Timetable
Day 1
Time | Topic | Description |
09:00 | 1. Review of DC motor model. (5s)
(4.3 min/slide) |
A review of the DC motor and its components is given. This is followed by a derivation of its model in the form of interconnected linear differential equations. |
2. Review of Background Theory, Dynamical Systems, Linearity and Laplace transforms (6s)
(4.3 min/slide) |
First some common control system terminology is reviewed. Then the properties of dynamical systems relevant to control system design are discussed. The Laplace transform is reviewed. | |
3. Review of Transfer Function and Block Diagram Models (10s)
(4.3 min/slide) |
The application of the Laplace transform to form transfer function and block diagram models of linear, time invariant plants is reviewed, exemplified using the DC motor model of Topic 1. This is used to derive transfer functions between the variables. These transfer functions are then used to form a block diagram model of the DC motor. | |
10:30 | Simulation 1 (30 min) | DC Motor Models |
11:00 | Coffee break. | |
11:15 | 4. Review of Background Theory: Dynamic Character and Modes (8s)
(4.4 min/slide) |
Analysis of the dynamic character of a system in terms of its modes and pole locations is reviewed. |
5. Model Order Reduction
(16s) (4.4 min/slide) |
Three methods are presented for obtaining simpler plant models for control system design, exemplified using the DC motor model of Topic 1. The first method entails analysis of the modes via partial fraction expansion of the transfer function to determine modes that can be ignored due to fast decay of their impulse responses. The second method entails comparison of the intrinsic dynamics of the physical components of a plant, illustrated by ignoring the electrical time constant of the DC motor. The third method is specific to electric drives, consisting of the introduction of a hysteresis current control loop using power electronics to eliminate the dynamic lag associated with inductive circuits. | |
13:00 | Buffet lunch | Opportunity for informal discussion and feedback. |
Time | Topic | Description |
14:00 | Simulation 2 (60 min) | a) Model order reduction by elimination of Fast Component Dynamics
b) Model Order Reduction by Cascade Control |
15:00 | 6. Traditional Controllers: Model Based Design: General (7s)
(4.3 min/slide) |
First the traditional PID controller and the purposes of the three terms are reviewed. Then a proportional controller is applied to the reduced order model of the DC motor for speed control and the closed loop transfer function derived. This is used to demonstrate the design of first order linear control systems to achieve a specified settling time, in contrast to gain adjustment by trial and error. Then the steady state error limitations with a constant external load torque are demonstrated, leading to the introduction of an integral term. At this juncture, Mason’s formula for transfer function derivation is presented for use directly with linear time invariant (LTI) control system block diagrams to avoid having to convert them to signal flow graphs. Then this is used to attain prescribed undamped natural frequency and damping ratio for the second order speed control system resulting from use of the PI controller. Then a short derivation of the Dodds settling time formula for LTI systems of arbitrary order with coincident closed loop poles is given in preparation for the design of control systems of third and higher order. |
15:30 | Coffee break | |
15:45 | 6. Traditional Controllers: Model Based Design: Speed Control Part 1 (17s)
(4.4 min/slide) |
The settling time formula for second order underdamped linear systems is applied to the speed control system using the PI controller. The zero introduced by the PI controller and its potential effect of causing a single overshoot of the step response, despite real closed loop poles, are demonstrated. The related IP controller that eliminates the zero is then introduced and designed. |
17:00 | End of day one. |
Day 2
Time | Topic | Description |
09:00 | 6. Traditional Controllers: Model Based Design: Speed Control Part 2 (8s)
(5 min/slide) |
The settling time formula for second order underdamped linear systems is applied to the speed control system using the PI controller. The zero introduced by the PI controller and its potential effect of causing a single overshoot of the step response, despite real closed loop poles, are demonstrated. The related IP controller that eliminates the zero is then introduced and designed. |
09:40 | Simulation 3 (40 min) | Speed Control of a DC Motor Based Drive with Negligible Friction |
10:20 | 6. Traditional Controllers: Model Based Design: Position Control (12s)
(5 min/slide) |
Position control of the same DC motor based electric drive is considered, commencing with a simple proportional controller and explaining why this will not work, moving on to a PD controller and showing the zero it introduces in the closed loop transfer function. Elimination of this zero by changing to a DP controller is then demonstrated. The steady state error caused by a constant external load torque is then eliminated with an IPD controller. The resulting third order position control system is designed with the aid of the settling time formula. The IPD controller is then replaced by a traditional PID controller and it is shown that this introduces two zeros in the closed loop transfer function, which can potentially cause an overshoot and an undershoot of the step response even with real closed loop poles, thereby demonstrating the reason for having an IPD controller. |
11:20 | Coffee break | |
11:35 | 7. The State Space Model, State Feedback Control and Observers (8s) (5 min/slide) | The concept of state is first discussed and then the derivation of state space models from transfer function block diagrams is explained, exemplified by the reduced order DC motor model of Topic 5. This is followed by the principal reason for having state space models, i.e., state feedback control. This is applied to the reduced order DC motor model and its equivalence to the DP controller highlighted. |
12:15 | Simulation 4 (30 min) | Linear State Feedback Position Control of a DC Motor Based Drive |
12:45 | Buffet lunch | Opportunity for informal discussion and feedback. |
13:45 | 7. The State Space Model, State Feedback Control and Observers (continued)
(5s) (5 min/slide) |
Observers that are often needed to render state feedback control practicable are introduced, again exemplified using the DC motor model. |
14:10 | Simulation 5 (45 min) | Observer for LSF Position Control of a DC Motor Based Drive |
14:55 | Coffee break | |
15:10 | 8. Torque Estimation (4s) (5 min/slide) | The DC motor state space model derived in Topic 7 is extended to include a constant load torque. An observer using this model is then formed, the principal purpose of which is load torque estimation in this application. Design of the observer by pole placement that also enables a time varying load torque to be estimated using the settling time formula is then presented. |
15:30 | Simulation 6 (45 mins) | Position Control with Augmented Observer for Torque Estimation and Compensation |
16:15 | 9. Modelling of the Mechanical Load (9s)
(5 min/slide) |
The inverse dynamic load representation is introduced that can be combined with a model of the motor alone to form a plant model. This is demonstrated for a balanced rigid body load and justified by agreement with the plant model of Topic 5 modified by simply adding the load moment of inertia to the rotor moment of inertia. Then the same method is used to model a DC motor driving a rigid body load via a flexible coupling. Attention is drawn to the complex conjugate zeros in the transfer function, which is between the electromagnetic torque and the rotor speed, that would also appear in the closed loop transfer function resulting from the application of any controller, bearing in mind that pole-zero cancellation is prohibited due to the zeros lying on the imaginary axis of the s-plane. It is then demonstrated that these zeros do not arise if the angular velocity of the mechanical load is measured rather than that of the rotor. |
17:00 | End of day two. |
Day 3
Time | Topic | Description |
09:00 | 10. Introduction to the Polynomial Controller (22s) (4.8 min/slide) | First, the reasons for introducing this simple but versatile controller are stated. Motivation is then provided by a demonstration of the inability of traditional controllers to be designed by pole assignment for position control of the drive with the flexible coupling introduced in Topic 9. The structure of the polynomial controller is then developed by first considering a general controller for LTI plants comprising a linear state feedback control law aided by an observer, which permits pole assignment design. Further motivation is then provided through the resulting general transfer function block diagram structure being much simpler than that of the linear state feedback and observer shown separately. Then the polynomial controller applied to the generic third order LTI plant is studied. This is used to develop a convenient pole placement design equation in matrix form that can be readily programmed on a computer, the extension to plants of any order being self-evident. Then a controller implementation block diagram based on realisable blocks is presented for the third order generic plant, which has a recognisable structure that can be extended to plants of any order. Then the method is clarified by application to the DC motor model of Topic 5. This is followed by the introduction of an integral term by plant augmentation to eliminate steady state errors due to constant external load torques. Finally, application to the drive with the flexible coupling introduced in Topic 9 is presented. |
10:45 | Coffee Break | |
11:00 | Simulation 7 (60 min) | Polynomial Control of the Load Speed of an Electric drive with a Flexible coupling |
12:00 | 11. The Permanent Magnet Synchronous Motor (PMSM), its Model and Vector Control (14s)
(5 min/slide) |
The basic structure of a PMSM motor is described. Then the production of a rotating magnet field by means of three phase stator currents and the torque production process through interaction with the permanent magnets are explained. This is followed by the representation of the stator current and the magnetic flux by two-dimensional rotating vectors (referred to as phasors more commonly in the past). The Park transformation is then introduced, which is a rotational transformation enabling the observer to view the current and flux vectors in the d-q (direct-quadrature) frame of reference rotating with the rotor instead of viewing them from the a-b frame of reference fixed in the stator. The fact that the vectors are two dimensional enables the three-phase machine to be modelled as an equivalent two-phase machine. The Clarke transformation from three phase currents to two phase vector components in the stator fixed a-b frame is introduced for this purpose. Then a general block diagram for vector control of a PMSM drive based on the Clarke-Park transformations and inverse transformations is presented. Then the PMSM d-q model, comprising a set of differential equations, upon which controller designs may be based, is presented and its similarities with the DC motor model noted. It is pointed out that vector control is not a feedback control technique but is usually defined as the process of controlling the current vector so that it is at right angles to the magnetic flux vector and therefore maximises the electromagnetic torque magnitude for a given stator current magnitude. Vector control is usually based on the d-q model that is the closest possible approach to the DC motor model. This enables PI controllers to be applied butt is also pointed out that the product nonlinearity in the model of a salient PMSM (with unequal quadrature axis and direct axis inductances) precludes model based control system design to achieve a specified closed loop performance. Then it is pointed out that for a salient PMSM, mutual orthogonality of the current and flux vectors is not optimal and an alternative approach yielding optimal vector control is presented. |
13:10 | Buffet Lunch | Opportunity for informal discussion and feedback |
14:10 | 12. Direct Torque Control of PMSM Drives and Feedback Linearising Control (7s)
(5 min/slide) |
It is first pointed out that the hysteresis current controllers enable a time varying electromagnetic torque demand to be realised without dynamic lag, which is already direct torque control, but that this simple approach does not allow sufficient control of the power electronic switching frequencies to guarantee smooth operation free of undesirable vibrations and noise. So, this provides a motivation for using the stator voltages as control variables with pulse modulators for ensuring the power electronic switching frequencies are not so high that the switching power loss is too great and not low enough to cause undesirable vibrations, while accepting a higher order motor model. It is stated that such drives are traditionally serviced by PI controllers that must be tuned by trial and error when the order of the system exceeds the number of adjustable gains and when nonlinearities are present. Under these circumstances the dynamics of the resulting closed loop system may fall short of the ideal performance and not be known in the form of a transfer function. |
14:45 | Coffee | |
15:00 | 12. Direct Torque Control of PMSM Drives and Feedback Linearising Control (continued) (4s)
(5 min/slide) |
It is emphasised that these problems are easily solved through the alternative of feedback linearising control. The technique is then fully explained, commencing with the determination of the relative degrees with respect to the controlled plant outputs, exemplified through its application to direct torque control of a PMSM drive with the stator voltages as the control variables. Finally, dynamic lag pre-compensation is introduced to enable the electromagnetic torque to be controlled to precisely follow a time varying demand without dynamic lag. |
15:20 | Simulation 8
(100 mins) |
Direct Torque Control of PMSM Drive using Feedback Linearising Control |
17:00 | End of course |
Delivery Methods
The lectures will be supported by an animated PowerPoint presentation designed for easy assimilation of the concepts. The attendees will be stimulated throughout the course by being involved through discussion, being encouraged to ask questions at any point and also to answer questions posed to them as a group. Most topics will be followed by simulation sessions in which the attendees carry out Matlab-Simulink simulations to reinforce their understanding. Individual attention will be available to provide guidance needed by any attendee as and when necessary while working through the simulations.
The attendees will also be provided with an electronic copy of the tutor’s textbook ‘Feedback Control: Linear, Nonlinear and Robust Techniques and Design with Industrial Applications’, Springer 2015, ISBN 978-1-4471-6674-0’ that serves as a course handbook and may subsequently be used as a reference aiding project development.