in search of “uniform and equable...
TRANSCRIPT
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In search of “Uniform and Equable Motion”
Three term (PID) control
A history and some thoughts on current practice
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The story
1. Early engines were “batch” devices with sequence controls
2. The interest in displacing water and wind mills meant an interest in “uniform and equable motion”
3. The engine governor emerged and exposed the principals of dynamics and of proportional and integral action
4. Meanwhile P+I enters the process world 5. The application to IC engines is widespread 6. How about PID control for research?
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Mentions
1. Stuart Bennett (Sheffield) - https://www.shef.ac.uk/acse/staff/sb
2. Colleagues (for use of slides and material) – Zhijia Yang – Dezong Zhao – Ed Winward
3. Our students and particularly Tyson Cheung. 4. Our Loughborough technical team
A parallel form
https://en.wikipedia.org/wiki/PID_controller
Newcomen’s environment
http://www.geevor.com/ (Geevor mine web site)
World of intense commercial activity Water was the nuisance factor in mining and agriculture Cornish mines presented a substantial market The principal energy source was South Wales coal Harveys of Hayle was to become one of the great steam engine suppliers
Henry Ford Museum Guide
Newcomen’s 1712 engine
Watt’s double acting engine
Windmills and making flour
8 gerald-massey.org.uk/windmills
The centrifugal governor
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• Already established –patented in 1787 by Thomas Mead
• Matthew Boulton suggested to James Watt to use as an engine governor
• Watt had already been working on the new throttle actuator
gerald-massey.org.uk/windmills
Watt’s governor
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A proportional controller – and known to suffer from offset
Stuart Bennett, A History of Control Engineering, 1800-1930
en.wikipedia.org/wiki/Lap_Engine
Pump governors
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• Pump regulators may have been invented by James Watt
• Continued to be developed in spite of the progress of the mechanical governor
• Two particular aspects – Integral action – Actuating force
Stuart Bennett, A History of Control Engineering, 1800-1930
Siemens and Maxwell
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• The Siemens brothers arrive in the UK to commercially exploit William Siemen’s new governor
• Maxwell motivated by the dynamics of governor behaviour
• Governors were also needed in scientific equipment
• Maxwell’s analysis presented in February 1868 – not as lucid as usual
• Routh picks up the ideas and works with them
Stuart Bennett, A History of Control Engineering, 1800-1930
Maxwell’s analysis
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• William Siemens already used the terms:
• Moderator for a governor demonstrating droop
• Governor for an isochronous (constant speed) controller.
𝑑𝑑𝑑𝑑𝑑𝑑
𝑀𝑀𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
= 𝑃𝑃 − 𝑅𝑅 − 𝐹𝐹𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
− 𝑉𝑉
In the steady state, left hand side vanishes so that
𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
= 𝑉𝑉 +𝑃𝑃 − 𝑅𝑅𝐹𝐹 Moderator
𝑑𝑑𝑑𝑑𝑑𝑑 𝐵𝐵
𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 𝐹𝐹
𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 − 𝑉𝑉
In the steady state,
𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
= 𝑉𝑉 Governor
J C Maxwell, On Governors, Philosophical Transactions of the Royal Society (20th February 1868)
Maxwell and Jenkins
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• In 1861 the British Association for the Advancement of Science appointed a committee to establish electrical standards
• Resistance measurements required the precise rotation of a coil
• Fleeming-Jenkins governor was conceived for this purpose
• It includes integral action.
Fleeming Jenkin’s governor (Stuart Bennett, History of Control Engineering
Mid 20th century and the diesel governor
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• Medium speed diesel engines in a wide variety of applications – marine, power railway traction
• Seminal paper on governors came from an unusual source.
• Paxman, AEC, University of Cambridge (Donald Welbourn)
• Main impact appears to have been a substantial reduction in test bed usage
D. B. Welbourn, D. K. Roberts, and R. A. Fuller, Governing of Compression-Ignition Oil Engines, Proceedings of the IMechE, June 1959 173: 575-604
The Ardleigh Type 222 governor
16 Ardleigh – later Regulateurs Europa
1𝑝𝑝0𝐷𝐷2 +
2𝜁𝜁𝑝𝑝0𝐷𝐷 + 1 𝑑𝑑 = 𝜇𝜇𝑔𝑔 sin𝑝𝑝𝑑𝑑
Achieving PI functionality
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−𝜇𝜇𝑔𝑔𝑇𝑇𝑁𝑁
𝑇𝑇𝑁𝑁 + 𝑇𝑇𝑆𝑆+
1𝑇𝑇𝑁𝑁 + 𝑇𝑇𝑆𝑆
.1𝐷𝐷
𝜔𝜔𝑒𝑒
𝑇𝑇𝑁𝑁 =𝑟𝑟
𝛾𝛾𝑟𝑟3𝑟𝑟2 𝑇𝑇𝑆𝑆 =
1𝛼𝛼𝑟𝑟1
𝑟𝑟1, 𝑟𝑟2, 𝑟𝑟3 are all geometrical factors
𝛼𝛼, 𝛾𝛾 are respectively flow co-efficients with dimensions 𝐿𝐿𝑇𝑇𝐿𝐿�
𝛼𝛼
𝛾𝛾
The PID controller in a process setting
𝑢𝑢 𝑠𝑠 = 𝐾𝐾 1 +1𝑠𝑠𝑇𝑇𝑖𝑖
+ 𝑠𝑠𝑇𝑇𝑑𝑑 𝑒𝑒 𝑠𝑠
• Actuating devices scaled 0-100%
• 1/K is defined as the proportional band (PB)
• N% PB gives N% output for a 100% error
• Reset time is the time taken for a 100% error to create 100% output
Clarke, D W PID algorithms and their computer implementation Trans Inst MC, 6, 6, 1984
𝑢𝑢 𝑠𝑠 = 𝐾𝐾 1 +1𝑠𝑠𝑇𝑇𝑖𝑖
𝑒𝑒 𝑠𝑠
Usually a PI controller will stabilise a plant so long as 𝑇𝑇𝑖𝑖 is chosen well
Early developments • The earliest process control
devices emerged in the early 20th century
• The late 19th century had seen a growing demand for instrumentation and automation
• Foxboro and Taylor Instrument both emerged during this period
• The modern pneumatic control is a testimony to the early beginnings
Foxboro 43AP controller
The pneumatic PI controller
20 Modern Control Engineering Ogata, 1997
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Practical considerations and the D term
1. Adding feedforward 2. Bump-less transfer from manual to
automatic control 3. Avoidance of derivative (and proportional)
kick 4. Measurement conditioning and filtering 5. Rate limits for start-up of a new process
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The D term
1. Provide phase lead 2. Simply adding a D term could improve phase
margin and decrease gain margin
3. Apply a term, 𝑠𝑠𝑇𝑇𝐷𝐷1+𝛼𝛼𝑠𝑠𝑇𝑇𝐷𝐷
where typically 𝛼𝛼 = 0.1
4. Velocity feedback – apply derivative of feedback alone
5. Set-point filter – progressively apply a set point change
Li, Y, Ang, K H, Chong, G C Y, PID Control Systems Analysis and Design, IEEE Control Systems Magazine 26 (1), pp32-41
A PI loop with feedforward
𝑒𝑒 𝑠𝑠
𝐾𝐾𝑓𝑓 𝑣𝑣 𝑠𝑠
𝐾𝐾
11 + 𝑠𝑠𝑇𝑇𝑖𝑖
+
+
Control, 𝑢𝑢 𝑠𝑠 = K 1 + 1𝑠𝑠𝑇𝑇𝑖𝑖
𝑒𝑒 𝑠𝑠 …
With feedforward, control, 𝑢𝑢 𝑠𝑠 = K 1 + 1𝑠𝑠𝑇𝑇𝑖𝑖
𝑒𝑒 𝑠𝑠 + 𝐾𝐾𝑓𝑓𝑣𝑣 𝑠𝑠
The PID controller A practical implementation
K
11 + 𝑠𝑠𝑇𝑇𝑖𝑖
1 + 𝑠𝑠𝑇𝑇𝑑𝑑
Limiter
- + +
+ 𝑤𝑤 𝑠𝑠
𝑢𝑢𝑢 𝑠𝑠
𝑑𝑑 𝑠𝑠
𝑓𝑓 𝑠𝑠
Total action on 𝑑𝑑 𝑠𝑠 is 𝑢𝑢 𝑠𝑠 = −K 1 + 1𝑠𝑠𝑇𝑇𝑖𝑖
1 + 𝑠𝑠𝑇𝑇𝑑𝑑 𝑑𝑑 𝑠𝑠
Clarke, D W
PID algorithms and their computer implementation Trans Inst MC, 6, 6, 1984
Zeros at −1𝑇𝑇𝑖𝑖
, −1𝑇𝑇𝑑𝑑
𝑢𝑢 𝑠𝑠
Non-interacting form of controller
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L1 provides de-saturation L2 provides a secondary limit
Clarke, D W PID algorithms and their computer implementation Trans Inst MC, 6, 6, 1984
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Tuning the PID controller • Tuning methods emerged from
the efforts made by the instrumentation suppliers
• Ideas were emerging in the 1930s
• Fulscope 100 from Taylor was the first PID controller (announced in 1939)
• Most significant result came from the Taylor Instrument Company
• Ziegler and Nichols papers (1942 and 1943)
• Cohen and Coon (1953)
J G Ziegler and N B Nichols, Optimum Settings for Automatic Controllers, Trans ASME, 64, 1942
The effect of tuning
Clarke, D W PID algorithms and their computer implementation Trans Inst MC, 6, 6, 1984
Increasing gain for simple plant, 1
1+𝑠𝑠3
Using the Z-N PI tuning rule Place zero at −1.2
𝑇𝑇𝑢𝑢= −0.33
• Clarke suggests following PI tuning rules produces a “too-lively” controller
• But the PID rules place two zeros on the real axis and guarantee stability.
Effect of tuning on disturbance rejection
Richard Dorf Modern Control Systems 12th Edition
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PID - a short summary
1. Form of control that emerged in both the engine and process sectors in a quite different form
2. Much of PID literature is process oriented but there is substantial carry-over
3. The PID algorithm is well documented, well supported by tuning rules and can be adapted to multi-variable situations
The diesel engine
Mass air
flow meter
Catalyst
Electronic control unit
Common rail
Exhaust manifold
Fuel tank
EGR cooler
Inlet manifold
Turbocharger
Inlet air temperature
Injectors
Intercooler
VNT control
EGR valve
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The Challenge
1. Rapid iteration 2. Relative ease of adjustment and tuning 3. Reliable known algorithms with understood
properties 4. Open to analytical and experimental
approaches
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Diesel Fuel Path Control(1)
Two loop engine speed control system with CA50 and Alpha control system (control structure)
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Diesel Fuel Path Control(1)
Two loop engine speed control system with CA50 and Alpha control system
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2I2O Pmax and IMEP Control Control structure
Diesel Fuel Path Control (2)
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2I2O Pmax and IMEP Control Control authority Engine Speed: 1100rpm
Variation of Pmax and IMEP with the two inputs: m1 and m2
Diesel Fuel Path Control (2)
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2I2O Pmax and IMEP Control System Control results: step response
Engine Speed: 1100rpm
Cylinder 2 IMEP step response Cylinder 2 Pmax step response
Diesel Fuel Path Control (2)
Evaluation of ETA hardware and controls
• The benefits of ETA
• Easily integrated into existing Turbine or VGT system
• Ability to improve the engine response to fast load increase
• Facilitates conversion of extra exhaust energy into electricity
• Some of the hallenges
• Thermal management • Control scheme • Mechanical installation
Number 1 Engine Lab at Loughborough
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Air Path System (with ETA)
Implementation of control system of the test engine
Three control inputs: 1) EGR valve position 2) VGT vane position 3) ETA torque demand
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ETA and Fuel Consumption(1) Delta Pressure = EXP-MAP
1) Delta pressure increases with the increased VGT vane position 2) Delta pressure drops a little at low VGT vane position and then evolves to increase a little at high VGT vane position when ETA swept from generation to motoring
1) Fuel rate strongly correlated to delta pressure 2) At low VGT vane position, small increase of
delta pressure caused by ETA generating mode results in bigger fuel rate increment (deteriorated efficiency)
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ETA and Fuel Consumption(2) When EGR valve open (engine test results-delta pressure & MAP)
When EGR valve open, the area of the shape of delta pressure related to MAP formed by different ETA and VGT becomes smaller because of the reduced delta pressure (it can be remembered as like a paper falling to the desk)
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ETA and Fuel Consumption(3) When EGR valve open (engine test results-fuel rate & delta pressure)
1) When EGR valve opened wider, the maximum delta pressure becomes smaller and the fuel rate is more sensitive to delta pressure change which is caused by ETA especially when ETA is generating
2) The total area formed by fuel rate and delta pressure for all feasible VGT, EGR and ETA combinations can be simplified as a triangular area
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3I3O Square Control Structure Selection (2)
Complete multi-variable controller
Three PI Controllers
Decoupler
Such structure is the simplest decoupling multi-variable controller
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3I3O Control System Implementation (1)
Use looptune or systune MATLAB function
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3I3O Control System Implementation (2) Gain-scheduling techniques were used for engine transient
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3I3O Control System Engine Test Results (1) Setpoint tracking performance
Three control variables
Three controlled variables
Engine Torque
Fast Starting Strategy
• Investigated by Toyota Motor Corporation to improve stop-start quality
• Utilises compressed cylinder for restart
• 50% Starting energy reduction
• 63% Cranking time reduction
• 10dB sound reduction
Assisted Direct Start Strategy
Kenji Kataoka, Kimitoshi Tsuji, Crankshaft Positioning Utilizing Compression Force and Fast Starting with Combustion Assist for Indirect Injection Engine, SAE Technical Paper 2005-04-11, DOI: 10.4271/2005-01-1166
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Deploying a PID Controller
Constant Deceleration Trajectory Speed Profile
𝜔𝜔 𝑑𝑑 =−𝜔𝜔𝑡𝑡𝑑𝑑𝑡𝑡𝑑𝑑𝑠𝑠𝑡𝑡𝑠𝑠𝑠𝑠
𝑑𝑑 + 𝜔𝜔𝑡𝑡𝑑𝑑𝑡𝑡
Crank angle profile
𝜃𝜃 𝑑𝑑 = −𝜔𝜔𝑡𝑡𝑑𝑑𝑡𝑡
2 ∗ 𝑑𝑑𝑠𝑠𝑡𝑡𝑠𝑠𝑠𝑠𝑑𝑑2 + 𝜔𝜔𝑡𝑡𝑑𝑑𝑡𝑡 ∗ 𝑑𝑑
Implementing a PID Controller
• Control Input
𝑢𝑢 𝑑𝑑 = 𝐾𝐾𝑠𝑠𝑒𝑒 𝑑𝑑 + 𝐾𝐾𝑖𝑖 � 𝑒𝑒 𝜏𝜏 𝑑𝑑𝜏𝜏 + 𝐾𝐾𝑑𝑑𝑑𝑑𝑒𝑒 𝑑𝑑𝑑𝑑𝑑𝑑
𝑡𝑡
0
• Application – Dynamic Lookup table for reference
speed – Low speed control loop
• Controller Gains Tuned iteratively – Increasing 𝐾𝐾𝑠𝑠 increased tracking
performance – 𝐾𝐾𝑑𝑑 and 𝐾𝐾𝑖𝑖 were found to decrease
performance – Initiation Speed reduced the peak control
input - engine initially has high angular momentum
The results – a benchmark assessment
0
2
4
6
8
10
12
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Stop Time[s]
Error[Degrees]
MaxAccelerationFluctuation[100 rad/s]
Peak Torque[daNm]
Net EnergyUsage
[kJ]
PID Controlled Shutdown
Uncontrolled PID
Observations and question
• PI and PID controls • Simple • Available • “Practical aspects”
covered • Tuning rules developed • Heuristics • Provides a benchmark
• Open source – Why so few examples? – Github - Arduino - RPi
• What makes for a good real-time “kit”?1
• Could we easily modularise both algorithms and tuning?
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1 LI, Y, Ang, K H and Chong, G C Y PID Control Analysis and Design IEEE Control Systems Magazine (26) 1,pp32-41
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Summary (1)
1. The origins of control for “uniform motion” echo in today’s control solutions
2. PID is widely understood – familiar to most engineers - but not that well understood
3. But surrounded by a kind of folklore – that has been dispelled in at least some of the process industry
4. Provides an excellent basis for quick implementations - tuning accessible – heuristics accessible
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Summary (2)
1. How do we access the best practice? 2. How do we choose the right analytical
approach? – Through modelling – Through experiment and design
3. By making the controller “easier” we emphasise the physical solution
4. Understanding the dynamics remains our central interest
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A concluding thought
“The change from ponderous beams to uniform and equable motion was not the result of a process of gradual improvement … Rather it was through the use and adaptation of existing devices to meet the practical problems as they arose …”
Stuart Bennett The Search for 'Uniform and Equable Motion': a study of the early methods of the control of the steam engine University of Sheffield, Department of Automatic Control, Research Report No 20. (Can be downloaded.)