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An apparatus and method to generate and evolve canonical form expressions representing a characteristic of a given system. Static and dynamic behavior of non-linear electrical circuits can be modeled. Searching of canonical form expressions can use evolutionary algorithms, simulated annealing and Ta

An apparatus and method to generate and evolve canonical form expressions representing a characteristic of a given system. Static and dynamic behavior of non-linear electrical circuits can be modeled. Searching of canonical form expressions can use evolutionary algorithms, simulated annealing and Tabu searching.

대표청구항 ▼

1. A tangible computer-readable medium having recorded thereon statements and instructions for execution by a computer of a method to generate at least one mathematical expression describing a performance characteristic of a system, the system associated with variables and with pre-determined data r

1. A tangible computer-readable medium having recorded thereon statements and instructions for execution by a computer of a method to generate at least one mathematical expression describing a performance characteristic of a system, the system associated with variables and with pre-determined data related to the performance characteristic of the system, the method comprising steps of: generating at least one initial mathematical expression having a pre-defined canonical form and being a function of the variables, the at least one initial mathematical expression having operators operating on the variables, the operators being selected from a pre-defined group of operators, the at least one initial mathematical expression describing the performance characteristic of the system;wherein, the variables of the system are representable as a vector {right arrow over (x)} and the canonical form of an expression F({right arrow over (x)}) is representable as F⁡(x)=woffset+∑i=0n⁢wi×fi⁡(x)×NLi⁡(x), “n” being an integer, woffset, being an offset value, wi being weights, fi(x) including at least one of a polynomial function of the variables and a rational function of the variables, and NLi({right arrow over (x)}) being a non-linear function of the variables, with NL0(x)=1; generating calculated data using the at least one initial mathematical expression;calculating an output of a goal function in accordance with the pre-determined data and the calculated data;determining that the goal function is outside a pre-defined range; anditeratively performing the following steps a-c until an additional output of the goal function is within the pre-defined range:a. modifying at least one input mathematical expression in accordance with a search algorithm to produce at least one modified mathematical expression having the canonical form and being a function of the variables, the search algorithm to search at least the pre-defined group of operators to identify operators with which to modify the at least one input mathematical expression, the at least one input mathematical expression being the at least one initial mathematical expression in a first iteration of steps a-c, the at least one input mathematical expression being the at least one modified mathematical expression in subsequent iterations of steps a-c;b. generating additional calculated data using the at least one modified mathematical expression; andc. calculating the additional output of the goal function based on the additional calculated data and the pre-determined data. 2. The tangible computer-readable medium of claim 1 wherein, the goal function is a single objective goal function for minimizing error. 3. The tangible computer-readable medium of claim 1 wherein, the goal function is a multi-objective goal function for minimizing error and for minimizing complexity. 4. The tangible computer-readable medium of claim 1 wherein, the goal function is a multi-objective goal function for minimizing error and for constraining complexity. 5. The tangible computer-readable medium of claim 1 wherein, the goal function is a multi-objective goal function for constraining error and for constraining complexity. 6. The tangible computer-readable medium of claim 1 wherein, the goal function is at least for minimizing a normalized root mean square error between the pre-determined data and either the calculated data generated using the at least one initial mathematical expression or the additional calculated data generated using the at least one modified mathematical expressions. 7. The tangible computer-readable medium of claim 1 wherein, the step of modifying the at least one input mathematical expression in accordance with a search algorithm includes modifying the at least one input mathematical expression in accordance with an evolutionary search algorithm to produce the at least one modified mathematical expression. 8. The tangible computer-readable medium of claim 1 wherein, the step of modifying the at least one input mathematical expression in accordance with a search algorithm includes modifying the at least one input mathematical expression in accordance with a simulated annealing search algorithm to produce the at least one modified mathematical expression. 9. The tangible computer-readable medium of claim 1 wherein, the step of modifying the at least one input mathematical expression in accordance with a search algorithm includes modifying the at least one input mathematical expression in accordance with a Tabu search algorithm to produce the at least one modified mathematical expression. 10. The tangible computer-readable medium of claim 7 wherein, the goal function is an evolutionary multi-objective goal function. 11. The tangible computer-readable medium of claim 1 wherein, a step of simplifying the at least one modified mathematical expression is performed before generating additional calculated data. 12. The tangible computer-readable medium of claim 11, wherein the step of simplifying is done in accordance with a predicted sum of squares statistic process. 13. The tangible computer-readable medium of claim 7 wherein, the at least one initial mathematical expression and the at least one modified mathematical expressions are representable as trees, the trees having a pre-determined depth and a pre- determined branchiness, some of the at least one initial mathematical expression and the at least one modified mathematical expressions having introns. 14. The tangible computer-readable medium of claim 7 wherein, the step of generating at least one initial mathematical expression and the step of modifying the at least one input mathematical expression are performed in accordance with genetic programming principles. 15. The tangible computer-readable medium of claim 1 wherein, the system is an electrical system. 16. The tangible computer-readable medium of claim 1 wherein, the variables associated with the system include at least one of design variables, environmental variables and random variables. 17. A tangible computer-readable medium having recorded thereon statements and instructions for execution by a computer of a method to model an electrical circuit, the electrical circuit associated with circuit variables, with pre-determined data, with at least one pre-determined transient input waveform and with at least one pre-determined transient output waveform the method comprising steps of: generating at least one initial mathematical expression dependent on at least one of at least one of the variables, at least one of the at least one pre-determined transient input waveform and at least one of the least one pre-determined transient output waveform, the at least one initial mathematical expression having a pre-defined canonical form, the at least one initial mathematical expression having operators operating on the variables, the operators selected from a pre-defined group of operators, the at least one initial mathematical expression describing a transient behavior of the electrical circuit;wherein, the variables of the system are representable as a vector {right arrow over (x)} and the canonical form of an expression F({right arrow over (x)}) is representable as F⁡(x)=woffset+∑i=0n⁢⁢wi×fi⁡(x)×NLi⁡(x) “n” being an integer, woffset, being an offset value, wi being weights, fi({right arrow over (x)}) including at least one of a polynomial function of the variables and a rational function of the variables, and NLi({right arrow over (x)}) being a non-linear function of the variables, with NL0({right arrow over (x)})=1; generating calculated data using the at least one initial mathematical expression;calculating an output of a goal function in accordance with the pre-determined data and the calculated data;determining that the goal function is outside a pre-defined range: and iteratively performing the following steps a-c until an additional output of the goal function is within the pre-defined range:a. modifying at least one input mathematical expression in accordance with a search algorithm to produce at least one modified mathematical expression dependent on at least one of at least one of the variables, at least one of the at least one pre-determined transient input waveform and at least one of the least one pre-determined transient output waveform, the at least one modified mathematical expression having the canonical form, the search algorithm to search at least the pre-defined group of operators to identify operators with which to modify the at least one input mathematical expression, the at least one input mathematical expression being the at least one initial mathematical expression in a first iteration of steps a-c, the at least one input mathematical expression being the at least one modified mathematical expression in subsequent iterations of steps a-c;b. generating additional calculated data using the at least one modified mathematical expression; andc. calculating the additional output of the goal function based on the additional calculated data and the pre-determined data. 18. The tangible computer-readable medium of claim 17 wherein, the pre-determined data includes at least one of steady-state electrical circuit data, transient electrical circuit data and noise electrical circuit data. 19. The tangible computer-readable medium of claim 17 wherein, the electrical circuit is nonlinear. 20. The tangible computer-readable medium of claim 17 wherein, the at least one initial mathematical expression and the at least one modified mathematical expression are independent of a connectivity topology of the electrical circuit. 21. The tangible computer-readable medium of claim 17 wherein, the at least one initial mathematical expression and the at least one modified mathematical expression are symbolic models of the electrical circuit. 22. The tangible computer-readable medium of claim 17 wherein, the at least one initial mathematical expression and the at least one modified mathematical expression are behavioral models of the electrical circuit.