Method of linear active disturbance rejection control with a fractional order integral action

A set point tracking system with fractional order integral linear active disturbance rejection control, comprising:

• a first multiplier having an input configured to receive a reference signal, r(t) and a first negative feedback signal and generate a feedback modified reference signal, ε(t);
• a set point tracking controller connected to the first multiplier, the set point tracking controller including:
• a first circuitry and a first processor having first program instructions to perform fractional order integration by applying a fractional order integral operator to the feedback modified reference signal, ε(t), to generate a fractional order reference signal, Zr(t);
• a first amplifier configured to amplify the fractional order reference signal, Zr, by a first integrated gain, kr, to generate a gain modified fractional order reference signal, Z′r(t);
• a second multiplier connected to the set point tracking controller, the second multiplier configured to multiply the gain modified fractional order reference signal, Z′r(t) by a series of second negative feedback signals to generate a modified signal, u′0(t);
• and
• a third multiplier connected to the second multiplier, the third multiplier configured to multiple the modified signal, u′0(t), by a third negative feedback signal to generate a feedback modified signal, u0(t);
• a second amplifier connected to the third multiplier, the second amplifier configured to divide the feedback modified signal, u0(t) by a static gain, b0, to generate a process control variable, u(t);
• a plant connected to the second amplifier, the plant having a first input configured to receive the process control variable, u(t), a second input configured to receive a disturbance, d;
• and an output configured to generate an output signal, y(t);
• an extended state observer, ESO, controller having a first input connected to the output signal, y(t), and a second input connected to the process variable, u(t), wherein the ESO controller includes second circuitry and a second processor having second program instructions configured to:
• estimate the disturbance, d;
• generate a series of n estimated disturbance canceling feedback signals, Z1(t), Z2(t), . . . Zn(t);
• generate a third negative feedback signal, Zn+1(t), based on an estimate of the process variable, u(t);
• a third amplifier having an input connected to receive the series of n estimated disturbance canceling feedback signals, Z1(t), Z2(t), . . . Zn(t) and multiply the series of n estimated disturbance canceling feedback signals, Z1(t), Z2(t), . . . Zn(t) by a state feedback vector, KS, to generate the series of second negative feedback signals, KSZ1(t), KSZ2(t), . . . KSZn(t);
• a first control input configured for tuning the set-point tracking controller;
• a second control input configured for tuning the ESO, wherein tuning the set-point tracking controller and the ESO is configured to eliminate the disturbance, d, from the output signal, y(t).

A fractional order integral linear active disturbance rejection control method for set-point tracking of a process variable, comprising:

• receiving, at a first multiplier, a reference signal;
• multiplying the reference signal by a first feedback signal;
• applying, by a set-point tracking controller, fractional order integration to the reference signal;
• amplifying the reference signal;
• multiplying the reference signal by a series of second feedback signals and a third feedback signal;
• dividing the reference signal by a static gain, b0, and generating a process control variable, u(t);
• inputting the process control variable, u(t), to a plant;
• applying a disturbance, d, to the plant;
• outputting, by the plant, an output signal, y(t);
• feeding back the output signal, y(t), as the first feedback signal;
• generating the series of second feedback signals and a third feedback signal by an extended state observer;
• and
• tuning the set-point tracking controller and the ESO to eliminate the disturbance, d, from the output signal, y(t).