Dsp implementation of nonlinear differentiators

A log-sign nonlinear differentiator for signal tracking, comprising:

• an analog-to-digital converter configured to receive an analog input signal and convert the analog signal to a digital signal, u(t);
• a first adder configured to receive a filtered first state signal, x1(t), subtract a filtered first state signal, x1(t), from the digital signal, u(t), and generate an error signal, v(t);
• a first log-sign differentiator configured to receive the error signal, v(t), and estimate a first derivative, {dot over (x)}1(t), of the filtered first state signal, x1(t);
• a second log-sign differentiator configured to receive the error signal, v(t), and generate an estimate of a second derivative, {dot over (x)}2(t), of the input signal, u(t);
• a first integrator connected in series with the second log-sign differentiator, wherein the first integrator is configured to integrate the second derivative, {dot over (x)}2(t), and generate a second state signal, x2(t), wherein the second state signal, x2(t), represents an estimate of a first derivative, {dot over (u)}(t), of the input signal, u(t);
• a second adder configured to add the first derivative, {dot over (x)}1(t), of the filtered first state signal, x1(t), to the second state signal, x2(t), thus generating a summed signal;
• a second integrator configured to integrate the summed signal and generate the filtered first state signal, x1(t);
• a first digital to analog converter configured to convert the filtered first state signal, x1(t) to an estimate of the input signal, u(t);
• and
• a second digital to analog converter configured to convert the second state signal to an estimate of the first derivative, {dot over (u)}(t), of the input signal, u(t), such that a first tracked output, y1(t), of the log-sign nonlinear differentiator, is an estimate of the input signal, u(t), tracked by the log-sign nonlinear differentiator, and a second output, y2(t) equals the first derivative, {dot over (u)}(t) of the input signal, u(t), indicating a tracked direction of the input signal, u(t).

An adaptive gain log-sign differentiator for signal tracking, comprising:

• an analog-to-digital converter configured to receive an analog input signal and convert the analog signal to a digital signal, u(t);
• a first adder configured to receive a filtered first state signal, x1(t), subtract the filtered first state signal, x1(t), from the digital signal, u(t), and generate an error signal, v(t);
• a deadzone function calculator configured to receive the error signal, v(t) and a first gain parameter, λ, where λ>0, and multiply a deadzone function, Dε(|u(t)−x1(t)|)), by the first gain parameter, λ, to generate a weighted deadzone function;
• a first integrator configured to integrate the weighted deadzone function and generate a gain value, γ(t), of the deadzone function, where γ(0)>0 and γ(t) is an increasing positive function of time, t;
• a first log-sign differentiator configured to receive the error signal, v(t), and the gain value, γ(t), and estimate a first derivative, {dot over (x)}1(t), of the filtered first state signal, x1(t);
• a second log-sign differentiator configured to receive the error signal, v(t), and the gain value, γ(t), and generate an estimate of a second derivative, {dot over (x)}2(t), of the input signal, u(t);
• a second integrator connected in series with the second log-sign differentiator, wherein the second integrator is configured to integrate the second derivative, {dot over (x)}2(t), and generate a second state signal, x2(t), wherein the second state signal, x2(t), represents an estimate of a first derivative, {dot over (u)}(t), of the input signal, u(t);
• a second adder configured to add the first derivative, {dot over (x)}1(t), of the filtered first state signal, x1(t), to the second state signal, x2(t), thus generating a summed signal;
• a third integrator configured to integrate the summed signal and generate the filtered first state signal, x1(t);
• a first digital to analog converter configured to convert the filtered first state signal, x1(t), to an estimate of the input signal, u(t);
• and
• a second digital to analog converter configured to convert the second state signal to an estimate of the first derivative, {dot over (u)}(t) of the input signal, u(t), such that a first tracked output, y1(t), of the adaptive log-sign differentiator, is an estimate of the input signal, u(t), and a second tracked output, y2(t), of the adaptive log-sign differentiator, equals the first derivative, {dot over (u)}(t) of the input signal, u(t), indicating a tracked direction of the input signal, u(t).

The adaptive gain log-sign differentiator, further comprising:

• a first input connected to the first log-sign differentiator, the first input configured to receive the gain value, γ(t), wherein the first log-sign differentiator is configured to estimate the first derivative, {dot over (x)}1(t), of the filtered first state, x1(t), by calculating:
• x
• ˙
• 1
• (
• t
• )
• =
• x
• 2
• (
• t
• )
• -
• γ
• 2
• ⁢
• ln
• ⁡
• (
• 1
• +
• ❘
• "\[LeftBracketingBar]"
• x
• 1
• (
• t
• )
• -
• u
• ⁡
• (
• t
• )
• ❘
• "\[RightBracketingBar]"
• )
• ⁢
• sign
• ⁡
• (
• x
• 1
• (
• t
• )
• -
• u
• ⁡
• (
• t
• )
• )
• ,
• and
• a second input connected to the second log-sign differentiator, the second input configured to receive the gain value, γ(t), wherein the second log-sign differentiator is configured to estimate the first derivative, {dot over (x)}2(t), of the second state, x2(t), by calculating:
• {dot over (x)}2(t)=−γ sign(x1(t)−u(t)).