1. Department of Mechanical Engineering The University of Strathclyde, Glasgow Hybrid Systems: Modelling, Analysis and Control Yan Pang Department of Mechanical Engineering The University of Strathclyde ypang@mecheng.strath.ac.uk

2. 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

3. 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.

4. Department of Mechanical Engineering The University of Strathclyde, Glasgow Hybrid Systems FA=(Q, Σ, δ) Q={1,2,3,4,5} Σ={a,b,c} δ=Q  Σ  Q Computer Science Finite state machines Control theory Continuous dynamical systems Hybrid Systems

5. Department of Mechanical Engineering The University of Strathclyde, Glasgow Simple Example 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

6. Department of Mechanical Engineering The University of Strathclyde, Glasgow Hybrid Automata Hybrid Automaton State space Input space Vector field Guard set Invariant set Remarks: and countable, Can add outputs, etc. (not needed here)

7. Department of Mechanical Engineering The University of Strathclyde, Glasgow Model of Thermostat Hybrid automaton Model A possible trajectory guard Invariant Vector field guard

8. Department of Mechanical Engineering The University of Strathclyde, Glasgow Current Research Directions 1. Modelling & Simulation 2. Analysis & Verification 3. Controller Synthesis 4. Applications

9. 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.

10. 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 }

11. Department of Mechanical Engineering The University of Strathclyde, Glasgow Air Traffic Management Systems Two aircraft joining and collision avoidance scheme

12. 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 linear programming and quantifier elimination techniques

13. Department of Mechanical Engineering The University of Strathclyde, Glasgow 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)