Normed Distances and Their Applications in Optimal Circuit Design

Transcription

1 Optimization and Enineerin, 4, , 2003 c 2003 Kluwer Academic Publishers. Manufactured in he Netherlands Normed Distances and heir Applications in Optimal Circuit Desin ABDEL-KARIM S.O. HASSAN Department of Enineerin Mathematics and Physics, Faculty of Enineerin, Cairo University, Giza 12211, Eypt aso hassan@ Received December 13, 2000; Revised Auust 8, 2002 Abstract. A new eometric method for optimal circuit desin is presented. he method treats the optimal desin problem throuh the concept of normed distances from a feasible point to the feasible reion boundaries in a norm related to the probability distribution of the circuit parameters. he method treats directly the nonlinear feasible reion boundaries without any reion approximation. he normed distances are found throuh the solution of a nonlinear optimization problem. he sufficient optimality conditions for this optimization problem are established and an ordinary explicit formula for the normed distance is also derived. An iterative boundary search technique is used to solve the nonlinear optimization problem concernin the normed distances. he converence of this technique is proved. Practical circuit examples are iven to test the method. Keywords: optimization optimal circuit desin, convexity based optimization, worst case desin, desin centerin, yield 1. Introduction he oal of optimal circuit desin is the circuit s robustness aainst the unavoidable statistical variations, which occur durin the manufacturin process. hese variations may cause some of the manufactured circuits to violate the desin specification bounds. he percentae of the manufactured circuits, which satisfy the desin specifications, is called the production yield. In eneral, optimal circuit desin treats the problem of yield maximization prior to the manufacturin process. For example, desin centerin is an important optimal desin problem, which seeks the optimal circuit parameter values that maximize the yield. he methods dealin with the optimal circuit desin can be classified into two approaches, in eneral. he first approach is based upon the Monte Carlo method for the estimation of the yield values durin the optimization process (Sinhal and Pinel, 1981; Hocevar et al., 1984; Styplinski and Oplaski, 1986; Yu et al., 1987; Zaabab et al., 1995). he Monte Carlo based methods require lare computational effort and suffer from lack of reliable information for yield maximization. he second approach is classified as eometric approach (Bandler and Abdel-Malek, 1978; Bandler and Chen, 1988; Director et al., 1978; Abdel-Malek and Hassan, 1991; Abdel-Malek et al., 1997, 1999; Antreich et al., 1994; Sapatnekar et al., 1994; Seifi et al., 1999; Hassan and Rabie, 2000), and it treats the problem in a eometric manner under the assumption that the feasible reion in the circuit parameter space where the desin specifications are satisfied is bounded and convex. he simplicial approximation

2 198 HASSAN approach of Director et al. (1978) approximates the feasible reion by a polytope, and then the center of the larest hypersphere that can be inscribed within the polytope is used as a desin center. Bandler and Abdel-Malek (1978) used hyperboxes to describe the tolerance reion. he feasible reion is approximated by linearizin the constraints at the vertices of the hyperboxes. he yield is estimated throuh the evaluation of the hypervolume of the feasible part of the hyperboxes. he ellipsoidal technique of Abdel-Malek and Hassan (1991) approximates the feasible reion by a hyperellipsoid, which is determined by eneratin a sequence of decreasin volume, different shape and center hyperellipsoids. he desin center is the center of the final hyperellipsoid. Several sinificant modifications to the ellipsoidal technique have been made (Abdel-Malek et al., 1997, 1999; Hassan and Rabie, 2000). he Sapatnekar, Vaidya and Kan method (Sapatnekar et al., 1994), updates an initial polytope containin the feasible reion by eneratin a set of tanential hyperplanes to the feasible reion boundary. he center of the polytope is then found by maximizin a certain potential function. In this context, the concept of normed distances from a feasible point in the parameter space to a feasible reion boundary has been frequently used to develop eometrical techniques for optimal circuit desin (Brayton et al., 1980; Antreich et al., 1994; Seifi et al., 1999). hese distances are measured in a well defined norm that corresponds to certain probability distribution of the circuit parameters. his probability distribution simulates the statistical variations that affect the circuit parameters durin the manufacturin process. he normed distances are found, in eneral, throuh the solution of a nonlinear optimization problem. he solution process leads to the location of a certain boundary point. he distance from a iven feasible point to the located boundary point in the norm used ives the required normed distance. he problem can be simplified by approximatin the feasible reion with a polytope (Brayton et al., 1980; Seifi et al., 1999). Antreich et al. (1994) treated the nonlinear optimization problem throuh an optimization packae. However the solution process is facin some problems like numerical ill-conditionin, effects of simulator inaccuracies and the solution localization. In this paper, a new method to obtain the normed distances is introduced and exploited in optimal circuit desin. he method treats directly the nonlinear feasible reion boundaries without any reion approximation. he normed distances are found, in eneral, throuh the solution of a nonlinear optimization problem. he sufficient optimality conditions for this optimization problem are used to derive the conditions to be satisfied by the solution point which is in the same time a boundary point. An iterative boundary search technique is used to locate the required boundary (solution) point. Hence, the optimization problem can be solved and the normed distances can be found. he converence properties of the boundary search technique are proved. Practical circuit examples are iven to test the method. he paper is oranized as follows: in Section 2 the fundamental concepts of the optimal circuit desin problem are introduced. Also, the normed distances are defined throuh the solution of nonlinear optimization problem. In Section 3, the sufficient optimality conditions for the optimization problem concernin the normed distances are established. An ordinary explicit formula for the normed distance is also derived. A boundary search technique is presented in Section 4 to locate the boundary points required for findin the normed distances. Based on the normed distances, the desin centerin problem is formulated and treated in Section 5. Practical circuit examples are iven in Section 6.

3 NORMED DISANCES AND HEIR APPLICAIONS Normed distances and optimal circuit desin In optimal circuit desin, the circuit is manipulated throuh some desinable parameters x R n. Circuit performance properties p R p are functions of circuit parameters. In eneral, the performance of a iven parameter set has to be evaluated by numerical circuit simulations: p : x p(x) (1) Desin specifications are characterized by specifyin bounds on the circuit performance properties. Generally, these desin specifications define a reion in the parameter space called the feasible (constrained) reion R C and can be defined as: where R C ={x R n f (x) 0} (2) f = ( f 1, f 2,..., f m ):R n R m For optimal circuit desin, the circuit parameters are assumed to be randomly distributed with joint probability density function (pdf) φ(x, v) where v is the distribution parameters, e.. various means, variances and correlation coefficients. Usin the probability density function of the circuit parameters, the production yield can be defined as: Y (v) = φ(x, v) dx (3) R C For desin centerin, the pdf is assumed to be dependent only on the nominal parameter vector x 0 R n. he desin centerin problem can be formulated as: max x 0 [ ] Y (x 0 ) = φ(x, x 0 ) dx R C With reards to practical requirements, circuit parameters are assumed to be normally distributed with pdf iven by: φ(x, x 0 ) = 1 (2π) n/2 det C exp[ 0.5(x x0 ) C 1 (x x 0 )] (5) where x 0 R n is the nominal parameter vector and C is (n n) covariance matrix which is positive definite. Other distributions like the unimodal are commonly approximated by normal pdf s. In particular, skew distributions are approximated by lonormal pdf s which are transformed into normal pdf s (Antreich et al., 1994). (4)

4 200 HASSAN here exists a correspondence between the level contours of a iven probability density function and a particular norm (Brayton et al., 1980). For example, the level contours of the normal pdf iven by (5) can be described usin the l 2 -norm such that: x = x C 1 x (6) where C is the covariance matrix and x R n. Also, the l -norm can be used to describe the uniform distribution. Assume that the circuit parameters are normally distributed, hence by usin the pdf norm iven by (6), the distance between the two points x 1, x 2 R n will be iven by: d(x 1, x 2 ) = x 1 x 2 = (x 1 x 2 ) C 1 (x 1 x 2 ) (7) Let x 0 R n be a feasible point, i.e., x 0 R C where R C is the feasible reion, which is iven by (2). Usin the norm defined by the normal pdf, the distance between the point x 0 and the feasible reion boundary ( f i (x) = 0, i = 1, 2,...,m) is iven by Brayton et al. (1980) and Antreich et al. (1994): β i = min x {d(x 0, x) f i (x) = 0} (8) he normed distance β i is denoted in Antreich et al. (1994) as a worst case distance as it is the minimal distance from x 0 to violate the constraint boundary f i (x) = 0 in the norm defined by the pdf. Several optimal desin problems can be formulated by the aid of normed distances (see Antreich et al., 1994). For example, he desin centerin problem can be eometrically formulated by usin the normed distances as follows: ( ) max min β i (9) x 0 i=1,2,...,m 3. Mathematical basis for findin the normed distances By usin Eq. (7) and droppin the suffix i for simplicity, the distance β between the point x 0 and the hypersurface f (x) = 0 is formulated as: β = min (x x 0 ) C 1 (x x 0 ) (10) x such that f (x) = 0 i.e., β will be the optimal value of the objective function of the constrained optimization problem (10).

5 NORMED DISANCES AND HEIR APPLICAIONS 201 he followin proposition ives the conditions to be satisfied by the solution of problem (10). hese conditions are derived on the basis of the second-order sufficient optimality conditions (Rao, 1984; Fletcher, 1990). Proposition 3.1. x solves problem (10) if: (i) f (x ) = 0 (11) (ii) x x 0 = where, βc C (12) = f (x ), (13) and (x x 0 ) β = C (14) (iii) he matrix L iven by L = 1 β C 1 1 C H (15) is positive definite where H is the Hessian matrix o (x) evaluated at x. Proof: Condition (i) of the proposition states that x is a feasible point for problem (10). By usin the method of Larane multipliers, the laranian function will be: L(x,λ) = (x x 0 ) C 1 (x x 0 ) + λf (x) (16) he first-order necessary conditions for the existence of a local minimizer at the feasible point x will be (Rao, 1984; Fletcher, 1990): x L x=x = 0 C 1 (x x 0 ) (x x 0 ) C 1 (x x 0 ) Postmultiplyin (17) by C and rearranin we et: + λ = 0 (17) (x x 0 ) λ = C (x x 0 ) C 1 (x x 0 ) (18)

6 202 HASSAN Substitute for λ in (17) and rearranin, then: C 1 (x x 0 ) = (x x 0 ) C (19) Premultiply both sides by (x x 0 ), then: Hence: (x x 0 ) C 1 (x x 0 ) = [ (x x 0 )] 2 β = C (x x 0 ) C 1 (x x 0 ) = (x x 0 ) C (20) (21) It is to be noticed that i (x 0 ) < 0 and f (x) is convex, then (x x 0 ) > 0, then: β = (x x 0 ) C (22) Without loss of enerality we will assume that f (x 0 ) < 0 and we assume that f (x) is convex then β will be iven by (22). By usin Eqs. (19) and (22) we et: hen: C 1 (x x 0 ) = β (23) C (x x 0 β ) = C (24) C Hence condition (ii) of Proposition 3.1 is proved. Condition (iii) of Proposition 3.1, will be derived by usin the second-order sufficient condition (Rao, 1984; Fletcher, 1990), where, x ( 2 x L x=x ) x > 0 (25) x = 0 (26)

7 NORMED DISANCES AND HEIR APPLICAIONS 203 We first find the Hessian matrix of the laranian function, ( x 2L x=x ). By usin Eq. (17), then: 2 x L x=x = C 1 (x x 0 ) C 1 (x x 0 ) [C 1 (x x 0 )] [C 1 (x x 0 )] [(x x 0 ) C 1 (x x 0 )] λh (27) where H is the Hessian matrix o (x) evaluated at x. By usin Eqs. (21) and (22), the value of λ iven by Eq. (18) will be: 1 λ = C (28) hen substitute into (27) usin (23) and (28): 2 x L x=x By usin (29) and (26), then: x ( 2 x L x=x = C 1 β β C 1 H (29) C ) x = x [ ] C 1 β 1 H x (30) C If the matrix L = [ C 1 β 1 H] is positive definite, then (30) will be positive and C condition (25) will be satisfied. Hence, the proof is complete. Based on Proposition 3.1 the followin remarks are made: (i) From the second condition of the proposition, it is evident the radient of the function f (x) evaluated at the solution point x must be parallel to the vector C 1 (x x 0 ). (ii) he expression for the normed distance iven by (22), i.e. β = (x x 0 ) is similar to C the ordinary expressions used by Brayton et al. (1980) and Seifi et al. (1999). 4. A search technique for the normed distances In this section an iterative search technique will be used to solve the optimization problem (10). he technique exploits the work presented in Abdel-Malek et al. (1997) to enerate a sequence of boundary points {x k } such that f (x k ) = 0. he sequence {x k } converes to the point x which solves problem (10).

8 204 HASSAN he technique starts with a point x 1 R n such that f (x 1 ) = 0. Startin from x 1, a point x C R n is found and is iven by: x C = x 1 + γ C 1 1 C 1 (31) where and 1 = f (x 1 ) γ = θβ 1, θ (0, 1) β 1 = (x 1 1 x0 ) C 1 1 (32) In eneral f (x C ) 0. After x C is obtained a line search startin from x C in the (x 0 x 1 ) direction is carried out to find a point x 2 R n such that f (x 2 ) = 0. his iteration is repeated until converence occurs and a point x f is found. An iteration of the technique can be iven by the followin steps: C x C = x k + γ k C k k x k+1 = x C + µ k (x 0 x k ) f (x k ) = f (x k+1 ) = 0 where µ k is the step of the line search startin from x C in the (x 0 x k ) direction. hus a sequence {x k } is enerated where x k R n and f (x k ) = 0. he eneration of this sequence can be obtained by the followin equation: (33) C x k+1 = x k + γ k + µ k (x 0 x k ) (34) C k k Equation (34) can be written as: x k+1 = F(x k ), k = 1, 2,... (35) where F is a transformation from R n into itself. A fixed point x f R n for this transformation F is defined as Josi et al. (1985): x f = F(x f ) (36)

9 NORMED DISANCES AND HEIR APPLICAIONS 205 Lemma 4.1. he fixed point x f of the transformation F iven by (34) and (35) satisfies: x f x 0 = βc f f C f (37) where β = f (x f x 0 ) f C f (38) Proof: he fixed point x f satisfies x f = F(x f ), hence usin (34): x f = x f + γ C f C + µ f (x 0 x f ) (39) hen γ C f µ f C + (x 0 x f ) = 0 (40) Multiply both sides by f we et: γ C µ f C + f (x0 x f ) = 0 (41) or γ (x x 0 ) = µ f C = β (42) Substitute into (40), hence: (x f x 0 β ) = C C f. (43) It is to be noticed that the fixed point x f satisfies the first and the second conditions of Proposition 3.1 in the previous section. his implies that the fixed point of the transformation F iven by (34) and (35) is a KK point for problem (10) i.e. it satisfies the Kaursk- Kuhn- ucker necessary conditions (Bazaraa et al., 1993; Fletcher, 1990)

10 206 HASSAN Lemma 4.2. Assume that the constraint function f (x) is convex, then for the enerated sequence of boundary points in the boundary search technique iven by (33), the value of the line search step µ k is bounded by the relation µ k γ k C k k (x k x0 ) where γ is iven by (32) (44) Proof: From (34), the chane of the position vector at the kth iteration is iven by x k = γ C k + µ k (x 0 x k ) (45) C k k Multiply both sides by k we et: k x k = γ k C k + µ k k (x0 x k ) (46) hus we have x C k k (x k k x0 ) = γ k k (x k k x0 ) µ k (47) As f (x) is convex and assumin that f (x 0 ) < 0, then k (x k x0 ) > 0 and k x k 0. Hence, (47) implies that µ k γ k C k k (x k x0 ). (48) At the fixed point of the boundary search technique we have x k = 0, hence (48) is satisfied as equality. his case arees with Eq. (42). In the followin proposition we will show that the boundary search technique will convere only to a fixed point which satisfies the third condition of Proposition 3.1. Hence, the technique will convere only to a point that satisfies the second-order sufficient conditions of problem (10). Proposition 4.1. Startin from a point x 1 R n in the neihborhood of the fixed point x f where f (x 1 ) = 0, and assumin that the constraint function f (x) is quadratic and convex in this neihborhood such that the matrix C H is positive definite where H is the Hessian matrix o (x), then the sequence {x k } enerated by Eq. (34) converes to the fixed point x f which is the solution of the optimization problem (10).

11 NORMED DISANCES AND HEIR APPLICAIONS 207 Proof: Banach s condition principle (Josi et al., 1985) states that if a mappin F is a contraction mappin on a complete metric space (X, d) into itself, then F has a unique fixed point in X say x f. Moreover, if x 1 is an arbitrary point in X and the sequence {x k } is defined by: then x k+1 = F(x k ), k = 1, 2,... lim k x k = x f Under the assumption of a quadratic behavior of the function f (x) in the neihborhood of the fixed point x f, then: k = H x k + η, η R n (49) Substitute into (34), then x k+1 = I µ k I + γ C H C k k x k + µ k x 0 + γ Cη k C k (50) where I is the identity matrix. Usin the result of Lemma 4.2, we may assume that in the neihborhood of the fixed point x f the value of ( µ k ) is approximated by: µ k γ C C k k (x k k x0 ) γ (x x 0 ) γ β, (51) and the value of C is almost constant in a small neihborhood of x k k f, thus (50) can be written as: or x k+1 = I γ β I + γ C H C x k + γ β x0 + γ Cη f C f x k+1 = B x k + b (53) where B = I γ β I + γ C H C (54) (52)

12 208 HASSAN and b = γ β x0 + γ Cη f C f (55) Let F(x 1 ) = B x 1 + b F(x 2 ) = B x 2 + b hen F(x 1 ) F(x 2 ) = B x 1 x 2 (56) For F to be a contraction mappin, then B < 1, i.e., the spectral radius of B should be less than 1 (Gill et al., 1991), or the eienvalues of the B matrix are inside the unit circle. he matrix C H is symmetric and positive definite and can be diaonalized such that: C H = Q λ Q, Q Q = I (57) where λ is a diaonal matrix whose entries are the eienvalues of the matrix C H. Substitutin into (54) we et: B = Q I γ β I + γ λ C Q (58) he eienvalues of the B matrix are iven by: λi B = 1 γ β + γλ i C, i = 1, 2,...,n (59) where λ i is the correspondin eienvalue of the C H matrix. Hence, for F to be a contraction mappin, then: λ i B < 1, i = 1, 2,...,n (60) or 1 < 1 γ β + γλ i C < 1 (61)

13 NORMED DISANCES AND HEIR APPLICAIONS 209 i.e., 0 <γ < 2 1 β λ i C (62) Hence there exists a rane for γ to convere if: 1 β λ i f C f > 0 (63) his condition is satisfied if the fixed point x f satisfies also the third condition of Proposition 3.1, i.e., the matrix: L = 1 H β C 1 C (64) is positive definite. Hence the matrix: C L = 1 β I C H C (65) can be diaonalized such that: C L = Q 1 β I λ f C f and its eienvalues are positive such that: λ CL i = 1 β λ i f C f Q, Q Q = I (66) > 0, i = 1, 2,...,n where λ i is the correspondin eienvalue of the the matrix C H matrix. his completes the proof. 5. Desin centerin usin the normed distances As previously stated, the desin centerin problem seeks the nominal vector of the circuit parameters x 0 R n, which maximizes the yield. he desin centerin problem is eometrically formulated by usin the normed distances as follows (Antreich et al., 1994): ) max min (67) ( β i x 0 i=1,2,...,m

14 210 HASSAN where, β i, i = 1, 2,...,m is the nomed distance from x 0 to the constraint boundary f i (x) = 0 in the norm defined by the normal probability distribution (7). he desin centerin problem (67) can be solved by transformin it into a nonlinear prorammin problem by introducin an additional variable z and the problem will be: max z x 0,z such that (68) z β i 0, i = 1, 2,...,m o deal with problem (68), the normed distances (β i, i = 1, 2,...,m) from the feasible point x 0 to the feasible reion boundaries must be found. o find the value of β i a line search from x 0 in the C i direction, (where i = f i (x 0 )) can be made to locate a point x 1 R n on the boundary f i (x) = 0. he direction C i maximizes the linearization of the function f i (x) in the norm defined by the C matrix (Rao, 1984). his enables to start the boundary search technique (33) with a point x 1 near to the required boundary point (Abdel-Malek et al., 1997). Also, for the processin of the search technique (33), a suested value of γ is iven by: γ = θβ 1,θ (0, 1) where β 1 is iven by (32). his suested value is a ood choice of γ, since from (62) it is clear that the upper bound of the value of γ is reater than 2β. he value of θ used in the followin examples is Rractical examples 6.1. Desin of a two-section transmission line he iven approach of desin centerin by usin the normed distances is applied in the optimal desin of a two-section 10 : 1 quarter wave transmission line (Bandler and Chen, 1988). he constraint functions are iven by the reflection coefficient sampled at 11 frequencies normalized with respect to 1 GHz: {0.5, 0.6,...,1.5}. he manitude of the reflection coefficient at these frequencies should not exceed he characteristic impedences Z 1,Z 2 are the desinable parameters. he initial desin is iven by (1.71, 4.36) which is a feasible point. he desinable parameters are assumed to be normally distributed with covariance matrix iven by C = 1 [ ] he desin centerin process is performed by transformin problem (67) into a nonlinear prorammin problem (iven by (68)), and is solved by the aid of the optimization routines supplied by the IMSL packae. A FORRAN routine which implements the boundary search technique proposed in Section 4 is used. he solution of the desin centerin problem (67) was (2.396, ), which is considered as the desin center. o evaluate this desin center, the initial and the final yield values correspondin to the initial desin and the proposed desin center are estimated via the Monte Carlo method. he Monte Carlo method is performed usin 500 sample points assumin that the circuit parameters are normally (69)

15 NORMED DISANCES AND HEIR APPLICAIONS 211 able 1. Results of the optimal desin of the transmission line. Covariance matrix Initial yield (%) Final yield (%) C C/ C/ distributed with covariance matrices of different scales of the matrix C iven by (69). he results are iven in able Desin of a CMOS operational amplifier A CMOS operational amplifier (Gray and Meyer, 1984; Sapatnekar et al., 1994; Hassan and Rabie, 2000) is optimally desined usin the iven approach. he operational amplifier is shown in fiure 1. he widths w 1,w 5,w 6,w 7,w 8 of the transistors M 1,M 5,M 6,M 7,M 8 in addition to the biasin current I B and the compensation capacitor C c are considered as the desin parameters. he feasible reion is defined by the followin constraints Low frequenc ain 98 db Gain Bandwidth product 17 Mrad/sec Power dissipation 0.65 mw Area = w 1 + w 5 + w 6 + w 7 + w µm All the physical and eometrical constants necessary for the calculations are iven in Gray and Meyer (1984). he initial values for the circuit parameters are (100 µm, 100 µm, 100 µm, 100 µm, 100 µm, 20 µa, 5 pf). he desinable parameters are assumed to Fiure 1. A CMOS operational amplifier.

16 212 HASSAN able 2. Results of the optimal desin of the CMOS operational amplifier. Covariance matrix Initial yield (%) Final yield (%) C C/ be normally distributed with covariance matrix C = dia (12, 12, 12, 12, 12, 1.5, 1.5). he same procedures implemented in the previous example are used here. he solution of the desin centerin problem (67) was ( µm, µm, µm, µm, µm, µa, pf), which is considered as the desin center. o evaluate this desin center yield values are estimated for both the initial desin and the final desin via the Monte Carlo method with 1000 sample points. he results are iven in able Conclusions In this paper a new method for optimal circuit desin is presented. he method treats the optimal desin problem throuh the concept of the normed distances from a feasible point to the feasible reion boundaries in a norm related to the probability distribution of the circuit parameters. he evaluation of the normed distances is based on the solution of a nonlinear optimization problem. he new method put the mathematical basis for the solution of this nonlinear optimization problem in addition to derivation of an ordinary explicit formula to evaluate the normed distances. he solution process depends on the location of certain boundary points on the feasible reion boundary. hese boundary points are located by implementin a converent boundary search technique. he new method is applied to optimally desin practical circuit examples. he results show the effectiveness of this method in optimal desin of circuits. Nomenclature φ ν β λ γ θ η λ Joint probability density function of the circuit parameters. Vector of distribution parameters. Normed distance. Larane multiplier parameter. Constant. Constant. n-dimensional vector. (n n) diaonal matrix. References H. L. Abdel-Malek and A. S. O. Hassan, he ellipsoidal technique for desin centerin and reion approximation, IEEE ransaction on Computer-Aided Desin, vol. 10, pp , 1991

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