Controllo e Stabilizzazione di Sistemi Quantizzati

1. Bruno Picasso b.picasso@sns.it Scuola Normale Superiore-Pisa c/o Politecnico di Milano PhD Student in Applied Mathematics Thanks to:A. Bicchi Università di Pisa Controllo e Stabilizzazione di Sistemi Quantizzati

2. Outline • 1. Introduction • Quantized Control Systems • The stabilization problem: practical stability • 2. Discrete-time linear systems • Lyapunov-based analysis of practical stability • Geometric construction of controlled-invariant sets • Control synthesis for stabilization • 3.An application to continuous-time linear systems: • stabilization of an inverted pendulum • 4. Conclusions

3. States evolve in a continuous space; Outputs map states to a discrete set; Control take values in a finite symbol alphabet; Continuous State-Space Discrete Input set Discrete Output set Quantized Control Systems

4. Example: Linear Systems with Quantized Input and/or Output

5. linear velocity angular velocity stop straight line motions rotations “on the spot” turns right of given curvature turns left … Example: Monocycle with Quantized Actuators

6. Plant 0 1 1 0 1 0 1 1 0 1 Controller Why Quantized Control Systems The control loop is closed over a finite capacity communication channel (e.g., a digital bus): information must be encoded using finite alphabets

7. Why Quantized Control Systems The actuator is inherently quantized (e.g., a stepper motor)

8. Why Quantized Control Systems Quantization may also be an inherent characteristic of the control system as in the rolling polyhedra problem

9. Sensor / Controller Why Quantized Control Systems Control of Distributed Networked Systems

10. Sensor / Controller Shared communication channel of finite capacity 0 1 0 1 1 0 1 0 … Why Quantized Control Systems Control in the Presence of Communication Constraints

11. Low cost sensors / actuators Communication constraints Typical situation: control of complex systems, as e.g., distributednetworked systems (embedded systems) sharing communication resources • Quantization can not be neglected • How can communication resources be effectively shared? Why Quantized Control Systems

12. Goal: design a controller, based on quantized measurements and taking quantized input values stabilizing the system. The Stabilization Problem for Quantized Linear Systems

13. A sector enclosing the nonlinearity must be of the type with , namely, should be stable ! A Quantizer as a non-linear Characteristic Are absolute stability criteria useful?

14. Not, unless is stable ! A Quantizer as a non-linear Characteristic Is small-gain theorem useful?

15. Stability: a delicate matter Stabilization of quantized systems is a delicate matter A more sofisticated analysis is necessary

16. Quantized feedback Closed loop piecewise affine system Open loop unstable system NON LINEAR ! Effects of Quantization on Dynamical Systems

17. Is still possible to guarantee • asymptotic stablity? Even in a discrete-time linearized model neither the formulation of the problem nor its solution is evident How to deal with Quantization? Toy problem: stabilization of an inverted pendulum using a stepper motor

18. Our goal is the Stabilization of QCS: Practical Stability As classical Lyapunov asymptotic stability can not be achieved, practical stability notions are introduced for QCS A. Find a small neighborhood of the equilibrium which can be made invariant, B. Find a large set for which can be made attractive, and C. Find a quantized feedback law that implements A. & B.

19. Outline • 1. Introduction • Quantized Control Systems • The stabilization problem: practical stability • 2. Discrete-time linear systems • Lyapunov-based analysis of practical stability • Geometric construction of controlled-invariant sets • Control synthesis for stabilization • 3.An application to continuous-time linear systems: • stabilization of an inverted pendulum • 4. Conclusions

20. The Model: Reachable Single-Input Discrete-TimeLinear Systems Hypothesis: denotes the corresponding system without state quantization ( )

21. Controlled-Invariant Sets: Given , • What kind of stability properties • can be achieved? • How to synthesize the control law? Def: The set is controlled-invariant iff such that Problem- construction of bounded controlled-invariant sets : Lyapunov techniquesDirect geometric method ellipsoidsHypercubes General technique but Of limited application but conservative somehow optimal

22. The theory applies to the more general case of quantized multi-input systems Extensions are possble to non-linear dynamics Lyapunov Technique: Construction of Invariant Ellipsoids

23. Continuous-time Discrete-time Lyapunov StabilityTheorem

24. Continuous-time Discrete-time Lyapunov Equation for Linear Systems

25. Analysis of the Closed-loop System Persistent bounded “noise” affecting the system

26. Invariant Ellipsoids

27. Invariant Ellipsoids

28. [S] Proposition:Let , where [S] Invariant Ellipsoids

29. System Controller (deadbeat) Closed-loop dynamics Example

30. The range of application is limited to linear dynamics and provides the best results for quantized single-input systems Geometric Method Starting point: change the coordinates in order to find an equivalent representation of the system in which the problem is easier

31. Change of coordinates in the state-space Hypothesis: The Controller-form

32. Controlled-Invariant Hypercubes: geometrical parameters effectively represents the resolution of the quantized control set.

33. Theorem 1: Remark: Hypothesis: Main results: Controlled-Invariant Hypercubes

34. Proposition : The qdb-controller Classical deadbeat controller: Quantized deadbeat controller: (qdb-controller)

35. Theorem 2: Remark: We provide only a sufficient condition for attractivity starting from but the condition is tight, i.e., none of the inequalities can be weakened. Main results: Attractivity analysis

36. Examples Unifolmly quantized controls

37. Logarithmically quantized controls Examples

38. Theorem : is a computable parameter related to the resolution of the quantized sensor modelled by Quantized state - quantized input case As far as attractivity is concerned, the conditions are the same with the weak inequalities replaced by the strict ones.

39. Example 2 -2 -1 1

40. System Controller Closed-loop dynamics Minimality properties of Hypercubes can be proved: Hypercubes vs of arbitrary shape EllipsoidsvsHypercubes

41. Outline • 1. Introduction • Quantized Control Systems • The stabilization problem: practical stability • 2. Discrete-time linear systems • Lyapunov-based analysis of practical stability • Geometric construction of controlled-invariant sets • Control synthesis for stabilization • 3.An application to continuous-time linear systems: • stabilization of an inverted pendulum • 4. Conclusions

42. Example:Stabilization of an Inverted Pendulum with a Finite Control Set (Stepper Motor) • Design parameters: • Sampling interval • Displacement for unit control Discrete time model: where

43. Aim:Confiningtheevolution of the pendulum within a given neighborhood of the (unstable) equilibrium. Remark: If (no friction), there are no cubic invariant sets, irrespective of any parameter of the system. Inverted Pendulum (continuation) We look for the maximal invariant at varying and This is done by explicitly calculating and applying the results concerning uniform quantized sets

44. Simulation result Controls applied (quantized-deadbeat) Trajectory

45. Conclusions • We have introduced Quantized Control Systems • and focused on the stability problem. • Need for practical stability notions • Controlled invariance analysis: Lyapunov-based method • vs • Geometric approach • We focused on systems under assigned quantization. • Other topics: reachability is an interesting problem involving sophisticated mathematical tools (e.g, algebraic number theory) and presenting strange phenomena (such as discrete anolonomy, fractals…) b.picasso@sns.it