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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. Outline. 1. Introduction Quantized Control Systems
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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
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
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
linear velocity angular velocity stop straight line motions rotations “on the spot” turns right of given curvature turns left … Example: Monocycle with Quantized Actuators
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
Why Quantized Control Systems The actuator is inherently quantized (e.g., a stepper motor)
Why Quantized Control Systems Quantization may also be an inherent characteristic of the control system as in the rolling polyhedra problem
Sensor / Controller Why Quantized Control Systems Control of Distributed Networked Systems
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
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
Goal: design a controller, based on quantized measurements and taking quantized input values stabilizing the system. The Stabilization Problem for Quantized Linear Systems
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?
Not, unless is stable ! A Quantizer as a non-linear Characteristic Is small-gain theorem useful?
Stability: a delicate matter Stabilization of quantized systems is a delicate matter A more sofisticated analysis is necessary
Quantized feedback Closed loop piecewise affine system Open loop unstable system NON LINEAR ! Effects of Quantization on Dynamical Systems
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
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.
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
The Model: Reachable Single-Input Discrete-TimeLinear Systems Hypothesis: denotes the corresponding system without state quantization ( )
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
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
Continuous-time Discrete-time Lyapunov StabilityTheorem
Continuous-time Discrete-time Lyapunov Equation for Linear Systems
Analysis of the Closed-loop System Persistent bounded “noise” affecting the system
[S] Proposition:Let , where [S] Invariant Ellipsoids
System Controller (deadbeat) Closed-loop dynamics Example
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
Change of coordinates in the state-space Hypothesis: The Controller-form
Controlled-Invariant Hypercubes: geometrical parameters effectively represents the resolution of the quantized control set.
Theorem 1: Remark: Hypothesis: Main results: Controlled-Invariant Hypercubes
Proposition : The qdb-controller Classical deadbeat controller: Quantized deadbeat controller: (qdb-controller)
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
Examples Unifolmly quantized controls
Logarithmically quantized controls Examples
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.
Example 2 -2 -1 1
System Controller Closed-loop dynamics Minimality properties of Hypercubes can be proved: Hypercubes vs of arbitrary shape EllipsoidsvsHypercubes
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
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
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
Simulation result Controls applied (quantized-deadbeat) Trajectory
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