Dynamic Regression for speeding the calculation time of the aileron drive system

Source: https://anton-chernykh.ru

Introduction

If you want to speed up your dynamic system several times, just consider the Dynamic Regression algorithm for solving time-consuming issues. Before this solution, I tried to apply different types of recurrent neural networks like GRU and LSTM, but that approach didn’t show a good performance and there were critical errors in the strong oscillatory process. Let’s take another way and drop the data science approach for a while!

Let’s go straight to the point!

Let’s take a look at the aircraft aileron drive control system in MATLAB/SIMULINK. This model is shown below.

Picture 1. Aircraft aileron drive control system

Here you can see the control block (Electronic Actuator Unit) that calculates the control signals for the main object by using feedback from sensors. The actuator takes control signals, pressure and resistance constants, and resistance torque produced by the environment.

In this paper, I want to replace block «Actuator» to DR (Dynamic Regression) with parameters online estimation that is shown in the paper [1] (a paper about DR ITMO) where the authors mentioned one in the context of adaptive systems and sensorless control.

The Calculating variable parameters of the Actuator model depend on a got error by the difference between reference and prediction signals and other input signals.

Picture 2. Replacing the functional block of the system with a model with dynamic linear regression

So, our DR is created to replace the object given in the form of differential equations with initial conditions (Cauchy form):

The replacement to Dynamic Regressor:

Regression model equation

The model regressor includes control signals from the controller C1(t), C2(t) constants P, R, which are responsible for the operating mode of the drive and the external total torque M(t):

In the current task, we know the control signal Reference, which the controlled object must work out, minimizing the deviation between its output and the standard (we assume that the control object is made correctly and has only negative feedback).

Basic error function

In the last equation, the output signal from the control object (which is described by differential equations) y(t) is replaced by a model with a dynamic regressor output.

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Then the error can be rewritten as follows:

It remains only to determine the vector of unknown constants. No magic, DR uses a gradient parameter estimation method for this, a quadratic criterion as an error function. Let us calculate the estimates of the unknown parameters:

The estimates obtaining

At each moment of time, a new vector of estimates of the linear regression parameters is formed and adaptation takes place to the nature of the reference signal R(t).

It is important to consider the conditions under which the dynamic regressor gives an adequate estimate of the model parameters when building models. For the view model:

The estimate vector converges to its true values or the errors of the parameters converges to zero asymptotically if the condition of persistent excitation is satisfied, which is imposed on the regressor:

Persistent excitation

where there are positive numbers for which the given inequality holds.

Experimental results

Picture 3. Calculation of the displacement of the aileron drive link

As you get closer, the DR-model output follows the Reference signal very tightly — well, nothing is broken here.

Picture 4. Simulation start time

And finally, let’s compare the results in time period with strong dynamics:

Picture 5. Period with strong oscillation

Both in the section with fluctuations and in the section with a sharp jump, the DR model works out well.

Picture 6. The final section of the simulation time

Conclusion

The dynamic regression control system showed good performance! The maximum relative error was 7%. The calculation process is accelerated by 5 times (7.34 sec instead of 36.6 sec). The advantage of the developed technique is online learning or adaptation to new estimates of the parameters of the linear model in the simulation process.

References

1. Adaptive Observer of Rotor Position and Flux for Salient Synchronous Motor. Bazylev, D.N., Pyrkin, A.A., Bobtsov, A.A. ITMO University, St. Petersburg, 197101, Russian Federation.
2. ETMC Exponenta. The Center for Engineering Technologies and Modeling in Russia.
3. ITMO University. It’s St. Petersburg’s best university and one of the top universities in the world.