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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 l.jpg

Dynamic Systems And Control

Course info.

Introduction (What this course is about)


Course home page l.jpg
Course home page

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

Lavi Shpigelman, Dynamic Systems and control – 76929 –


Course info l.jpg
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 –


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Intro – Dynamical Systems

  • What are dynamic systems?

Lavi Shpigelman, Dynamic Systems and control – 76929 –

Physical things with

states that evolve in

time


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(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 l.jpg
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 l.jpg
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 –


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Control Signals

Task Goal

Controller

Plant

Observations

Modeling (making assumptions)

Graphical representation (information flow)

Lavi Shpigelman, Dynamic Systems and control – 76929 –

Mathematical relationships


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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 l.jpg

Control Signals

Neural Pattern

Task Goal

Brain

controller

Handplant

Observations sensory Feedback

Modeling Motor Control

Lavi Shpigelman, Dynamic Systems and control – 76929 –

Details…


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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 –


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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


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State Estimation – step 1

  • Open loop estimate (w/o feedback)

Lavi Shpigelman, Dynamic Systems and control – 76929 –


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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 –


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Context Estimation (Adaptive Control)

Lavi Shpigelman, Dynamic Systems and control – 76929 –


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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 l.jpg

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 l.jpg

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)


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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


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Other Modeling Issues* States

  • 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


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Rough course outline States

  • 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 –