Cyber-Physical Systems Modeling Physical Dynamics IECE 553/453 Fall - - PowerPoint PPT Presentation

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Cyber-Physical Systems Modeling Physical Dynamics IECE 553/453 Fall - - PowerPoint PPT Presentation

Apr 22, 2023 245 likes •538 views

Cyber-Physical Systems Modeling Physical Dynamics IECE 553/453 Fall 2020 Prof. Dola Saha 1 Modeling Techniques Models that are abstractions of system dynamics (how system behavior changes over time) Modeling physical phenomena

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Cyber-Physical Systems Modeling Physical Dynamics

IECE 553/453– Fall 2020

  • Prof. Dola Saha
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Modeling Techniques

Ø Models that are abstractions of system dynamics

(how system behavior changes over time)

§ Modeling physical phenomena – differential equations § Feedback control systems – time-domain modeling § Modeling modal behavior – FSMs, hybrid automata, … § Modeling sensors and actuators –calibration, noise, … § Hardware and software – concurrency, timing, power, … § Networks – latencies, error rates, packet losses, …

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Modeling of Continuous Dynamics

Ø Ordinary differential equations, Laplace

transforms, feedback control models, …

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Example CPS System

Ø Helicopter Dynamics

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Modeling Physical Motion

Ø Six Degrees of Freedom § Position: x, y, z § Orientation: roll (!!), yaw (!"), pitch (!#)

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Notation

Ø Functions of this form are known as continuous-time signals

Orientation can be represented in the same form

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Notation

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Newton’s Second Law

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Orientation

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Torque: Angular version of Force

Just as force is a push or a pull, a torque is a twist. Units: newton-meters/radian, Joules/radian

Ty(t ) = r f (t )

angular momentum, momentum

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Rotational Version of Newton’s Law

If the object is spherical, this reluctance is the same around all axes, so it reduces to a constant scalar I (or equivalently, to a diagonal matrix I with equal diagonal elements I).

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For a spherical object

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Simplified Model

Ø Model-order Reduction

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Simplified Model of Helicopter

Ø the force produced by the tail rotor must counter the

torque produced by the main rotor

Ø Assumptions:

§ helicopter position is fixed at the origin § helicopter remains vertical, so pitch and roll are fixed at zero

Ø the moment of inertia reduces to a scalar that

represents a torque that resists changes in yaw

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Feedback Control Problem

Control system problem: Apply torque using the tail rotor to counterbalance the torque of the top rotor. A helicopter without a tail rotor, like the one below, will spin uncontrollably due to the torque induced by friction in the rotor shaft.

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Linear System

Ø Block with mass M connected to a wall through spring Ø Velocity Ø Friction (static, Coulomb, Viscous) is not a linear function of Ø For simplicity, we only consider viscous friction Ø Spring force (linear in operating range ) Ø Thus we can model it as a linear system

˙ y = dy/dt

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˙ y

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k1 ˙ y(t)

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k2y

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Transfer Function

Ø Description of the input-output relation for a linear system Ø Ratio of the output of a system to the input of a system Ø Laplace domain considering its initial conditions and

equilibrium point to be zero

Ø Time domain representation is Impulse Function

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Transfer Function

Ø Newton’s Law Ø Applying Laplace Transform with zero initial condition Ø Transfer Function

m¨ y = u − k1 ˙ y − k2y

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¨ y = d2y(t)/dt2

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˙ y = dy(t)/dt

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ms2ˆ y(s) = ˆ u(s) − k1sˆ y(s) − k2ˆ y(s)

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State Space Representation

Ø Mathematical model of a physical system § Set of input, output and state variables § Related by first-order differential equations or difference equations

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State Space Equation

Ø Let’s consider displacement and velocity as state variables Ø Using Newton’s Law: Ø In Matrix form:

m¨ y = u − k1 ˙ y − k2y

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Actor Model

Ø Mathematical Model of Concurrent Computation Ø Actor is an unit of computation Ø Actors can § Create more actors § Send messages to other actors § Designate what to do with the next message Ø Multiple actors may execute at the same time

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Actor Model of Systems

ØA system is a function that

accepts an input signal and yields an output signal.

ØThe domain and range of the

system function are sets of signals, which themselves are functions.

ØParameters may affect the

definition of the function S.

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Actor Model of the Helicopter

Ø Input is the net torque of the

tail rotor and the top rotor. Output is the angular velocity around the y-axis.

Ø Parameters of the model are

shown in the box. The input and output relation is given by the equation to the right.

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Composition of Actor Model

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Actor Models with Multiple Inputs

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Modern Actor Based Platforms

Ø Simulink (The MathWorks) Ø Labview (National Instruments) Ø Modelica (Linkoping) Ø OPNET (Opnet Technologies) Ø Polis & Metropolis (UC Berkeley) Ø Gabriel, Ptolemy, and Ptolemy II

(UC Berkeley)

Ø OCP, open control platform

(Boeing)

Ø GME, actor-oriented meta-

modeling (Vanderbilt)

Ø SPW, signal processing worksystem

(Cadence)

Ø System studio (Synopsys) Ø ROOM, real-time object-oriented

modeling (Rational)

Ø Easy5 (Boeing) Ø Port-based objects (U of Maryland) Ø I/O automata (MIT) Ø VHDL, Verilog, SystemC (Various)

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Example LabVIEW Screenshot