Abdel-Karim S. O. Hassan

The ellipsoidal technique for design centering and region approximation

A technique for design centering and feasible region approximation that is based on generating a sequence of ellipsoids of decreasing volume and preserves the property of containing a bounded convex feasible region is introduced. The technique converges to an ellipsoid the center of which is the proposed design center. The ellipsoid matrix can be used to give what is called a preferable covariance matrix, assuming a multinormal distribution of parameters. This covariance matrix is preferred since it significantly increases the production yield for the feasible region under consideration. In addition, an ellipsoidal region approximation can be obtained by scaling the final ellipsoid, which allows an inexpensive yield estimate using the Monte Carlo method. Numerical and practical examples are considered. >

A boundary gradient search technique and its applications in design centering

A boundary gradient search technique is introduced. The technique generates a sequence of points on the boundary of the feasible region converging to a fixed point. The gradient of the boundary at this fixed point has a prespecified directional ratio. The boundary search technique is exploited in a modification of the ellipsoidal technique for design centering. This modification allows the use of double-sided ellipsoidal sections instead of single-sided ellipsoidal sections. It improves the speed of convergence of the ellipsoidal technique. Also, a computationally inexpensive technique of determining the gradient of the gain constraints is introduced. Practical examples are given to demonstrate the new technique.

Non-derivative design centering algorithm using trust region optimization and variance reduction

Fluctuations in manufactured integrated circuit parameters may dramatically reduce the parametric yield. Yield maximization can be formulated as an unconstrained optimization problem in nominal parameter values, which is known as design centering. The high expense of yield evaluations, the absence of any gradient information, and the presence of some numerical noise obstruct the use of the traditional derivative-based optimization methods. In this article, a novel design centering algorithm is presented, which consists of a non-derivative unconstrained optimizer coupled with a variance reduction estimator. The used optimizer combines a trust region mechanism with quadratic interpolation and provides efficient use of yield evaluations. The stratified sampling technique is used to develop a lower variance yield estimator that reduces the number of circuit simulations required to reach a desired accuracy level. Numerical and practical circuit examples are used to demonstrate the efficiency of the proposed algorithm with respect to other methods in the same field.

Surrogate-Based Circuit Design Centering

Circuit design centering is one of the most important problems concerning the optimal design of circuits. Circuit design centering seeks nominal values of designable circuit parameters that maximize the probability of satisfying the design specifications (yield function). Design centering can be performed geometrically by finding the center of the feasible region (region in the designable parameter space where the design specifications are satisfied), or by maximizing the yield function explicitly. For all cases, the high expense of circuit simulations required obstructs the design centering process, especially for microwave circuits. To overcome this, computationally cheap surrogate-based models (e.g., space mapping, response surfaces, kriging, and neural networks) can be used for approximating the response functions or the yield function itself. In this chapter the design centering problem is formulated as an optimization problem, and the estimation of the yield function through several sampling techniques is explained. The difficulties facing the design centering process, especially for microwave circuits, are discussed, and the role of surrogate-based models in overcoming these difficulties is demonstrated. Special interest is devoted to space mapping surrogates and microwave circuit design centering. Some of the important surrogate-based circuit design centering approaches are reviewed with an overview of their theoretical bases. Tutorial and practical circuit examples are given to show the effectiveness of these approaches.

Normed Distances and Their Applications in Optimal Circuit Design

A new geometric method for optimal circuit design is presented. The method treats the optimal design problem through the concept of normed distances from a feasible point to the feasible region boundaries in a norm related to the probability distribution of the circuit parameters. The method treats directly the nonlinear feasible region boundaries without any region approximation. The normed distances are found through the solution of a nonlinear optimization problem. The 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 concerning the normed distances. The convergence of this technique is proved. Practical circuit examples are given to test the method.

A new hybrid method for optimal circuit design using semi-definite programming

In this article a new method for yield optimization (design centring) is introduced. The method has a statistical-geometrical nature, hence it is called hybrid. The method exploits the semi-definite programming applications in approximating the feasible region with two bounding ellipsoids. These ellipsoids are obtained using a two phase algorithm. In the first phase, the minimum volume ellipsoid enclosing the feasible region is obtained. The largest ellipsoid that can be inscribed inside the feasible region is obtained in the second phase. The centres of these bounding ellipsoids are used as design centres. In the second phase, an additional polytopic region approximation is constructed. A comparison between the obtained region approximations is given. Saving in the number of circuit simulations needed for yield optimization is also considered. Practical examples are given to show the effectiveness of the new method.

Statistical microwave circuit optimization via a non-derivative trust region approach and space mapping surrogates

A novel yield optimization technique for microwave circuits is presented. Yield optimization of microwave circuits is obstructed by the high expense of electromagnetic simulations required in yield evaluations in addition to the absence of any gradient information. In the proposed technique, surrogates using the generalized space mapping (GSM) algorithm is incorporated with a derivative-free trust region optimization method (NEWUOA: new unconstrained optimization algorithm). Moreover, a variance reduction technique is implemented in yield estimation process. Practical examples to demonstrate this new technique are included showing its efficiency.

A new approach for the selection of test points for fault diagnosis

A new criterion and an efficient algorithm for the selection of test points in multifrequency fault diagnosis of linear circuits are presented. The proposed criterion exploits the biquadratic nature of the response in terms of circuit parameters instead of the common use of first order sensitivities. Accordingly it is capable of handling catastrophic faults. Employing the proposed criterion, an efficient two-phase fault diagnosis algorithm is introduced. The first phase selects a set of test points and characterizes the response for possible faults. This is done without the simulation of a preselected set of faults. The second phase efficiently isolates on-line actual faults using test points without any computation. A test example is presented to demonstrate the effectiveness of the proposed criterion and algorithm.

Design centering and polyhedral region approximation via parallel-cuts ellipsoidal technique

A new technique for constructing a polyhedral approximation of the feasible region and finding the associated design center through a parallel-cuts ellipsoidal technique is presented. The linearizations of the feasible region boundary required to implement the parallel-cuts ellipsoidal technique are saved from one iteration to another in a non-redundant form. These linearizations are used in the construction of the parallel cuts as well as in the generation of an exterior polyhedral approximation of the feasible region at no additional cost. Numerical and practical examples are given to demonstrate the effectiveness of the new technique.

A novel surrogate-based approach for optimal design of electromagnetic-based circuits

A new geometric design centring approach for optimal design of central processing unit-intensive electromagnetic (EM)-based circuits is introduced. The approach uses norms related to the probability distribution of the circuit parameters to find distances from a point to the feasible region boundaries by solving nonlinear optimization problems. Based on these normed distances, the design centring problem is formulated as a max–min optimization problem. A convergent iterative boundary search technique is exploited to find the normed distances. To alleviate the computation cost associated with the EM-based circuits design cycle, space-mapping (SM) surrogates are used to create a sequence of iteratively updated feasible region approximations. In each SM feasible region approximation, the centring process using normed distances is implemented, leading to a better centre point. The process is repeated until a final design centre is attained. Practical examples are given to show the effectiveness of the new design centring method for EM-based circuits.