Control System Using the Localization Method from A. S. Vostrikov. C Code and Octave Script.
1. Abbreviation
2. Introduction
Here I want to consider the localization method used in control systems developed by A.S. Vostrikov and his group at Novosibirsk State Technical University (NSTU). The localization method is used to control nonlinear and linear objects. In my article I will give a simple example for a linear system. For a detailed description of the application for nonlinear objects, see [1], [2].
Figure 21 shows the structure of the localization method for controlling a linear object. The method localizes (suppresses) the characteristics of the object and forms a control effect/signal on the control object, using the derivatives of the output values in the feedback (12a).
Figure 21: Analog Localization Control System
If K is a large quantity (or K => +∞) then:
Thus we obtain the desired characteristic of the transient process at large values of K, while the external disturbance m(t) is completely suppressed. It is recommended to choose a loop gain:
In this case the adjustment error of (20) for a static signal without disturbances m(t)=0 in steadystate mode s=0:
Thus the adjustment error is 5%…1%
A large loop gain can lead to instability of the system and in this case additional correction is needed to avoid the generation.
3. Analog Example
Let there be a secondorder control object described by a transfer function:
The reaction of the control object to the step signal has the form:
Figure 31: Step Response on the Control Object without Regulator
The control object has a strong overshoot, which does not meet the requirements of the transition process.
Now let’s choose the transfer characteristic of the desired transition process:
The choice of quality factor is described in the article “Digital Derivative Filter Using Integrators. C Code and Octave Script”
Step response on the required transform function is showed on the Figure 32
Figure 32: Required Step Response
Now let’s choose a filter that suppresses highfrequency interference in the feedback, using (15), (16):
Then:
Using (23) we select the loop gain:
Then from (26):
To estimate the stability reserves, we calculate the transfer characteristic of the system with open feedback:
Then magnitude and phase response of the open loop is showed on the Figure 33:
Figure 33: Open Loop Magnitude and Phase Response
Close Loop Step Response is showed on the Figure 34. The system has the desired transient response:
Figure 34: Close Loop Step Response
Note
At a loop gain value of 20, the control system has an adjustment error of 5%. See (24). This can be seen in Figure 34. The signal does not reach 1 (only 0.95). Increasing the loop gain will reduce the adjustment error, but may lead to instability, which will require additional correction.
4. Digital Regulator
Now we implement the regulator from our example in digital form. Let’s choose the sampling frequency based on 5˚ degrees = 0.0873 radians of uncertainty caused by sampling, at a cross frequency of 125 Rad/s (see (37)). In this case the stability reserves (38) will worsen for the digital control system by 5 degrees and will amount to 50˚ – 5˚ = 45˚ degrees, which is also sufficient for stable operation. The uncertainty of the phase is calculated by the formula:
Then:
The structure of the digital regulator is shown in Figure 41
Figure 41: Localization Control System with Digital Regulator
The digital controller itself is described using ztransform, and the object is controlled via DAC (Digital to Analog Converter), which is described using ZOH (Zero Order Hold). See a description of the ZOH in [3]. The output analog signal is read using an ADC (Analog to Digital Converter), which then goes to the digital regulator.
The digital controller is calculated on the basis of analog variant using bilinear conversion (6):
Analog:
To analyze and simulate the control system from our example, the whole control system was converted using a bilinear transformation (6) into digital form, including ZOH and Object.
The step response of the digital control system is shown in Figure 42.
Figure 42: Step Response of the Localization Control System with Digital Regulator
Thus the control system with a digital controller fully meets the specified requirements of the transition process.
5. Conclusions of the Localization Method

The method localizes (suppresses) the characteristics of the object, forms a control effect/signal on the object, using the derivatives of the output values in the feedback. The method works for controlling nonlinear and linear objects.

There is a control error that depends on the loop gain:
Gain 20 corresponds to error 5%
Gain 100 corresponds to error 1%

The transition process is completely determined by the developer, who chooses the desired transfer function himself.

In the case of a nonlinear object to ensure the stability of the control system, an analysis method is used based on the separation of the system’s own movements into slow and fast (see [1], [2]) that occur during transient processes when working out initial conditions. These differenttempo movements arise as a result of the choice (15). Thus two simpler subsystems are considered, and the subsystem of fast motions is described by linear differential equations.
6. Octave GNU file LocalizationControlSystem.m
The m script generates magnitude, phase and step responses, test signals and checks the analog/digital localization control systems. To do this, the following functions are defined in this file:
% Plotting the Step Response of the Analog Filter: (…+b1*s^2+b2*s+b3)/(…+a1*s^2+a2*s+a3)
% Input parameters:
% SB — numerator
% SA — denominator
% TimeVector – Time space
function plot_StepResponseAnalogFilter(sys, TimeVector, Text)
% Plotting the Step Response for the Digital Filter: (b1+b2*z^1+b3*z^2+…)/(a1+a2*z^1+a3*z^2+…)
% Input parameters:
% ZB — numerator
% ZA — denominator
% SampleNumber – Sample number of the step function
function plot_StepResponseDigitalFilter(ZB, ZA, SampleNumber, Text)
% Digital Close Loop: Localization Regulator
% Input parameters:
% K — gain
% ZB_object — numerator
% ZA_object — denominator
% ZB_feedback — numerator
% ZA_feedback — denominator
% x_input – input signal
function [y_output] = DigitalCloseLoopLocalizationRegulator(K, ZB_object, ZA_object, ZB_feedback, ZA_feedback, x_input)
% Float Table to File
% Support CCode
% Input parameters:
% ParameterVector – data array
% FileNameString – output file name
function FloatParamVector2file(ParameterVector, FileNameString)
7. C Language LocalizationControlSystem.c/h
The demo SW is realized the digital regulator of the Localization Control System. See the files LocalizationControlSystem.c/h
Functions:
/*
Control System Initialization
Clear the working arrays
Input: void
Return: void
*/
void ControlSystemInit(void)
/*
IIR Filter with Transposed Form II
Input
float input – filter input
Return
float – filter output
*/
float iir_filter(float input, float const* all_coef_ptr, float* all_history_ptr, uint8 sos_nmb)
/*
Regulator
Input
float require – require value
float object – object value
Return
float – control value
*/
float Regulator(float require, float object)
/*
Object Simulation
Input
float control – control value
Return
float – object output value
*/
float ObjectSimulation(float control)