Analog computing using dynamic amplitude scaling and methods of use

Abstract

An improved integrator for use in physical analog-computing systems is disclosed, featuring real-time dynamic amplitude scaling schemas that make use of an injected correction factor responsive to a contemporaneous change in an input dynamic-amplitude-scaling compensation factor. The injected correction factor is designed to reduce or eliminate transient output perturbations due to the amplitude scaling change. The disclosures discussed have real-world applications for physical analog computers and hybrid computers used to control and manage many types of industrial-control systems.

Claims

1. A method for operating a physical analog computer adapted to receive and process at least one input signal, the physical analog computer defining an integrator downstream of said at least one input signal and further defining an output downstream of said integrator, the method comprising the steps of: applying a first change to a scaling value at the at least one input signal responsive to an increase or a decrease in a magnitude of the at least one input signal; and in response to any change in the scaling value: dynamically calculating a first one-time correction factor based upon the magnitude of the first change to the scaling value; and at the integrator, injecting the first one-time correction factor to the integrator with an adder, the timing and magnitude of the first one-time correction factor selected to reduce any perturbation that would be observed at the output in the absence of the injection of the first one-time correction factor.

2. The method of claim 1 wherein the application of a first change to the scaling value at the at least one input signal is an increase in the scaling value in response to an increase in a magnitude of the at least one input signal.

3. The method of claim 1 wherein the application of a first change to the scaling value at the at least one input signal is a decrease in the scaling value in response to a decrease in a magnitude of the at least one input signal.

4. The method of claim 1 wherein the selection of the timing and magnitude of the first one-time correction factor are carried out by means of digital computation running in parallel with any analog computation carried out by the physical analog computer.

5. The method of claim 1 wherein the changing of the scaling value at the least one input signal is determined by means of digital computation running in parallel with any analog computation carried out by the physical analog computer.

6. The method of claim 1 further comprising the steps, carried out after the steps of applying a first change to a scaling value and injecting a first one-time correction factor, of: applying a second change to the scaling value at the at least one input signal; at the integrator, injecting a second one-time correction factor to the integrator with an adder, the timing and magnitude of the second one-time correction factor selected to reduce any perturbation that would be observed at the output in the absence of the injection of the second one-time correction factor.

7. The method of claim 6 wherein the second change to the scaling value is in a different direction than the first change to the scaling value.

8. The method of claim 6 wherein the second change to the scaling value is in the same direction as the first change to the scaling value.

9. A physical analog computer adapted to receive and process at least one input signal, the physical analog computer defining an integrator downstream of said at least one input signal and further defining an output downstream of said integrator, the analog computer further comprising: first means responsive to increases and decreases in a magnitude of the at least one input signal for applying a first change to a scaling value at the at least one input signal; second means, responsive to any change to the scaling value, for selecting a timing and magnitude of a one-time correction factor for injection at an output of the integrator, the timing and magnitude of the first one-time correction factor selected to reduce any perturbation that would be observed at the output in the absence of the injection of the first one-time correction factor, and an adder which injects the one-time correction factor at the integrator, wherein the first one-time correction factor is dynamically calculated based on the magnitude of the first change to the scaling value.

10. The analog computer of claim 9 wherein the selection of the timing and magnitude of the first one-time correction factor are carried out by means of digital computation running in parallel with any analog computation carried out by the physical analog computer.

11. The analog computer of claim 9 wherein the changing of the scaling value at the least one input signal is determined by means of digital computation running in parallel with any analog computation carried out by the physical analog computer.

12. A method for operating a physical analog computer adapted to receive and process at least one input signal, the physical analog computer defining an integrator downstream of said at least one input signal and further defining an output downstream of said integrator, the method comprising the steps of: applying a first change to a scaling value at the at least one input signal; at the integrator, injecting a first one-time correction factor to the integrator with an adder, the timing and magnitude of the first one-time correction factor selected to reduce any perturbation that would be observed at the output in the absence of the injection of the first one-time correction factor; wherein the application of a first change to the scaling value at the at least one input signal is an increase in the scaling value in response to an increase in a magnitude of the at least one input signal.

13. The method of claim 12 wherein the selection of the timing and magnitude of the first one-time correction factor are carried out by means of digital computation running in parallel with any analog computation carried out by the physical analog computer.

14. The method of claim 12 wherein the changing of the scaling value at the least one input signal is determined by means of digital computation running in parallel with any analog computation carried out by the physical analog computer.

15. The method of claim 12 further comprising the steps, carried out after the steps of applying a first change to a scaling value and injecting a first one-time correction factor, of: applying a second change to the scaling value at the at least one input signal; at the integrator, injecting a second one-time correction factor to the integrator, the timing and magnitude of the second one-time correction factor selected to reduce any perturbation that would be observed at the output in the absence of the injection of the second one-time correction factor.

16. The method of claim 15 wherein the second change to the scaling value is in a different direction than the first change to the scaling value.

17. The method of claim 15 wherein the second change to the scaling value is in the same direction as the first change to the scaling value.

18. A method for operating a physical analog computer adapted to receive and process at least one input signal, the physical analog computer defining an integrator downstream of said at least one input signal and further defining an output downstream of said integrator, the method comprising the steps of: applying a first change to a scaling value at the at least one input signal; at the integrator, injecting a first one-time correction factor to the integrator with an adder, the timing and magnitude of the first one-time correction factor selected to reduce any perturbation that would be observed at the output in the absence of the injection of the first one-time compensation factor; wherein the application of a first change to the scaling value at the at least one input signal is a decrease in the scaling value in response to a decrease in a magnitude of the at least one input signal.

19. The method of claim 18 wherein the selection of the timing and magnitude of the first one-time correction factor are carried out by means of digital computation running in parallel with any analog computation carried out by the physical analog computer.

20. The method of claim 18 wherein the changing of the scaling value at the least one input signal is determined by means of digital computation running in parallel with any analog computation carried out by the physical analog computer.

21. The method of claim 18 further comprising the steps, carried out after the steps of applying a first change to a scaling value and injecting a first one-time correction factor, of: applying a second change to the scaling value at the at least one input signal; at the integrator, injecting a second one-time correction factor to the integrator, the timing and magnitude of the second one-time correction factor selected to reduce any perturbation that would be observed at the output in the absence of the injection of the second one-time correction factor.

22. The method of claim 21 wherein the second change to the scaling value is in a different direction than the first change to the scaling value.

23. The method of claim 21 wherein the second change to the scaling value is in the same direction as the first change to the scaling value.

24. A method for operating a physical analog computer adapted to receive and process at least one input signal, the physical analog computer defining an integrator downstream of said at least one input signal and further defining an output downstream of said integrator, the method comprising the steps of: applying a first change to a scaling value at the at least one input signal; at the integrator, injecting a first one-time correction factor to the integrator with an adder, the timing and magnitude of the first one-time correction factor selected to reduce any perturbation that would be observed at the output in the absence of the injection of the first one-time correction factor, further comprising the steps, carried out after the steps of applying a first change to a scaling value and injecting a first one-time correction factor, of: applying a second change to the scaling value at the at least one input signal; at the integrator, injecting a second one-time correction factor to the integrator with the adder, the timing and magnitude of the second one-time correction factor selected to reduce any perturbation that would be observed at the output in the absence of the injection of the second one-time correction factor, wherein the second change to the scaling value is in the same direction as the first change to the scaling value.

25. The method of claim 24 wherein the application of a first change to the scaling value at the at least one input signal is an increase in the scaling value in response to an increase in a magnitude of the at least one input signal.

26. The method of claim 24 wherein the application of a first change to the scaling value at the at least one input signal is a decrease in the scaling value in response to a decrease in a magnitude of the at least one input signal.

27. The method of claim 24 wherein the selection of the timing and magnitude of the first one-time correction factor are carried out by means of digital computation running in parallel with any analog computation carried out by the physical analog computer.

28. The method of claim 24 wherein the changing of the scaling value at the least one input signal is determined by means of digital computation running in parallel with any analog computation carried out by the physical analog computer.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIGS. 1A and 1B depict one simplified example of an analog computer with and without dynamic amplitude scaling, respectively, where the scaling in FIG. 1B is shown by dividing inputs a.sub.0 and b.sub.0 by a factor of k and a multiplicative factor k is inserted in the integrator.

(2) FIGS. 2A and 2B collectively depict another simple example of an analog-computing system before any amplitude scaling is applied, with FIG. 2B providing a graph of various internal signals from the system in FIG. 2A that are not subjected to dynamic amplitude scaling.

(3) FIGS. 2C and 2D collectively depict the system in FIG. 2A, but with dynamic amplitude scaling applied, as shown by a change in the input values at key system points in FIG. 2C. FIG. 2D provides two graphs of various internal signals from the system in FIG. 2A and from the system of FIG. 2C, wherein the top graph reflects no dynamic amplitude scaling and the bottom graph reflects dynamic amplitude scaling.

(4) FIGS. 3A, 3B, and 3C depict simplified examples (and related output-characteristic curves), respectively, of a “fixed” (i.e., non-amplitude-scaled) analog-computing system, of a “blindly scaled” (i.e., amplitude-scaled at an arbitrary point in time) analog-computing system, and of a “properly” dynamically amplitude-scaled) analog-computing system.

(5) FIGS. 4A and 4B depict, respectively, a typical integrator to be considered for constant integrator input gain g under “0” initial conditions (with the system output y(t) generated as a function of g and g.sup.−1), and a graphical representation of the changing of constant g over time.

(6) FIG. 4C depicts one embodiment of an improved integrator (and related mathematical functions) that employs dynamic amplitude scaling as a function of the integrator output over time.

(7) FIG. 4D depicts another embodiment of an improved integrator (and related mathematical functions) that employs dynamic amplitude scaling as a function of the integrator output over time.

(8) FIG. 4E depicts one embodiment of a graph of a “blind” application of amplitude scaling) as a function of constant g, based on FIGS. 4C and 4D, wherein the output of the integrator is represented by w, and is continuous.

(9) FIG. 5A depicts one embodiment of an improved analog computer that features dynamic amplitude scaling, as well as a means to adjust the output via an additional one-time correction factor added to the analog-computer output y to counter transient perturbances.

(10) FIGS. 5B, 5C, and 5D each depict a graph of the improved analog computer (from FIG. 5A) output; specifically, where FIG. 5B shows the raw gain inputs (dynamic amplitude scaling), followed by FIG. 5C, which shows an embodiment of the calculation and graphical plot of a one-time transient correction factor to add to the output y of an improved analog computer output, followed by FIG. 5D, which provides an embodiment of a graph of the properly and dynamically amplitude-scaled output y of the improved analog computer. FIG. 5E shows a functional block diagram in which g is inserted at the input of the integrator and in which g.sup.−1 (the inverse) is inserted at the output thereof.

(11) FIG. 5F depicts one embodiment of an improved analog computer that features dynamic amplitude scaling, as well as a means to adjust the output via an additional one-time correction factor added to output w of the integrator of the improved analog-computer to counter transient perturbances.

(12) FIGS. 5G, 5H, and 5I each depict a graph of the improved analog computer (from FIG. 5F) output; specifically, where FIG. 5G shows the raw gain inputs (dynamic amplitude scaling), followed by FIG. 5H, which shows an embodiment of the calculation and graphical plot of a one-time transient correction factor to add to the output w of an improved integrator in an analog computer improved integrator output, followed by FIG. 5I, which provides an embodiment of a graph of the properly and dynamically amplitude-scaled output y of the improved analog computer.

(13) FIG. 6A depicts one in principle embodiment of one example schematic of an improved integrator for use in an improved analog computer, which was generated with a simulated program for integrated circuits emphasis (SPICE) with various enumerated points used for analysis.

(14) FIGS. 6B, 6C, and 6D provides various graphs of concurrent integrator measurements versus time, each graph over the same time scale. FIG. 6B provides a graph of an embodiment of an improved integrator (based on FIG. 6A) voltage output 104 versus time, based on a 20-kHz unity-gain frequency, a constant g=1, and a unit step input. FIG. 6C provides a graph of an embodiment of an improved integrator (based on FIG. 6A) voltage gain adjustments g, g−1 used for dynamic amplitude scaling versus time. FIG. 6D provides a graph of an embodiment of an improved integrator (based on FIG. 6A) capacitance voltage buildup w versus time during dynamic amplitude scaling.

(15) FIGS. 6E, 6F, 6G, and 6H provides various graphs of concurrent integrator measurements versus time, each graph over the same time scale, after a “jump” or system perturbation is introduced. FIG. 6E provides a graph of an embodiment of an improved integrator (based on FIG. 6A) voltage output 104 versus time, based on a 20-kHz unity-gain frequency, a constant g=1, and after a sudden dynamic amplitude scaling adjustment. FIG. 6F provides a graph of an embodiment of an improved integrator (based on FIG. 6A) voltage gain adjustments g, g−1 used for dynamic amplitude scaling versus time. FIG. 6G provides a graph of an embodiment of an improved integrator (based on FIG. 6A) capacitance voltage buildup w versus time during dynamic amplitude scaling, as well as a gain adjustment at point 111 to help deal with the “jump.” Finally, FIG. 6H provides a graph of an embodiment of an improved integrator (based on FIG. 6A) depicting a largely corrected (with a minor momentary “glitch” in the center) integrator-output 104 voltage profile after application of a one-time correction factor in input amplitude responsive to an imminent change in said input dynamic-amplitude-scaling compensation factor (gain).

(16) FIG. 6I depicts one embodiment of a simplified diagram of an improved analog computer that incorporates an improved integrator.

(17) FIG. 7A depicts one embodiment of a dual-integrator analog computer (AC).

(18) FIG. 7B depicts one embodiment of a simplified representation of a fixed analog computer (AC).

(19) FIG. 7C depicts one embodiment of simplified representations of a fixed analog computer (AC) (from FIG. 7B) and of an adjustable analog computer (AC) for purposes of comparison.

DETAILED DESCRIPTION

I. Overview

(20) Amplitude scaling can help internal signals avoid exceeding their allowable range and being buried in noise. In a sense, amplitude scaling optimizes the “dynamic range” of an analog computer. The idea is to maximize the signal-to-noise ratio for the analog paths.

(21) The inventive disclosures described herein pertain to improved physical analog computers and integrators that employ dynamic amplitude scaling in order to reduce or eliminate analog-computer output distortions and input-signal-to-noise ratios in real-world applications. The basic schemas provide for detecting when an input signal range is not optimum for the analog-computing environment, then strategically introducing an input dynamic-amplitude-scaling compensation factor in response to an input-signal while the physical analog computer is in service in order to ensure that said input signal's range is constrained to be within the design limits of said physical analog computer, whereby the introduction of said dynamic-amplitude-scaling compensation factor reduces system-output distortion and the signal-to-noise ratio. The schema also provides for introducing an output dynamic-amplitude-descaling compensation factor at the output of said physical analog computer in order to prepare the analog computer output for presentation to a system user. In variations, the schema incorporates an improved integrator that is adapted to receive a one-time correction factor in input amplitude responsive to an imminent change in the input dynamic-amplitude-scaling compensation factor, which is designed to counteract any transient output perturbations due to the introduction of a dynamic-amplitude-scaling compensation factor and to ensure that the output of the improved physical analog computer is better than without said one-time correction factor.

(22) It may be helpful to clarify what “one-time” means in this context. The insertion of a one-time correction factor at an integrator, to correct for a contemporaneous change in a scaling factor at an input, is a correction factor that is inserted one time in response to the scaling factor change. It does not mean that such insertion of a correction factor happens only one time during a period of time during which analog computation is taking place. Indeed the teachings of the invention contemplate that during a time when analog computation is taking place, for a particular input to the analog computer a scaling factor change might happen at one time and another scaling factor change might happen at another time. The teachings of the invention also contemplate that during a time when analog computation is taking place, for a first input to the analog computer a scaling factor change might happen at one time and for a second input to the analog computer another scaling factor change might happen, at the same time or at a different time.

II. General Technical Description of Dynamic Amplitude Scaling in an Improved Physical Analog Computer

(23) This Section II generally describes the principles underlying the use of dynamic amplitude scaling in an improved physical analog computer. Refer to FIGS. 1A through 7C.

(24) Amplitude scaling can help internal signals avoid exceeding their allowable range and being buried in noise. In a sense, amplitude scaling optimizes the “dynamic range” of an analog computer. FIGS. 1A and 1B provides a simple example: In FIG. 1A, it is assumed that the coefficients a.sub.0 and b.sub.0 may have values such that the output z ends up in an overload condition or ends up buried in “noise.” A simple amplitude scaling of the inputs by a factor k can correct this problem. It involves those inputs, a.sub.0/k and b.sub.0/k, and the input of the first integrator, k∫, as shown in FIG. 1B. The overall behavior at circuit points x (input) and y (output) behavior remains the same; however, the size of the affected internal signals has been changed.

(25) FIGS. 2A through 2D provide another simple example: The original system is depicted in FIG. 2A. As depicted in the graphs in FIG. 2B, let the forcing function x be a ramp that reaches 100 A in 0.5 s and remains 100 A thereafter. Some variables have very small magnitude and thus can be corrupted by noise. In the example contained in FIGS. 2A and 2B:
|x|.sub.max=100A
|ÿ|.sub.max=55.7A
|{dot over (y)}|.sub.max=14.8A
|y|.sub.max=5.8A

(26) The above information, using the techniques described herein, results in the amplitude-scaled system shown in FIG. 2C. The behavior of the new amplitude-scaled system is compared to the behavior of the original system in FIG. 2D. As can be observed, all of the variables in the amplitude-scaled system have absolute values close to the maximum-allowed value (80 A), and can be expected to be well-above “noise” levels.

(27) Following computation, the amplitude-scaled variables should be de-scaled before presentation to the system user. For example, the amplitude-scaled solution for variable y depicted in FIG. 2C and in the graphs in FIG. 2D should be divided by 13 for presentation to a system user.

(28) The alert reader will appreciate that different types of input signals (e.g., different amplitudes or frequency content) may ideally require different scaling. If the types of input signals are known ahead of time, then the amplitude scaling can be adjusted beforehand.

(29) However, if a change in amplitude scaling is needed while a computer is in service, then, while the “before” and “after” scaling may be correct, the transition from the old to the new amplitude-scaling factors (gains) can cause disturbances at the output, thus disrupting system operation. For example, FIG. 3A depicts a simplified system being controlled by an analog computer with an accompanying graph of the analog computer output versus time that does not use amplitude scaling. FIG. 3B depicts a simplified system being controlled by an analog computer with an accompanying graph of the analog computer output versus time that does use amplitude scaling, though the amplitude scaling is “blindly” scaled at some arbitrary time t.sub.k. It should be noted that at the point/moment that some scaling is applied, the system gives rise to potentially large perturbations/glitches, which might be solved by momentarily injecting an equal and opposite perturbation at the output, as shown in FIG. 3C, which depicts a simplified system being controlled by an analog computer with an accompanying graph of the analog computer output versus time that does use “proper” amplitude scaling at some time t.sub.k. As another example, FIG. 4A depicts a typical integrator to be considered for constant g under “0” initial conditions, with FIG. 4B showing a graphical representation of the changing of constant g over time. The system output y(t) is generated as a function of g and g.sup.−1 as follows:
y(t)=g.sup.−1∫.sub.0.sup.1gx(τ)δτ=∫.sub.0.sup.1x(τ)dτ

(30) The output of the integrator is represented in FIGS. 4C, 4D, and 4E (with FIG. 4E depicting a graph of a “blind” application of amplitude scaling) as a function of constant g. The output of the integrator is represented by w, and we have:

(31) $w\left(t_k^{+}\right)=w\left(t_k^{-}\right)=q_{1}y\left(t_k^{-}\right)y\left(t_k^{+}\right)=q_2^{-1}w\left(t_k^{+}\right)=\frac{q_1}{q_2}y\left(t_k^{-}\right)$

(32) Thus, the output jumps by:

(33) $y\left(t_k^{+}\right)-y\left(t_k^{-}\right)=\frac{g_1-g_2}{g_2}y\left(t_k^{-}\right)$

(34) Such system disruptions (i.e., output jumps) has been observed in filters. In the prior-art literature, it has been addressed by updating the values of capacitor voltages (for example, see U.S. Pat. No. 5,541,600 to Blumenkrantz). However, a better and more feasable solution is as follows.

(35) Refer to FIGS. 5A through 5E. In this solution, the opposite of the jump is added directly to the output (which is relatively easy, if the signals are currents).

(36) The above requires sampling the output around the time it jumps. To avoid possible complications, the output of the integrator, w, is sampled instead:

(37) Since:
w(t.sub.k.sup.−)=g.sub.1y(t.sub.k.sup.−)

(38) Then the following compensation for the disruption can be used to add to the output:

(39) $\Delta =\frac{g_1-g_2}{g_2}y\left(t_k^{-}\right)=\left(g_1^{-1}-g_2^{-1}\right)w\left(t_k^{-}\right)$

(40) Referring to FIGS. 5F through 5H, in an alternative solution, the compensation for the disruption can be added (E) to the output of the integrator instead.

(41) FIG. 6A depicts one example integrator schematic developed with a simulated program for integrated circuits emphasis (SPICE) representing the integrator in an improved analog computer that features dynamic amplitude scaling, with various enumerated points (101, 103, 104, 105, 106, 107, 109, 110, and 111) used for study.

(42) FIG. 6B graphs the integrator output 104 versus time, based on a 20-kHz unity-gain frequency, a constant g=1, and a unit step input.

(43) FIGS. 6C through 6E depict the integrator with implementing variable gain g, g.sup.−1. It should be noted that the gains are given significant rise time, in order to observe the effect at nodes 103, 104, and 106 and on the integrator output w and system output y.

(44) A “jump” correction E may be added, as depicted in FIG. 5F. Output profiles at various enumerated nodes (103, 104, 105, 106, 111 for the integrator output w and system output y) are depicted in FIGS. 6F through 6H. The gains are given significant rise time. However, it should be noted in FIG. 6H that the “glitch” (or perturbation) in the integrator voltage output profile shown at t=1.0 μs is due to the nonzero rise time of the gains g, g.sup.−1. The glitch can be managed by adjusting the various time delays. In addition, such glitches are present also due to the digital-to-analog converter (DAC). A corrective technique requires knowledge of w (or y) a moment before switching the gain, in order to generate the correction term E. To retain, and operate on, the sample of w, the control box (see FIG. 6I) needs to contain a sample-and-hold circuit or an analog-to-digital-converter (ADC)—digital-to-analog-converter (DAC) combination. While multiplier factors (gains) can be used that are inverses of each other as an example, this does not have to be the case, if it is desired to change the overall gain constant. The aforementioned technique described deals with a way to change the output coefficient w without disturbances, independent of what the input coefficient x is.

(45) The above-discussed principles can be generalized to linear analog computers with: Arbitrary topologies; Gains that depend not only on one, but on multiple integrator outputs; and Arbitrary g shapes (even continuously varying).

(46) Such implementations can be accomplished by applying to the analog computer earlier results as discussed in the following prior-art references, which were created by the same Inventor as for the present patent application, and which are hereby incorporated by reference: Y. Tsividis, “Externally linear, time-invariant systems and their application to companding signal processors”, IEEE Transactions on Circuits and Systems, Part II, vol. 44, no. 2, pp. 65-85, February 1997; and U.S. Pat. No. 6,389,445, “Methods and Systems for Designing and Making Signal-Processor Circuits With Internal Companding, and the Resulting Circuits,” Yannis Tsividis.

(47) Refer to FIGS. 7A through 7C. As a preliminary matter, the state of the integrator outputs need to be determined. If those are known, then all of the other variables can be evaluated. The first derivatives of the states and the output are expressed as follows:
{dot over (x)}.sub.1=−a.sub.1x.sub.1−a.sub.2x.sub.2+u
{dot over (x)}.sub.2=x.sub.1
y=c.sub.1x.sub.1+c.sub.2x.sub.2+d.sub.1u

(48) This can also be expressed in matrix form:

(49) $\left[\begin{array}{c}{\stackrel{.}{x}}_1\\ {\stackrel{.}{x}}_2\end{array}\right]=\left[\begin{array}{cc}-a_1& 1a_2\\ 1& 0\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}1\\ 0\end{array}\right]u$$y=\left[c_1{c}_2\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+d_{1}u$

(50) These are of the form:
{dot over (x)}=Ax+Bu
y=Cx+Du Where:

(51) $x=\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$$A=\left[\begin{array}{cc}-a_1& -a_2\\ 1& 0\end{array}\right]$$B=\left[\begin{array}{c}1\\ 0\end{array}\right]$$C=\left[c_1{c}_2\right]$$D=d_1$

(52) As was developed in the two references cited earlier, in general, any linear implementation on the analog computer can be described by state equations:
{dot over (x)}(t)=Ax(t)+Bu(t)
y(t)=Cx(t)+Du(t) Where: x is the vector of all integrator outputs; u is the vector of all inputs; y is the vector of the analog computer's outputs; and A, B, C, and D are appropriate matrices.

(53) These state equations can be applied to dynamic amplitude scaling, and can also be written for nonlinear systems. In FIGS. 7A, 7B, and 7C, w varies with the varying gains, but y does not. This can be done by appropriately choosing the hatted matrices.

(54) This is accomplished by starting with a linear-time-invariant (LTI) prototype:
{dot over (x)}(t)=Ax(t)+Bu(t)
y(t)=Cx(t)+Du(t)

(55) Consider a linear-time-varying (LTV) system:
{dot over (w)}(t)=Â(t)w(t)+{circumflex over (B)}(t)u(t)
ŷ(t)=Ĉ(t)w(t)+{circumflex over (D)}(t)u(t)

(56) The actual physical analog-computing system is required to have, for the same input, the same output as the prototype:
ŷ(t)=y(t),allt

(57) Whereas, its state variables are “scaled” according to a gain matrix G(t):
w(t)=G(t)x(t)

(58) Direct substitution shows that for the above equations to be satisfied the following is required:
Â(t)=Ġ(t)G.sup.−1(t)+G(t)AG.sup.−1(t)
{circumflex over (B)}(t)=G(t)B
Ĉ(t)=CG.sup.−1(t)
{circumflex over (D)}(t)=D

(59) These are linear transformations that convert the original, time-invariant analog computer to a time-varying one. This allows the internal waveforms to be amplitude-scaled, without any transients at the output. However, unfortunately, practical implementation is difficult. This technique is valid for linear equations only. Scaling for nonlinear cases is tricky and case-dependent.

CONCLUSIONS

(60) Gain adjustments are important for optimizing the input-output performance of analog-computer circuits that have, by themselves, severe linearity and noise limitations. When it is attempted to vary gains while the analog computer is in service, output disturbances occur. Such disturbances can be large, and are likely to interfere with proper operation in the case of real-time control. A simple technique for eliminating such disturbances in the case of an integrator has been presented in the above discussion. In addition, related results have been adapted from linear systems theory and have been reviewed. Gain adjustment has the potential of drastic power reduction for a given signal-to-noise ration (SNR), if implemented successfully.

III. Alternative Embodiments and Other Variations

(61) The various embodiments and variations thereof described herein, including the descriptions in any appended Claims and/or illustrated in the accompanying Figures, are merely exemplary and are not meant to limit the scope of the inventive disclosure. It should be appreciated that numerous variations of the invention have been contemplated as would be obvious to one of ordinary skill in the art with the benefit of this disclosure.

(62) Hence, the alert reader will have no difficulty devising myriad obvious variations and improvements to the invention, all of which are intended to be encompassed within the scope of the Description, Figures, and Claims herein.