Automatic Design of Target Motor Control

Automatic Design of Controllers

The design method of section 8.4 can be applied to the target motor. The method introduced in this section, automatic design of controllers, sometimes called “automatic tuning,” is a technique for automatically determining a controller transfer function. There are a wide variety of automatic design methods. Some require a sufficiently accurate mathematical model of the plant dynamics, and others characterize those dynamics by introducing a known input to the plant and observing the response. From the plant dynamics, controller gains can be adjusted to optimize performance criteria, such as robustness. These criteria can be evaluated by means of simulation, experimentation, analysis, or some combination thereof.

Most methods of designing or tuning controllers can be automated. However, there is an important limitation to most automatic design tools. For a given controller type (e.g., PID), feedback architecture, and plant, there may not be a set of gains that meet the performance criteria. No amount of optimization can change that. It can be the case that the controller type, feedback architecture, or plant must be augmented to meet the design criteria. For instance, a controller with an integrator, rate feedback, a sensor with less noise, or an actuator with more power could be required. Although the design of many controller types can be automated, the automatic tuning of PID or similar controllers is most common. A further limitation of this design tool is that in most cases, there is no consideration for control effort; therefore, proper use of it requires checking that there is sufficient control effort available for a given design.

Perhaps the simplest tuning algorithms are the Ziegler-Nichols “step response” and “frequency response” methods for PID controllers (Åström and Hägglund 2006, sec. 6.2). These were originally developed for manual tuning, but they can be adapted for automatic tuning. These methods do not necessarily require a mathematical model of the plant, the dynamics of which can be estimated from response measurements. The step response method includes the measurement of the open-loop response of the plant to a step input. The frequency response method includes the closed-loop response measurement for increasing proportional gain \(K_P\) (\(K_I = K_D = 0\)) to determine at what gain the output becomes continuously sinusoidal. These tuning methods are relatively simple but can have severe performance limitations.

Most automatic design techniques were developed first for adaptive control, which adapts a controller to improve performance. Automatic design methods could be considered adaptive controllers that stop adapting once they have been tuned because, in applications for which automatic design is effective, the plant’s dynamics don’t change significantly. If the plant’s dynamics do change, the controller can be retuned.

With an accurate plant model, design (automatic or otherwise) can be performed in software, independent of the plant itself. Automatic design in a software environment can be performed with a custom algorithm or with an existing software library. The MATLAB Control Systems Toolbox has an automatic design function called pidtune(), which can tune a variety of PID controllers, including PIDF. For instance, for a transfer function model GP of a plant, a PIDF controller transfer function GC can be tuned with

wc = 30;                      % rad/s, controlled bandwidth
GC = pidtune(GP, 'PIDF', wc)  % tuned PIDF transfer function

The parameter wc is the open-loop \(0\) dB gain crossover frequency target value for the tuning algorithm. This is approximately equal to the closed-loop bandwidth, which is related to the closed-loop rise time and which will be discussed in the next section.

Controller bandwidth

The method that we will use here works best when we set a design controller bandwidth. Although we have not considered the frequency response control theory techniques required to fully understand how to choose a controller bandwidth, some basic background will suffice to motivate our bandwidth selection process.

A closed-loop bandwidth \(\omega_\text{BW}\) is a measure of the closed-loop system’s speed of response: a higher bandwidth corresponds to an ability to track higher frequencies. Higher speeds correspond to a lower rise time \(T_r\), lower peak time \(T_p\), and lower settling time \(T_s\). Two distinct bandwidths are relevant: that of the open-loop transfer function and that of the closed-loop transfer function. However, these are usually near each other, so adjusting one typically has a similar effect on the other.

A common definition of the bandwidth of a transfer function is the interval of frequencies within which the magnitude of its frequency response function is not less than \(-3\) dB relative to its DC (zero-frequency) magnitude. Consider the magnitude Bode plot of a transfer function \(L(s)\), shown in figure 8.8. This could be the open-loop or closed-loop transfer function. It passes frequencies within the bandwidth, but it attenuates higher frequencies. For first-order linear systems with time constant \(\tau\), the bandwidth corresponds to \(1/\tau\). For second-order linear systems, it is near the natural frequency, \(\omega_n\). This suggests that for higher bandwidths, a faster transient response will be observed.

Another closely related concept is the gain crossover frequency \(\omega_c\): for a system with open-loop transfer function \(L(s)\), the frequency at which the open-loop magnitude \(\left|L(s)\right|\) is 1 (\(0\) dB on a Bode plot) and is greater than 1 at lower frequencies. Those familiar with the gain margin and the phase margin, which are defined in Nyquist frequency-domain control theory, will recognize the \(\omega_c\) as the frequency where the phase margin is defined (Nise 2019, sec. 10.6). The gain crossover frequency and the maximum frequency in the bandwidth differ, but they tend to be close in value. For our purposes, assuming that they are similar is reasonable.

It can be shown that the closed-loop bandwidth \(\omega_\text{BW}\) is related to the closed-loop rise time \(T_r\) via the approximation¹ \[ T_r \approx \frac{2}{\omega_\text{BW}}. \qquad{(1)}\] This heuristic will help us design a PIDF controller, which we will discuss in the following section.

Using MATLAB to tune a PIDF controller

We are now ready to use MATLAB to tune a PIDF controller, discretize it, and export it to a C header file for use on the target system.

load(filename);

tau = p.J / p.B;                        % time constant
G = tf([p.Ka * p.Km / p.B], [tau, 1]);  % electromech. system
                                        % transfer function

s = tf([1, 0], [1]);            % make s a tf object
GP = G / s;                     % plant tf, Theta/I_S
H = tf([1], [1]);               % unity feedback

Tr = 0.04;   % s, design closed-loop rise time (conservative)
wc = 2 / Tr; % rad/s, control bandwidth (gain crossover frequency)
options = pidtuneOptions('DesignFocus', 'reference-tracking');
GC = pidtune(GP * H, 'PIDF', wc, options);
GCL = GC * GP / (1 + GC * GP * H);  % closed-loop tf Theta/U_a

T = 5e-3;                                 % s, sample period (BTI)
GCd = c2d(GC, T, 'Tustin');               % using Tustin's method
GPd = c2d(GP, T, 'Tustin');
Hd = c2d(H, T, 'Tustin');
GCLd = GCd * GPd / (1 + GCd * GPd * Hd);  % discrete closed-loop tf

R_deg = 25;                     % deg, command angular position
R_rad = R_deg * pi / 180;       % rad
t = 0:T:0.2;
Theta = R_rad * step(GCL, t);
ThetaT = R_rad * step(GCLd, t);

U_R = GCd / (1 + GCd * GPd * Hd);  % V, control effort cl tf, ampl. in.
u = R_rad * step(U_R, t);          % V, amplifier input voltage
u_c = u * p.Ka;                    % A, amplifier output current

u_v = p.R * u_c(1:end - 1) + ...   % amplifier output voltage
      p.L * diff(u_c)/T + ...
      p.Km * diff(ThetaT)/T;       % OmegaT = diff(ThetaT)/T