Scuola di ingegneria
Corso di laurea magistrale in Ingegneria Aerospaziale
M
ASTER THESISDEVELOPMENT OF METHODOLOGIES FOR THE
CONTROL LAWS DESIGN AND THE OPTIMIZATION OF
PERFORMANCES OF ELECTROMECHANICAL AEROSPACE
ACTUATORS
Supervisor:
Prof. Gianpietro DI RITO
Author:
Gesualdo TRAMONTANA Corporate tutors:
Ing. Giulio PISPOLA Ing. Nicola PORZI
II
IV
Abstract
The aim of this thesis is to investigate and define methods for the design and
the optimization of the control laws of electromechanical actuators primary
flight surfaces, with particular reference to an actuator designed and
manufactured by UMBRAGROUP Spa for the primary flight surfaces of an
unmanned aerial vehicle. The control laws design is carried out by using a
linearized model of the actuator, aiming to identify the most suitable structure
of control laws for the enhancement of performances in terms of position
tracking, dynamic stiffness and current absorption. Starting from the control
design results, an optimization on the actuator's mechanical/electrical
parameters is performed, with the objective of minimizing tracking error and
current draw during typical transient responses.
VI
Contents
1 STATE-OF-ART OF ELECTROMECHANICAL ACTUATION SYSTEM IN AEROSPACE ... 1
1.1 Basic functional requirements of Fly-by-Wire systems ... 1
1.2 On-board systems electrification: the “more-electric aircraft” concept ... 4
1.3 Electromechanical actuators for flight controls ... 7
1.3.1 Advantages of electrical power for flight controls ... 7
1.3.2 Basic architectures ... 8
1.3.3 Linear actuators ... 10
1.3.4 Rotary actuator ... 15
1.3.5 Critical issues ... 16
2 REFERENCE SYSTEM DESCRIPTION: “DE-RISK ACTUATOR” ... 17
2.1 System architecture and design features ... 17
2.1.1 Mechanical section ... 19
2.1.2 Motor and sensors ... 20
2.1.3 Electro-magnetic brake ... 20
2.1.4 Electronic section ... 20
2.2 Concerns on closed-loop dynamics ... 22
3 PRELIMINARY DESIGN OF THE ACTUATOR CONTROL LAWS ... 23
3.1 Types and method ... 23
3.2 Linearized model of the open-loop actuator dynamics ... 24
3.2.1 State-space model ... 24
3.2.2 Equilibrium point definition ... 26
3.2.3 System transfer functions ... 27
3.3 Effects of control strategies on closed-loop actuator performances ... 27
3.3.1 Control strategies’ test cases ... 27
3.3.2 Definition of normalized transfer functions ... 28
3.3.3 Position regulator (PID) ... 29
3.3.4 Position (PID) & Current (PI) regulator ... 37
3.3.5 Position (PID) & Speed (PI) regulator ... 48
VII
3.3.7 Position(PI), Speed(PI) & Current(PI) regulator ... 71
3.4 Definition of closed-loop performances requirements ... 85
3.4.1 Tracking frequency responses ... 85
3.4.2 Dynamic stiffness/compliance responses ... 87
3.4.3 Current draw responses ... 89
3.4.4 Time responses ... 91
3.5 Comparison of closed-loop control strategies ... 93
3.6 Performance sensitivity to model uncertainties ... 95
3.6.1 Friction effects ... 95
3.6.2 Aerodynamic loading effects... 98
4 OPTIMIZATION OF MACHINE PARAMETERS ... 101
4.1 Parameters and constraints ... 101
4.2 Optimization tool ... 103
4.3 Optimization results system parameters ... 106
4.3.1 Introduction to optimization ... 106
4.3.2 Optimal set of system parameters ... 109
4.3.3 Frequency domain results ... 117
4.3.4 Time domain results ... 120
CONCLUSION AND FUTURE WORKS ... 123
References ... 125
VIII
List of tables
Table 1-1 Standard forces exerted by the pilot [1] ... 2
Table 1-2 Qualitative comparison ballscrews vs rollerscrews ... 12
Table 2-1 Mechanical section data ... 19
Table 2-2 Normal Operative Mode ... 19
Table 2-3 Electric Motor data ... 20
Table 3-1 Transfer functions Open Loop ... 27
Table 3-2 Legend of normalized transfer function ... 28
Table 3-3 Control strategies ... 93
Table 3-4 Legend control strategies ... 93
Table 3-5 Parameter values ... 95
Table 4-1 De-Risk initial values ... 101
Table 4-2 Parameters for system optimization (Position & Current control) ... 114
X
List of figures
Figure 1.1 Actuation needs on a commercial aircraft [1] ... 5
Figure 1.2 Swashplate actuation [1] ... 5
Figure 1.3 Geared Configuration ... 8
Figure 1.4 Direct Drive Configuration ... 8
Figure 1.5 Magnets mounted on the Ballnut ... 10
Figure 1.6 New sealing concept for the ballscrews ... 11
Figure 1.7 Electric Motors [7] ... 13
Figure 2.1 De-Risk actuator layout ... 18
Figure 2.2 Power and signal flows in the De-Risk actuator ... 18
Figure 2.3 Electronic section ... 21
Figure 3.1 De Risk Actuator ... 26
Figure 3.2 Bode diagram x/V – Position regulator ... 29
Figure 3.3 Root Locus, x/Ԑ – Pos. regulator ... 31
Figure 3.4 Bode input "Position" to output "Displacement" – Position regulator ... 32
Figure 3.5 Bode - Aerodynamic compliance, Pos. control ... 33
Figure 3.6 Bode- Current draw of Position Regulator ... 34
Figure 3.7 Bode i/F - Position control ... 35
Figure 3.8 Time responses - Position regulator ... 36
Figure 3.9 Bode diagram i/V ... 37
Figure 3.10 Root Locus - i/Ԑ_curr – with Curr. reg. (PID+PI CONTROL) ... 39
Figure 3.11 Bode diagram i/ii – with current regulator ... 40
Figure 3.12 Bode diagram x/ii – with current regulator ... 41
Figure 3.13 Root locus x/Ԑ with Pos. + Curr. reg. – (PID+PI CONTROL)... 42
Figure 3.14 Bode diagram- close-loop x/xi - (PID+PI CONTROL) ... 43
Figure 3.15 Bode x/F Aerodynamic compliance (PID + PI CONTROL) ... 44
Figure 3.16 Bode- current draw of Position +Current Regulator ... 45
Figure 3.17 Bode i/F (PID+PI CONTROL) ... 46
Figure 3.18 Time response of Position + Current Regulators ... 47
Figure 3.19 Bode diagram V/xdot ... 48
Figure 3.20 Root locus - xdot/Ԑ_speed – with Speed regulator (Pos + Speed Control) ... 49
Figure 3.21 Bode diagram – xdot/xdot_i with speed reg.- (Pos + Speed Control) ... 50
Figure 3.22 Bode diagram, x/xdot with speed reg.- (Pos + Speed control) ... 51
Figure 3.23 Root locus, x/Ԑ – with Pos + Speed reg. (Pos + Speed control) ... 52
Figure 3.24 Bode diagram, closed-loop x/xi - Pos + Speed control ... 53
Figure 3.25 Bode x/F - Aerodynamic- compliance – (Position + Speed control) ... 54
Figure 3.26 Bode, current draw – (Position + Speed control) ... 55
Figure 3.27 Bode i/F - Position + Speed control ... 56
Figure 3.28 Time response - Position + Speed control ... 57
XI
Figure 3.30 Root locus, i/Ԑ_current – with current reg. (PI+P+PI control) ... 60
Figure 3.31 Bode diagram, with current regulator – (PI+P+PI CONTROL) ... 61
Figure 3.32 Root locus xdot/Ԑ_speed with Curr. + Speed control – (PI+P+PI Control) ... 62
Figure 3.33 Bode diagram, xdot/xdot_i (PI+P+PI CONTROL) ... 63
Figure 3.34 Bode diagram, x/xdot_i (PI+P+PI CONTROL) ... 64
Figure 3.35 Root locus, x/Ԑ - with pos. + speed + curr. regulator – (PI+P+PI CONTROL) ... 65
Figure 3.36 Bode diagram, closed-loop x/xi – (PI+P+PI CONTROL) ... 66
Figure 3.37 Bode x/F Aerodynamic compliance- Pos(PI) + Speed (P) + Curr(PI) control ... 67
Figure 3.38 Bode i/xi Current draw - Pos(PI) + Speed (P) + Curr(PI) control ... 68
Figure 3.39 Bode i/F - Pos (PI) + Speed (P) + Curr(PI) control ... 69
Figure 3.40 Time response of Pos(PI) + Speed(P)+ Curr(PI) control ... 70
Figure 3.41 Bode diagram, i/V ... 72
Figure 3.42 Root locus, i/Ԑ_current with current (PI) regulator (PI+PI+PI CONTROL) ... 73
Figure 3.43 Bode diagram, i/ii with current regulator - (PI+PI+PI CONTROL) ... 74
Figure 3.44 Root locus xdot/Ԑ_speed with Curr + Speed regulator – Pos(PI), Speed(PI) & Curr(PI) control 76 Figure 3.45 Bode diagram xdot/xdot_i, with Curr+ Speed regulator - Pos(PI), Speed(PI) & Curr(PI) control77 Figure 3.46 Bode diagram x/xdot_i with Curr+ Speed regulator - Pos(PI), Speed (PI) & Curr(PI) control... 78
Figure 3.47 Root locus, x/Ԑ with Pos + Speed +Curr regulator - Pos(PI), Speed (PI) & Curr (PI) control .... 79
Figure 3.48 Bode diagram x/xi, Position(PI), Speed (PI) & Current (PI) control ... 80
Figure 3.49 Bode x/F Aerodynamic compliance - Pos(PI), Speed (PI) & Curr (PI) control ... 81
Figure 3.50 Bode i/xi - Current draw - Pos(PI), Speed (PI) & Curr(PI) control ... 82
Figure 3.51 Bode i/F - Pos(PI), Speed (PI) & Curr (PI) control ... 83
Figure 3.52 Time response - Pos(PI) + Speed(PI) + Curr(PI) control ... 84
Figure 3.53 Comparison: Tracking frequency responses - Comparison ... 86
Figure 3.54 Dynamic stiffness/compliance response- Comparison ... 88
Figure 3.55 Current draw responses - Comparison ... 89
Figure 3.56 Current draw response in the time domain ... 90
Figure 3.57 Time response results - Comparison ... 92
Figure 3.58 Simulation without application of external loads (with friction), (1 of 2) ... 96
Figure 3.59 Simulation without application of external loads (with friction), (2 of 2) ... 97
Figure 3.60 Test results with different load application (1 of 2) ... 98
Figure 3.61 Test results with different load application (2 of 2) ... 99
Figure 4.1 modeFRONTIER workflow ... 105
Figure 4.2 Simulink, De-Risk actuator with PID(position) & PI(current) control ... 106
Figure 4.3 Simulink, De-Risk actuator with PI(position), PI(speed) & PI(current) control ... 106
Figure 4.4 Design table- PID+PI control - Inertia parameters optimization 1 of 2 ... 109
Figure 4.5 Design table- PID+PI control - Inertia parameters optimization 2 of 2 ... 110
Figure 4.6 Design table- PID+PI control - Back-EMF constant parameters optimization 1 of 2 ... 111
Figure 4.7 Design table- PID+PI control - Back-EMF constant parameters optimization 2 of 2 ... 111
XII
Figure 4.9 Design table- PID+PI control - Resistance parameters optimization 2 of 2 ... 112
Figure 4.10 Design table- PID+PI control - Inductance parameters optimization ... 113
Figure 4.11 Design table- PI+PI+PI control - Inertia parameters optimization ... 114
Figure 4.12 Design table- PI+PI+PI control - Back-EMF constant parameters optimization ... 115
Figure 4.13 Design table- PI+PI+PI control - Resistance parameters optimization ... 115
Figure 4.14 Design table- PI+PI+PI control - Inductance parameters optimization ... 116
Figure 4.15 Bode diagram - Position(PID) & Current(PI) optimized control with optimized parameters set118 Figure 4.16 Bode diagram - Position(PI), Speed(PI) & Current(PI) optimized control with optimized parameters ... 119
Figure 4.17 Time response - Position(PID) & Current(PI) optimized control with optimized parameters set ... 120
Figure 4.18 Time response - Position(PI), Speed(PI) & Current(PI) optimized control with optimized parameters set ... 121
XIV
Introduction
The work of this thesis was carried out at UMBRAGROUP Spa company in Foligno (Italy) through an internship. With offices in Germany, United States and Italy, the company has been designing and manufacturing ballscrews and high precision mechanical components for the aerospace market for more than 35 years, becoming world leader, finally, in recent years has become prominent protagonist in the design and development of electro-mechanical aerospace actuators. The internship and then the thesis project from beginning to end developed in close contact with the R&D Department of UMBRAGROUP Spa. The initial goal, widely achieved, was to develop methodologies for the design of control laws for an electromechanical actuator, called "De-Risk" and to perform an optimization both on control parameters, both on the electrical/mechanical parameters.
In recent years, the world of aeronautics is facing new and interesting challenges, the first of all is the design of totally electric aircraft. Precisely for this reason but not only, electromechanical actuators are excellent protagonists in this new scenario. The latter prove more reliable, less complex and more energy saving than the hydraulic actuators.
In the first chapter a description on the use of actuators in the aeronautical field is provided, their presence on board of aircrafts is supported by different and multiple advantages: reduction of efforts by the pilot, greater control in stabilization of the aircraft and an increase in flight performance, both with regards to flight envelope and to passenger comfort. The second chapter will introduce the architecture of the "De-Risk" actuator, which, as we have already said before, is an electromechanical actuator used for the handling of the primary flight surfaces of a UAV aircraft and will describe the components mechanical and electronic ones.
In the third chapter we will proceed with a preliminary design of the control laws of the actuator before with a linearization of the detailed model and finally introducing different types of control in closed loop. In this section the results of the various configurations will be compared in terms of: frequency response, temporal response, dynamic stiffness and current draw.
XV In the fourth chapter, the optimization of the Actuator machine parameters (mechanical and electrical) will be addressed. Optimization aims to provide, for future developments, possible design cues that are able to produce better performance. In the last part of the elaborate, an optimal design solution will be proposed.
1
1 STATE-OF-ART OF ELECTROMECHANICAL
ACTUATION SYSTEM IN AEROSPACE
1.1 Basic functional requirements of Fly-by-Wire systems
A function can be defined as the act of transforming matter, energy or data in time, shape or space. In practice, the perspective from which a function is viewed depends on the engineering task at hand. For instance, for the purpose of power scaling, the actuation function can be viewed as the transformation of power received at the source into power transmitted to the load; this transformation takes place both in shape (e.g. hydraulics toward translational mechanics) and in space (aspect of power transmission from point A to point B). In contrast, when designing flight controls, the actuation function is considered as the act of converting a signal (e.g. an electrical command for positioning a load) into another signal (current position of the load).[2]
The piloting of an aircraft, initially purely manual, developed actuation functions at both power and signal levels. This was due to some fundamental diverse needs, listed in the following sections, and which are closely related to the aircraft type, mission and the inherent nature of the actuation itself.
Reducing the control forces needed for piloting:
It is necessary to limit the forces to be exerted by the pilot and make them compatible with human capabilities with a view to reduce physical fatigue.
This is to ensure that the pilot can apply the necessary level of effort required in the worst piloting situations (during transient) or for longer periods (steady state conditions).
Table 1.1 gives some examples for the maximum levels of control forces conceivable, as defined by European standards. [1]
2
Table 1-1 Standard forces exerted by the pilot [1]
Reducing the intellectual burden on the pilot:
In order for the pilot to concentrate on the ongoing mission, it is important to reduce the intellectual effort associated with the flight and the aircraft. This is strictly related to the interaction with the flight controls:
- stabilize the aircraft (control increase system or CAS) by refusing the effects of various disturbances (for example gusts or cross winds);
- able to decouple, synchronize or coordinate the different commands to act only on the desired degrees of freedom;
- compensate for the checks to ensure balancing of the aircraft (for example, action on the angle of incidence of the horizontal stabilizer to ensure longitudinal equilibrium through the pitch covering);
- stay within the flight envelope by monitoring the margins with respect to the permitted limits (e.g. in terms of load or speed not exceeding).
Standard Force applied to the control wheel or pedals (N)
CS-25.143d Large airplanes [EUR 15] Pitch/Roll Yaw
Transient, one handed 222/111 667
Transient, both hands 334/222 667
Long duration 44.5/22 89
CS-29.397 Large rotorcraft [EUR 12] Longitudinal/Lateral Yaw
3 Allowing independent and automatic piloting:
Autopilot (AP) or an Automatic Flight Control System (AFCS), permit parts of the mission to be flown automatically without the assistance of a human pilot.
Increasing the performance of the aircraft:
With human piloting, the issuing of actuator control orders is limited by the accuracy and subjectivity of a pilot’s senses, his capacity for processing information in real time and finally by his ability to react. This latter limitation can be illustrated by studying the transfer functions of the pilot, that is to say, the mathematical model representing the transfer between perception and action. However, as the pilot “adapts” to the nature (to the transfer function) of the system under control, it is impossible to uniquely define his transfer function. The pilot can for example “correct” the dynamics of the controlled system by introducing a proportional, proportional-integral or lead-lag action. In any case, it is noted that the command applied by the pilot is tainted with a pure delay varying from 0.1 to 0.4 s typically.[2]
By pushing the limits of human control, the development of computer-controlled commands can significantly increase the performance of an aircraft, and at different levels:
– for the flight envelope the reduction of margins is permitted by the speed of monitoring and limiting reaction, timing and decoupling.
This allows, for example, limiting structural loads due to maneuvering and gusts or to increase passenger comfort by reducing the impact of accelerations;
– for the stability of the aircraft (Stability Augmentation System, or SAS) through the introduction of correctors in the flight control laws that prevent, for example, aerodynamic coupling, like the Dutch roll;
– for the flight qualities in order to improve aerodynamic efficiency, for example, by lowering the ailerons (aileron droop) when the flaps are deployed in the landing phase; – for the dynamics regarding the generation of flight control setpoints, for example, hyper-maneuverability, provided that the actuators have sufficient bandwidth.
4 However, as mentioned above, the actuation in the aeronautical field intervenes on different issues. The use of the actuators on the aircraft has therefore increased the reliability of the components, the safety of the passengers, their comfort and has incredibly brought advantages to the command and therefore to the piloting.
1.2 On-board systems electrification: the “more-electric
aircraft” concept
Reducing the power consumption and thus the fuel burn is a major target for the next generation of aircraft. Electrical actuation is viewed as a technology that can contribute to the reduction of the non-propulsive power because electromechanical actuators, when compared to the hydraulic actuators, rely on a power less subject to losses and lighter to distribute, besides presenting higher reliability and maintainability with a lower life-cycle cost.
Electrical actuation is a technological area that has been widely addressed in the past years, and technological progresses in electromechanical actuators (EMA) for flight control surfaces have been pursued in several research programmes. Over the last years, several industrial programmes initiated the concept of a More Electric Aircraft. [2]
About the idea of "more-electric aircraft" concept, the actuators play a major role, the following are the most important uses:
– Primary flight controls
The purpose of primary flight controls involves of aircraft trajectory control. On a conventional aircraft, as the one pictured in figure 1.1, they take the form of control surfaces responsible for controlling the three rotational degrees of freedom: the ailerons for roll, the rudder for yaw and the elevator for pitch.
5 As for helicopters, they offer four degrees of freedom. Helicopter flights are controlled by acting on the swashplate, figure 1.2.
Figure 1.2 Swashplate actuation [1]
6 – Secondary flight controls
Secondary flight controls make it possible to modify the aerodynamic configuration during particular flight phases. On conventional aircraft, slats and flaps increase the chord and curvature of the wings. This is done to increase the lift of wings at low speeds and therefore decrease the takeoff or landing speeds. Airbrakes (also called spoilers) reduce aircraft speed by increasing aerodynamic drag. Trim tabs, for instance the trimmable horizontal stabilizer, ensure the global equilibrium of the aircraft during the given rectilinear flight phases (e.g. climb, cruise or approach) so that primary flight controls operate around their neutral position on average.
– Landing gears
These require numerous actuation functions:
for raising or lowering landing gear by sequencing the opening or closing of doors, extending or retracting the gear and locking it in a raised or lowered position; for steering the wheels in order to ensure steerability on the ground during taxiing; for wheel braking in order to dissipate as heat part of the kinetic energy associated
with the horizontal speed of the aircraft (in addition to airbrakes and thrust reversers during landing). Left/right differential braking can also contribute to improve steerability on the ground;
also, worth mentioning, landing gear struts are autonomous hydropneumatic components. Upon touchdown, they absorb the kinetic energy associated with the vertical speed component of the aircraft with respect to the ground.
– Engines
Engines also rely on actuators to steer inlet guide vanes on the turbine stator, to deploy or stow thrust reversers, to operate maintenance panels, to modify the geometry of air intakes or nozzles, or even to control propeller blade pitch.[8]
7 – Utilities
Other actuators are also used, for example, to operate cargo doors (and passenger doors on new large aircraft such as Airbus A380 and Boeing 787), rotor brakes for helicopters, winches, weapon systems (aiming guns, raising or lowering the arresting hook, etc.) among other things.[2]
1.3 Electromechanical actuators for flight controls
1.3.1 Advantages of electrical power for flight controls
The technology able to reduce the non-propulsive power takes shape from the electric drive, the latter is considered the goal par excellence for the next generation of aircraft.
Over the past few years, there have been several industrial programs based all about the concept of producing an aircraft as much electrical as possible.
For this reason, the electromechanical actuators are increasingly taking part, which, unlike the hydraulic ones, are not subjected to leakage and have a higher reliability and maintainability with considerably lower costs.
The aero-equipment industry has launched several studies and developments on more electrical actuation with Electro Hydrostatic Actuators (EHA) and started to introduce EMA for auxiliary equipments or use of EMAs for some systems. Several collaborative research and development projects also started to develop the All-Electric Aircraft. EMA systems are consequently viewed as a promising candidate for the aircraft of the future (i.e. the All-Electric Aircraft) if we consider that they are:
Less complex at aircraft level because of the absence of centralized hydraulic system. Better suited to long term storage since there is no leak potential.
Energy saving with respect to hydraulic systems.[8] Easier to install and maintain (no filtration, no bleeding).
Easier to design for power distribution and management (power transmitted without mass transfer).
8 The presence of electromechanical components (in particular of the actuators) in the flight controls, to date, is not established as the hydraulic one but is in strong development. Surely in the years to come the hydraulic components will be replaced gradually.
1.3.2 Basic architectures
As far as the basic architecture is concerned, the actuators are generally divided into two large classes. To determine the differentiation between the actuators, is the presence (and therefore also their disposition) of the mechanical gears. The latter are a common element among those that compose a linear EMA, do not add any specific characteristic to the EMA, but can contribute to meet the requirements of mass and envelope.
The use or not of the gears determines two different EMA architectures:
A) Geared (Figure 1.3)
B) Direct drive (Figure 1.4)
Figure 1.3 Geared Configuration
9 The “geared” architecture can be further split in further two configurations:
off-set motor and in line motor
Between the two configurations there are obviously pros and cons. The actuator we will talk about in the next chapters is characterized by a configuration "Direct-Drive ". For this reason, the following description will focus more on this last configuration.
In the "Direct-Drive" configuration, the magnets are permanently glued to the ballnut or screw-shaft, so that either the magnetized ballnut or the magnetized screw-shaft are now the rotor of an engine: electric motor and ballscrew are combined in one piece so that we don't need a gear train. The benefits of configuring “Direct Drive” compared to those “oriented” are:
Low reflected inertia from the motor rotor to the load due to low gear ratio Small number of components for greater reliability and easier assembly High efficiency
Thermal stability Reduced maintenance Reduced noise
Increased accuracy
Reduced irreversibility load Lower backlash
Figure 1.5 shows a ballnut with the magnets glued onto the OD of the ballnut and the tube that keeps the magnets in position in case the glue fails.
In this configuration the internal ring of the bearings is integral to the ballnut.
The drawback of the “direct drive” configuration compared to the “geared” one could be the mass of the electric motor because a high torque/low speed motor weighs more than a low torque/high speed one. Pros and Cons of the two configurations must be evaluated case by case.
10
1.3.3 Linear actuators
There are few items which are “basic” in a linear EMA and many others which are “added” to meet the different requirements (i.e. performance, reliability and safety) the EMA must satisfy. The “basic” items are:
Ballscrew or roller screw (it converts rotary motion in linear and vice versa); Electric motor;
Electric Motor phases commutation devices (unless sensor-less controls are used);
Bearings; Housing;
Miscellaneous components/devices (static and dynamic seals, mechanical and electric connectors, screws);
Electronic power unit (EPU); Anti-rotation devices;
The “added” items provide features like:
• Output position (position sensors);
• Limitation of the max. applied load/torque (clutch); • Electrical end stroke sensor;
• Load holding (passive or active brakes); • Load sensing (load cell);
• Jamming and fault tolerant devices;
11 Ballscrew
For the EMAs, several industrial companies have introduced a new type of ballscrew which provides good sealing for the lubricant as well: the sealing surface is a rod that can be the output rod of the EMA (Figure 1.6). The ball recirculation device is part of the screwshaft – a more reliable configuration because of the lower number of parts – and has a smaller diameter and shorter length than other ballscrew configurations. The mechanical end stroke device is part of the ballscrew assy. Finally, mechanical stop can be either of hard or soft type.
The backlash of ballscrew can be set at “minimum functional”, it means the minimum to have a free fall of the screwshaft/ballnut under its own weight.
Such figure is, in general, below 0.01 mm.
The ballscrew can be preloaded but accurate thermal analyses must be performed in order to evaluate how the preload varies with temperature; the ballscrew preload increases the back-drive load.
12 Rollerscrew
Rollerscrews can be an alternative to ballscrews. Even if the latter are preferred in more than 95% of the aeronautical applications the former have some merits to be considered. In the table below the main characteristics of ballscrews and rollerscrews are compared from qualitative point of view.
Table 1-2 Qualitative comparison ballscrews vs rollerscrews
Qualitative comparison of ballscrews versus rollerscrews
Criterion Ballscrew Rollerscrew Comments
Load rating 0 + Due to larger contact area
Life 0 + Due to larger contact area
Output speed + 0
Positioning
accuracy/backlash + 0
More flexibility and easier implementation of preloaded nuts
Back drivability ++ 0
In order to have the same back driving load the pitch of the rollerscrew is bigger than that of the ballscrew carrying the same load
Stiffness 0 ++ Stiffness in the contact area only
Screw diameter 0 +
Nut diameter + 0
Reduction ratio 0 +
Efficiency ++ 0
Efficiency has an impact on motor sizing as well as temperature and wear
Cost ++ 0
13 Electric motor
The electric motors are divided into two macro-categories: DC (direct current) and AC (alternating current) motors. This subdivision is born from the type of feeding. The DC motors are divided into brush motors and brushless motors, while alternating current motors are divided into synchronous motors and asynchronous motors. In respect of the electric motor classification, partially shown in figure 1.7, the preferred configurations are those with the permanent magnets, either BLAC best power vs. volume ratio and the less weight as well. The motor can withstand different supply levels and types such as 28 Vdc or 270 Vdc or 540 Vdc or 115 Vac, 400-Hz, 3-phase. [7]
Synchronous Reluctance motor, even if, not so performant as the permanent magnet type, has some merits like to be magnetless on the rotor, but it can deliver less torque than permanent magnet motor at same envelope, mass and performances. In case of “direct drive” EMA, the most suitable motor configuration is generally the one which combines the lowest RPM with the highest motor torque constant. [2]
The selected motor type can be realized according to different architectures in respect of:
Rotor vs. Stator: internal or external rotor; Stator pack: ironless or slotless or slotted;
Control: Sensor (Hall sensors, resolver, encoder, ...) or sensorless; Magnet mounting:
o Interior or surface mounted; o Skewed or not skewed;
14 o Halbach array;
Magnetic flux direction: axial or radial or transverse;
Windings: distributed (single or double layer) or concentrated;
Position sensors
With regard to sensors, there are many types offered by the market according to characteristics and uses. There is in the market a consolidated technology based on inductive sensors as LVDT’s and resolvers. Safety requirements can be met through electrically redundant sensor arrangement (i.e. simplex, duplex and triplex arrangement); Higher safety demands might also require mechanical redundancy. The safety of such arrangements can be increased by further differing, thus mitigating common failure modes. The same approach can be used for mechanical redundancy – for example using two sets of mechanical components. As safety relevance increases further, sensors can be referenced externally and, in turn, both internally and externally.[9]
Load holding (brakes)
As for position sensors, also for load holding systems, different types are offered by the market. The load holding systems can either of passive or active type. Skewed roller no-back brakes are of passive type and used where a back-driving load must be counteracted; the friction torque generated by the brake is proportional to the torque applied by the external load. The ratio between the input and friction torque can be established by design. The combination of a skewed roller no-back and a Sprag clutch (Sprag clutch is a one-way freewheel clutch), provides a friction torque in the back-driving condition and no friction torque when the motor operates against the external load. The skewed roller no-back can be mono or bi-directional. The latent failure of the brake could be an issue but, in general, there are operating conditions which allow to verify if the brake is still performing well. Such type of break is of dissipative type in back driving condition and energy regeneration is not possible. On the other hand, it does not need any input to engage and disengage. The active brakes are based on a solenoid which lets friction/teethed disks to engage and hold the external loads. There are different arrangements for both the solenoid and the disks.
The brakes can be of power-on, power-off and bi-stable type. Monitoring of the disk’s engagement/disengagement can be realized through sensors.
15 Jamming-Tolerant Devices
Jamming of the mechanical components – mostly of the ballscrew – is matter of concern for EMA in systems whose failure can lead to hazardous/catastrophic conditions for the Aircraft. The jamming failure of the ballscrews is not an instantaneous phenomenon but is a progressive one with an incremental reduction of the efficiency, health monitoring methods can provide contribution to solve that issue. In general, there is no need to add any sensor to the EMA in order to monitor the ballscrews jamming. Nevertheless, the confidence is still low and disconnection devices are required.
Disconnection from the external load:
When disconnected the EMA, as minimum, moves freely under the action of the external load (free-wheel condition). The disconnection devices make the EMA “jam tolerant” (JT); in case more than an EMA operates an aerodynamic surface, as the failed EMA is disconnected, the redundant actuators are still capable to take over the surface control. The projects are focused on disconnection devices of reversible and irreversible types. The devices disconnect the ballscrew/output rod from the power drive train. The reversible device is usually actuated by either a solenoid or an electric motor, while the irreversible type is triggered by a pyrotechnic actuator.
1.3.4 Rotary actuator
The operation of the primary flight control surfaces has been obtained almost universally by using linear actuators connected to a point of the aerodynamic surface which then rotates around a hinge of the wing structure. A similar configuration comes to the landing gear. One of the main reasons for adopting this configuration was the use of hydraulic actuators as endorsers. It is in fact known that rotary hydraulic actuators (vane actuators), although in principle constitute an interesting solution, present the critical problem of large losses on both sides. Switching from hydraulic technology to electromechanical technology and to electromechanical actuators with rotating output, instead of linear output, could be considered for operation of primary flight control surfaces and landing gear.
The mechanical drive with rotating output is used to drive fins and flaps of different aircraft, in particular for those aircraft in which the movement of the aerodynamic surface is mainly the rotation, as typically occurs in the surfaces of the front edge of the wing.[2]
16 A rotary output can be obtained with different mechanical devices (planetary gearboxes, rotary gear actuators, harmonic drives, cycle drives), which all share the characteristics of sharing between multiple contact points the transmission of mechanical power through the stage exit where high torques prevail. Among all the different possible solutions, the rotary gear actuators (GRA) have been by far the most preferred because they offer the best torque/weight ratio and can therefore be considered the main candidates for rotary drive in primary flight control surfaces and landing gears.
1.3.5 Critical issues
Compared to hydraulic actuators or EHAs, technological barriers still persist for a wide adoption of EMAs, especially when considering these issues:
Envelope constraints created by the adoption of thin wings and associated flight control surfaces;
Sensitivity to certain single point of failures that can lead to mechanical jams, resulting in a reluctance to adopt EMAs for flight safety critical applications as solutions are heavy and costly (redundancy, fail safe behavior, etc.), thus creating difficulties for adoption and certification and impacts on costs;
Thermal balance at motor and power electronics levels as there is no more hydraulic fluid to take away the heat generated by power losses;
Motor rotor inertia effects due to the huge mass it reflects at the load level due to the high gear ratio (in case of geared configuration);
Difficulty to address the issue of energy regeneration.
Moreover, application of EMAs for continuous control of flight control surfaces requires a thorough optimization effort in terms of architecture and design of control law, materials and sensors in order to come up with a product which can be certified for the next generation of aircraft. Recent programmes encourage evaluation of EMA technology for aircraft flight critical applications, ranging from actuation of primary flight control and load alleviation surfaces to actuation of the landing gear. The promising perspectives of electrical actuation for flight control surfaces and landing gear need to be thoroughly investigated and verified in order to gain the necessary confidence and maturity level for moving to their implementation in a flying demonstrator.
17
2 REFERENCE SYSTEM DESCRIPTION: “DE-RISK
ACTUATOR”
2.1 System architecture and design features
“De-Risk” actuator (fig. 2.1 and fig. 2.2) consists of an electromechanical actuator (EMA) with a built-in electric control unit (ECU), which provides the load / hinging moment needed to control the primary flight surface. De-Risk is used for three different primary control systems: aileron, elevator and rudder.
The reference electro-mechanical actuator, manufactured by UMBRAGROUP, can be used in dual “active-active” redundancy for the actuation of primary flight controls. The system has linear output, and it is characterized by a compact “direct-drive” design, in which the electric motor is integrated in the nut of a low-pitch ballscrew assembly.
The actuator, supplied with 28 VDC power input, is basically made of:
Three-phase brushless DC motor; Ball-screw assembly;
Electronic Control Unit (ECU) box, assembled with the actuator cylinder; Electro-magnetic safety brake;
Three current sensors (one per phase);
18 The main features of the De-Risk actuator are many, citing some:
Reduced noise High efficiency
Small number of components for greater reliability and easier assembly Reduced maintenance
Reduced noise
Figure 2.2 Power and signal flows in the De-Risk actuator Figure 2.1 De-Risk actuator layout
19
2.1.1 Mechanical section
Thanks to the “direct-drive” design, the mechanical transmission of the actuator is very simple, and only made of a low-pitch ball-screw with rotating nut and translating screwshaft. The load path is obtained by steel balls recirculating on suitable paths on the screwshaft itself, and angular contact bearing inner rings directly realized on the nut.
The screwshaft anti-rotation is obtained by the elliptical coupling of the output shaft with the related hole in the actuator cylinder. As regards the mechanical characteristics of De-Risk actuator, these are shown in the table 2-1:
Table 2-1 Mechanical section data
Value Unit of Measure
Pitch 3,18E-03 [m]
Nut mass 5,80E-01 [Kg]
Ballscrew mass 1,82E-01 [Kg]
Ballscrew inertia equivalent 4,65E-08 [Kg*m^ 2]
Nut inertia 1,52E-04 [Kg*m^ 2]
Motor inertia 7,50E-05 [Kg*m^ 2]
Total inertia 2,27E-04 [Kg*m^ 2]
Actuator strokes in normal operative conditions are in accordance with values reported in table 2-2, depending on the application.
Table 2-2 Normal Operative Mode
Installation Operating Range[mm]- Normal Operative Mode
Aileron 20.5 ([-11.4; +9.1])
Elevator 48.5 ([-22.6; 25.9])
20
2.1.2 Motor and sensors
The actuator integrates a three-phase brushless DC motor, with phase commutation provided by Hall-effect sensors embedded in the stator windings. The permanent magnets of the motor rotor are directly mounted on the ball-nut, to obtain the compact “direct-drive” arrangement. Concerning the system sensors, output position and phase currents are made both available for the actuator closed-loop control.
As regards the motor characteristics of De-Risk actuator, these are shown in table 2-3:
Table 2-3 Electric Motor data
Parameters Definition Value Unit of Measure
R Resistance (single phase) 0.235 [Ω]
Ke Back-EMF constant 0.11 [V⋅s/rad]
L Inductance (single phase) 3E-04 [H]
Max V Max voltage supply ±28 [V]
Max i Max current ±33 [A]
Kt Torque constant 0.11 [Nm/A]
Re Equivalent Resistance 0.46 [Ω]
Le Equivalent Inductance 6E-04 [H]
2.1.3 Electro-magnetic brake
The actuator also includes an electro-magnetic “power-on” brake, used either to safely stop the output rod in case of failure, or to hold it at the system initialization. The braking action is obtained by coupling two teethed flanges, attached to the brake housing and the ball-nut respectively. With no voltage applied to the solenoid, the nut teethed flange is kept distant from the brake flange by a preloaded foil spring, and the rotor is free to rotate, and the output rod is free to move. When the solenoid is energized, the nut flange is attracted up to overcome the foil spring force, so that the teethed flanges couple and the ball-nut (i.e. the output rod) is blocked.
2.1.4 Electronic section
The actuator ECU box is provided with two interface connections (one for the power supply lines and chassis from aircraft, and the other for communication bus (RS422 lines), rigging lines and static brake commands), and it basically consists of two boards (Fig. 2.3):
Control Board used to implement the following basic functions:
21 digital closed-loop control algorithms;
health-monitoring algorithms;
communication with the flight control computer via RS422 lines; motor PWM drive;
Hall-effect sensors interface; LVDT supply and control; brake supply and control.
Power and Filter Board used to implement filtering and protection capabilities for the 28 VDC electrical supply of the Control Board as well as to command the electro-magnetic brake.
22
2.2 Concerns on closed-loop dynamics
The actuator closed-loop control is obtained by two nested loops, on output rod position and motor current. The position demand from the flight control computer is updated at 80 Hz, while the two controls are operated at 4 kHz and 800 Hz, with reference to current and position loops respectively. To facilitate the airworthiness certification of the actuator control software, regulators are both based on standard linear operators, with additional signal filtering, saturation, and windup protection. In particular, the current regulator is proportional-integrative, while the position regulator is proportional-integrative-derivative.
23
3 PRELIMINARY DESIGN OF THE ACTUATOR
CONTROL LAWS
3.1 Types and method
The closed loop control of the actuator is obtained from two nested loops, from the position of the output rod and from the motor current. With this type of control, several problems were encountered in the simulations. First, it emerged that with a control in position (PID) and current (PI), the actuator achieves poor results in terms of dynamic stiffness. Another type of problem encountered is the current absorption during a typical step test in position [1 mm]. Trying to solve and avoid these problems that restrict the actuator also in terms of frequency response with unacceptable performances, we decided to consider alternative controls and in case of proceeding to comparisons with the control above mentioned, in order to observe clearly, possible improvements.
Five different types of control will be considered based on the position of the rod, on the current supplied to the motor and on the speed of the rod. It is implied the possibility of obtaining position, speed and current with the aid of appropriate instruments of measurement. The following strategies for the actuator control laws have been investigated: 1. Position Regulator (PID);
2. Position Regulator (PID) + Current Regulator (PI); 3. Position Regulator (PID) + Speed Regulator (PI);
4. Position Regulator (PI) + Speed Regulator (P) + Current Regulator (PI); 5. Position Regulator (PI) + Speed Regulator (PI) + Current Regulator (PI).
The redesign of De-Risk actuator control laws stems from the need to achieve significantly higher performance than the current ones. Trying to optimize the already existing control or to completely replace it with a control able to achieve much higher performance. In addition, in the redesign it has been considered to improve the mechanical architecture and therefore the parameters of the machine.
The optimization has been performed at two levels: Optimization of regulators parameters Optimization of the machine parameters
24 With regards to the optimization of the regulators’ parameters, control strategies have been considered. Each regulator of each control structures has been optimized to achieve the best position tracking performance in terms of:
Null asymptotic error (equal to zero) Minimum Over-shoot
Bandwidth ≈ 4.5 Hz
After the definition of the regulators’ parameters, the control strategies have been compared in terms of:
Dynamic compliance frequency response Current draw frequency response
Response in time domain
3.2 Linearized model of the open-loop actuator dynamics
3.2.1 State-space model
As a first step, the detailed model of the actuator has been linearized, by neglecting sliding friction, and by assuming that the 3-phase brushless motor dynamics can be represented by an equivalent mono-phase DC motor with adequate circuit characteristics, Eq. (1).
𝑖 is the motor current, 𝜃 represent the angle of rotation of the nut, 𝜃̇ and 𝜃̈ respectively the angular speed andthe angular acceleration. 𝐹 represents the external force and 𝑉 is the supply voltage.
25 The standard state-space form of the linearized model of the actuator has been finally obtained by Eq. (2).
𝑥 = 𝐴𝑥 + 𝐵𝑢
̇
𝑦 = 𝐶𝑥 + 𝐷𝑢
Where:A=
−𝑅 /𝐿
0
−𝐾 /𝐿
0
0
1
𝐾
−𝐾 /𝐽
−𝐵/𝐽
; B=
0
1/𝐿
0
0
( ∗ )0
;
C=
0
1/𝐾
0
0
0
1/𝐾
1
0
0
; D=[0];
𝒚 =
̇
𝑖
𝒖 = [𝐹 𝑉]
𝒙 = 𝜃 𝜃̇ 𝑖
Once the A, B, C, D matrices have been obtained, referring to the standard state-space form, it is possible to derive the transfer functions that relate the inputs with the respective outputs. In particular, the developed model has as input, the external force (considered as an external disturbance) and supply voltage, instead, as outputs: the displacement of the rod, the speed of the rod and the motor current, fig. 3.1.
26
3.2.2 Equilibrium point definition
By definition of the equilibrium point, we have:
𝒙
𝒆𝒒̇ = 𝜃
̇
𝛳 ̈
𝚤̇̇
=0
If we substitute the equilibrium conditions in the governing equations, we obtain:
Now, considering that the equilibrium point is to be found at 𝜃 = 0, we have
27
3.2.3 System transfer functions
All the open-loop transfer functions of the model are expressed in the table 3-2.
3.3 Effects of control strategies on closed-loop actuator
performances
3.3.1 Control strategies’ test cases
The strategies that have been chosen are based on a common objective that consists in the use of a limited number of control parameters. In fact, a regulator can be defined more or less simple depending on the number of parameters inserted inside it. Given the types of control that have been mentioned, we have chosen to use a limited number of parameters that varies between 4 and 6. The regulators that have been chosen, have within a number of parameters ranging from 4 to 6. All regulators must comply with the bandwidth requirement equal to 4.5 Hz and asymptotic error equal to zero.
Input “Force”, Output “Displacement”: Input “Voltage”, Output “Displacement”:
𝐱 𝐅𝐞 = 𝟎, 𝟎𝟎𝟏𝟏𝟐𝟐𝟖 ⋅ (𝐬 + 𝟕𝟖𝟑, 𝟑) 𝐬 ⋅ (𝐬 + 𝟔𝟒𝟔, 𝟏) ⋅ (𝐬 + 𝟏𝟑𝟕, 𝟐) 𝐱 𝐕= 𝟒𝟎𝟕, 𝟑𝟓 𝐬 ⋅ (𝐬 + 𝟔𝟒𝟔, 𝟏) ⋅ (𝐬 + 𝟏𝟑𝟕, 𝟐)
Input “Force”, Output “Current”: Input “Voltage”, Output “Current”:
𝐢𝐦 𝐅𝐞 = −𝟒𝟎𝟕, 𝟑𝟓 (𝐬 + 𝟔𝟒𝟔, 𝟏) ⋅ (𝐬 + 𝟏𝟑𝟕, 𝟐) 𝐢𝐦 𝐕 = 𝟏𝟔𝟔𝟔, 𝟕 ⋅ 𝐬 (𝐬 + 𝟔𝟒𝟔, 𝟏) ⋅ (𝐬 + 𝟏𝟑𝟕, 𝟐)
Input “Force”, Output “Speed”: Input “Voltage”, Output “Speed”:
𝒙̇ 𝐅𝐞 = 𝟎, 𝟎𝟎𝟏𝟏𝟐𝟐𝟖 ⋅ (𝒔 + 𝟕𝟖𝟑, 𝟑) (𝐬 + 𝟔𝟒𝟔, 𝟏) ⋅ (𝐬 + 𝟏𝟑𝟕, 𝟐) 𝒙̇ 𝐕= 𝟒𝟎𝟕, 𝟑𝟓 (𝐬 + 𝟔𝟒𝟔, 𝟏) ⋅ (𝐬 + 𝟏𝟑𝟕, 𝟐)
28 After selecting the appropriate parameters to meet the abovementioned requirements, the performance that this type of control is able to achieve will be shown, in particular for each control will be calculated:
• Bode diagram (position) • Current draw
• Dynamic compliance • Time response
The results that will be observed will be normalized according to appropriate values that will be described later.
With regard to the method used in the choice of parameters, this is often a pole/zero cancellation that allows to simplify the function of open cycle transfer of the system and to control more easily the system. Finally, all types of control, with the same bandwidth, will be compared.
3.3.2 Definition of normalized transfer functions
Among the many transfer functions, four of these will be represented in the form of frequency response and all four are normalized with respect to some multiplicative factors illustrated below:
𝐻
=
;
𝐻
=
⋅
∗;
𝐻
=
𝑥
𝐹
⋅ 𝐾
𝐻 =
𝑖
𝐹
⋅
2𝜋𝐾
𝑝𝑖𝑡𝑐ℎ
;
Table 3-2 Legend of normalized transfer function
Factor Value 𝑥∗ 1mm 𝑖 33A 𝐾 250N/mm 𝐾 0.11V⋅s 𝑝𝑖𝑡𝑐ℎ 3.18 mm
29
3.3.3 Position regulator (PID)
The open-loop transfer function with Voltage input and Displacement output is:
𝑥/𝑉 =
407,35
𝑠 ⋅ (𝑠 + 646,1) ⋅ (𝑠 + 137,2)
30 With a control only in position, a PID will be used. The PID transfer function is:
The transfer function in open-loop with the PID control will be:
𝑃𝐼𝐷 =
12
⋅ (𝑠 + 1,9) ⋅ (𝑠 + 646)
𝑠
x
Ԑ
=
407,35 ⋅ (s + 1,9) ⋅ (s + 646)
s ⋅ (s + 646,1) ⋅ (s + 137,2)
31 The parameters that have been chosen for the PID allow the cancellation of the high frequency pole with the respective zero. In the root-locus is highlighted the spot where the two poles (which in open-loop are found in the origin) will move with a proportional gain equal to +12. (Fig. 3.4)
32 The close-loop transfer function is:
It’s possible to note that the control has been chosen in such a way as to obtain a bandwidth of 4.5 Hz, this value will be common to all the controls that follow, as satisfying the initial requirement.
𝑥/𝑥 =
4888,8 ∗ (𝑠 + 646)(𝑠 + 1,9)
(𝑠 + 646,1)(𝑠 + 2,012)(𝑠 + 135,2𝑠 + 4615)
Bandwidth: Wb (-45°) = 4,5 Hz
33
Dynamic compliance
The closed-loop transfer function (Input: Force Output: Displacement) represents the behavior of the actuator, in particular its displacement when it is energized by external forces and thus defines its dynamic compliance. The transfer function in question was normalized with the factor 𝐾 , where 𝐾 = 250N/mm and represents the stiffness of the actuator. In this way, in the Bode diagram, at the 0dB, it means that the actuator stiffness is exactly equal to its dynamic compliance.
Figure 3.5 Bode - Aerodynamic compliance, Pos. control
0𝑑𝐵 =
| |34
Current Draw
The closed loop function represents the current absorption by the motor when the system is energized by the attainment of a defined position. Again, in the Bode diagram, the result is normalized by a factor .
This multiplication factor is equal to 𝑥 = 1𝑚𝑚 and 𝑖 = 33𝐴, (therefore we impose a maximum current absorption constraint equal to 10% of that imposed in the technical specification (30 A).
In this way, at 0db we have a maximum possible current absorption (33 A). Exceeded the 0db, the absorption is higher than permitted.
Figure 3.6 Bode- Current draw of Position Regulator
0𝑑𝐵 =
| |35 An interesting result is the closed-loop transfer function . The following plot represents the current draw of the motor when there is an external force input. The result is normalized by the factor , where 𝐾 is the back-electromotive force constant of the motor (𝐾 = 0.11 𝑉, 𝑝𝑖𝑡𝑐ℎ = 3.175𝑒 ).
Figure 3.7 Bode i/F - Position control
0𝑑𝐵 =
| |36
Time response
Below, the time responses for a position step command. One response represents the condition without loads, the other instead, the condition with an aerodynamic load constant
equal to 𝐾 = 0.0638 𝑁𝑚/𝑟𝑎𝑑, which corresponds to elastic load, with constant 𝐾 = 250000 .
Figure 3.8 Time responses - Position regulator
37
3.3.4 Position (PID) & Current (PI) regulator
The open-loop transfer function with Voltage input and Current output is:
𝑖
𝑉
=
1666,7 ⋅ 𝑠
(𝑠 + 646,1) ⋅ (𝑠 + 137,2)
38 First of all, let's start with the definition of the current control, choosing a PI (proportional-integrative). The transfer function of the current control will be:
The transfer function in open-loop with the PI control will be:
𝑃𝐼 =
1,1
⋅ (𝑠 + 646,1)
𝑠
i
Ԑ
=
1666,7 ⋅ (s + 646,1)
(s + 646,1) ⋅ (s + 137,2)
39 The parameters that have been chosen for the PI allow the cancellation of the high frequency pole with the respective zero. In the root-locus is highlighted the spot where the pole (which in open-loop is found ≈ -137,2) will move with a proportional gain = +1,1. It will move reaching the value ≈-1971 (Fig. 3.11).
40 The close-loop transfer function is:
It is possible to note that the current control has been chosen to obtain a bandwidth of +312 Hz.
i
i
(Cur. reg) =
1833,3 ⋅ (s + 646,1)
(s + 1971) ⋅ (s + 646,1)
Bandwidth: Wb (-3dB) = 312,8 Hz41 Once described the current control, it must be define the position control. First of all, consider the open loop transfer function (Input: Current, Output: Displacement) of the system:
Regarding the position regulator, we have been chosen PID, its transfer function is:
𝑃𝐼𝐷 =
200
⋅ (𝑠 + 18) ⋅ (𝑠 + 2)
𝑠
x
i
=
448,09 ⋅ (s + 646,1)
s ⋅ (s + 1971) ⋅ (s + 646,1)
42 With the position regulator newly defined, the open-loop transfer function has three poles in the origin:
x Ԑ=
448,09 ⋅ (s + 646,1) ⋅ (s + 2) ⋅ (s + 18) s ⋅(s + 1971) ⋅ (s + 646,1)
43 The parameters that have been chosen for the PID, give us an open loop transfer function with three poles in the origin and two low frequency zeros respectively at -2 and -18. In the root-locus is highlighted the spots where the poles (which in open-loop are found in the origin) will move with a proportional gain = +200. One of them will erase the zero place at 2, on the other hand, the other two , will form a pair of complex conjugate poles.
The close-loop transfer function is:
It’s possible to note that the control has been chosen in such a way as to obtain a bandwidth of 4.5 Hz, this value will be common to all the controls that follow, as satisfying the initial requirement.
x
x
=
89618(s + 646,1)(s + 18)(s + 2)
(s + 1924)(s + 646,1)(s + 1,989)(s + 44,09s + 842,8)
44
Dynamic compliance
The close-loop transfer function (Input: Force Output: Displacement) represents the behavior of the actuator, in particular its displacement when it is energized by external forces and thus defines its dynamic compliance. The transfer function in question was normalized with the factor 𝐾 , where 𝐾 = 250 N/mm and represents the stiffness of the actuator. In this way, in the Bode diagram, at the 0db, it means that the actuator stiffness is exactly equal to its dynamic compliance.
Figure 3.15 Bode x/F Aerodynamic compliance (PID + PI CONTROL)
0𝑑𝐵 =
| |45
Current Draw
The closed loop function represents the current absorption by the motor when the system is energized by the attainment of a defined position. Again, in the Bode diagram, the result is normalized by a factor ∗ .
This multiplication factor is equal to 𝑥∗ = 1𝑚𝑚 and 𝑖 = 33𝐴, (therefore we impose a maximum current absorption constraint equal to 10% of that imposed in the technical specification (30 A). In this way, at 0db we have a maximum possible current absorption (33 A). Exceeded the 0db, the absorption is higher than permitted.
Figure 3.16 Bode- current draw of Position +Current Regulator
0𝑑𝐵 =
| |46 An interesting result is the closed-loop transfer function . The following plot represents the current draw of the motor when there is an external force input.
The result is normalized by the factor , where 𝐾 is the back-electromotive force constant of the motor (𝐾 = 0.11 𝑉, 𝑝𝑖𝑡𝑐ℎ = 3.175𝑒 )
Figure 3.17 Bode i/F (PID+PI CONTROL)
0𝑑𝐵 =
|𝑖|
|𝐹|
=
𝑝𝑖𝑡𝑐ℎ
2𝜋 ⋅ 𝐾
47
Time response
Below, the time responses for a position step command. One response represents the condition without loads, the other instead, the condition with an aerodynamic load constant
equal to 𝐾 = 0.0638 𝑁𝑚/𝑟𝑎𝑑, which corresponds to elastic load, with constant 𝐾 = 250000 . F ig ur e 3. 18 T im e re sp on se o f P os it io n + C ur re nt R eg ul at or s N O T E : K a= 0, 06 38 [N m ] 𝐾 = 0 .0 6 3 8 𝑁 𝑚 ≈ 2 5 0 0 0 0
48
3.3.5 Position (PID) & Speed (PI) regulator
The open-loop transfer function with Voltage input and Speed output is:
𝑉
𝑥̇
=
407,35
(𝑠 + 646,1) ⋅ (𝑠 + 137,2)
49 The speed control transfer function is now defined. We will use a PI (proportional-integrative) control.
The transfer function in open-loop with the PI control will be:
The parameters that have been selected in the speed controller guarantee a pole/zero cancellation (both positioned at -646,1). There will then two other poles: one in the origin and another located at -137,2.
𝑃𝐼 =
52
⋅ (𝑠 + 646,1)
𝑠
𝑥̇
Ԑ
=
407,35 ⋅ (𝑠 + 646,1)
𝑠 ⋅ (𝑠 + 646,1) ⋅ (𝑠 + 137,2)
50 In figure 3.21, the root locus is represented, and we can see where the remaining two poles will move with a proportional gain equal to +52. It is possible to see that the proportional effect will make the two poles, as conjugated complexes.
To follow, the close-loop transfer function with PI speed control:
Once defined the speed control parameters, must be define the position control. First of all, consider the open loop transfer function (Input: Speed, Output: Displacement) of the system:
𝑥 𝑥̇= 21182 ⋅ (𝑠 + 646,1) 𝑠 ⋅ (𝑠 + 646,1) ⋅ (𝑠 + 137,2 ⋅ 𝑠 + 2,11𝑒04)
𝑥̇
𝑥̇
=
21182 ⋅ (𝑠 + 646,1)
(𝑠 + 646,1) ⋅ (𝑠 + 137,2𝑠 + 2,118𝑒04)
51 Regarding the position regulator, the one chosen is a PID, its transfer function is:
The transfer function in open-loop with the PID position control will be:
𝑃𝐼𝐷 =
0,2
⋅ (𝑠 + 144) ⋅ (𝑠 + 2)
𝑠
𝑥
Ԑ
=
21182 ⋅ (𝑠 + 144) ⋅ (𝑠 + 646,1) ⋅ (𝑠 + 2)
𝑠 ⋅ (𝑠 + 646,1) ⋅ (𝑠 + 137,2𝑠 + 2,118𝑒04)
52 The transfer function has two poles in the origin and a pair of complex conjugate poles, the effect of the proportional gain (equal to +0.2) will maintain the pair of complex conjugated poles (but at a higher frequency) and, as regards the two poles located in the origin will have: one of them will go to "delete" the effect of zero in -2, and the other, instead, will move to a frequency of ≈ -25.
53 The close-loop transfer function will be:
Note that even in this type of control the bandwidth is fixed at 4,5 Hz.
𝑥
𝑥
=
4236,5 ⋅ (𝑠 + 144) ⋅ (𝑠 + 646,1) ⋅ (𝑠 + 2)
(𝑠 + 25,24) ⋅ (𝑠 + 646,1) ⋅ (𝑠 + 2,163) ⋅ (𝑠 + 109,8𝑠 + 2,235𝑒04)
54
Dynamic compliance
The closed-loop transfer function (Input: Force Output: Displacement) represents the behavior of the actuator, in particular its displacement when it is energized by external forces and thus defines its dynamic compliance. The transfer function in question was normalized with the factor 𝐾 , where 𝐾 = 250 N/mm and represents the stiffness of the actuator. In this way, in the Bode diagram, at the 0dB, it means that the actuator stiffness is exactly equal to its dynamic compliance.
Figure 3.25 Bode x/F - Aerodynamic- compliance – (Position + Speed control)
0𝑑𝐵 =
| |55
Current Draw
The closed loop function represents the current absorption by the motor when the system is energized by the attainment of a defined position. Once more, in the Bode diagram, the result is normalized by a factor .
This multiplication factor is equal to 𝑥 = 1𝑚𝑚 and 𝑖 = 33𝐴, (therefore we impose a maximum current absorption constraint equal to 10% of that imposed in the technical specification (30 A). In this way, at 0db we have a maximum possible current absorption (33 A). Exceeded the 0db, the absorption is higher than permitted.
Figure 3.26 Bode, current draw – (Position + Speed control)
0𝑑𝐵 =
| |56 An interesting result is the closed-loop transfer function . The following plot represents the current draw of the motor when there is an external force input. The result is normalized by the factor , where 𝐾 is the back-electromotive force constant of the motor (𝐾 = 0.11 𝑉, 𝑝𝑖𝑡𝑐ℎ = 3.175𝑒 )
Figure 3.27 Bode i/F - Position + Speed control
0𝑑𝐵 =
|𝑖|
|𝐹|
=
𝑝𝑖𝑡𝑐ℎ
2𝜋 ⋅ 𝐾
57
Time response
Below, the time responses for a position step command. One response represents the condition without loads, the other instead, the condition with an aerodynamic load constant
equal to 𝐾 = 0.0638 𝑁𝑚/𝑟𝑎𝑑, which corresponds to elastic load, with constant 𝐾 = 250000 .
Figure 3.28 Time response - Position + Speed control
58
3.3.6 Position(PI), Speed(P) & Current(PI) regulator
In this sub-paragraph and the next one, two types of controls will be described, both with internal closed-loops in current, speed and position. The two types of controls differ, as we shall see shortly, on the number of parameters that will be used. The control that will be described in this part, contains position and current controllers, both PI (proportional-integrative), and a purely proportional speed controller (in total there are 5 control parameters). As regard to the last control, which will be described successively, it has, for all three closed-loops, PI (proportional-integrative) controllers, with a total of 6 parameters to select. The benefits of adding an extra parameter will be discussed later.
As previously done, we start by defining the open-loop function i/V:
