Motor velocity control

Introduction

In this exercise, a closed-loop control system for the DC motor will be developed. This system is similar to actuators used in many types of positioning systems. The primary drive of one axis of an automated machine tool or of one axis of motion of an industrial robot is often a computer-controlled DC motor.

Our system will control the motor velocity, as shown in figure 7.24. The control algorithm will repeatedly compare the measured velocity of the motor \(\Omega_J\) with the desired reference velocity \(\Omega_R\), and automatically alter the applied control current to correct any differences. Although this is not a trivial computer control task, we have developed nearly all the required elements in the preceding lab exercises.

The quadrature encoder (through the field-programmable-gate array (FPGA)), the DAC (connected to the current amplifier), and the periodic timer interrupt will be combined to control the DC motor.

As in lab 6, a separate timer thread will produce an interrupt at the end of each basic time interval (BTI), also called the “sample period.” The ISR will load the IRQTIMERWRITE and IRQTIMERSETTIME registers to schedule the next BTI, and then call functions to do the following:

1. Read the encoder counter and compute the velocity.
2. Execute the motor control algorithm.
3. Save the results as necessary.

The control system will be “table-driven.” That is, the parameters used by the control algorithm (reference speed, system gains, and BTI length) will be kept in a special table of values. Through the keypad and display user interface (UI), the values of the parameters in the table will be altered (interactively) by a “table editor” function called from the main program thread. The only tasks of the table editor will be to change the table values in response to commands from the keypad and to display performance information.

This table-driven structure will allow the user to change any of the control parameters, at any time, without stopping the execution of the control algorithm. It will appear as though two programs, the table editor and the control algorithm, are executing simultaneously.

You will not write the table editor. The required function, ctable2(), is described in section D.2. It has been included in the T1 C library. Although you will not write this function, it uses the basic keypad/display algorithms that you developed in lab 1, lab 2, lab 3. The prototype for ctable2() is in ctable2.h.

Following the controller design discussion of section 7.6, the control block diagram is shown in figure 7.25. The motor will be controlled using a PI control law, as shown inside the dashed box.

The PI control law relates the error \(e(t)\) to the output control signal \(u_a(t)\) using the gain constants \(K_P\) and \(K_I\).

Applying Tustin’s method to the continuous controller transfer function \[ G_C(s)=K_P+\frac{K_I}{s}, \] the corresponding discrete transfer function is \[ G_C(z) = \frac{U_a(z)}{E(z)}=\frac{b_{0}+b_{1}z^{-1}}{a_{0}+a_{1}z^{-1}}, \qquad{(1)}\] where \[ \begin{aligned} a_{0}&=1, &b_{0}&=K_P+\frac{1}{2}K_I T, \\ a_{1}&=-1,\text{ and} &b_{1}&=-K_P+\frac{1}{2}K_I T, \end{aligned} \] where \(T\) is the sample time, and the error is \[ E(n)=\Omega_R(n)-\Omega_J(n). \] See section 6.4 on discrete time systems. You will implement the corresponding difference equation using the general-purpose algorithm that you developed in lab 6.

Writing the program

Drawing on your previous work, write two threads to (1) communicate with the user and (2) control the motor.

double *Omega_R = &((threadResource->a_table + 0)->value);
double *Omega_J = &((threadResource->a_table + 1)->value);
// etc.

Laboratory exploration and evaluation

In the following, let the base set of controller parameters be a sample time or BTI of \(T = 5\) ms and the controller gains \(K_P\) and \(K_I\) from eq. ¿eq:base-set-gains?, repeated here: \[ \begin{aligned} K_P &= -\frac{B + 2 J \Re(\psi)}{K_a K_M}\text{ and} \\ K_I &= \frac{Z \left(B + 2 J \Re(\psi)\right)}{K_a K_M}. \end{aligned} \] The T1a target system parameters can be found in subsection L7.4² and the design point \(\Re(\psi)=-20\) rad/s, which corresponds to \(T_S\approx 0.2\) s, and \(Z = -20\) rad/s. See subsection L7.4 for a control systems model and associated transfer functions.

Appendix: DC Motor Control Model

The model shown in figure 7.26 is a continuous approximation of the discrete control of the DC motor implemented in this lab exercise. Three assumptions are made:

1. The sampling frequency is much larger than the system natural frequency.
2. Time delays caused by computation are insignificant.
3. The control value does not saturate. Parameters are shown in table 7.2.

Typically, the effect of the mechanical damping \(B\) is small compared to the effect of the proportional term in the controller. Also, during transient response, the damping torque is small compared to the inertial torque. Therefore, for design purposes, we assume that \(B = 0\) N\(\cdot\)m\(\cdot\)s/rad.

The following three transfer functions are obtained from the block diagram:

1. The transfer function relating the reference input \(\Omega_R(s)\) to the output angular velocity of the load \(\Omega_J(s)\) is \[ \frac{\Omega_J(s)}{\Omega_R(s)}=\frac{(\tau s+1) {\omega_n}^2}{s^2+2\zeta \omega_n s+{\omega_n}^2}. \qquad{(2)}\]
2. The transfer function relating the disturbance torque input \(T_d(s)\) to the output angular velocity of the load \(\Omega_J(s)\) is \[ \frac{\Omega_J(s)}{T_{d}(s)}=\frac{s K_d {\omega_n}^2}{s^2+2\zeta \omega_n s+{\omega_n}^2}. \qquad{(3)}\]
3. The transfer function relating the reference input \(\Omega_R(s)\) to the DAC output voltage \(U_a(s)\) is \[ \frac{U_a(s)}{\Omega_R(s)}=\frac{s (\tau s+1) K_u {\omega_n}^2}{s^2+2\zeta \omega_n s+{\omega_n}^2}. \qquad{(4)}\]

Let \(K=K_a K_M\). Then the natural frequency and damping ratio are \[ \omega_n=\sqrt{\frac{K_I K}{J}} \quad\text{and}\quad \zeta=\frac{K_P}{2}\sqrt{\frac{K}{J K_I}} \] where \[ \tau=\frac{K_P}{K_I},\quad K_d=\frac{1}{K_I K},\text{ and}\quad K_u=\frac{J}{K}. \]