EWDTS 2009, Moscow, Russia

1. EWDTS 2009, Moscow, Russia Analysis of the Control Vector Optimal Structure for a Minimal-Time Circuit Optimization Process A.M. Zemliak1,2, M.A. Torres1, T.M. Markina2 1Puebla Autonomous University, Mexico 2National Technical University of Ukraine, Kiev, Ukraine

2. Outline • 1. Introduction • 2. Time-optimal design problem formulation • 3. Acceleration effect • 4. Lyapunov function definition • 5. Optimal strategy prediction • 6. Conclusions

3. 1. Introduction The problem of the computer time reduction of a large system design is one of the essential problems of the total quality design improvement. The traditional design approach for the system design has two determined parts: system analysis and optimization procedure. The new idea was defined by means of the redefinition of the circuit optimization problem as the controllable dynamic process.

4. 2. Time-optimal design problem formulation Any original dependent parameter can be defined as independent or dependent. The control vector is the key of new methodology. a) The model of the system: (1) b) The parametric optimization procedure: (2) where The control functions: ,

5. Continuous form • The optimization procedure: • (3) • The model of the system: • (4)

6. The functions are determined for the gradient method as: • (5)

7. Discrete form • Gradient method • (6) • (7) • Where • (8) • where

8. 3. Acceleration effect 3.1 Two dimensional problem Fig. 1. Topology of a simplest electronic circuit The non-linear element has the following dependency :

9. Optimization procedure: (9) Model of circuit: (10)

10. (a) (b)Fig. 2. Trajectories for the traditional strategy (solid line) and for the modified traditional strategy (dash line) for: (a) Xin=(1,1); (b) Xin=(1,-1) Negative value of coordinate X2 is the special condition to obtain an additional acceleration effect.

11. TABLE 1 Structural basis of design strategies for initial vector = (1, 1) TABLE 2 Structural basis of design strategies for initial vector = (1, -1)

12. 3.2 N - dimensional problemPassive circuits (a) (b) Fig. 3. Optimal strategy potential computer time gain. 1 - Gradient method, 2 - Newton method, 3 - DFP method. (a) without an additional acceleration effect; (b) with an additional acceleration effect.

13. Active circuits Fig. 4. Circuit topology for three-stage transistor amplifier.

14. (a) (b) Fig. 5. Optimal strategy potential computer time gain. 1 - Gradient method, 2 - DFP method. (a) without an additional acceleration effect; (b) with an additional acceleration effect.

15. The acceleration effect serves as the first principal element of the time-optimal design algorithm construction The second problem is the optimal switch point position between different design strategies

16. 4. Lyapunov function definition What is the structure of the optimal control vector U ? The minimal time problem is correlated with the more general problem of stability of each trajectory. There is a well known idea to study of any dynamic process stability by means of the Lyapunov direct method. The design process is defined now as a dynamic controllable system. It is proposed to use a Lyapunov function of the design process for the optimal algorithm structure revelation, in particular for the optimal switching points searching. There is a freedom of the Lyapunov function choice because of a non-unique form of this function.

17. Let us define the Lyapunov function of the system design process • (1)-(2) by the following expression: • (11) • where is the stationary value of the coordinate • Let us define other variables . • In this case the formula (11) can be rewritten as: • (12) • This function satisfies all of the conditions of the standard Lyapunov function definition. • i) V(Y) >0, • ii) V(0)=0, and • when .

18. We need to construct other form of the Lyapunov function that doesn’t depend on the unknown stationary point. Let us define two new forms of the Lyapunov function by the next formulas: (13) (14) where F(X,U) is the generalized cost function of the optimization procedure. The last formula can be used when the general cost function is non negative and has zero value at the stationary point a. i) V(X,U)>0 ii) = 0 in the stationary point iii) ? However we can consider, from the practical experience, that the function V(X,U) increases in a sufficient large neighborhood of the stationary point.

19. 5. Optimal strategy prediction We can minimize the time of the transition process by means of the special choice of the right hand part of the principal system of equations • It is necessary to change the functions by means of the control vector U selection to obtain the maximum speed of the Lyapunov function decreasing (the maximum absolute value of the Lyapunov function time derivative ). • we can define now more informative function as a time derivative of Lyapunov function relatively the Lyapunov function: (15)

20. Examples Fig. 6. Three-node nonlinear passive network.

21. Fig. 7. Behavior of the functions V(t) and W(t) during the design process for seven different switch points (from 6 to 12).Control vector structure (111); (000). TABLE 3Number of iterations and computer time for strategies with different switch points

22. Fig. 8. Four-node nonlinear passive network.

23. Fig. 9. Behavior of the functions V(t) and W(t) during the design process for seven different switch points (from 30 to 36).Control vector structure (1111); (0000).(The function W(t) for 4th graph has a maximum absolute value leading of the 54th integration step) TABLE 4Number of iterations and computer time for strategies with different switch points

24. Fig. 10. One-stage transistor amplifier.

25. Fig. 11. Behavior of the functions V(t) and W(t) during the design process for seven different switch points (from 33 to 39).Control vector structure (111); (000).(The function W(t) for 4th graph has a maximum absolute value leading off the 25th integration step) TABLE 5Number of iterations and computer time for strategies with different switch points

26. Fig. 12. Two-stage transistor amplifier.

27. Fig. 13. Behavior of the functions V(t) and W(t) during the design process for seven different switch points (from 7 to 13).Control vector structure (11111); (00000); (11111).(The function W(t) for 4th graph has a maximum absolute value leading off the 55th integration step) TABLE 6Number of iterations and computer time for strategies with different switch points

28. Fig. 14. Three-stage transistor amplifier.

29. TABLE 7. Data of some strategies with different switch points TABLE 8. Data of some strategies with different switch points Fig. 15. Dependence of time derivative of Lyapunov function for different switch points

30. Fig. 16. Transistor amplifier

31. TABLE 9 Data of some strategies with different switch points Fig. 17. Time derivative of Lyapunov function dependency

32. 6. Conclusions • The problem of the minimal-time design algorithm construction can be solved adequately on the basis of the control theory. The design process in this case is formulated as the controllable dynamic system. • The Lyapunov function of the design process and its time derivative include the sufficient information to select more perspective design strategies from infinite set of the different design strategies that exist into the general design methodology. • The Lyapunov function and a special function W(t) were proposed to predict the structure of a time optimal design strategy. • These function can be used as the main instrument to construct the optimal sequence of the control vector switch points.

33. Thanks for your attention. Questions ?