Essay: Control of a Twin Rotor MIMO System

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CONTENTS
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Chapter 1
Introduction
1.1 Motivation
Helicopters are extremely useful pieces of flying equipment, with the ability to both take off and land vertically. This along with being able to hover in a given place and fly in any direction, including backwards and to either side, means that they have a huge amount of uses worldwide and in many different environments, which include but are not limited to rescue operations, observation platforms, firefighting and military purposes.
Helicopters get their power from rotors or blades. They do not look much like a helicopter, but the rotors blades have an aerofoil shape when spinning, much like the wings of an airplane. This means that as the rotors turn, air flows more quickly over the tops of the blades than it does below which creates the lift required for flight.
http://www.boeing.com/companyoffices/aboutus/wonder_of_flight/heli.html Accessed on 25/11/14
The same aerodynamic principles allow helicopters to fly as well as any other aircraft. The force needed to keep any aircraft in the air is ‘Lift’, which is produced by air flowing over the wings.
Wings are shaped so that the air flowing over the top surface has to travel further, therefore travelling quicker than the air under the wing. This in turn causes pressure above the wing to be lower, meaning that the lower part of the wing is sucked up by the lower pressure. This therefore makes any aircraft fly and helicopters are no different.
A helicopter however is a far more complex machine than an aeroplane although the fundamental principles of flight are the same. The rotor blades of a helicopter are identical to the wings of an aeroplane, in where air is blown over them, lift is produced. The major difference between an aeroplane and a helicopter however is that the flow of air is produced by rotating the wings (rotor blades), rather than by moving the entire aircraft. When a helicopters rotor blades start to rotate, the air flowing over them produces lift which causes the helicopter to rise into the air. Thus it is seen that the engine is used to turn the blades which produce the required lift, rather than using an engine to produce thrust to lift an aeroplane into the air.
There is more required from a helicopter however as it is required to lift into the air when wanted, rather than rise up as soon as the engine is started. In a helicopter, for various reasons, the blades need to be turned at the same speed all the time so a different way to control the amount of lift produced is needed.
The magnitude of this lift is actually changed by altering the angle at which the rotor blades meet the air blowing over them. This is known as increasing or decreasing the pitch angle of the rotor blades, which changes the amount of lift produced. This enables helicopters to be able to hover when the pilot decides to, by increasing the pitch on the wings.
However, increasing the lift also causes more drag to appear in the blades. This means the engine needs to produce more power in order to prevent the rotor blades slowing down. In the past this had to be operated by the pilot, however in present day helicopters this is operated automatically. This massively improves the safety aspect and takes more responsibility away from the pilot, allowing him to concentrate on other vital aspects of flying.
As mentioned earlier, helicopters are able to move in all directions. This is done by changing the pitch angle of the rotor blades, which differs between each blade individually and by a different amount. To turn the helicopter left or right however, the pitch angle of the tail rotor only ‘ the small rotor at the end of the helicopter is altered.
This is a basic outline of how helicopters achieve flight and there are various other factors which have not been mentioned here, such as different air to flow over each blade.
http://www.decodedscience.com/how-do-helicopters-fly/20418/2, accessed on 25/11/14
There are a variety of helicopters in the market and the design has changed over the years, however nowadays the most popular type is one with a lift blade and tail blade.
The main rotor of a helicopter can have anywhere from two blades to six depending on the design which rotates on the horizontal plane. The main rotor blades are hinged to the rotor heads in such a way that they have limited movement up or down and also the pitch, or angle of incidence, can be altered.
Root: The inner end of the blade where the rotors connect to the blade grips.
Blade Grips: Large attaching points where the rotor blade connects to the hub.
Hub: Sits atop the mast, and connects the rotor blades to the control tubes.
Mast: Rotating shaft from the transmission, which connects the rotor blades to the helicopter.
Control Tubes: Push/pull tubes that change the pitch of the rotor blades.
Pitch Change Horn: The armature that converts control tube movement to blade pitch.
Pitch: Increased or decreased angle of the rotor blades to raise, lower, or change the direction of the rotors thrust force.
Jesus Nut: Is the singular nut that holds the hub onto the mast.
http://www.helicopterpage.com/html/terms.html, accessed on 25/11/14
The tail rotor of a helicopter has anywhere from two to five small blades mounted on the tail at the rear of the aircraft. It rotates on the vertical plane and turns the helicopter by changing the pitch in the direction desired.
This project is based on using the twin rotor multiple-input multiple-output system (TRMS) available at the laboratories of the School of Engineering and Built Environment at Glasgow Caledonian University (GCU). The (TRMS) is a piece of educational equipment used for control experiments designed and manufactured by Feedback Instruments Ltd. Its behaviour represents that of a helicopter in certain aspects, however the angle of attack of the rotors is not able to be altered and the aerodynamic forces are controlled by varying the speeds of the motors.
Significant cross-coupling is observed between the actions of the rotors, with both rotors each influencing angle positions. This along with the fact that some of its states and outputs cannot be measured creates a challenging engineering problem with the TRMS.
The control objective of most systems is to make it achieve its desired output quickly and accurately and the TRMS aim is no different. The users target is to make sure that the TRMS beams moves rapidly and with high accuracy to the desired position, with respect to the pitch and yaw angles. Although the TRMS system is a two-degree-of-freedom (2-DOF) device it is often considered as a two separate one-degree-of-freedom (1-DOF) systems. (L. Osikibo 2012).
As a starting point for this project, the model provided in the manual and some additional references will be used.
1.2 Aim of Study
To design, implement and analyse complex logic controllers for the TRMS angles
1.3 Objectives
Develop independent dynamic models for the pitch and yaw angles in 1-DOF
Design two complex logic controllers for both the pitch and yaw dynamic models in 1-DOF
Implement two complex logic controllers in both simulation and real experiements in 1-DOF using various set-point profiles.
Compare performances of the complex controllers with PID controllers supplied by the manufacturer in 1-DOF control.
Implement 2-DOF control of the TRMS.
Compare performance of the complex controllers and the PID controllers in 2-DOF TRMS control.
Investigate on complex controllers robustness and compare with PID controllers robustness in 2-DOF TRMS control.
Discuss and critically analyse the comparative results in a logical manner.
Make engineering conclusions and relevant recommendation.
1.4 Project Structure
The work is divided into six chapters and the contents of each chapter are briefly described as follows:
CHAPTER 1: introduced the project with the motivation of the study and thereafter, aim of the study, objectives to achieve and the thesis layout.
CHAPTER 2: gives general background studies of the TRMS, review of existing literature on modelling and control of the TRMS.
CHAPTER 3: presents general methodology which unfolds with TRMS system description, modelling of1-DOF TRMS angles, complex controller design, controller implementation in simulation and real experiments, comparative studies on performance of the complex controllers and PID controllers.
CHAPTER 4: presents the results of the simulation and real time experiments.
CHAPTER 5: presents discussion and critical analysis of results.
CHAPTER 6: presents project conclusion and recommendation for future work.
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Chapter 2
Literature Review
2.1 Background Studies on TRMS
The TRMS is a considered by researchers as a challenging engineering problem due to the high nonlinearity, significant cross-coupling between its axes and also because some of its states and outputs are inaccessible for measurements. (L. Osikibo 2012). (Juang et al., 2008b & Rahideh et al., 2008b). All of this means that this experimental system gives a high-order nonlinear and complex MIMO system from a control perspective, which is reported in Lie et al. (2006) and Ahmad et al. (2000a). The system can also be a good benchmark for implementing and conducting various control strategies and has been used many times for MIMO system laboratory experiments over the years as reported in Mahmoud et al. (2010) and Tahersima & Fatehi (2008). An image of this platform and aerodynamic test-rig TRMS is shown below, giving details of what each part is.
The control objective for this experimental system is to make the beam of the TRMS move as quickly and as accurately as possible to the desired positions in terms of its pitch and yaw angles. According to Rahideh & Shaheed (2006) and Juang et al. (2008b) this is the case when considering the TRMS as being decoupled between its vertical (pitch) and horizontal (yaw) axes, however in Juang et al. (2011), the authors objective although similar, considered stabilizing the TRMS in a coupled condition.
Because of the high nonlinearity of the TRMS and the significant cross coupling, a lot of researchers have deemed it necessary to decouple the system into its vertical and horizontal subsystems, especially when deriving the plant dynamic models. Once this is done they are then put together in order to get the complete system model. For example, in Wen & Lu (2008), the authors identified the nonlinear problem of the system and decoupled the TRMS into two single-input single-output (SISO) systems, and considered the cross-coupling effects as disturbance to the system. In the same year Juang & Liu (2008) also presented the case for doing this. A similar approach has also been taken by Ahmed et al. (2009b) and is seen in the report that the authors have used hadamard weights to effectively decouple a twin rotor system into its vertical and horizontal plane to give a desired performance in terms of control. The main objective in this research is to reduce cross-coupling effects in the twin rotor dynamics. In this they have generally stated that all twin rotor aircrafts have high cross-coupling in all degrees of freedom (DOF) and that the yaw or azimuth dynamics prevents maneuvers by an operator. This then requires compensation for cross-coupling which incurs additional cost on the system control. However in Ahmed et al. (2009a) the writes have listed some decoupling techniques which include declaring coupling as a disturbance and designing a robust controller to handle it. Also listed is integrating a decoupling sequence and treating coupling explicitly or integrating decoupling as part of system dynamics and with the appropriate compensation scheme in place, state the space approach and the linear matrix inequalities.
The TRMS possesses some similar characteristics as a conventional helicopter according to some research reports and the system manuals but there are a few major differences. As declared earlier in this research project and also in the system manual (Feedback Instruments Ltd, 1998), helicopter aerodynamic force is controlled by changing the angle of attack on the blades, thus changing the flight direction whereas in the case of the experimental TRMS which has a fixed angle of attack, the aerodynamic force is controlled by varying the speed of rotation of the DC motors which produce the propulsive force in the rotors. In Toha et al. (2010) the writes have stated the similarity between the TRMS and a standard helicopter which is that both systems possess strong cross-coupling between their collective (main rotor) and tail rotor. The authors however created a table listing the differences between the two which is outlined below. A similar table has also been produced in Rahideh & Shaheed (2008b).
System TRMS Helicopter
Location of pivot point Midway between the two rotors The main rotor head
Lift generation of vertical control Speed control of the main rotor. Collective pitch control.
Yaw is controlled by Tail rotor speed Pitch angle of the tail rotor blades*
Cyclical control No Yes for directional control
*Pitch angles of the blades of the main rotor are also changed but this is at constant rotor speed, hence not affecting control of the yaw.
2.1.1 TRMS Configuration
The physical structure or unit of the TRMS Is provided with locking screws which mechanically restrict movement in any particular axis of rotation (Feedback Instruments Ltd., 1998). This means that the system is able to be experimented on in either one-degree-of-freedom (1-DOF) or two-degree-of-freedom (2-DOF) as reported in Wen & Lu (2008) and Subudhi (2009). For 1-DOF experiments, this means that the TRMS is decoupled into its vertical and horizontal planes respectively by the main and tail rotors as shown in Rahideh et al. (2008b) and Wen & Lu (2008). The system manual provided by the manufacturer (Feedback Instruments Ltd., 1998) explains that the TRMS is considered a 2-DOF system when its angle or positions are simultaneously under the control of two controllers without any physical restriction of rotation in any plane (i.e. when either of the locking screws are in place) and it is under 1-DOF when the subsystem of the TRMS, either horizontal or vertical, is controlled independently of the other part.
The twin rotor MIMO system is said to be a 2-DOF system with two inputs and two outputs in Karimi et al. (2006), as is also the case in Liu et al. (2006) with the same definition for the TRMS. However the authors presented both 1-DOF and 2-DOF TRMS systems and emphasized that without the cross-coupling between the vertical and horizontal axes that the TRMS would represent two 1-DOF TRMS respectively for both the vertical and horizontal planes. In Juang & Liu (2008) and Wen & Lu (2008) a similar account is suggested with the authors presenting 1-DOF and 2-DOF TRMS with the aim of making the TRMS follow a desired trajectory or reference position. However in Islam et al. (2003), the authors view about the TRMS degree-of-freedom is actually that of a three-degree-of-freedom but is later considered as a system with 2-DOF in their work. Although interesting that the writer viewed it as a 3-DOF TRMS it was not stated if this requires three controllers or has three axes of rotation or control objectives.
Other literature has addressed other degrees of freedom in twin rotor helicopters. In Khan & Iqbal (2004), the design of an optimal regulator for 3-DOF was reported. In this case, three areas of control were listed which were azimuth(yaw), elevation(pitch) and height. In the paper, ‘dynamic modelling of a TRMS using analytical and empirical approaches’ by Rahideh et al. (2008b) the case of modelling and control of a four rotor (quad rotor) system is mentioned while in Ahmad et al. (2001a), and Krishnamruthy et al. (2008), the configuration of a 6-DOF dynamic model of a twin rotor aircraft has been looked at. Ahmad et al. (2001a) however related that to the estimation of the parameters of a linearized 6-DOF equations from the flight data of a classical fixed and rotary wing aircraft in a way of stating the role of system identification in parametric modelling, but the authors actually worked on a 2-DOF twin rotor MIMO system. In Krishnamurthy et al. (2008), the authors not only address the 6-DOF but worked on the design, modelling and control of a 6-DOF twin rotor aircraft which was operating on a shipboard environment with the aim of attaining robustness and good disturbance attenuation in the midst of severe aerodynamic disturbances.
2.1.2 System Variables
A multiple-input multiple-output nonlinear system is one that has more than one input and output and has some sort of nonlinearity included within it. The twin rotor laboratory equipment is considered as a nonlinear MIMO system because of this, with the high nonlinearity generated by the cross-coupling of the axes. As mentioned earlier in this project, some of its states and outputs are non-measurable contributing to the complexity of this system.
This means that the TRMS is basically not a linear system in that the response caused by a forcing function is directly dependent and related on the forcing function, and which is normally modelled by a transfer function. However MIMO systems can be linear or non-linear, time-variant or time-invariant and are applicable to modern control theory. (Ogata, 2002). As also shown in Dorf & Bishop (1998), it was stated that non-linear, time-varying and multivariable systems can be analysed using time-domain techniques and that a multivariable system is that with various input and output signals.
However, using a generalised time-domain approach referred to as state-space, MIMO systems can be modelled analyses and designed if they are linear time-varying systems (Burns, 2001). Also in his book, the state of a system has been defined as: ‘the set of variables (called the state variables) which at some initial time t= t_0, together with the input variables completely determine the behaviour of the system at a future time t’ t_0′. However in Dorf & Bishop (1998), the state of the system was defined as ‘a set of variables such that the knowledge of these variables and the input functions will, along with the equations describing the dynamics, provide the future state and output of the system’.
According to Burns (2001), state variables are the smallest set of states which are required to describe the dynamic behaviour of the system and that being measurable or not should not be required. Ogata (2002) also added that state variables need not to be physically measurable or observable quantities and in choosing state variables for a system those that do not represent physical quantities, are non-measureable and cannot be observed may be chosen. For conveniences and ease however simply measurable quantities for the state variables could be chosen because certain control laws require feedback of all state variables with suitable weighting.
Ogata (2002) has also explained the contrast between a state vector and state-space. He assumed the number of state variables needed to completely describe the behaviour of a given system to be n. These n state variables are considered as the n components of a vector x called the state vector. A state vector according to him therefore is that which determines uniquely the system state x(t) for any time t ‘ t_0, once the state at t= t_0 is given and the input u(t) for t ‘ t_0 is specified. Then he defines state-space as ‘the n-dimensional space whose coordinate axes consist of the x_1 axis, ‘, x_n axis, where x_1, x_2, ‘, x_n are the state variables’, and that state-space analysis is concerned with three types of variables and state variables. However in Nise(2007), the system variables of a controlled system have been divided into four categories: input variables, output variables, state variables and disturbance variables.
According to Nise(2007), input variables are usually the inputs to the system which are operated and manipulated by humans usually. Disturbance variables are normally inputs which are put of the control of human operators of systems and they tend to deflect the actual outputs from the desired outputs.
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Equations of Motion
In mathematical physics, equations of motion are equations that describe the behaviour of physical systems in terms of its motion as a function of time. Specifically the equations of motion describe the behaviour of a physical system as a set of mathematical functions in terms of dynamic variables: normally spatial coordinates and time are used, but others are also possible, such as momentum components and time. The most general choice is generalized coordinates which can be any convenient variables characteristic of the physical system. The functions are defined in a Euclidean space in classical mechanics, nut are replaced by curved spaces in relativity. If the dynamics of a system is known, the equations are the solutions to the differential equations describing the motion of dynamics.
There are two main descriptions of motion: dynamics and kinematics. Dynamics is general, since momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term refers to the differential equations that the system satisfies (e.g., Newton’s second law or Euler-Lagrange equations), and sometimes to the solutions to those equations.
However kinematics is simpler as it concerns only spatial and time-related variables. In circumstances of constant acceleration, these simpler equations of motion are usually referred to as the ‘SUVAT’ equations, arising from the definitions of kinematics quantities: displacement (S), initial velocity (U), final velocity (V), acceleration (A), and time (T).
Equations of motion can therefore be grouped under these main classifiers of motion. In all cases, the main types of motion are translations, rotations, oscillations or any combination of these.
Historically, equations of motion initiated in classical mechanics and the extension to celestial mechanics, to describe the motion of massive object. Later they appeared in electrodynamics, when describing the motion of charged particles in electric and magnetic fields. With the advent of general relativity, the classical equations of motion become modified. In all these cases the differential equations were in terms of a function describing the particle’s trajectory in terms of space and time coordinates, as influenced by forces or energy transformations. However the equations of quantum mechanics can also be considered equations of motion, since they are differential equations of the wave function, which describes how a quantum state behaves analogously using the space and time coordinates of the particles. There are analogs of equation of motion on other areas of physics, notably waves. (http://en.wikipedia.org/wiki/Equations_of_motion)
Plant modelling always features at the beginning of any control project, thus as much information as possible must be obtained about the process itself. The mechanical-electrical model of the Twin Rotor Multi-Input Multi-Output System is presented as below from Feedback Instruments.
Models are normally nonlinear, which means at least one of the states (i ‘ rotor current, ‘?? position) is an argument of a nonlinear function. To present a model such as this as a transfer function, a form of linear plant dynamics representation that is used in control engineering, it has to be linearised.
The following momentum equations for the mechanical unit can be derived for the vertical movement, taken from feedback instruments manual.
Where,
The motor and the electrical control circuit is approximated by a first order transfer function thus in Laplace domain the motor momentum is described by:
Similar equations refer to the horizontal plane motion:
Where,
Mr is the cross reaction momentum approximated by:
Again the DC motor with the electrical circuit is given by:
The phenomenological model parameters have been chosen by Feedback Instruments more or less experimentally, which actually makes the TRMS nonlinear model a semi-phenomenological model. The following table again culled from the manual gives their approximate values.
The bound for the control signal is set to [-2.5V .. +2.5V].
The TRMS is a MIMO plant ‘ multiple input multiple output. The below figure presents a simplified schematic of the TRMS.
The TRMS is controlled with two inputs the u1 and u2. The dynamics cross couplings are one of the key features of the TRMS (Figure 4). The position of the beams is measured with the means of incremental encoders, which provide a relative position signal. Thus every time the Real-Time TRMS simulation is run one must remember that setting proper initial conditions is important.
Getting Started
The twin rotor MIMO system (TRMS) is a laboratory set-up designed for control experiments. In certain aspects its behaviour resembles that of a helicopter. From the control point of view it exemplifies a high order non-linear system with significant cross-couplings. A schematic diagram of the laboratory set-up is shown below.
The TRMS consists of a beam pivoted on its base in such a way that it can rotate freely both in the horizontal and vertical planes. At both ends of the beam there are rotors (the main and tail rotors) driven by DC motors. A counterbalance arm with a weight at its end is fixed to the beam at the pivot. The state of the beam is described by four process variables: horizontal and vertical angles measured by position sensors fitted at the pivot, and two corresponding angular velocities. Two additional state variables are the angular velocities of the rotors, measured by tachogenerators coupled with the driving DC motors.
In a normal helicopter the aerodynamic force is controlled by changing the angle of attack. The laboratory set-up of the TRMS is so constructed that the angle of attack is fixed. The aerodynamic force is controlled by varying the speed of the rotors. Therefore, the control inputs are supply voltages of the DC motors. A change in the voltage results in a change of the rotation speed of the propeller which results in a change of the corresponding position of the beam.
However, significant cross-couplings are observed between the actions of the rotors; each rotor influences both position angles.
The design of stabilising controllers for such a system is based on de-coupling. For a de-coupled system an independent control input can be applied for each co-ordinate of the system.
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References
http://www.boeing.com/companyoffices/aboutus/wonder_of_flight/heli.html, accessed on 25/11/14
http://www.decodedscience.com/how-do-helicopters-fly/20418/2, accessed on 25/11/14
http://www.helicopterpage.com/html/terms.html, accessed on 25/11/14
(L. Osikibo 2012.)
(Juang et al., 2008b & Rahideh et al., 2008b).
Mahmoud et al. (2010) and Tahersima & Fatehi (2008).
Lie et al. (2006) and Ahmad et al. (2000a).
Rahideh & Shaheed (2006) and Juang et al. (2008b)
however in Juang et al. (2011),
Wen & Lu (2008),
Juang & Liu (2008)
Ahmed et al. (2009b)
Ahmed et al. (2009a)