AO - 5A1 P(p,0) is a system matrix and det P(p,0) = det(pI - AC) has r non-zero roots since AC has rank r. This shows that con- dition 4 Of Definition 3.0.1 holds. Condition 1 holds for any choice of input and output matrices independent Of 3. Conditions 2 and 3 are satisfied only for certain choices of input and output matrices. If (3.3.18) is not satisfied, then Al(e) is important, as shown by (3.3.20) and (3.3.21). ”-1 0 0] exl = 0 -e 1 x1 (3.3.20) I0 0 -e_ :1. O 0 2x2 = 0 0 1 x2 (3.3.21) [0 -€ 0 . . (3.3.20) has eigenvalues -1, -1, and --%—, satisfying condition 4 Of Definition 3.0.1. (3.3.21), however, has eigenvalues -—%— and tj//ET Example 3.3.3: (High Frequency Oscillatory Modes) Given in [ 7] (without inputs and outputs) is the 66 second order system (3.3.22) where Q4 has simple, positive eigenvalues. x + Bx + Qx = Lu y = Cx x1 B1 32 where x = 18 = Ixzj I33 B4_ C = [c 1 c J L — 1 “7T 2 The system matrix (3.3.23) can be written. n 2 , s I + sBl + Ql P(s,u) = 3B3 + Q3 .01 b (3.3.22) ' Q2 I Q1 7 Q = Q4 [Q3 73?; L1 dim x1 = n1 L2 dim x2 = n2 usz + Q L j 2 2 l “2821 + usz4 + Q4 L2 -C2 0 (3.3.23) The slow and fast characteristic polynomials are shown in (3.3.24) and (3.3.25). ' 2 s I + sB1 + Q1 det T(s,0) det sB3 + Q3 Q2Q4-1(SB3 + Q37] det Q4 - det[sZI + 331 + Q1 Q2 Q4 d (3.3.24) 67 -2 I p1 Q2 det T(p,0) = det 0 p21 + Q4 L . = pznl . det(pZI + Q4) (3.3.25) Again, the high frequency oscillations are obvious from (3.3.25). IV. APPROXIMATION OF TWO FREQUENCY SCALE RATIONAL MATRICES The main result Of this chapter may be regarded as a frequency domain version Of (1.6). Even though this result holds for more general cases, most practical applications will require stability conditions on the slow and fast sub- systems. This parallels the stability requirements for (1.6) tO hold. I Robustness and sensitivity results for linear feedback systems typically involve properties of stable rational matrices along the imaginary axis, e.g. [13, [16, [18]. The next theorem shows that under certain conditions, the values Of Hs(s) and HF(p) along the imaginary axis determine a uniform 0(6) approximation Of H(s,e) along the imaginary axis. If such a rational matrix represents a signal gain, then HS(jw) and HF(jew) are approximate signal gains for low and high frequency sinusoidal inputs. The reciprocal Of singular value graphs used for robustness evaluation can be approximated from HS(s) and HF(p). Note, however, that the 0(8) approximation may be lost here. Ifg[HS(jw)] goes to zero at some specific value ml Of w, it cannot be concluded that l/g[H(jwl,e)] is infinity. It can only be concluded that 1/g[H(jwl,e)] is large when a is small. 68 69 Lemma 4.1.1: Let h(s,e) be a two frequency scale scalar rational function. Let h(s,e) be expressed as the ratio of two polynomials with analytic coefficients: h(S,e) = mn(s,€) S,€ Let the denominator have the expansion d(S.€) = f(€) (8 - 81(6)) (68 - bj(e)) "=17: H ||=11'* H i 3' Let D be the imaginary axis. Suppose that for all i and j, ai(0) and bj(0) are not on D. Then sup h(s,e) - hs(s) - hF(es) + w = 0(6) (4.1.1) seD where W’= hS(w) = hF(0) and e is restricted to be real. Proof: Let h(S,€) be expressed as guaranteed by Lemma 3.2.5. h(S,e) = hl(s,e) + h2(eS,e) + A(e) Then hS(s) = hl(s,0) + h2(0,0) + A(0) hF

= h1<-§—.e>l,=o + h2 + A h2 + 4(0) (4.1.1) can be written: 70 sup Ih(s.e> - hs(s) - hF(es) + w I seD = sup Ih1(s,€) - hl(s,0) + h9(€s,€) - h2(€s,0) 56D ‘ + 4(a) - A(o>| :Ssup Ihl(s,e) - h1(s,0)I + supI h2(es,e) seD seD - h2(es,0)I + IA(€) - A(0)I Since A(e) is analytic, A(€) - A(0) = 0(6). Let hl(s,€) be expressed as n1(s,€) h1 = BITE??? where nl and d1 are polynomials in s with analytic coeffi- cients. Then sup hl(s,€) - hl(S’0) = sup a(S,€) 36D 36D 31(s,e) 31(S,O) where a(s,€) = nl(s,€) dl(s,0) - nl(s,0) dl(s,€). Let deg d1(s,€) = K. Then deg dl(s,0) = K and deg nl(s,e)£ K - 1. Thus, deg a(s,e) 5 2K - 1. Since a(s,0) a O, d(s,g) = e - B(s,e) where B is a polynomial in s with ana- lytic coefficients, and having degree 3 2K -1. Combining, suP h1(S.€) - hl(S.0)I = IEISUP 865.6) 36D 36D dl(s,€)-d1(s,0) The sup on the right hand side is uniformly bounded for sufficiently small 8 by Corollary 4.2.2. 71 Proceeding in a like manner, h2(€S,€) can be expressed: n2(es,e) h2<€$~€> = W We then have sue h (28.6) - h (85,0) =Iel- sup V(es.e> SGD I 2 2 I seD dE(es,e)d2(es,0) The rational function on the right hand side is two frequency scale and has no slow or fast poles on the imaginary axis. Theorem 4.2.5 shows that the sup on the right hand side is uniformly bounded for sufficiently small real 8. (Note that the requirement Of real 6 appears only at this point.) This proves the lemma. 0 Lemma 4.1.2: Suppose that all of the elements Of a rational matrix H(s,e) satisfy the conditions of Lemma 4.1.1. Then sup lIH(s,e) - HS(s) - HF(es) + W H = 0(8) seD . where W = HS(w) = HF(O)’ II ' II is some matrix norm, and D is the imaginary axis. Proof: By the norm equivalence theorem, there is a constant B such that 'IIAII <13 IIAIIM where H A IIM = ma? IAijI 72 For brevity, let H(s,e) - HS(s) - HF(es) + W = A (3,5) sup I A(s,e) 'Ig sup B I‘ A(8,2) Il SED SED M = sup B max A..(S,e) seD i,j 13 I = B max sup A ..(S,g) i,j seD I 13 I By Lemma 4.1.1, there exists 6* and constants Cij such that for E e[- 5*, 8*], B max C.. 1.3 13 E E B max sup Ai.(s,e) 5 B max Ci’ i,j 36D 3 i,j J Cl Theorem 4.1.1: Let H(s,s) be a two frequency scale rational matrix. Suppose that HS(s) and HF(p) have no pure imaginary poles and that H(s,e) has no pure imaginary lost poles. Then sup ’H(s,e) -HS(S)'HF(es) + W H = 0(6) seD where W = Hs(w) = HF(O), II ' II is some matrix norm, and D is the imaginary axis. Proof: Let hij(s,e) = nij(s,e)/dij(s,e) be any term of H(s,e). Then hij clearly satisfies the first three condi- tions of Definition 3.0.1. The characteristic polynomial q(s,e) of H(s,e) can be expressed as (4.1.4). K L q(s,e) = f(€) It (s - a (5)) N (as - b (5)) (4.1.4) =1 n m n m=l 73 where f(0) # O and for all 1 s n s K and l s m s L, an(0) and bm(0) do not lie on the imaginary axis. Since each dij(s,s) divides q(s,e), it is seen that all hypothesis of Lemma 4.1.2 are met. D IV.2. Appendix--Bounds on Rational Functions This appendix shows in detail the boundedness of the rational functions as needed in the proof of Lemma 4.1.1. Theorem 4.2.1: Let g(s) = n(s)/d(s) be a proper rational function over c with n(s) and d(s) polynomials. Suppose that none of the zeros of d(s) lie in the closed set D. Then d(s) is uniformly bounded on D. Proof: g(s) can be expressed K K-l Ks + nK_ls + ... + no K K-l sz + dK-ls + ... + do n g(S) = where the nj's and di's are complex. Rewrite g(s) —l -K g(s) = nK + nK_ls + ... + nos -1 -K dK + dK_1s + ... + dos Choose R large enough so that the values nK-lR-l , ..., -K -1 -K noR ’ dK-lR ’ "" doR are all less than gk . min (G, IdKI)' where G = max (anI, l). 5‘6 %|dxl then for Isl > R. Ig(s)I < 74 The set DFNISIS S R} is closed and bounded and therefore compact. g is analytic and therefore continuous on this set. Since the continuous image of a compact set is compact, g is uniformly bounded on DID ISISJSRI. Thus, g is bounded on all of D. 0 Theorem 4.2.2: Let g(s,8) = n(s,8)/d(s) where g is proper, d is a polynomial with complex coefficients, and n(s,8) is a polynomial with coefficients analytic at 8 = 0. Suppose that none of the zeros of d lie in a closed set D. Then there exists 8* >0 such that g(s,8) is uniformly bounded for all (8,8) 6 D x {8] I8] < 8*}. Proof: Choose 8* so that all coefficients of n(s,8) are analytic in the disc E = {8| |8|s 8*}. Let n(s,€) = nK(8)sK + ... + no(8) where the nj's are analytic at 8 = 0. Let B. = max In.(8)l. 8£E J K n(8)s +...+n(€) |(se)|=K ° g ’ d(s) 0 such that g(s,g) is uniformly bounded for all (8,5)6 D x [-e*, 5*]. Proof: Choose 8* such that all coefficients of n are analy- tic on {8| |8Is 8*}. By changing variables, (83 8) _ n( ,8) supIg(s,8)l= sup BET—LT— — sup _d%—)— SGD seD as I pED p I Boundedness follows from Theorem 4.2 2. Theorem 4.2.4: Let D be the imaginary axis and let d(s,8) = dl(S,€) + SKd2(€S,E) where l. d1(s,8) and d2(p,8) have coefficients analytic in 8 at 8 = O; 2. deg dl(s,8) = deg d1(s,0) = K; 3. d2(0,€) = 0 Define dS(s) = d1(s,0) 1 dF(P) ='—;— [X + d2(P.0)] where x is the leading coefficient of dl(s,0). Let (1S and dF have no roots in D. Let r(s,€) = dS(s)dF(es)/d(s,€). Then lim r(s,€) = l uniformly for s E D. 8+0 real In particular, there exists 8* > 0 such that r(s,€) is uniformly bounded for (3,6) GDXE -€*, 8*]. 76 Proof: Using previously derived expressions, K L d(s,8) = f(8) - H (s-ai(8)) II (as - b-(8)) i=1 j=l 3 where f is analytic at 8 = 0, all ai(8) and bj(€) approach finite limits as 8 + O, and all bj(0) are non-zero. Also, "21% H ll:1[-* H dS(s) dF(€S) = f(0) (s - ai(0)) (8S - b.(0)) j J i _ f(0> 1; s - 31(0) , }{ es - bi(0) r(s,8) _ 8 _j i 1 S - ai(€) j=1 es - b3(€7 If s e D, then ai(0) - ai(8) s - ai(0) s - ai(8) s - ai(0) - ai(0) - ai(8) Re 31(0) -*0 as 8-+ O. This limit is uniform in s. s - ai(0) Therefore, + l as 8 + O uniformly in s. S - 31(8) Similarly, as - bj<0> + l as 8 + 0 uniformly in s. 88 - ij8) The theorem follows using standard limit theorems. 0 Corollary 4.2.1: Let D be the imaginary axis and let d(s,8) be a polynomial with coefficients analytic in 8 at 8 = 0. Suppose that the leading coefficient of d(s,8) does not vanish when 8 is set to zero, and that d(s,0) has no roots in D. Define r(s,8) = d(s,O)/d(s,8). Then 77 1im r(s,8) = l uniformly for s e D. 8+0 Proof: Easily extracted from proof of Theorem 4.2.4. 0 Theorem 4.2.5: Let h(s,8) be two frequency scale. Suppose that the limits of the roots of the characteristic polyno- mial do not lie on the imaginary axis D (as in Lemma 4.1.1). Then there exists E*>’O such that h(s,8) is uniformly bounded for (s,8)e D x [-8*,8*]. Proof: Ih(S,€)I can be written |h(s,8)| = nl(S 8) + s Kn 2(83, 8) dl(S,€) + S Kd2(€S,€) where the right hand side is the form guaranteed by Theorem 3.1.1. nl(S,€) + SKn2(€S,€) dS(S) dF(8S) lh(S,€)I -| dS(s) - dF(€S) dl(s’€) + SKd2(ES,€) The right hand factor is bounded by virtue of Theorem 4.2.4. The left hand factor can be rewritten nl(s,8) + SKn2(€S,€) dSCS) dF(ES) =n1(s, 8) SK n2(8s, 8) as (s 5 dFie 3) +3: (3 5 d_(Es) The theorem follows from application of Theorems 4.2.2 and 4.2.3 to the individual functions on the right hand side.EJ 78 Corollary 4.2.2: Let h(s,8) be a rational, proper function of s with coefficients analytic in 8 at 8 = 0. Let h be expressed h(s, e) = mn(:,:) where d(s,8) satisfies the hypothesis of Corollary 4.2.1. Then there exists 8* > 0 such that h(s,8) is uniformly bounded for all imaginary s and 18|< 8*. Proof: Similar to proof of Theorem 4.2.5. 0 V. CLOSED LOOP SYSTEMS This chapter investigates what can be said about a two frequency scale system when a feedback loop is closed around it. A cascade of two two frequency scale systems with feed- back applied will be considered. It is shown that under cer- tain conditions, knowledge of the open loop lost poles and the poles of the closed loop slow and fast subsystems is suffi- cient to approximate all of the closed loop poles. Figure 5.1 shows the inputs and outputs and the manner in which the loop is closed. Figures 5.2 and 5.3 show the closed loop slow and fast subsystems, respectively. Figure 5.1 + I + 47 l H2(s,e) 79 80 Figure 5.2 f‘ .Hls(S) v + \L . HZS(S) L J}. + Figure 5.3 ,9 H1F(P) H2F(p) 5):..— + V « The following notation is introduced for convenience. Let i = 1,2. l. Hi(S,€) has a least order polynomial system matrix representation Pi(S,€) in MFD form such that Pi(s,0) is a polynomial system.matrix. 2. Hi8 has a least order polynomial system matrix Pis derived from Pi(s,0) by extracting all output decoupling zeros. 3. Hi(—§—,E) has a least order polynomial system matrix representation Pi(p,g) in MFD form such that Pi(p,0) is a polynomial system matrix. 8l 4. HiF has a least order polynomial system matrix PiF derived from Pi(p,0) by extracting all output decoupling zeros. 5. PCL(S,€) is formed by inserting the blocks of Pi(s,e) (i = l, 2) into (2.1.8). This is a can- didate polynomial system matrix for Figure 5.1. 6. PCLS(S) is formed by inserting the blocks of PiS(S) (i = l, 2) into (2.1.8). This is a can- didate polynomial system matrix for Figure 5.2. 7. PCL(p,g) is formed by inserting the blocks of Pi(p,8) into (2.1.8). This is a candidate fre- quency scaled polynomial system matrix for Figure 5.1. 8. PCLF(p) is formed by inserting the blocks of PiF into (2.1.8). This is a candidate polynomial system matrix for Figure 5.3. 9. y is a generic output decoupling polynomial. 10. q is a generic lost polynomial. Theorem 5.1: Let H1(S,€) and H2(s,€) be two frequency scale rational matrices. Suppose that (5.1) and (5.2) hold. det(I + Hls(w)HZS(m)) ¢ 0 (5.1) det(I + H1F(m)H2F(m)) # O (5.2) Then (with the above and previous notation), PCL(s,€) and PCL(p,€) are polynomial system matrices for Figure 5.1, 82 and PCLS(s) and PCLF(p) are polynomial system matrices for Figures 5.2 and 5.3, respectively. The relations (5.3) and (5.4) for limiting forms of the closed loop characteristic polynomial hold. det TCL(s,O)aqlLS(s) . qZLS(S) - det TCLS(S) (5.3) ~ K1+K2 det TCLCP.0)GP ' qlLF(P)q2LF(p) det T (5.4) CLF(p) where Ki = deg det Ti(s,0), i = 1,2 . Furthermore, the lost poles of Figure 5.1 are the open loop lost poles of Hl(S,€) and H2(s,8). Proof: By Theorem 2.1.5, (5.1) and (5.2) guarantee that PCLS(S) and PCLF(p) are polynomial system matrices for the closed-loop slow and fast subsystems, respectively. From Corollary 3.1.1, det(I + Hl(m,€)HZQ”fi)) 6 =0 = det(I + H1F(w)H2F(w)) ¢ 0 Thus, det(I + Hl(m,8)H2(m,8)) # 0. Applying Theorem 2.1.5 shows that PCL(S,€) and PCL(p,8) are polynomial system matrices for the closed-100p system of Figure 5.1. Since Pi(s,0) is in MFD form, it follows that det Ti(s,0)ayis(s) - det TiS(S) where yiS(s) is the output decoupling polynomial of Pi(s,0). 83 Moreover, YiS = qiLS' Theorem 2.1.3 shows that det TCL(s,0)adet Tl(s,0)det T2(s,0)det(l + HlS(S)HZS(S)) aqlLS(s)q2LS(s)det TlS(s)det TZS O‘qu.s(S)qus(S)det TCLS(S) which proves (5.3). (5.4) is proved similarly. The last statement in the theorem follows from Theorem 2.1.4. Consider the lost slow poles of Figure 5.1, i.e , the output decoupling polynomial TS of PCL(s,O) (Note that the input decoupling polynomial is unity.) Then FS(S) = Yls(S)Y28(S) Theorem 5.2: With the hypothesis and notation of Theorem 5.1, all closed loop poles in Figure 5.1 can be approximated by either (5.5) or (5.6). 31(8) = si0 + Ai(€), l i i 5 K1 + K2 (5.5) p. + D.(8) Sj(8) = JO 8 ,J , 5 J 5 L1 + L2 (5.6) where Kc = deg det Ta(s,0) a = 1,2 La = deg det Ta(p.0)- deg det Ta(s,0) a = 1,2 Ai(8) + o as e + o, 1 s i 5 K1 + K2 Dj(€) + 0 as e + O, l S j 5 L1 + L2 84 K1 + K2 pjO is a root of gF(p), l S j L1 + L i IA IA 310 is a root of gS(s), l M 2 gS(S) = qlLS(S)q2LS(S)det TCLS(S) (5.7) gF(p) = qlLF(p)q2LF(p)det TCLF(p) (5‘8) grggfz Since the roots of a polynomial vary continuously with its coefficients, for every si0 satisfying (5.7) there is a root 31(5) of det TCL(s,g) satisfying (5.5). Similarly, for every pjo # O satisfying (5.8), there is a root pj(e) of det TCL(p,e) satisfying (5.9). If sj(e) = pj(e)/€, sj(e) satisfies (5.6). It remains to be shown that all roots of det TCL(s,e) = 0 satisfy (5.5) or (5.6). This is done by showing that deg det TCL(s,e) is the sum of the degrees of gs and gF. By Theorem 5.1, deg gS(s) deg det TCL(s,O) deg det T1(s,0) + deg det T2(s,0) + K 1 2 K +K 1 2 deg p gF(p) deg det TCL(p,O) deg det Tl(p,0) + deg det T2(p,0) Kl + K2 + Ll + L2 This shows that deg gF(p) = L1 + L2 85 Together, (5.7) and (5.8) determine Kl + K2 + Ll + L2 roots of det TCL(s,e) = 0. By Corollary 3.1.1, det(I +.Hl(w,e) H2(m,e)) # O for sufficiently small 8, and Theorem 2.1.5 can be applied to the system of Figure 5.1. deg det TCL(s,e) deg det Tl(s,e) + deg det T2(s,e) deg det Tl(p,0) + deg det T2(p,0) K1+K2+L1+L2 D In words, Theorem 5.2 shows that for sufficiently small a the closed-loop poles of the system of Figure 5.1 can be approximated by the closed-loop poles of the slow and fast subsystems and the lost poles. Corollary 5.1: Suppose Hl(s,e) and H2(s,e) satisfy the hypothesis of Theorem 5.1. Let G(s,e) be any point to point transfer matrix in Figure 5.1. Then G(s,a) is two frequency scale. Proof: The four conditions of Definition 3.0.1 are easily verified. D This corollary shows that Theorem 4.1.1 can be applied to the point to point transfer matrix G(s,e) when both slow and fast closed-loop subsystems as well as all lost poles are stable. Corollary 5.2: Suppose H1(s,e) and H2(s,a) satisfy the hypothesis of Theorem 5.1, and both the slow and fast 86 closed—loop subsystems as well as all lost poles are stable. Then, for sufficiently small positive 8, the closed-loop system of Figure 5.1 is stable. D The results of this section are illustrated with two simple examples. Example 5.1: Consider the transfer function of ExampleliOfll with unity feedback applied. Entire closed-loop transfer function: h (3,8) = s + 1 CL s + l + (E + 2)(ts + l) _ s + l €32 + (2 + Ze)s + 3 This has two poles: one which approaches - —%—, and one which asymptotically approaches - a Slow closed-loop transfer function: _ s + l hCLS(s) ‘ 25‘1‘3 This has a pole at - —%—. Fast closed-loop transfer function: h <>=—-zl CLF P p + This has a pole at -2. 87 Example 5.2: Consider the transfer function of Example.3.0.2 with unity feedback applied. 1 hCL(S’€) = I + (s + 1)(es + T) 1 (5.10) 832 + (1 + E)s + 2 This has two poles: one which approaches -2, and one which asymptotically appraoches - —%—. Slow closed-loop transfer function: _ 1' hCLS(S) ‘ s—+Z This has a pole at -2. The fast closed-loop transfer function (like the fast open- loop transfer function) is zero. However, there is a lost fast pole at p = -1. Note that -l is also a lost fast pole of (5.10). VI. APPLICATIONS VI 1. Steady State LQG Controller for a Singularly Perturbed System This section provides an interpretation of the time domain solution of an output feedback regulator problem for a singularly perturbed system [Id]. The solution employs the usual division of the problem into slow and fast sub- problems, followed by synthesis of the two subproblem solu- tions into a composite controller. A number of matrix manipulations are involved, making it not intuitively obvious why the solution works. The problem statement is: find a control law for the system. + A + B u + G w (6.1.1) 5‘1 = A11x1 12x2 1 1 8X2 A21Xl + A22x2 + Bzu + G2w y Hlxl + H2x2 + v such that the performance criterion J in (6.1.2) is minimized 1 tf T T J=lim t-t E f [zz+uRu]dt (6.1.2) t+‘°° f O t 0 o tf++-co where z = Clxl + sz2 The following conditions hold: 88 For quadratic-Gaussian problem [e.g., 89 All coefficient matrices (capital letters) in (6.1.1) are independent of t. F, Cl’ and 02 are constant and R is positive definite. w and V are independent, zero mean, stationary, white Gaussian noise processes with intensities I and V respectively. V is positive definite.- The input u and output y are available for feed- back purposes. fixed values of 5, this is a standard linear- ll]. It can be shown, however, that a slightly sub-optimal solution can be found by using a time scale decomposition. The slow subproblem is: find a control law for the system x3 = ons + Bens + Gd” yS = Hoxs + SouS + Now + u where A = A - A A '1 A 0 ll 12 22 21 _ -1 Bo ‘ B1 ' A12A22 B2 _ -1 Ho ‘ H1 ’ H2A22 A21 - -1. So _ -H2A22 B2 N = -H A '1 G I o 2 22 2 _ -l Go ‘ G1 A12A22 Gz 90 which minimizes the criterion . 1 f T J = 11m -————-- E j; ([C x + D u ] tf-+ 00 [C x8 + DO u ] + uS I‘Ru o S S) dt h C = c - C A '1A W ere o 1 2 22 21 _ -1 Co ‘ ’C2A22 B2 The fast subproblem is: find a control law for the system exf = A22Xf + 3211f + GZW yf = Hzxf + v which minimizes the criterion _ . 1 T Jf - 11m E—Z—t— E 1;: (X; C; szf + 11f Ruf) dt to+-w f’ o tf+ on It can be shown that the solution for the slow subprob- lem has the form us 'Fs x3 x3 = ons + Bens + Qs [Vs ’ Sous ‘ Hoxs] Similarly, it can be shown that the fast subproblem solution can be written 91 uf 'fof The matrices FS’ Ff, QS’ and Qf are found by least squares methods. We assume that stabilizing solutions to the two subproblems exist. After the above solutions have been computed, a com- posite control is formed: uC = -Fst - fof (6.1.3) XS = ons + BouC + QS [y - Souc - HoxS] E”‘f ' (A22 ' BZFf ' QfHZ) Xf + Qf [Y + Sosts‘ Hoxs] A block diagram for the entire closed-loop system is shown in Figure 6.1.1. It can be shown that the relative error in the criterion (6.1.2) for the system (6.1.1) with the controller (6.1.3) applied asymptotically approaches zero as e + 0. We assume here that the fast filter has no poles at the origin. Let Hl(s,e) be the transfer matrix from a to d with the fast filter removed in Figure 6.1.1. It can be shown that Hls(s) = S H1F(P) = H2(pI ’ A22) 32 92 Figure 6.1.1 -—————fi u \l Plant Y + -- XS Slow / F 4 . ‘ (2 %+ S .‘ - 1 filter < 11+ I H O L ... S 1.. b. > ° —:@ .(,, g Fast ‘Ff e 5 filter 4 0 ~ c d Let H2(s,€) be the transfer matrix for the fast filter by itself, that is, from c to b in Figure 6.1.1. Then st(s) = 'Ff ('A22 + BZFf + Qsz)-l Qf which is a constant independent of 3. Theorem 5.2 shows that all of the closed 100p slow poles are the lost slow poles of H1(s,e) which are clearly the poles of the slow design sub- system shown in Figure 6.1.2 (in addition to the lost slow poles of the plant). 93 Figure 6.1.2 Slow part of plant N) FS Slow 4——————— filter .m\ Also, the fast poles are the lost fast poles of the plant together with the poles of the system of Figure 6.1.3, which is the fast design subsystem. Figure 6.1.3 Fast part of plant -F Fast f filter The case when the fast filter has a pole at the origin can be treated in a similar way after H1(s,s) above is split into exact slow and fast components using Theorem 3.2.1. VI.2. Feedback Design Strategies Theorems 5.2 and 4.1.1 can be used to derive various feedback design strategies. Although there are many possi- bilities, only two will be discussed here. 94 Design Strategy #1: Let H(s,e) be a two frequency scale transfer matrix which has no unstable lost poles. Step 1: Design slow and fast cascade and feedback compensators so that the closed loop systems in Figure 6.2.1 and 6.2.2 have desirable characteristics including asymptotic stability. Figure 6.2.1 + CS(s) HS(s) F3(3). 'Figure 6.2.2 CF(p) ' HF

FF(p) The designs are subject to the constraints 6 (6.2.1) W> O m A 8 v II cF<0) "D f (6.2.2) '11 U) A 8 V l FF(0) Step 2: Form composite cascade and feedback compensa- tors in one of two different ways: 95 IK> F(s,e) FS(s) + FF( 3) - F or F(s,e) FS(s) F-l FF(es) if F is invertable Similarly, C(S,e) is computed from CS(s) and CF(p). These are combined to form the composite control of Figure 6.2.3. Figure 6.2.3 C(s,e)} ] H(s,e) + n ij .—.—___..——_—_ i F(S,e) I Result: The poles of the system in Figure 6.2.3 can be approximated from the lost poles of H(s,e) and from the poles of the systems in Figures 6.2.1 and 6.2.2. Further- more, any point to point transfer matrix in Figure 6.2.3 can be approximated along the imaginary axis from the cor- responding transfer matrices in Figures 6.2.1 and 6.2.2. Proof: Follows by direct application of Theorems 5.2 and 4.1.1. Note that the compensators have no lost poles. The main difficulty with the above strategy is the con- straints (6.2.1) and (6.2.2) which make the subproblems nonstandard. Although not-as satisfying theoretically, the next strategy circumvents this problem. 96 Design Strategy #2: Let H(s,e) be a two frequency scale transfer matrix with no unstable lost poles. Step 1: Design a controller PS(s) to stabilize Hs(s) and to shape the transfer matrix from a to b in a desirable way, as shown in Figure 6.2.4. Let T(s) be the transfer matrix from a to b. Figure.6.2.4 + a——————* >44 ' HS(S) b PS(S) Step 2: Design PF(p) to stabilize the system of Figure 6.2.5. Figure 6.2.5 T(s) HF(p) _ + Ps(w) é____ —_———- b-——————— 4* PF(p) < 97 Step 3: Form the composite control of Figure 6.2.6. Figure 6.2.6 .; T(s) L a ! i ':i{+ ; H(s,e) ¢' >b' ; I “f- ”: i 1 a * . a ! i ‘ I PS(S) ;, I 5 i --—--‘ ' PF(es) §< Result: Pole approximation holds as for Design Strategy #1. The transfer matrix from a' to b' is approximated at low frequencies by T(s). Proof: Direct application of Theoream 5.2 and 4.1.1 suffices. Theorem 3.2.1 is required if PF(p) has poles at the origin. This method is similar to the explanation of the LQG design in the previous section. In this case, however, the stable transfer matrix T(s) must be realized separately since the states of PS(s) do not replicate the states of Hs(s) in general. 98 V1.3. Numerical Examples If a numerically described transfer matrix is given, these methods cannot be applied unless a is introduced in some manner. This can be done quite freely in the single variable case (that is, for lxm or mxl transfer matrices). The next example will show that the problem is nontrivial for the multivariable case. Example 6.3.1: Suppose a system is described by the trans- fer matrix (6.3.1). Then the characteristic polynomial is (s—l). Suppose we perturb one of the numerators by e as in (6.3.2). The characteristic polynomial becomes (s-1)2. fi(s) = 3‘31 g._-2.1 (6.3.1) 2 2 s—-T s—-I H - H(s,e) = gff EgI (6.3.2) 2 2+6 :1 3:1 It might be argued that only existing poles have been doubled, but if unity feedback is placed around (6.3.1), then CLCP(s) OLCP(s) ~ det (1 + fi(s)) (6.3.3) (s-l) ' (£172 ' [(s+l)2 - 4] = s + 3 99 where CLCP and OLCP stand for ”closed loop characteristic polynomial" and "open loop characteristic polynomial", respectively. Note that (6.3.3) is a restatement of (2.1.10). Unity feedback around (6.3.2) gives the closed loop characteristic polynomial CLCP(s) = (9,-1)2 - 3:132 - [(s+l)(s+l+e) - 4] = 82+(2+e)s+e- 3 (6.3.4) (6.3.4) has two roots: -3 + 0(a) and l + 0(a). In other words, 1 is a lost (slow) pole of (6.3.2) and the system formed by placing unity feedback around (6.3.2). Thus, introducing a has created new poles in this case. V Given a transfer function h(s), there are two obvious ways of introducing e if the roots of the denominator seem to fall into fast and slow groups. Let h(s) be written * _ n(s) h(s) - ds(s) dF(s) (6.3.5) By equating coefficients and solving the resulting set of linear equations, (6.3.5) can be rewritten A 118(3) nF(S) h(S) = W + W + a We can now choose 51 > 0 arbitrarily and write 113(3) nF(-§-S) h(S, E) ‘-" W + ml-T + a 100 Now h(s,el) = h(s). If 61 is chosen so that it represents an "average" of the ratio of magnitudes of slow and fast poles, then dS(s) and dF(7§f) will have zeros which are of approximately the same magnitude. For compu- tational simplicity, however, it is best to choose 81 = 1. This method can be applied to a transfer matrix of arbitrary dimensions without encountering the problem of Example 6.3.1. Suppose that the zeros of the numerator of h(s) also fall fast and slow groups. Let h(s) be written A nS(s) nF(s) h(s) = dSZ§) dFTs) Here we must have deg nS 5 deg dS' a can be introduced: as ns(s) nF(?I) es h(s,8) Again, 81 is arbitrary and h(s) = h(s,81). This cannot be applied to multivariable transfer matrices because of the pole duplication problem. Note that all fast poles will be lost poles if nS/dS is strictly preper so that this method tends to give more simple reduced models. Also, the compu- tations are more simple than with the previous method. The next example applies the sum method to a nontrivial numerical system. Example 6.3.2: The transfer matrix fi(s) below is for an open loop unstable chemical reactor and C(s) is the contrtfller design derived in [12]. 101 . a (s) 3 (s) “(5) _ A t . 11 12 N21(s) 322(3) where Nll(s) = 29.24 + 263.3 fi12(s) = -3.146s3 - 32.6232 - 89.833 - 31.81 fiZI(s) = 5.67933 + 42.6732 - 68.843 - 106.8 fi22(s) = 9.433 + 15.15 _ 4 . 3 2 1(3) — s + 11.673 + 15.753 - 88.313 + 5.514 (s - .06318)(3 - l.99l)(s + 5.057)(s + 8.666) The four roots of A(s) are the poles of this system. They are all simple because the determinant N11N22 - N12N21 has 4(3) as an exact division. . 1 0 208 + 18 (3(3) = __ S -(203 + 42) -16 Note that C(s) is second order. The exact characteristic polynomial of the closed loop system is 32 A(s) - det [I I H(s) 0(3)] 1 2 = -K(3) - [447315 - 59914503 - 163897003 3 4 5 6 + 27313603 + 11118303 8 + 199.8439 + 31°] - 121805003 7 - 4253093 + 1750713 + 11114.053 102 The roots of the polynomial in brackets are .06321 * -8.666 * 1.991 * -4.9128 * -1.0098 : j.075002 -2.6739 -4.l631 -62.567 -116.89 The four roots marked with asterisks are cancelled by A(S), leaving the remaining roots as the closed loop poles. The slight discrepancies are caused by round-off errors. We now factorize A(S) as 68(3) AF(s) where 18(3) = (s - .06318)(s - 1.991) = s2 - 2.054183 + .12579 AF(S) = (3 + 5.057)(s + 8.666) 52 + 13.7233 + 43.824 By equating coefficients of like powers of s and solving the resulting systems of linear equations, A _ 5.98648 - 8.573393 7.5399 + .8573393 3 _ -.710036 - 2.082773 -5.51215 - 1.063233 ‘&2(S) ‘ 18(5Y + AF(S) (6-3°6b) ‘ _ -2.55133 + 1.0815473 39.82326 + 4.597453 103 ‘ _ .349037 + .0515488 -1.16232 - .0515483 H22(S) - AS(S) + AF(Sj (6.3.6d) H(s,€) is formed as described above. For simplicity, El is taken as 1. HS(s) can now be computed (H(s,€) need not be witten out): 5.98648 - 8.573393 7 5399 “311(3) = 18(3) + 437824 = .1720532 - 1.210763 + 6.11227 43(8) Similarly, H (S) = -.125779s2 - 1.8244s - .725858 812 05(8) H (S) = .90871s2 - .7851063 - 2.43702 $21 03(8) H (S) = -.026szzss2 + .106033 + .345701 322 AS(s) It is easily seen that HF(p) consists of the right hand terms of the sums (6.3.6) with 3 replaced by p. For.example, . _ 7.5399 + .857339p hF11(P) ‘ AF(p) ' The compensator C(s) has no fast poles. Thus, Cs(s) = C(3) and _ - O 20 CF(p) = -20 0 104 The slow and fast closed loop characteristic polynomials are computed in the same manner as for the entire system. The slow closed loop characteristic polynomial is: 3248(3) - det [I + HS(s) CS(s)] K—%§7 - [43.893436 + 796.15255 + 1491.5434 3 - 2333.14s3 - 6660.68s2 - 3163.983 + 257.77] The roots of the polynomial in brackets are: .070699‘ * 1.98732 * -.856861 -l.9ll92 -1.61333 -15.842 Again, the roots marked by asterisks are cancelled (with some round-off discrepancy) by 48(3). The fast closed loop characteristic polynomial is: AF(p) - det [I + HF(p) CF(p)] l 2 + 4673.89p + 45126.2p + 125955] The roots of the polynomial in brackets are: 105 -5.02555 * -8.66673 * -29.7445 -97.2232 Table 6.3.1 gives a comparison of exact and approximate poles. 'Note that, as may be expected, for poles of intermediate magnitude. Table 6.3.1 the approximation is worst Exact roots Appoximate roots -1.0098 ij.075002 -.856861, -l.61333 -2.6739 -1.91192 -4.l63l -15.842 -62.567 -29.7445 -116.89 .2232 An example of the product method is included for completeness. Example 6.3.3: Let h(s) = (s-I§(:+10)' « + h(s) = §:% §£10 . Then _ 3+1 1 MS”) ‘ s—-'I'Es_+IU where El = 1. _ 1 3+1 hS(S) - I0" s-I h()= 1 PP p—+TU A Factorize h as 106 If static negative feedback of 20 is placed around h(s) then the characteristic polynomial is 2 (s-l)(s+10) + 20(s+1) = s + 293 + 10, yielding poles at -.34903 and -28.65. The slow closed loop pole is computed: s - l + 20 ° f% (3+1) = 3s + l, which gives a slow pole at s = -.3333. Similarly, the fast closed loop pole is given by: p + 10 + 20 = p + 30 = 0. The approximate poles are in good agreement with the exact poles. V11. CONCLUSION Some advantages of this approach are now apparent. The LQG regulator structure of Section V1.1 became intuitively obvious because internal structures for the plant and con- troller blocks were removed. Similarly for the design strategies of Section VI.2. The sum form decomposition of Section V1.3 corresponds to an exact block diagonalization of a system in state space form into slow and fast blocks. Again, no internal model is needed. There are some basic questions which still remain, however. The first is one which should be easily answered. It was shown in Section 111.3 that a transfer matrix described by a set of singularly perturbed state space equations is two frequency scale. The converse statement that any two frequency scale transfer matrix has a singularly perturbed state space description is most likely true. Approaching the problem through MFD methods runs into some technical difficulties. Theorem 3.2.1 reduces the question to that of realizing a "regularly perturbed" transfer matrix with a state space system whose matrices are analytic in E. To see this, let a two frequency scale rational matrix H(s,e) be written as guaranteed by Theorem 3.2.1: H(S,€) = H1(S,e) + H2(es,e) + D(e) (7.1) 107 108 Suppose H1(S,€) = Cl(e)(sI - Al(e))'lBl(e) and H2(p E) = C2(€)(PI ' A2(€))-le(e) where the matrices Ai’ Bi’ and Ci are analytic at e = 0. Then H(s,e) = c‘18Q(s)1 (7.2) A number of questions arise now, the most important being under what conditions the two factors in (7.2) are proper. This is not guaranteed by having 88* and SF* proper. Given that the two factors are proper, one can proceed as in V1.3 to introduce c. There will be no additional poles created in this case. Once the question of properness is answered, a "best" choice of the matrices P, Q, and R in (7.2) is needed. 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