MY WAY -

HOW TO TEACH COMPUTER

SIMULATION

Heikki Koivo

Aalto University

ESPOO, FINLAND

Objectives

LEARN BASICS OF

• Modelling

• Simulation

• Automatic control

• Optimization

• Tools: MATLAB, SIMULINK

Beginning of COMPUTER SIMULATION course

1983 Slides about contents

Course Content in 19831 MODELLING

– Basic philosophy

– How to determine the model structure from experimental results

– Parameter estimation

– Examples: paper machine, heat recovery system for a TMP line,

industrial glass oven, plastics extruder, flotation circuit,

magnetic levitation vehicle, main control loops of a nuclear

power plant, training simulator

2 NUMERICAL METHODS FOR SOLVING DIFFERENTIAL EQUATIONS

– Integration methods

– Difficulties: Functions given as tables, discontinuities, delays in

differential equations, selection of step size and its automatic

control

– Library programs (e.g. IMSL, NAG)

Course Content in 1983, cont.

3 INTERACTION

– Interaction with computer

– Software solutions, example given

– Graphics

4 PROGRAMMING THE SIMULATOR

– Structure

– General principles for software design and implementation

– Special features of FORTRAN

– Documentation

Course Content in 1983, cont.

5 HARDWARE USED IN SIMULATION

– Parallel processors

– Personal computers (PC’s)

– Real-time simulation

6 FURTHER EXAMPLES OF SIMULATED SYSTEMS

– Elevator

– TMP (Thermo Mechanical Pulping) process

– Cancerous cell

– National economy

Lectures, exercises in microcomputer class,

take-home exam

Course Content in 2006

7

Computer tools

Basics in simulation

• Mathematical model building– Use of the model

– Profitability in mathematical modelling

– Model users

– What is computed and how accurately ()

– How detailed a model is needed

Basics in simulation

• Different phases in model building– Preplanning

– Selection of variables

– Model structure selection

– Planning experiments, if needed

– Performing the experiments

– Analyzing the results from experiments

– Finalizing the model and testing it

Basics in simulation

• Use of model

• E.g. paper machine

Short cycle of a paper machine

Microfluidics simulation

MS MS

PSTN

M S C

RNCRNC RNC

BS BS

Wireless radio network – Radio resource

management

Chernobyl

• Sometimes, even with computer-controlled processes thingsgo wrong

• In Chernobyl, engineers in early morning hours of 1986, in ’wolf’s hour’, started making experiments.

• They turned the main controllers off, and started making whatcould be called (standard) step experiments thinking that theycould manually control the process.

• Unfortunately the process was inherently unstable, and the rest we know.

• Once it had happened, my PhD student reminded me of ourRussian visitor – visit in 1983.

16

Chernobyl

16

MIR Submersible – Institute of Oceanology,

Russian Academy of Sciences

17

• Used in filming TITANIC

• Finnish built and designed

• Max diving depth 6000 m

• Reserve power generation

simulation (our conribution)

Model structures

• Static models

• Dynamic models

• SIMULINK – starters

• Stochastic processes to generate noise

Model structures

• STATIC MODELS

Linear, scalar

y = k u + b

y output; u input; k gain; b bias

EXAMPLE: OHM’s Law

U=RI

U = voltage (output)

I = current (input)

R = resistor

Nonlinear, scalar

y = f(u; a),

a is a parameter

OHM’s Law with nonlinear resistor

R = R(I) = R0 I2 ; Resistor depends on current

0

3

; ; ;

( ; )

y U u I a R

y f u a au

2 3

0 0( )U R I I R I I R I

Model structures

• STATIC MODELS

– Linear, multi-input multi-output (MIMO)

y = Au + b

y is output vector, u input vector, b constant vector

Model structures

• DYNAMIC MODELS

– Linear, scalar

0 0, ( )dx

x ax bu x t xdt

a and b are constants

Model structures

• DYNAMIC MODELS - EXAMPLE

– 1798: Model of population dynamics (Malthus, 1766-1834) :

Population year 1960 was 3.04.109 and its growth 2% .

Compute how population evolved up to year 2008.

90.02 , (1960) 3.04 10x x x

0.02( 1960)

0

9 0.02(2008 1960) 9

( )

(2008) 3.04 10 7.9 10

tx t x e

x e

MODEL STRUCTURESYear Population

Average

annual

growth

rate (%)

Average annual

population change

1960 3,039,669,330 1.33 40,792,172

1969 3,632,780,614 2.05 75,286,491

1970 3,708,067,105 2.07 77,587,001

1971 3,785,654,106 2.01 76,694,660

1972 3,862,348,766 1.95 76,183,283

1973 3,938,532,049 1.90 75,547,218

2000 6,085,478,778 1.21 74,220,528

2001 6,159,699,306 1.18 73,002,863

2002 6,232,702,169 1.16 72,442,511

2003 6,305,144,680 1.14 72,496,962

2004 6,377,641,642 1.13 72,578,164

2005 6,450,219,806 1.12 72,540,568

2006 6,522,760,374 1.10 72,466,183

2007 6,595,226,557 1.09 72,442,792

2008 6,667,669,349 1.08 72,368,570

2009 6,740,037,919 1.07 72,210,364

Model structures

• MATLAB – SIMULINK SOLUTION

• Instead of differential equation –

integrate it resulting in:

• In SIMULINK, integral operator is

9

0.02 ,

(1960) 3.04 10

x x

x

0 0 0

0( ) ( ) ( )

t t t

t t t

dxx t dt dt dx x t x t

dt

( )x t x(t)

𝑥(𝑡0

Model structures

9

0.02 ,

(1960) 3.04 10

x x

x

( )x t x(t) 0.02x(t)

• What about x(t0)? Click the the integrator block open.

In the space Initial condition type: 3.04*10^9 ;

• Until now, we have handled the left-hand side of the equation.

In order to have balance in the equation, we need to complete the SIMULINK

diagram with the term 0.02x(t).

𝑥(1960 = 3.04 ⋅ 109

Model structures

9

0.02 ,

(1960) 3.04 10

x x

x

( )x t x(t)

0.02x(t)

• MATLAB – SIMULINK SOLUTION –

SOLUTION DISPLAYED WITH A SCOPE

Simulation result

Model structures• DYNAMIC STATE-SPACE MODEL

– Nonlinear, multi-input multi-output (MIMO)

0( ( ), ( ), ), ( )t t t t

0x f x u x x

𝐲(𝑡 = 𝐡(𝐱(𝑡 , 𝐮(𝑡 , 𝑡

𝑡 has been droppedto simplify notation

State space model

First order derivatives on the

left hand side, no derivatives

on the right hand side

Output equation

Remark: Discrete time state space equations are similar

Model structures

• DYNAMIC STATE-SPACE MODEL

– Nonlinear, multi-input single-output (MISO)

23 , Drop the argument x x x u t

y x

2( ) 3 ( ) ( ) ( )

( ) ( )

x t x t x t u t

y t x t

1

2

x x

x x

To get state - space model

solve for the highest order

derivative and define

1 2

2

2 1 2

1

3

x x

x x x u

y x

1 1 21 22

2 1 22 1 2 1

, ,= ,

, ,3

( , )

f x x ux x

f x x ux x x u

u

y x h x u

x f x u

u

y

Model structures

• LINEAR STATE-SPACE MODEL MODELS

– Linear, multi-input multi-output (MIMO)

A (nxn), B (nxm),

C (pxn) and D (pxm) real, constant

,x Ax Bu

y Cx

TRANSFER FUNCTION

GU Y

Y = G*U

Linear, time-invariant state equation

,x Ax Bu

y Cx

Corresponding MIMO (multi input - multi output) transfer function

G( ) C( I A) B1s s continuous ,

CONTINUOUS CASE

Scalar transfer

function

To generate noisy data for measurements:

WHITE NOISE or RANDOM NUMBERS

White noise: Current state

depends

neither on history nor future

values

Noise spectrum is constant

Random number generator:

Uniformally or Normally distributed

Noisy data generation, example

Numerical solution of

differential equations

Summary of solvers

Initial value problem

0

( ( ), ), 0

(0)

x f x t t t

x x

1 1

1 1( ) ( ) ( ( ), ) or ( ) ( ) ( ( ), ) m m

m m

t t

m m m m

t t

x t x t f x s s ds x t x t f x s s ds

Let

Integrating both sides

0

0

( ) ( ( ), ) ,

t

x t x f x s s ds

Initial value problem (scalar case)

Assume f depends only on x

t (time)1mt mt

1( )

mf x

( )m

f x

Area

1m mh t t

Rectangular

approximation

of the integral

ErrorKnown Unknown

h integration step

1( )mf x h

𝑡𝑚−1

𝑡𝑚

𝑓(𝑥(𝑠 𝑑𝑠 ≈ 𝑓 𝑥 𝑡𝑚−1 𝑡𝑚 − 𝑡𝑚−1 = 𝑓(𝑥(𝑡𝑚−1 ℎ

Euler method

t (time)0t 4t

0 0( , )f x t4 4( , )f x t

( , )f x t

4 ( )p t

4 4

0 0

4( ) ( ) , m=4

t t

t t

f s ds p s ds

Euler method-First order method:

Global error (error at a given time) is proportional to the step size.

1 1 1( , ) , ( )m m m m m mx x f x t h x x t

Can use higher order

polynomials for better

approximation:InterpolationExtrapolationLagrangeNewton’s differenceor orthogonal polynomialsLegendreTscebychevJacobiLaguerreHermite

Runge-Kutta methods (open)

• Runge-Kutta methods use

only function values – not

derivatives

• General form of Runge-Kutta

method1

1

1

1

( , )

v

m m i i

i

i

i m ij j m i

i

x x w k

k hf x a k t c h

Approximates derivatives

• Classical 4. order Runge-Kutta

1 1 1 2 2 3 3 4 4

1

2 21 1 2

3 31 1 32 2 3

4 41 1 42 2 43 3 4

( , )

( , )

( , )

( , )

m m

m m

m m

m m

m m

x x w k w k w k w k

k hf x t

k hf x a k t c h

k hf x a k a k t c h

k hf x a k a k a k t c h

1 4 2 3 2 3 4

21 31 32 41 42

1 1 1, , , 1

6 3 2

1, 0

2

w w w w c c c

a a a a a

Predictor-corrector method

• General form

1 1 2 1 1

0 1 1 1

m m m k m k

m m k m k

x x x x

h x x x

1 1 2 1 1

0 1 1 1 1( , )

m m m k m k

m m m k m k

x x x x

h f x t x x

Predictor Euler – Corrector trapezoidal

1. Compute xm+1 with predictor formula

2. Compute corrected xm+1 from corrector formula

Requires starting values, e.g. Runge-Kutta

Multiple step method

e.g. Adams-Moulton

1ˆ

m m mx x hf

1 1 1ˆ( , ) ( , )2

m m m m m m

hx x f x t f x t

Stiff equations

• Definition

Equation is stiff (in linear, time-invariant case) when

max

min

Re1

Re

Stiff equations

• Example

• Characteristic equation

• Roots of polynomial

• roots([1, 100, .9999]) ans =-99.9900 -0.0100

• Solution

100 0.9999 0x x x

(0) 1; (0) 0x x

2 100 0.9999 0r r

99.99 0.001( ) 0.0001 1.001t tx t e e

Fast mode disappears

quickly, cannot be seen

in the figure

Slow

mode

dominates

Slow mode

Fast mode

Errors in numerical integration

• Truncation error

– Only finite number of terms

are taken into account

• Rounding error

– Floating point does not represent

real number accurately, because

computers have finite word lengths.

Error grows, when step size

decreases

• Cumulative error

• Truncation error – e.g. Taylorin

series

1

2 3

1 1 1 1

( ) ( )

( ) ( ) ( ) ( )2! 3!

m m

m m m m

x t x t h

h hx t x t h x t x t

11( )

( 1)!

kkhT h x

k

Truncation error

Bottlenecks in simulation

Torque T

Speed n

Field voltage e

Two-phase induction servomotor

torque-speed characteristics.

Torque function characteristics

not given explicitly as a function.

It is given as a graph or in table

form (very common in practice).

Dependence is often nonlinear.

A system component creates a

nonlinear dynamic equation.

How to solve these problems in

simulation?

0

Linearization example

Pendulum (assume no friction)

L is constant,

mass point mass m

mg

T (external

torque)

mg sin mg cos

L

m

Pendulum - equation of motion

L

TmgorcesfmL sin

Assuming constants to simplify this

sin u

Define state variables:

angle α and angular

velocity

1

2

:

:

x

x

1 2

2 1

:

: sin

x x

x x u

1 2 1 1 2

2 1 2 1 2

: ( , , )

: sin ( , , )

x x f x x u

x x u f x x u

Remark: State definition not unique!

Could define state variables as

State equation becomes

2

1

:

:

x

x

Pendulum (u=0) - Equilibrium state

Exact solution

0 0

1 2

0 0

2 1

0

0 sin

x x

x x

0 0( ) x 0 f x

Equilibrium state

satisfies

0

1

0

2

, 0, 1, 2,...

0

x n n

x

Pendulum

Assume u=0

for simplicity

1 2 1 1 2

2 1 2 1 2

: ( , )

: sin ( , )

x x f x x

x x f x x

Equilibrium state – SIMULINK/ TRIM

TRIM Finds steady state parameters

for a Simulink system.TRIM solves for steady-state parameters that

satisfy certain input, output and state conditions.

Set initial condition.

Is there another equilibrium point?

One has to make another initial guess.

Set initial guess to be (1,1)

[X,U,Y,DX]=TRIM(' Pendulum ',[1 1]')

X =1.0e-12 * [ 0 0.2922]’

U = Empty matrix: 0-by-1

Y = Empty matrix: 0-by-1

DX = 1.0e-12 * [ 0.2922 0]’

Remark: SIMULINK uses states

2

1

:

:

x

x

TRIM(‘Pendulum‘,[2 2]’)

X = 0 angular velocity

3.1416 angle α

𝛼

Linearized state space model corresponding to

computed equilibrium states

MATLAB Command

[A,B,C,D]=LINMOD('SYS',X,U)

[A,B,C,D]=LINMOD(‘Pendulum', [0 0])

x Ax Bu

y Cx Du

[A,B,C,D]=LINMOD('Pendulum', [0 0])

A = 0 -1.0000

1.0000 0

B = Empty matrix: 2-by-0

C = Empty matrix: 0-by-2

D = []

[A,B,C,D]=LINMOD('Pendulum', [0 pi ])

A = 0 1.0000

1.0000 0

Difficulties in simulation

• Models not given

• Use parameterized, empirical models

• Parameters determined from

measurements

Examples of parameterized models

• STATIC - LINEAR– Guess the model

y = a u+b

Determine parameters a and b, when a measurement (ni noise) is available.

– Form cumulative error (constant weights)

i i iy a u b n

2

1

1( , )

2

Ni i

i

J a b y au b

Minimize cost function w.r.t. a and b.

Linear -multivariable

2

1

1( ) ( ; )

2

Ni i

i

J

a y f u a Minimize cost function w.r.t. a.

Curve fitting – fit polynomial

• Data generation

• MATLAB or SIMULINK

20.3+2sin(t)+0.1ty

>> [t y]ans =

0 0.3000

0.5000 1.2839

1.0000 2.0829

1.5000 2.5200

2.0000 2.5186

... ...

8.5000 9.1220

9.0000 9.2242

9.5000 9.1747

10.0000 9.2120

Table

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

x

y

Graph

POLYFIT Fit polynomial to data.POLYFIT(X,Y,N) finds the coefficients of a polynomial P(X) of degree N that fits the data,

P(X(I))~=Y(I), in a least-squares sense.

Result of 3rd order polynomial fit

[p,s]=polyfit(t,y,3)

ans =-0.0152 0.3768 -1.4204 2.2505

y1=polyval(p,t);

e=y-y1;

3 2 1 0 3 2 1 0

2213 2

3 2 1 0 3 2 1 0, , , , , ,

1

min ( , , , ) min i i i ia a a a a a a a

i

J a a a a y a t a t a t a

3 2

3 2 1 0y a t a t a t a

0 1 2 3 4 5 6 7 8 9 10-4

-2

0

2

4

6

8

10

12

Error

Data

Not a good fit!

Result of 6th order polynomial fit

p=polyfit(t,y,6);

y1=polyval(p,t);

e=y-y1;

Much better!0 1 2 3 4 5 6 7 8 9 10

-2

0

2

4

6

8

10

Do not fit one high-order polynomial

globally but rather many low order

polynomials locally. WHY?

tt = 0:.25:10;

yy=spline(t,y,tt);

SPLINE Cubic spline data interpolation.

YY = SPLINE(X,Y,XX) uses cubic spline interpolation to find

YY, the values of the underlying function Y at the points in the

vector XX. The vector X specifies the points at which the data

Y is given. If Y is a matrix, then the data is taken to be vector-

valued and interpolation is performed for each column of Y and

YY will be length(XX)-by-size(Y,2).

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

p=polyfit(t,y,6) yy=spline(t,y,tt)

Error

Excellent fit!

Polynomial fit on noisy data

• Noisy data, repeat curve fitting

>> plot(tout,y)

[p,s]=polyfit(tout,y,6);

y1=polyval(p,t);

plot(tout,y,tout,y1,tout,y-y1)

Data

Polyno-

mial fit

polyfit(tout,y,6)

Common mistake: Model fit very good

but model follows noise.

Use prefiltering and lower order

models.

Look-up table in simulation

Look-up table can be used in interpolation of dataSimulate the system

with different initial conditions, when is given in the previous example.

Show the result in phase plane.

. ( )x x f x 0 6 y f x ( )

Previous noisy data was generated in MATLAB

using sampling interval 1.

x=[-10:1:10]; y=0.3+2*sin(x)+0.1*x.^2;

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

x

y

Graph

Look-up table in simulation

EXAMPLE 2.

Simulate the system

with different initial conditions,

when is given in the

previous example.

Show the result in phase plane.

. ( )x x f x 0 6

y f x ( )

Look-up

table

Look-up table simulation,

initial condition (1,1)States x1(t) and x2(t)

0 5 10 15 20-1.5

-1

-0.5

0

0.5

1

1.5

Time (s)

x1

x2

-1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

x1

x2

Phase plane (x1,x2)

-6 -4 -2 0 2 4 6-5

-4

-3

-2

-1

0

1

2

3

x1

x2

Phase plane plot with different initial conditions

Discontinuity in time

• State equation

0

( ( ), ( ), ),

( )

t t t

t

0

x f x u

x x

u(t)

ttk tdisctk+1

tdisc-tk

tdisc known

• Discontinuity in time (or in

known input)

• Example: relay in the system

0

( ( ), ( ), ),

( )

t t t

t

0

x f x u

x x

60

Discontinuity in time –

not a big problem

Dynamics of a satellite in space , one-degree of freedom

Input u(t) assumes values -1, 1 at relay output.

u

Discontinuity in state can generate problems

• State equation

0

( ( ), ( ), ),

( )

t t t

t

0

x f x u

x x

u(t)

ttk tdisctk+1

tdisc-tk

tdisc not known

dx/dt=f1(x,u,t)

Old dynamics

New dynamics

dx/dt=f2(x,u,t)

Discontinuity in state -Backlash

62

Automatic step size

Choose e.g. smaller stepsize

Algebraic loop

2 3 1 3

3 2 1

1 2 34, 2, 0.5

x u k k x k k x

u k k x k x

k k k

Same variable on both sides of equation

• E.g. derivative of state – not a proper state equation

x f (x , x, u, t ), x(t0 ) x 0

Algebraic loop

2 3 1 3

2 3 1 3

1 3

2 3 2 3

1

1

1 1

x u k k x k k x

k k x u k k x

k kx x u

k k k k

Fixed

Find the minimum of J.

Static optimizationCost function

1 2( ) ( , , , )nJ J x x xx Global maximum

1

min ( ),

[ , , ]

nR

n

J

x x

xx

x

Global minimum

Example[x,y,z]=peaks(40);meshc(x,y,z)

Several local max and min

Numerical optimization methods -

unconstrained

• ’Gradient’ methods

– Gradient

– Conjugate gradient

– Davidon-Fletcher-Powell

– Broyden-Fletcher-Goldfarb-Shanno

– Newton or Gauss-Newton

– Marquardt-Levenberg

Need gradientinformation,

Hessian approximation!

X=fminunc(FUN,X0,OPTIONS)

Choice of quasi-Newton methods

FUN = Cost function; XO=Initial guess

Numerical optimization methods

• All ’gradient’ methods can be written in the form

1i i i i x x s

step size

search direction

Numerical optimization methods –

unconstrained direct search

• Direct search methods– Hooke-Jeeves

– Rosenbrock

– Simplex (Nelder-Mead)

Need only cost function values!

• X =

fminsearch(FUN,X0,OPTIONS)

(Nelder-Mead)

• FUN = Cost function;

• XO=Initial guess

Uses a polytope of N+1

vertices in N dimensions

Computer controlled traffic –

streets, (intelligent) highways

MATLAB Optimization

Test exampleUNCONSTRAINED: Gradient method - Steepest Descent

DOES NOT CONVERGE! THERE IS NOT ENOUGH CHANGE IN THE

GRADIENT IN THE BOTTOM OF THE VALLEY!

REMARK: In feedforward

neural networks:

Backpropagation algorithm

is a gradient algorithm –

Is not a very efficient

method.

MATLAB Optimization

Test example[x,fval,exitflag,output] = fminunc({f,GRAD},x,OPTIONS);

• Broyden-Fletcher-Goldfarb-Shanno quasi-Newton method

Value of the function at the solution: 8.98565e-009

Number of function evaluations: 105

MATLAB Optimization

Test exampleChoose any of the following methods to minimize the banana function

UNCONSTRAINED:

4) Simplex Search (NELDER-MEAD) – uses only function

values

Dynamic systems and optimization

– Combining SIMULINK simulation and optimization

– PID controller optimization

Principle of feedback

Temperature control

Block diagram

Temperature control of sauna

PI(D)Resistor

wires

Sauna

temperature

SensorFeedbackMeasured

output

Error

e(t)Set point Output

+

Control Actuator Sauna

PID control

• ”Textbook” version of PID

• Control variable u is a sum of:

– P-term (proportional to error e) - present

– I-term (proportional to integral of error e) - history

– D-term (proportional to derivatice of error e) – future

• Cost function

– Many cost functions can be used to optimize PID control

parameters.

( )( ) ( ) ( )

t

p i d

o

de tu t K e t K e d K

dt

0

=

, , ; , ,p i d p i d

ITAE Integral Time Absolute Error

J K K K t e t K K K dt

yref

e uy

2

Out2

1

Out1

Transport

Delay

5s+1

0.4

Transfer FcnStep

Product

1

s

Integrator1

1

s

Integrator

Ki

Gain1

Kp

Gain

Clock

|u|

Abs

MATLAB realization – System and cost

function configuration in SIMULINK

Cost function

realization

0

( , ) ; ,p i p iJ K K t e t K K dt

Replace with known, large enough T

• fminsearch (N-M)

• fminunc (BFGS)

MATLAB realization – Optimized step

responses

• fminsearch (N-M) the cost function has reached steady-state faster than

• fminunc (BFGS), which still searches for the optimal values.

0 5 10 15 20 25 30 35 40 45 500

0.2

0.4

0.6

0.8

1

1.2

1.4Step responses

fminsearch

fminunc

0 20 40 60 80 100 120 1400

20

40

60

80

100

120

140Cost function values

fminsearch

fminuncfminsearch

fminunc

Further comments

• Toolboxes have been used in my courses on

– Automatic Control: Analog and Digital control

– System Identification

– Neuro-Fuzzy Systems

– Wireless Automation

• PiccSim simulator has been developed for

(wireless) networked control systems

– MATLAB/Simulink and ns-2.

– PiccSIM stands for Platform for integrated

communications and control design, simulation,

implementation and modeling.

PiccSim

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