System and method for generating discrete-time model (DTM) of continuous-time model (CTM) for a dynamical system

• US 8,805,910 B2
• Filed: 03/04/2011
• Issued: 08/12/2014
• Est. Priority Date: 03/04/2011
• Status: Active Grant

• Current Assignee: Kylowave Incorporated
• Original Assignee: Kylowave Incorporated
• Inventors: Pimentel, Julio C. G., Gad, Emad
• Primary Examiner(s): Malzahn, David H
• Application Number:
  US13/040,624
• Publication Number:
  US 20120226726A1
• Time in Patent Office: 1,257 Days
• Field of Search: None
• US Class Current: 708/446
• CPC Class Codes: G05B 17/02 electric
• CPC Class Codes: G06F 2119/12 Timing analysis or timing optimisation
• CPC Class Codes: G06F 30/327 Logic synthesis; Behaviour synthesis, e.g. mapping logic, HDL to netlist, high-level language to RTL or netlist
• CPC Class Codes: G06F 30/3312 Timing analysis
• CPC Class Codes: G06F 30/343 Logical level

Abstract

Method and apparatus for processing continuous-time models (CTM) in a digital processing architecture is disclosed. The discrete state-space technique maps the CTM into the discrete-time model (DTM) and stores the states of the system in a sample time independent discrete state space set of matrices. The resulting state-space matrices can be processed in software or directly in hardware. The method disclosed is particularly suited to be used in automatic synthesis algorithms where a digital circuit is generated from an algorithm described in a high level language or model representation such as, for example, data flow or bond graph into a hardware description language (HDL), and the HDL model can be synthesized using system specific tools to generate an application specific integrated circuit (ASIC) or an FPGA configuration.

Claims

1. A method for execution by a processor, the method comprising:

retrieving from the memory a continuous time model (CTM) defining a continuous-time electrical or mechanical system;

determining a sampling time and CTM state-space matrices [A, B, C, D] for the CTM defined for a dynamical system ordinary differential equation mathematical formulation;

determining an order of approximation [i,j,k] parameters of the coefficients of an Obreshkov integration formula, to be used in determining a discrete time model (DTM) [A_dm, B_dm, C_dm, D_dm] to provide an approximation of accuracy and to meet A- or L-stability requirements defined for the dynamical system; and

discretizing the CTM into the DTM using the selected order of approximation [i,j,k], and the CTM state-space matrices [A, B, C, D] and sampling time (T) by calculating Obreshkov-coefficient and matrices [A_dm, B_dm, C_dm, D_dm] using Obreshkov-based polynomial matrices; and

storing the DTM representing the continuous-time electrical or mechanical system to the memory;

wherein subscript d in the state-space matrix refers to a matrix for the DTM and m subscript refers to the parameter k of the order of the discretization such that 0<

=m<

=k and the DTM is a fixed point or a floating point representation.

2. The method of claim 1 wherein the sampling time and the order of approximation [i,j,k] parameters are selected to meet A-stability requirements when the parameter i is set equal to j and L-stability requirements when condition i=j+1 or i=j+2 is satisfied.

3. The method of claim 1 further comprising computing the output signals y(nT) at the sampling time using the calculated Obreshkov coefficients and matrices [A_dm, B_dm, C_dm, A_dm], inputs u(nT) and its k-order derivatives u^(k)(nT).

4. The method of claim 2 wherein the DTM can then be implemented in hardware or software using the value of the inputs (u(nT)) and its k-order derivatives (u^(k)(nT)) at the sampling time to compute the output signals (y(nT)).

5. The method of claim 3 wherein the DTM state-space representation is obtained through an Obreshkov-based mapping to map the state-space matrices of the CTM into matrices of the DTM system where a CTM is described by the following state-space representation:

6. The method of claim 3 wherein the parameters (i, j, k) of the Obreshkov coefficients are used to configure a high-order approximation (order=i+j) for the CTM state transition matrix such that the approximation is unconditionally stable if and only if j−

2≦

i≦

j and where the parameter k, controls which derivatives of the input signal vanishes in the approximation and, therefore, independently controls accuracy of the forced response.

7. The method of claim 3 where in representation of a continuous-time system such as electrical system or mechanical system is provided as the transfer function representation defined by

8. The method of claim 3 wherein determining the DTM is implemented using fixed point representation the method further comprising:

scaling each element of the zero-order DTMs by a multiplication factor to increase the accuracy by a defined multiple; and

quantizing each element of the zero-order DTMs to remove floating point decimal components.

9. The method of claim 8 wherein scaling of the zero-order DTM matrices is scaled by, for example, a L2-norm dynamic range scaling method to increase accuracy of the model by ensuring the entries of the A, B, and C matrices are homogenous in magnitude and the DTM characteristics are insensitive to small perturbations in A, B, and C in comparison to their norms, and quantization is performed by truncation or rounding of the elements of the DTM matrices.

10. The method of claim 8 wherein an order of approximation [i,j,k] parameters of the coefficients of an Obreshkov integration formula is selected to be used in optimizing the tradeoffs between the accuracy requirements and the number of bits of the fixed-point representation used for the scaling of the DTM matrices.

11. The method of claim 3 wherein the order of approximation can be increased by instantiating zero-order DTMs of [A_dm, B_dm, C_dm, A_dm] where m can vary from 0 up to k.

12. The method of claim 11 wherein the output of each zero-order DTM is combined to as a parallel combination of k structures of zero-order DTMs where an output of each zero-order DTM (0 to k) is summed.

13. The method of claim 12 wherein the Obreshkov-based polynomial matrices [L_m, H_m] and [Q_m, G_m] are defined by:

14. The method of claim 13 wherein discretizing the CTM into the DTM further comprises:

determining [A_dm, B_dm, C_dm, A_dm] where

A_dm=L_m(AT)H_m<sup>−

1</sup>(AT)−

Q_m(AT)G_m<sup>−

1</sup>(AT)

B_dm=Q_m(AT)G_m<sup>−

1</sup>(AT)H_m(AT)(BT

C_dm=G_m<sup>−

1</sup>(AT)

D_dm=−

G_m<sup>−

1</sup>(AT)H_m(AT)(BT).

15. The method of claim 14 wherein the discrete-time state variable x*_m(k, T) at the sampling time are defined by:

w_m(n+1,T)=A_dmw_m(n,T)+B_dmu(n,T)

x*_m(n,T)=C_dmw_m(n,T)+D_dmu(n,T)

where DTM output is defined by;

16. An apparatus the apparatus comprising:

a processor for executing instructions stored in a memory coupled to the processor, the instructions for performing;

retrieving from the memory a continuous time model (CTM) defining a continuous-time electrical or mechanical system;

receiving a discrete time model (DTM) output and determining a sampling time (T) and CTM state-space matrices [A, B, C, D] defined for a dynamical system ordinary differential equation mathematical formulation;

determining an order of approximation [i,j,k] parameters of the coefficients of an Obreshkov integration formula, to be used in determining the DTM [A_dm, B_dm, C_dm, D_dm] to provide an approximation of accuracy and to meet A- or L-stability requirements defined for the dynamical system;

discretizing the CTM into a DTM using the selected order of approximation [i,j,k], and the CTM state-space matrices [A, B, C, D] and sampling time by calculating Obreshkov coefficients and matrices [A_dm, B_dm, C_dm, D_dm] using Obreshkov polynomial matrices; and

storing the DTM representing the continuous-time electrical or mechanical system to the memory;

wherein subscript d in the state-space matrix refers to a matrix for the DTM and m subscript refers to the parameter k of the order of the discretization such that 0<

=m<

=k and the DTM is a fixed point or a floating point representation.

17. The apparatus of claim 16 further comprising computing the output signals y(nT) at the sampling time using the calculated Obreshkov-based coefficients and matrices [A_dm, B_dm, C_dm, D_dm], inputs u(nT) and its k-order derivatives u^(k)(nT).

18. The apparatus of claim 16 wherein the order of approximation can be increased by instantiating zero-order DTMs of [A_dm, B_dm, C_dm, D_dm] where m can vary from 0 up to k.

19. The apparatus of claim 18 wherein the output of each zero-order DTM is combined to as a parallel combination of k structures of zero-order DTMs where the output of each zero-order DTM (0 to k) is summed.

20. The apparatus of claim 16 wherein the Obreshkov-based polynomial matrices [L_m, H_m] and[Q_m, G_m] are defined by:

21. The apparatus of claim 20 wherein discretizing the CTM into the DTM further comprises:

determining [A_dm, B_dm, C_dm, D_dm] where

A_dm=L_m(AT)H_m<sup>−

1</sup>(AT)−

Q_m(AT)G_m<sup>−

1</sup>(AT)

B_dm=Q_m(AT)G_m<sup>−

1</sup>(AT)H_m(AT)(BT

C_dm=G_m<sup>−

1</sup>(AT)

D_dm=−

G_m<sup>−

1</sup>(AT)H_m(AT)(BT).

22. An apparatus for executing a discrete time model (DTM) the apparatus comprising:

a processor coupled to a memory, the memory containing instructions which when executed by the processor performing;

retrieving from the memory a continuous time model (CTM) defining a continuous-time electrical or mechanical system;

computing an approximation for output signals y (nT) at the sampling time (T) comprising;

w_m(n+1,T)=A_dmw_m(n,T)+B_dmu(n,T)

x*_m(n,T)=C_dmw_m(n,T)+D_dmu(n,T)

where a DTM output is defined by;

23. The apparatus of claim 22 wherein determining the DTM output is implemented using fixed point representation the method further comprising:

scaling each element of the zero-order DTM given by [A_dm, B_dm, C_dm, D_dm] by a multiplication factor to increase the accuracy by a defined multiple; and

quantizing each element of the zero-order DTM to remove floating point decimal components.

24. The apparatus of claim 22 wherein scaling of the zero-order DTM matrices is scaled by a L2-norm dynamic range scaling method to increase accuracy of the model by ensuring the entries of the A, B, and C matrices are homogenous in magnitude and the DTM characteristics are insensitive to small perturbations in A, B, and C in comparison to their norms, and quantization is performed by truncation or rounding of the elements of the DTM matrices.

25. The apparatus of claim 24 wherein the DTM can then be implemented in hardware or software using the value of inputs (u(nT)) and its k-order derivatives (u^(k)(nT)) at the sampling time to compute the output signals (y(nT)).

26. The apparatus of claim 22 wherein the DTM state-space representation is obtained through an Obreshkov-based mapping to map the state-space matrices of the CTM into matrices of the DTM system where a CTM is described by the following state-space representation:

27. The apparatus of claim 22 wherein the parameters (i, j, k) of the Obreshkov coefficients are used to configure a high-order approximation (order=i+j) for the CTM state transition matrix such that the approximation is unconditionally stable if and only if j−

2≦

i≦

j and where the parameter k, controls which derivatives of an input signal vanishes in the approximation and, therefore, independently controls accuracy of the forced response.

28. The apparatus of claim 22 where in representation of a continuous-time system such as electrical system or mechanical system is provided as the transfer function representation defined by

29. The apparatus of claim 22 wherein the sampling time and the order of approximation [i,j,k] parameters are selected to meet A-stability requirements when the parameter i is set equal to j and L-stability requirements when condition i=j+1 or i=j+2 is satisfied.