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Hybrid Systems: Modelling, Analysis
and Control
Yan Pang
Department of Mechanical Engineering
The University of Strathclyde
[email protected]
Department of Mechanical Engineering
The University of Strathclyde, Glasgow
Outline
• Why Hybrid Systems?
• What Are Hybrid Systems?
• A Thermostat Example
• Hybrid Models : Hybrid Automata
• Current Research Areas
• Applications: The Batch Evaporator and Air Traffic Control System
• Conclusions
Department of Mechanical Engineering
The University of Strathclyde, Glasgow
Motivation
•Computer Electronics
•Office Automation
•Automobiles
•Industry Process Plants
•Air Traffic Management Systems
•...
Hybrid models describe systems
composed of both continuous and
discrete components, the former
typically associated with dynamical
laws (e.g., physical systems), the
latter with logic devices, such as
switches, digital circuitry, software
code.
Department of Mechanical Engineering
The University of Strathclyde, Glasgow
Hybrid Systems
Computer Science
Finite state machines
Control theory
Continuous dynamical systems
Hybrid Systems
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FA=(Q, Σ, δ)
Q={1,2,3,4,5}
Σ={a,b,c}
δ=Q ΣQ
Department of Mechanical Engineering
The University of Strathclyde, Glasgow
dx ( t )
dt
 Ax  Bu
y (t )  Cx  Du
x  Rn ,u  Rm , y  R p
Simple Example
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Model of room, heater and thermostat
x : room temperature
Thermostat switches heater on between 68 & 70
Thermostat switches heater off between 80 & 82
Room warms and cools exponentially
Discrete states :{on,off}
Continuous variable : x
Department of Mechanical Engineering
The University of Strathclyde, Glasgow
Hybrid Automata
Hybrid Automaton
H  (S ,V , f , G, Inv)
S  Q X
V  VC  VD
f : X V  Rn
G : Q  VD  Q  X
State space
Input space
Vector field
Guard set
Invariant set Inv : Q  X
Remarks:
Q and VD countable, X  Rn VC  R m
Can add outputs, etc. (not needed here)
Department of Mechanical Engineering
The University of Strathclyde, Glasgow
Model of Thermostat
Hybrid automaton Model
guard
Vector field
guard
Invariant
Department of Mechanical Engineering
The University of Strathclyde, Glasgow
A possible trajectory
Current Research Directions
1. Modelling & Simulation
2. Analysis & Verification
3. Controller Synthesis
4. Applications
Department of Mechanical Engineering
The University of Strathclyde, Glasgow
The Batch Evaporator
The Evaporation Process
1. The solution is filled into tank T1 and the solvent is
evaporated until the desired concentration is
reached. During evaporation the condenser collects
the steam.
2. When the desired concentration is reached, the
material is drained from T1 to T2 as soon as T2 is
available, i.e. emptied from the previous batch.
Department of Mechanical Engineering
The University of Strathclyde, Glasgow
The Batch Evaporator Simulation
Initial set: I = q1×X1, where X1 = {0.2m  x1  0.22m  0.28m  x2  0.3 m  x3=373K}
Target set: F = q3×X3, where X3 = {0.01m  x1  0.04m  0.08m  x2  0.18 m  338K  x3  373K}
Invariant sets: INV(qi) = {0  x1  0.04m  0  x2  0.4 m  338K  x3  400K }
Department of Mechanical Engineering
The University of Strathclyde, Glasgow
Air Traffic Management Systems
Two aircraft joining and
collision avoidance scheme
Department of Mechanical Engineering
The University of Strathclyde, Glasgow
Optimal Control
Control Objectives: Aircraft 2 is joining the path of Aircraft 1, in approach to the airport,
and it is up to the controller to ensure that a minimum separation between the two aircraft is
maintained and also minimize all the cost during this procedure.
Optimal trajectory : By
Department of Mechanical Engineering
The University of Strathclyde, Glasgow
linear programming and quantifier elimination techniques
Conclusions
The interest in hybrid systems has grown over in last few
years
• Theoretical Challenges : computer science + control theory
• Impact on Application: automotive power train control, batch
system, bioengineering process, mixed integer process control,
embedded software verification.
Current and Further Research
• Control Synthesis : Reachability Analysis, Under approximation
reachable sets. (linear, non-linear)
• Optimal Control: Linear Programming(LP), Quantifier
Elimination(QE) techniques
• Model Predictive Control (MPC) : Mixed Integer Programming
(MIP)
Department of Mechanical Engineering
The University of Strathclyde, Glasgow