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Dynamic Systems And Control. Course info. Introduction (What this course is about). Course home page. Home page : http://www.cs.huji.ac.il/~control. Course Info. Home page : http://www.cs.huji.ac.il/~control

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dynamic systems and control

Dynamic Systems And Control

Course info.

Introduction (What this course is about)

course home page
Course home page
  • Home page: http://www.cs.huji.ac.il/~control

Lavi Shpigelman, Dynamic Systems and control – 76929 –

course info
Course Info
  • Home page: http://www.cs.huji.ac.il/~control
  • Staff: Prof. Naftali Tishby (Ross, room 207)Lavi Shpigelman (Ross, room 61)
  • Class:Sunday, 12-3pm, ICNC
  • Grading
    • 40% exercises, 60% project
  • Textbooks:
    • Chi-Tsong Chen, Linear System Theory and Design, Oxford University Press, 1999
    • Robert F. Stengel, Optimal Control and Estimation, Dover Publications, 1994
    • J.J.E. Slotine and W. Li, Applied nonlinear control, Prentice Hall, Englewood cliffs, New Jersey, 1991
    • H. K. Khalil, Non-linear Systems, Prentice Hall, 2001

Lavi Shpigelman, Dynamic Systems and control – 76929 –

intro dynamical systems
Intro – Dynamical Systems
  • What are dynamic systems?

Lavi Shpigelman, Dynamic Systems and control – 76929 –

Physical things with

states that evolve in

time

optimal control
(Optimal) Control

Objective: Interact with a dynamical system to achieve desired goals

  • Stabilize nuclear reactor within safety limits
  • Fly aircraft minimizing fuel consumption
  • Pick up glass without spilling any milk

Lavi Shpigelman, Dynamic Systems and control – 76929 –

...Measures of optimality

example prosthetics bionics
Example:Prosthetics  bionics
  • Problem:Make a leg that knows when to bend.
  • Inputs:
    • Knee angle.
    • Ankle angle.
    • Ground pressure.
    • Stump pressures.
  • Outputs:
    • Variable joint stiffness and damping

Lavi Shpigelman, Dynamic Systems and control – 76929 –

example robotics reinforcement learning
Example: Robotics, Reinforcement Learning
  • How do you stand up?
  • How do you teach someone to stand up?
  • Reinforcement learning: Let the controller learn by trial and error and give it general feedback (reinforce ‘good’ moves).
  • Training a 3 piece robot to stand up:
    • Start of training:
    • End of training:

Lavi Shpigelman, Dynamic Systems and control – 76929 –

modeling making assumptions

Control Signals

Task Goal

Controller

Plant

Observations

Modeling (making assumptions)

Graphical representation (information flow)

Lavi Shpigelman, Dynamic Systems and control – 76929 –

Mathematical relationships

control example motor control
Control Example: Motor control

Plant (controlled system): hand

Controller: Nervous System

Control objective:Task dependent (e.g. hit ball)

Plant Inputs: Neural muscle activation signals.

Plant Outputs: Visual, Proprioceptive, ...

Plant State: Positions, velocities, muscle activations, available energy…

Controller Input: Noisy sensory information

Controller Output: Noisy neural patterns

plant

cont-roller

Lavi Shpigelman, Dynamic Systems and control – 76929 –

modeling motor control

Control Signals

Neural Pattern

Task Goal

Brain

controller

Handplant

Observations sensory Feedback

Modeling Motor Control

Lavi Shpigelman, Dynamic Systems and control – 76929 –

Details…

optimal movements
Optimal Movements
  • Control Objective: Reach from a to b.
  • Fact: more than one way to skin a cat...
  • How to choose: Add optimality principle
  • E.g. optimality principle: Minimum variance at b.
  • Modeling assumption(s): Control is noisy: noise/ ||control signal||
  • Control problem: find the “optimal” control signal.
  • Note: No feedback (open loop control)

Lavi Shpigelman, Dynamic Systems and control – 76929 –

modeling motor control details
Modeling Motor Control - Details

sensory - motor control loop

Lavi Shpigelman, Dynamic Systems and control – 76929 –

Wolpert DM & Ghahramani Z (2000) Computational principles of movement neuroscienceNature Neuroscience 3:1212-1217

state estimation step 1
State Estimation – step 1
  • Open loop estimate (w/o feedback)

Lavi Shpigelman, Dynamic Systems and control – 76929 –

state estimation step 2
State Estimation – step 2
  • Step 1: Control signal & a forward dynamics model (dynamics predictor) updates the change in state estimate.
  • Step 2:Sensory information & forward sensory model (sensory predictor) are used to refine the estimate

Lavi Shpigelman, Dynamic Systems and control – 76929 –

context estimation adaptive control
Context Estimation (Adaptive Control)

Lavi Shpigelman, Dynamic Systems and control – 76929 –

adaptive control generation
Adaptive Control generation
  • An inverse model learns to translate a desired state (sequence) into a control signal.
  • A non-adapting, low gain feedback controller does the same for the state error. Its output is used as an error signal for learning the Inverse model.

Lavi Shpigelman, Dynamic Systems and control – 76929 –

simple st dynamical system example

u

m

External

force (u)

Contraction

(y)

Shock Absorber

Simple(st) Dynamical System Example
  • Consider a shock absorber.
  • We wish to formulate a dynamical system model of the mass that is suspended by the absorber.
  • We choose a linear Ordinary Differential Equation (ODE) of 2nd order

Lavi Shpigelman, Dynamic Systems and control – 76929 –

net force

damping force

spring force

external force

y

elements of the dynamic system

Observable ProcessOutputs y

Controllable inputsu

Observationsz

ObservationProcess

Dynamic Process

State x

Process noise w

Observation noise n

Plant

Elements of the Dynamic System

Lavi Shpigelman, Dynamic Systems and control – 76929 –

State evolving with time (differential equations)

controllability observability of the dynamic process states
Controllability & Observability of the Dynamic Process States

Main issues:stabilitystabilizability

Controllable

Controllable inputsu

Observable

controlled observed

ObservableOutputs y

Lavi Shpigelman, Dynamic Systems and control – 76929 –

Disturbance (noise) w

Uncontrolled Unobserved

Dynamic Process

States x

other modeling issues
Other Modeling Issues*
  • Time-varying / Time-invariant
  • Continuous time / Discreet time
  • Continuous states / Discreet states
  • Linear / Nonlinear
  • Lumped / Not-lumped (having a state vector of finite/infinite dimension)
  • Stochastic / Deterministic

More:

  • Types of disturbances (noise)
  • Control models

Lavi Shpigelman, Dynamic Systems and control – 76929 –

* All combinations are possible

rough course outline
Rough course outline
  • Review of continuous (state and time), Linear, Time Invariant, state space models.
    • Linear algebra, state space model, solutions, realizations, stability, observability, controllability
  • Noiseless optimal control (non linear)
    • Loss functions, calculus of variations, optimization methods.
  • Stochastic LTI Gaussian models
    • State estimation, stochastic optimal control
  • Model Learning
  • Nonlinear system analysis
    • Phase plane analysis, Lyapunov theory.
  • Nonlinear control methods
    • Feedback linearization, sliding control, adaptive control, Reinforcement learning, ML.

Lavi Shpigelman, Dynamic Systems and control – 76929 –

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