Fundamentals of Feedback Control using Microcontroller [Advanced]

Among the advanced controls, robust control will be explained in this article.

Robust control refers to a control system in which the control output is not easily affected by disturbances on the input side of the control target P(s), such as modeling errors and load fluctuations when a physical control model such as a DC motor is converted into a mathematical expression, and disturbances on the output side of the control target, such as sensor noise.

In robust control, the compensator C(s) is designed so that the transfer function G_ry(s) from input r to output y has a set characteristic (e.g., 1st-order lag with time constant T_r) even in the presence of disturbance d, observation noise n, and some model error ⊿.

Among robust control methods, if it is as simple as PID control, I would like to actively apply it to small applications. In this part, I would like to verify such practical applications that can be handled with relative ease.

Conventional PID control adjusts the output by setting the proportional, integral, and derivative gains of the PID compensator in the feedback loop, respectively. Each gain is often determined by trial and error, with values based on experience.

The advantage of PID control is that it can be set easily and sensibly without theoretical understanding, but the gain can only adjust the response, not fundamentally improve the characteristics against external disturbances.

In addition, conventional feedback control such as PID control can improve stability and response by increasing the gain in the loop (amplification), but at the same time, it can also amplify modeling errors, noise, and other unwanted effects, which may result in failure to achieve the expected performance. The There is a limit to how much improvement in accuracy and responsiveness can be hoped for with mere feedback control.

To achieve robust control, there is a control method called a 2-degree of freedom control system in which the response characteristics and disturbance suppression can be set independently. There are other robust control methods, such as a disturbance observer method that estimates disturbances and modeling errors and cancels the changes, but this part deals with 2-degree of freedom control.

There are various types of 2-degree of freedom control, but the one I will be dealing with here is the simplest, easiest to verify by anyone, and the easiest to realize through programming.

The block diagram from input r to output y takes the form of the figure below. In addition to the usual feedback control loop, information branched from the input is added.

Rearranging the block diagram above yields the equivalent block diagram shown below. What this means is that it consists of a feedback section and a feedforward section. Disturbance suppression is improved by the feedback compensator C(s), and response characteristics are improved by G_ry(s) and P_n(s) in the feedforward section. Since these can be set independently without interference, they are called 2-degree of freedom.

In "Fundamentals of Feedback Control using Microcontroller [Application]", I explained that the motor rotation speed can be adjusted by changing the input voltage. Since the rotation speed is almost proportional to the input voltage when there is no load, the motor can be approximated by a 1st-order lag as shown in the figure below, where u is the input and y is the output.

The approximate model gain K_m and time constant T_m are obtained by actually measuring the rotational speed y when given the actual input. This is called a kind of parameter identification, but these are approximate model parameters rather than physical parameters such as the actual moment of inertia J or viscous friction D of the DC motor.

This model is more practical because the physical parameters from input u to output y are concentrated in K_m and T_m. The input can be either a voltage or a current, and the parameters K_m and T_m are correspondingly valued. K_m is the gain of the transfer function, but it may be easier to understand as a conversion factor from input to output.

Speed control using a 2-degree of freedom control system with a DC motor as the control target will be explained.

The control object P(s) is a 1st-order lag model with input u as voltage or current and output y as motor rotation speed. This real model has parameters K_m and T_m that may vary, but let P_n(s) be this nominal model.

The parameters of the nominal model P_n(s) are the parameters K_m and T_m obtained from the parameter identification of the real model P(s) as the normative model parameters K_n and T_n, respectively, which are used in the feedforward compensator.

The feedforward compensation section consists of the reference response characteristic G_ry(s) between input and output after characteristic improvement and the inverse system Pn (s) ^-1 of the nominal model, and the feedback compensation section contains the gain C (constant this time depending on the control target) that determines the robust characteristic.

The reference response characteristic G_ry(s) between input and output after characteristic improvement is assumed to be the 1st-order lag of the time constant T_m2, but the time constant T_m2 must be set at a level that can be realized.

Another feature of the type of feed-forward compensation set up in this case is that the real and nominal model parameters cancel each other out even when there are fluctuations.

Inevitably, to make the output match the target value completely, a PI feedback compensation loop is added to eliminate the steady-state deviation on the outside and adjust the appropriate proportional and integral gains.

The key to implementation is to process the robust control section as fast as possible, and process the servo compensation loop with PI feedback slower than that to eliminate the effects of mutual interference.

This is the response when a DC motor is subjected to a pulse-like load disturbance at the step input in open-loop condition. It can be seen that even a small pulse-like load has a large effect on the output.

This is the response when a step-like load disturbance is applied. The torque is balanced with the torque generated by the load at the point where the load has dropped significantly from the command value.

Further increasing the robust compensator gain C improves the suppression of fluctuations in disturbances and modeling errors.

This is the response when the disturbance is step-loaded after increasing only the robust compensator gain C to 3. Again, the output maintains the target response characteristic G_ry(s), and the effect of the robust control can be seen as the effect of the disturbance is almost nonexistent on the output. This is a good example of the effect of 2-degree of freedom control, which allows the response characteristic and disturbance suppression characteristic to be set independently.

A simple method to improve speed control characteristics is the High-gain feedback method. This is a very simple method in which the output side is fed back through gain C₁, and the gain between input and output can be brought close to 1 while suppressing disturbances by combining the values of gains C₁ and C₂.

To improve the disturbance characteristics, the disturbance suppression feedback gain C₁ can be increased, which is called High-gain feedback because it relies on the magnitude of C₁.

The response characteristic can be obtained by adjusting C₂ so that the overall gain becomes 1, but it depends on the size of the disturbance suppression gain C₁ and cannot be obtained arbitrary response characteristics. In addition, since it relies on high gain for disturbance suppression, care must be taken to ensure that it is not affected by noise from the output-side sensor.

Simulation results of the response to a pulsed disturbance load. The disturbance pulse is suppressed by feedback gain C₁, but at the same time the response is affected, resulting in over-response depending on the gain and modeling parameters.

Disturbance suppression and response characteristics are determined by the feedback gain C₁, but the High-gain feedback method is the simplest and most practical if it can be set within a feasible range.

If the speed control model is modeled with a reference response characteristic G_ry(s) that is not affected by disturbances or modeling errors, it can easily be developed into a positioning tracking control. This section describes the Acceleration command type positioning tracking control, which is different from the type that consists of a position control feedback loop outside of a velocity control feedback loop, which is often seen in industrial applications.

The time constant T_m2 of response G_ry(s) should be set small enough to be feasible.

In order to generate acceleration command values, the target values of the following trajectory for position θ₀, velocity θ^'₀, and acceleration θ^"₀ are created in advance. The acceleration command value θ^"ref is consist of the acceleration reference value θ^"₀ used as the feedforward term, and the error between the velocity θ^'₀ and the position θ₀ reference value and the actual values θ and θ^' multiplied by the gains K_v and K_p, respectively, as feedback terms.

The gains K_v and K_p can be easily determined with reference to the response of the 2nd-order lag system, taking into account the quick response and damping of the target. When the error at startup is converged by the response of the set 2nd-order lag system, Δθ(=θ₀ -θ) is 0, which means that the actual position θ follows the reference value θ₀ without delay.

The above equation is valid because robust control is applied to the velocity system and the velocity command value θ^'ref ≈ velocity θ^', which means that the acceleration command value θ^"ref ≈ acceleration θ^".

Looking at the above equation, it seems that the gains K_V and K_P can be determined arbitrarily, but if the gains are selected appropriately, the response may be disturbed. This is because these gains are used to construct a 2nd-order lag system normative model.

When viewed as a position command system, if only the target value θ_o is given as the command value, the transfer function from the target value θ_o to the output θ is K_P/(s²+K_Vs+K_P), the normative model of the 2nd-order lag system, and the output θ follows with a delay.

By including acceleration θ^"ref and velocity θ^'ref in the command values for the normative model, they play a feed-forward role in the position command system, eliminating steady-state deviation and allowing the system to follow without delay.

In other words, the normative model must be stable as a system, and if we construct an inverse system of a normative model with unstable zeros to a normative model with unstable poles, there seems to be no problem since they cancel each other out in the equation, but in reality, there are always lag factors, modeling errors, disturbances, etc., so they do not cancel each other out and remain unstable.

I simulated the tracking performance of the reference response characteristic G_ry(s) when the time constant T_m2 is 10ms and the gains K_v and K_p are K_v=16 and K_p=100, assuming quick response ω_n=10 and damping 0.8. The simulation results show that the position can be tracked even if the time constant is large, since the command value is relatively slow.