Prerequisite

Install Matlab Simscape App

Please make sure the Simscape and Simscape Multibody add-on has been installed in Matlab as shown in the following pictures. You can find the information and install this add-on by clicking “Add-Ons” button.

Simscape™ enables you to rapidly create models of physical systems within the            Simulink®  environment. With Simscape, you build physical component models based on physical connections that directly integrate with block diagrams and other modelling    paradigms. You model systems such as electric motors, bridge rectifiers, hydraulic       actuators, and refrigeration systems, by assembling fundamental components into a    schematic. Simscape Multibody™ provides a multibody simulation environment for 3D   mechanical systems.

Introduction

A motor is an electromechanical component that produces a displacement output for a voltage input, i.e., a mechanical output generated by an electrical input. Applications for systems with electromechanical components are robot controls, sun and star trackers, and computer tape and disk-drive position controls.

In this lab, we will firstly derive the transfer function for one particular kind of electromechanical system, the armature-controlled DC servomotor. Then we will put this motor into a feedback control system and design a controller using root locus    method (in Matlab) to make sure the transient and steady-state responses of the complete system meets the design requirements. Finally, we will verify the feedback control system design in Simulink.

Modelling

The schematic of the armature-controlled DC servomotor is shown in Figure 1 (a), and the transfer function we will derive is shown in Figure 1 (b) [1].

Figure 1

In Figure 1, a magnetic field is developed by stationary permanent magnets or a stationary electromagnet called the fixed field. A rotating circuit called the armature, through which current ia (t) flows, passes through this magnetic field at right angles and feels a force, F = Blia (t) , where B is the magnetic field strength and l is the    length of the conductor. The resulting torque turns the rotor, the rotating member of the motor.

In this electromechanical system, the electric circuit can be modelled by applying Kirchoff's laws as we covered in Lecture 3: Mathematical Modelling, and the mechanical part of the system, the rotor, can be modelled by applying Newton’s laws for Rotational mechanical systems with the consideration of some electromagnetic effects.

Rotational mechanical systems are handled the same way as translational    mechanical systems (as you have learned in high school), except that torque replaces force, angular displacement replaces translational displacement and the     term associated with the mass is replaced by inertia. Table 1 shows the components along with the relationships between torque and angular velocity, as well as angular displacement [1]. The values of K, D, and J are called spring constant, coefficient of viscous friction, and moment of inertia, respectively.

Table 1

There is another phenomenon that occurs in the motor: A conductor moving at right angles to a magnetic field generates a voltage at the terminals of the conductor       equal to e = Blv , where e is the voltage and v is the velocity of the conductor normal to the magnetic field. Since the current-carrying armature is rotating in a magnetic    field, its voltage is proportional to speed. This voltage is called back electromotive    force (back emf), which is given by

vb (t) = Kb (1)

where Kb  is a constant of proportionality called the back emf constant.

By applying Kirchhoff's voltage law and writing a loop equation around the armature circuit shown in Figure 1, we have

Ra ia (t) + L + vb (t) = ea (t)             (2)

The torque developed by the motor is proportional to the armature current. So we have

Tm (t) = Kt ia (t)         (3)

where Tm  is the torque developed by the motor, and Kt  is a constant of proportionality, called the motor torque constant, which depends on the motor and  magnetic field characteristics. In a consistent set of units, the value of Kt  is equal to the value of Kb . So let Kt  = Kb  = K .

By apply Newton’s 2nd law and referring to Table 1 for the relationships between torque and angular velocity and inertia, we have

Tm (t) = J + Dm

(4)

Design

The physical parameters of the motor used in this lab are summarised in Table 2.

Table 2

Design requirements

The most basic requirement of a motor system is that it should rotate at the desired   speed. In this lab, the steady-state error of the motor angular velocity is desired to be less than 1.5%. Another performance requirement for the motor system is that it        must accelerate to its steady-state speed as soon as it switches on. In this lab, the    desired settling time is less than 2 seconds. Additionally, since a speed (angular        velocity for the motor) faster than the reference speed may damage the equipment,   we want an overshoot of less than 5% is desired for a step response.

In summary, for a unit step command ( 1-rad/sec step reference) in motor speed, the control system's output should meet the following requirements.

1. Settling time less than 2 seconds.

2. Overshoot less than 5%.

3. Steady-state error less than 1.5%.

In order to meet the design requirement, a unity feedback control system with the structure shown below should be designed in this lab.

r          +        e t                   ea

C(s)

e.(t)

一

Figure 2

Control System Designer

The Control System Designer app lets you design single-input, single-output (SISO) controllers for feedback systems modelled in MATLAB® or Simulink® (requires        Simulink Control Design™ software).

The documentation of the Control System Designer app can be found in “Control System Toolbox->Control System Design and Tuning->Classical Control Design->Control System Designer”

For design via root locus, the app is suggested to be launched using command “controlSystemDesigner('rlocus', plant_transfer_function)” .

Design via Gain Adjustment

Firstly, let us try to design the feedback control system shown in Figure 2 using root locus technique with only gain adjustment and investigate the system performance.

Lag Compensator Design

Now let us design a lag compensator to improve the steady-state error while also meet the requirement for the transient response.