Constructing and Verifying Cyber Physical...

Marcus Völp

Constructing and Verifying Cyber Physical SystemsStability and Root Locus Method

Summer 2015 Constructing and Verifying Cyber Physical Systems - Marcus Völp 143

Overview

Math

FeedbackControl

RTOS

Verification

Physics

Introduction

Mathematical Foundations (Differential Equations and Laplace Transformation)

Control and Feedback

Transfer Functions and State Space Models

Poles, Zeros / PID Control

Stability, Root Locust Method, Digital Control

Mixed-Criticality Scheduling and Real-Time Operating Systems (RTOS)

Coordinating Networked Cyber-Physical Systems

Program Verification

Differential Dynamic Logic and KeYmaera X

Differential Invariants

CPS


Overview

Observability

Sensor Fusion

Sensitivity

Stability

K. Åström, R. Murray“Feedback Systems –An Introduction for Scientists and Engineers”Princeton University Press

G. Franklin, J.D. Powell, A. Emami-Naeini“Feedback Control of Dynamic Systems”7th ed. Pearson

Root Locus Design Method

Digital Control

Advanced Topics(non-linearity, state-space design, …)

Brian Douglas Control Systems Lectures on YouTube


Sensitivity

How sensitive is the system to unanticipated changes of the plant (𝐺 → 𝐺 + 𝜕𝐺)?

Sensitivity of system transfer function T:

𝑆𝐺𝑇 =

𝜕𝑇𝑇𝜕𝐺𝐺

=𝐺 𝜕𝑇

𝑇 𝜕𝐺

Open Loop: 𝑇𝑂𝑝𝑒𝑛 + 𝜕𝑇𝑂𝑝𝑒𝑛 = 𝐷𝑂𝑝𝑒𝑛 𝐺 + 𝜕𝐺

= 𝐷𝑂𝑝𝑒𝑛𝐺 + 𝐷𝑂𝑝𝑒𝑛𝜕𝐺

= 𝑇𝑂𝑝𝑒𝑛 + 𝐷𝑂𝑝𝑒𝑛𝜕𝐺

𝜕𝑇𝑂𝑝𝑒𝑛

𝑇𝑂𝑝𝑒𝑛=

𝐷𝑂𝑝𝑒𝑛𝜕𝐺

𝐷𝑂𝑝𝑒𝑛𝐺=

𝜕𝐺

𝐺=> 𝑆𝐺

𝑇 = 1

Open loop systems have no means to compensate plant changes: any error in G is immediately reflected in T


Sensitivity

How sensitive is the system to unanticipated changes of the plant (𝐺 → 𝐺 + 𝜕𝐺)?

Sensitivity of system transfer function T:

𝑆𝐺𝑇 =

𝜕𝑇𝑇𝜕𝐺𝐺

=𝐺 𝜕𝑇

𝑇 𝜕𝐺

Closed Loop: 𝑇𝐶𝑙𝑜𝑠𝑒𝑑 + 𝜕𝑇𝐶𝑙𝑜𝑠𝑒𝑑 =𝐷𝐶𝑙 𝐺 + 𝜕𝐺

1 + 𝐷𝐶𝑙 𝐺 + 𝜕𝐺

Closed loop systems compensate plant changes the better the higher the gain. e.g. for 1 + 𝐺𝐷𝐶𝑙 = 100, a 10% change results in only 0.1% steady state gain change.

𝑆𝐺𝑇 =

𝐺 𝜕𝑇𝐶𝑙𝑇 𝜕𝐺

=𝐺 𝑑𝑇𝐶𝑙𝑇 𝑑𝐺

=𝐺

𝐺𝐷𝐶𝑙1 + 𝐺𝐷𝐶𝑙

1 + 𝐺𝐷𝐶𝑙 𝐷𝐶𝑙 − 𝐷𝐶𝑙(𝐺𝐷𝐶𝑙)

1 + 𝐺𝐷𝐶𝑙2

=1

1 + 𝐺𝐷𝐶𝑙


Stability

Roots in the right half of the s-plane cause instability.

Open Loop: 𝑇𝑂𝑝𝑒𝑛 = 𝐺𝐷𝑂𝑝𝑒𝑛 =𝑏 𝑠 𝑐(𝑠)

𝑎 𝑠 𝑑(𝑠)

𝐺 𝐷𝑂𝑝𝑒𝑛

Tempting to cancel instable root in a(s) with zero in c(s), but root remains in presence of slightest variance of plant.=> don’t use open loop control on instable plants

Closed Loop: 𝑇𝐶𝑙 =𝐺𝐷𝐶𝑙

1+𝐺𝐷𝐶𝑙1 + 𝐺𝐷𝐶𝑙 = 0

1 +𝑏 𝑠 𝑐(𝑠)

𝑎 𝑠 𝑑(𝑠)= 0

𝑎 𝑠 𝑑(𝑠) + 𝑏 𝑠 𝑐(𝑠) = 0

Cancelling with zero in c(s) is still sensitive to plant change, but it is still possible to construct stable systems from unstable plants.


Stability

𝐼,𝑚𝑝

𝑚𝑡 𝐹

𝜃

𝑥

𝑚𝑡 +𝑚𝑝 𝑥 + 𝑏 𝑥 + 𝑚𝑝𝑙 𝜃 cos 𝜃 + 𝑚𝑝𝑙 𝜃2 sin 𝜃 = 𝐹

𝐼 + 𝑚𝑝𝑙2 𝜃 + 𝑚𝑝𝑔𝑙 sin 𝜃 = −𝑚𝑝𝑙 𝑥 cos 𝜃

sin 𝜃 ≈ 𝜃cos 𝜃 ≈ 1

𝜃2 ≈ 0

𝑚𝑡 +𝑚𝑝 𝑥 + 𝑏 𝑥 + 𝑚𝑝𝑙 𝜃 = 𝐹𝐼 + 𝑚𝑝𝑙2 𝜃 + 𝑚𝑝𝑔𝑙𝜃 = −𝑚𝑝𝑙 𝑥

eliminate x

𝐼 + 𝑚𝑝𝑙2 𝑚𝑡 +𝑚𝑝 −𝑚𝑝

2𝑙2 𝜃 − 𝑚𝑝𝑔𝑙 𝑚𝑡 +𝑚𝑝 𝜃 = −𝑚𝑝𝑙𝐹

ℒΘ(𝑠)

𝐹(𝑠)=

−𝑚𝑝𝑙


2𝑙2 𝑠2 −𝑚𝑝𝑔𝑙 𝑚𝑡 +𝑚𝑝

𝑙


Stability

𝐼,𝑚𝑝

𝑚𝑡 𝐹

𝜃

𝑥

𝐺 𝑠 =Θ(𝑠)

𝐹(𝑠)=

1

𝑠2 − 1=

1

(𝑠 + 1)(𝑠 − 1)

Θ(𝑠)

𝐹(𝑠)=

−𝑚𝑝𝑙



𝐷𝐶𝑙 𝑠 =𝐾(𝑠 + 𝛾)

(𝑠 + 𝛿)

𝑎 𝑠 𝑑(𝑠) + 𝑏 𝑠 𝑐(𝑠) = 0

(𝑠 + 1)(𝑠 − 1)(𝑠 + 𝛿) + 𝐾(𝑠 + 𝛾) = 0 𝛾 = 1

(𝑠 − 1)(𝑠 + 𝛿) + 𝐾 = 0

𝑠2 + 𝛿 − 1 𝑠 + (𝐾 − 𝛿) = 0 𝑠2 + 2𝜁𝜔𝑛𝑠 + 𝜔𝑛2 = 0

𝛿 = 2𝜁𝜔𝑛 + 1 𝐾 = 𝜔𝑛2+2𝜁𝜔𝑛 + 1

Im(s)

Re(s)

sin−1 𝜉

𝜔𝑛

x

𝑙

cancel stable pole using zero

Often systems are parametric in at least one parameter!

e.g., 𝐾𝑃 of P controller

(temperature) variations in plant



1 + 𝐾 𝐺 𝑠 = 1 + 𝐾𝑃(𝑠)

𝑄(𝑠)= 0

Plant

𝐺 𝑠 =𝑃(𝑠)

𝑄(𝑠)

𝑅(𝑠)Σ

+

-

(PD) Control𝐾(𝑠 + 1)

Open Loop Transfer Function

How do poles of the overall system move with K?

𝑠2 + 3𝑠 + 𝐾𝑠 + 4𝐾 + 3 = 0 (𝑠2 + 3𝑠 + 3) + 𝐾(𝑠 + 4) = 0 1 + 𝐾(𝑠 + 4)

(𝑠2 + 3𝑠 + 3)= 0

Rules to draw the paths of the roots of the characteristic equation:

(MatLab “rlocus” does it but remembering the rules gives insight in system behavior)



Rule 1) There are n-lines (loci) where n is the maximum degree of Q or P, whichever is greater

Rule 2) As K increases from 0 to ∞, the roots move from the poles of G(s) to the zeros of G(s) (They attract zeros from infinity)

Rule 3) When roots are complex they occur in conjugate pairs

Rule 4) At no time will the same root cross over its path

1 + 𝐾𝑃(𝑠)

𝑄(𝑠)= 0

𝜁 ≥ 0.5

x

x

x

x

ox

x









1 + 𝐾𝑃(𝑠)

𝑄(𝑠)= 0

x ox∞

∞ o xo

o x

x

x









1 + 𝐾𝑃(𝑠)

𝑄(𝑠)= 0

x

x

x

xx x

OKwon’t happen





1 + 𝐾𝑃(𝑠)

𝑄(𝑠)= 0

Rule 5) The portion of the real axis to the left of an odd number of open loop poles and zeros are part of the loci

Rule 6) Lines leave (break out) and enter (break in) the real axis at 90°

Rule 7) If there are not enough poles or zeros to make a pair, then the extra lines go to or come from infinity.

Rule 8) Lines go to infinity along asymptotes. The angles of the asymptotes are:

Φ𝐴 =2𝑞+1

𝑛−𝑚180° where q = 0, 1, …, (n-m-1)

The centroid of the asymptotes are: 𝑓𝑖𝑛𝑖𝑡𝑒 𝑝𝑜𝑙𝑒𝑠 − 𝑓𝑖𝑛𝑖𝑡𝑒 𝑧𝑒𝑟𝑜𝑠

𝑛−𝑚

xxx ox123

12345





1 + 𝐾𝑃(𝑠)

𝑄(𝑠)= 0





Φ𝐴 =2𝑞+1

𝑛−𝑚180° where q = 0, 1, …, (n-m-1)


𝑛−𝑚

xxx ox123

12345

∞

∞

∞

xxo∞



Constructing and Verifying Cyber Physical Systems - Marcus Völp 156


1 + 𝐾𝑃(𝑠)

𝑄(𝑠)= 0





Φ𝐴 =2𝑞+1

𝑛−𝑚180° where q = 0, 1, …, (n-m-1)


𝑛−𝑚

xx

-1-2∞

∞

Φ𝐴0 =2∙0+1

2180° = 90°

Φ𝐴1 =2∙1+1

2180° = 270° = −90°

2 lines (q={0,1})

Summer 2015

𝑐𝑒𝑛𝑡 =−2−1

2= -1.5



Constructing and Verifying Cyber Physical Systems - Marcus Völp 157


1 + 𝐾𝑃(𝑠)

𝑄(𝑠)= 0





Φ𝐴 =2𝑞+1

𝑛−𝑚180° where q = 0, 1, …, (n-m-1)


𝑛−𝑚

Summer 2015

xx

-1-2

Φ𝐴0 =2∙0+1

3180° = 60°

Φ𝐴1 =2∙1+1

3180° = 180°

3 lines (q={0,1,2})

x

-3

Φ𝐴2 =2∙2+1

3180° = -60°

𝑐𝑒𝑛𝑡 =−3−2−1

3= -2





1 + 𝐾𝑃(𝑠)

𝑄(𝑠)= 0

Rule 9) If there is at least two lines to infinity, then the sum of all roots is constant

Rule 10) K going from 0 to negative infinity can be drawn by reversing rule 5 and adding 180° to the asymptote angles.xx

-1-2

x

-3(-3) + (-2) + (-1) = -6 K=0(-3.52) + (– 1.24+0.86i) + (– 1.24-0.86i) = -6 K=2

xxx

stabilize 2 as fast

destabilize ½ as fastx

x

x





1 + 𝐾𝑃(𝑠)

𝑄(𝑠)= 0

Rule 9) If there is at least two lines to infinity, then the sum of all roots is constant

Rule 10) K going from 0 to negative infinity can be drawn by reversing rule 5 and adding 180° to the asymptote angles.

Plant1

(𝑠 + 1)(𝑠 − 2)

𝑅(𝑠)Σ

+

-

Controlo x x


Digital Control

Plant𝐺 𝑠

𝑟(𝑡) 𝑦(𝑡)Σ+

+

𝑊(𝑠)

disturbance

𝑢(𝑘𝑇)Σ

Σ

+

-

++

𝑒(𝑘𝑇) DifferenceEquation

D/A

A/D 𝑉(𝑠)

sensor noise

ZOH

analog-to-digital converter

digital-to-analog converterzero order hold

Sensor𝑦(𝑘𝑇)

A/D

sample rate 1

𝑇=> one sample every sample period T

=> one output every sample period T

Clock

computer

zero order hold: hold output constant until next clock tick

𝑒(𝑘𝑇)


Digital Control

Analog / Digital Conversion

Resolution Nyquist-Shannon Theorem

full scale voltage range ('span')

If a function x(t) contains no frequencies higher thanB Hz, it is completely determined by giving itsordinates at a series of points spaced 1/(2B) secondsapart.

Alias because signal frequency B’ is higher than sampling frequency 2B


Digital Control

Difference Equations

𝑢(𝑘𝑇)Σ

+

-


D/A

A/D

ZOH

𝑦(𝑘𝑇)

A/D

Clock

computer

𝑥 𝑘 + 1 = 𝑓 𝑥 𝑘 , 𝑢 𝑘

𝑦 𝑘 = ℎ(𝑥 𝑘 , 𝑢 𝑘 )

𝜕𝑥

𝜕𝑡≈𝑥 𝑘 + 1 − 𝑥[𝑘]

𝑡 𝑘 + 1 − 𝑡[𝑘]


Digital Control

React asynchronously

𝑢(𝑘𝑇)Σ

+

-


D/A

A/D

ZOH

𝑦(𝑘𝑇)

A/D

Clock

computer

Electrical signal every time wheel reaches a certain position

Core D/A

A/D

ZOHA/D

Clock

computer

Capture

Capture

Compare

r[k], tr[k]

y[k], ty[k]

capture:store y[k] and timestamp ty[k]trigger interrupt in core

interrupt handler:read y[k], ty[k];read r[k], tr[k];compute u[k] and signal time ts[k];end of interrupt

compare:signal u[k] if clock == ts[k]

varying interrupt delivery times• disabled interrupts• long instructions• voltage / frequency scaling

varying execution times• voltage / frequency scaling• caches• input dependent jitter

execution times

interrupt delivery

Topics we did not cover:

• Nonlinear systems

• Frequency-Response Design Method

• State-Space Design

• Digital Control: z-Transform

• Matlab / Simulink

• a lot of case studies and examples

Topics we will be covering next:

• How to construct the execution environment for digital controllers?

• How to construct controllers that work together for a greater goal?

• How to verify that the controllers we design and implement are safe and secure?


Advanced Topics

Nonlinear systems: Taylor series


Advanced Topics

sin 𝜃 ≈ 𝜃; cos 𝜃 ≈ 1 for 𝜃 ≈ 0 𝑓 𝑎 =

𝑛=0

∞𝑓 𝑛 𝑎

𝑛!(𝑥 − 𝑎)𝑛Taylor series:

𝑒𝑥 =

𝑛=0

∞𝑥𝑛

𝑛!

1

1 − 𝑥=

𝑛=0

∞

𝑥𝑛

cos(𝑥) =

𝑛=0

∞(−1)𝑛

2𝑛 !𝑥2𝑛

sin(𝑥) =

𝑛=0

∞(−1)𝑛

2𝑛 + 1 !𝑥2𝑛+1

Nonlinear systems: Linearization by nonlinear feedback


Advanced Topics

Plant

𝑇𝑐 = 𝑚𝑙2 𝜃 + 𝑚𝑔𝑙 sin 𝜃

𝑅(𝑠) 𝑌(𝑠)Σ+

+

𝑊(𝑠)

disturbance

𝑈(𝑠)Σ

Σ

+

-

++

𝐸(𝑠) Control

𝑚𝑙2 𝜃 = 𝑢

Compute𝑇𝑐 = 𝑚𝑔𝑙 sin 𝜃 + 𝑢

Compute𝑦 = 𝑇𝑐 −𝑚𝑔𝑙 sin 𝜃

𝑉(𝑠)

sensor noise

State-Space Design (modern control design):

• ODEs don’t need to be linear

• no transformation required

• make internal energy transfers explicit; extract required output

• works for multiple input / multiple output (here SISO only)


Advanced Topics

𝒙 = 𝑨𝒙 + 𝑩𝑢

𝑦 = 𝑪𝒙 + 𝐷𝑢

system matrix (nxn) input matrix (nx1)

output matrix (1xn) direct transmission term

State-Space Design (modern control design):


Advanced Topics



system matrix (nxn) input matrix (nx1)

output matrix (1xn) direct transmission term

𝑥 = −𝑏(1 +𝑚𝑝𝑙𝐶)

𝑚𝑡 +𝑚𝑝

𝑥 + 𝑚𝑝𝑙𝐶𝑔 𝜃 + [1 + 𝑚𝑝𝑙𝐶

𝑚𝑡 +𝑚𝑝

]𝐹

𝜃 = −𝑚𝑝𝑙𝑔 𝑚𝑡 +𝑚𝑝

𝑚𝑡 +𝑚𝑝 𝐼 + 𝑚𝑝𝑙2 −𝑚𝑝

2𝑙2𝜃 +

𝑚𝑝𝑙𝑏


2𝑙2 𝑥 −

𝑚𝑝𝑙


2𝑙2𝐹

C

𝑚𝑡 +𝑚𝑝 𝑥 + 𝑏 𝑥 + 𝑚𝑝𝑙 𝜃 cos 𝜃 + 𝑚𝑝𝑙𝜃2 sin 𝜃 = 𝐹

𝐼 + 𝑚𝑝𝑙2 𝜃 + 𝑚𝑝𝑔𝑙 sin 𝜃 = −𝑚𝑝𝑙 𝑥 cos 𝜃

𝜃 𝜃 𝑥

=

0 𝐶𝑔(𝑚𝑡 +𝑚𝑝) 𝐶𝑏

1 0 0

0 𝑚𝑝𝑙𝐶𝑔 −𝑏(1 + 𝑚𝑝𝑙𝐶)

𝑚𝑡 +𝑚𝑝

𝜃𝜃 𝑥

+

𝐶0

1 + 𝑚𝑝𝑙𝐶

𝑚𝑡 +𝑚𝑝

𝐹 𝜃 = 0 1 0

𝜃𝜃 𝑥

+ 0


Observability



some part of x may remain internal (i.e., not observable)

Definition: (Observability)A linear system is observable if for any T > 0 it is possible to determine the state of the system x(T) through measurements of y(t) and u(t) on the interval [0, T].

𝜕𝑦

𝜕𝑡= 𝑪

𝝏𝒙

𝝏𝒕= 𝑪𝑨𝑥

𝜕𝑥

𝜕𝑡= 𝑨𝒙, 𝑦 = 𝑪𝑥

neglect input

𝑦 𝑦 𝑦⋮

𝑦(𝑛−1)

=

𝐶𝐶𝐴𝐶𝐴2

⋮𝐶𝐴𝑛−1

𝒙

observability matrix: 𝑾𝟎

Theorem: (Observability rank condition)A linear system of the form 𝒙 = 𝑨𝒙 + 𝑩𝑢,𝑦 = 𝑪𝒙 + 𝐷𝑢 is observable if and only if the observability matrix 𝑾𝟎 is full rank.


Combination of sensory data or data derived from sensory data from disparate sources to reduce uncertainty of the result below that of each individual sensor.

Virtual Sensor = Sensor + Mathematical Model to derive value of interest


Sensor Fusion

more accurate, more complete, more dependable, view which emerges from the individual signals (e.g., stereoscopic vision)

Direct fusion: fusion from sensor data and history valuesIndirect fusion: use a-priori knowledge about environment

average of all sensorsdiscard sensor data if returned gravity < 9 or > 9.3

Digital Control: z-Transform


Advanced Topics

ℒ 𝑓 𝑡 = 𝐹 𝑠 = 0

∞

𝑓(𝑡)𝑒−𝑠𝑡 Z 𝑓 𝑘 = 𝐹 𝑧 = 𝑘=0∞ 𝑓 𝑘 𝑧−𝑘

Z 𝑓 𝑘 − 1 = 𝑧−1𝐹 𝑧

𝑓 𝑘𝑇 = 𝑒−𝑎𝑘𝑇

𝐹 𝑧 =𝑧

𝑧 − 𝑒−𝑎𝑇1

𝑠 + 𝑎

𝑧 = 𝑒𝑠𝑇

𝑧 = 𝑒𝑠𝑇 s = −𝜉𝜔𝑛 ± 𝑗𝜔𝑛 1 − 𝜉2

Matlab / Simulink


Advanced Topics

Programming Interface

Source: http://www.mathworks.com/help/

Tools (e.g., Control System Toolbox)

Plots


Overview

Observability

Sensor Fusion

Sensitivity

Stability


G. Franklin, J.D. Powell, A. Emami-Naeini“Feedback Control of Dynamic Systems”7th ed. Pearson


Digital Control

Advanced Topics(non-linearity, state-space design, …)


Overview

Math

FeedbackControl

RTOS

Verification

Physics

Introduction

Mathematical Foundations (Differential Equations and Laplace Transformation)

Control and Feedback

Transfer Functions and State Space Models

Poles, Zeros / PID Control

Stability, Root Locust Method, Digital Control

Mixed-Criticality Scheduling and Real-Time Operating Systems (RTOS)

Coordinating Networked Cyber-Physical Systems

Program Verification

Differential Dynamic Logic and KeYmaera X

Differential Invariants

CPS


Leftovers

Constraints for designing second order systems

settling time 𝜎 ≥4.6

𝑡𝑠(decay below 1%)

Im(s)

Re(s)

𝜎

settling time

x

𝑒−𝜁𝜔𝑛𝑡𝑠 = 0.01

𝜁𝜔𝑛𝑡𝑠 ≈ 4.6

𝑡𝑠 ≈4.6

𝜎

𝑡𝑠 ≤ 𝑡 ⇒4.6

𝜎≤ 𝑡 ⇒ 𝜎 ≥

4.6

𝑡

Instability of I-Controllers

assume 𝐺 𝑠 =1

(𝑠+1)(𝑠+1)

𝑇𝐹 =𝑘𝐼𝐺(𝑠)

𝑠 + 𝑘𝐼𝐺(𝑠)=

𝑘𝐼𝑠(𝑠 + 1)(𝑠 + 1) + 𝑘𝐼

=𝑘𝐼

𝑠3 + 2𝑠2 + 𝑠 + 𝑘𝐼


Leftovers

𝐼,𝑚𝑝

𝑚𝑡 𝐹

𝜃

𝑥

𝐺 𝑠 =Θ(𝑠)

𝐹(𝑠)=

1

𝑠2 − 1=

1

(𝑠 + 1)(𝑠 − 1)

Θ(𝑠)

𝐹(𝑠)=

−𝑚𝑝𝑙



𝐷𝐶𝑙 𝑠 =𝐾(𝑠 + 𝛾)

(𝑠 + 𝛿)

𝑎 𝑠 𝑑(𝑠) + 𝑏 𝑠 𝑐(𝑠) = 0

(𝑠 + 1)(𝑠 − 1)(𝑠 + 𝛿) + 𝐾(𝑠 + 𝛾) = 0 𝛾 = 1

(𝑠 − 1)(𝑠 + 𝛿) + 𝐾 = 0

𝑠2 + 𝛿 − 1 𝑠 + (𝐾 − 𝛿) = 0 𝑠2 + 2𝜁𝜔𝑛𝑠 + 𝜔𝑛2 = 0

𝛿 = 2𝜁𝜔𝑛 + 1 𝐾 = 𝜔𝑛2+2𝜁𝜔𝑛 + 1

Im(s)

Re(s)

sin−1 𝜉

𝜔𝑛

x

𝑙

cancel stable pole using zero

Date post:	22-May-2020
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