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METTI5 Spring School Tutorial 7 June 2011 METTI 5 Spring School T7 S. Malan & C. Greco 1 METTI 5 Spring School T7 S. Malan & C. Greco 2 Outline TUTORIAL 7 - REAL DATA Introduction IDENTIFICATION OF AN Plant

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METTI5 Spring School – Tutorial 7 June 2011

  • S. Malan & C. Greco - Politecnico di Torino

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TUTORIAL 7 - REAL DATA IDENTIFICATION OF AN ACTUAL RADIATOR-ROOM SYSTEM AIMED TO VIRTUAL SENSOR DESIGN

Stefano Malan & Cosimo Greco

  • Dip. di Automatica e Informatica

Politecnico di Torino

METTI5 Spring School – T7 – S. Malan & C. Greco

1

Outline

  • Introduction
  • Plant description
  • Models choice: SSV models; I/O models
  • Experimental setup and data
  • Estimation method: Operational aspects; Software tools
  • Identification and validation results
  • Virtual sensors: State space solution; Input output solution
  • Matlab scripts

METTI5 Spring School – T7 – S. Malan & C. Greco

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INTRODUCTION

METTI5 Spring School – T7 – S. Malan & C. Greco

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Introduction

Many reasons drives to energy management and saving in buildings

  • Reduce pollution (respect protocols target)
  • Reduce operating costs
  • Maximize user comfort
  • Fairly rearrange cost distribution on actual

consumption: right accounting

  • Give a sense of responsibility to final user
  • Create and maximize final user’s energy awareness
  • ...

METTI5 Spring School – T7 – S. Malan & C. Greco

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Introduction

Reduce monitoring and right accounting costs and complexity:

  • use virtual or soft sensors, based on a identified

model, to substitute actual sensors First step to design a Virtual Sensor:

  • identify a suitable mathematical model of the

system able to provide, as an output, the signal to be measured: open loop VS Second step to design a Virtual Sensor:

  • use Control Theory to design a closed loop VS

METTI5 Spring School – T7 – S. Malan & C. Greco

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PLANT DESCRIPTION

METTI5 Spring School – T7 – S. Malan & C. Greco

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METTI5 Spring School – Tutorial 7 June 2011

  • S. Malan & C. Greco - Politecnico di Torino

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Radiatore

CORRIDOR OTHER ROOMS EXTERNAL

N

ROOM UNDER STUDY RADIATOR WINDOW

Plant description

  • University office room, 12 square meters
  • Heating system: radiator connected to central heating unit
  • Input water temperature: set by the central heating unit
  • Water flow: can be regulated by means of a valve

METTI5 Spring School – T7 – S. Malan & C. Greco

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Plant description

Research aim:

  • set up a methodology for micro-accounting heating

energy consumption

  • reduce the need of physical sensors → virtual sensor
  • 3 measurements needed to compute Heating Power:

1.

water flow 𝑅

2.

water input temperature 𝑈

𝑛

3.

water output temperature 𝑈

𝑠

  • tutorial aim:

identify a model of the overall radiator-room system to suitably design a temperature 𝑈

𝑠 virtual sensor

METTI5 Spring School – T7 – S. Malan & C. Greco

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𝑄ℎ = 𝑅𝜍𝑑 𝑈

𝑛 − 𝑈 𝑠

Plant description

Inputs:

  • radiator input water

temperature 𝑈

𝑛

  • heating water flow 𝑅
  • external environment

temperature 𝑈

𝑓

Outputs:

  • radiator output water

temperature 𝑈

𝑠

  • room temperature 𝑈

𝑏

METTI5 Spring School – T7 – S. Malan & C. Greco

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Plant

(CT) Tm(t) Q(t) Te(t) Tr(t) Ta(t)

Overall continuous time (CT) system definition

Plant description

Further inputs:

  • surrounding rooms temperatures → negligible effects

State variable model:

  • physical states: homogeneous bodies temperatures
  • state number: model order
  • spatial discretization: great number of states
  • equations: heat exchanges equilibriums, physical

parameters  Simple models with few constraints Model order as a “black-box” identification parameter

METTI5 Spring School – T7 – S. Malan & C. Greco

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MODELS CHOICE

SSV models I/O models

METTI5 Spring School – T7 – S. Malan & C. Greco

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Models choice

The continuous time (CT) dynamic system is represented by a discrete time (DT) model due to inputs and outputs sampling given by the technological configuration ADC: number of bits 𝑂 sampling interval 𝑈

𝑡

sampling instant 𝑢𝑗 = 𝑗𝑈

𝑡 , with 𝑗 the DT variable

METTI5 Spring School – T7 – S. Malan & C. Greco

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Te(i) Tr(i) Ta(i) PLANT

(CT)

Tm(t) Q(t) Te(t) Tr(t) Ta(t)

ADC ADC

Tm(i) Q(i)

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METTI5 Spring School – Tutorial 7 June 2011

  • S. Malan & C. Greco - Politecnico di Torino

3 Models choice

DT dynamic model representing the CT dynamic plant Model characteristic:

  • linear
  • lumped parameters

Model classes:

  • SSV : state space variable
  • I/O : input-output

METTI5 Spring School – T7 – S. Malan & C. Greco

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Model (DT)

Tm(i) Q(i) Te(i) Tr(i) Ta(i)

Models choice: SSV models

Time domain model Z transform domain model In stationary conditions, for an asymptotically stable system, the initial condition is no more taken into account

METTI5 Spring School – T7 – S. Malan & C. Greco

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          ) ( condition initial the given ) ( ) ( ) ( ) ( ) ( ) 1 ( x i Du i Cx i y i Bu i Ax i x

        ) ( ) ( ) ( ) ( ) ( ) ( ) ( z Du z Cx z y zx z Bu z Ax z zx

Models choice: I/O models

Z transform domain model The elements of 𝐻 𝑨 are transfer functions all having the same denominator 𝐸𝑑 𝑨 of degree 𝑜 The elements of 𝑧0 𝑨 are rational proper functions all having the same denominator 𝐸𝑑 𝑨 of degree 𝑜 In stationary conditions, for an asymptotically stable system, the initial condition is no more taken into account

METTI5 Spring School – T7 – S. Malan & C. Greco

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) ( ) ( ) ( ) (

0 z

y z u z G z y   ) ( ) ( ) ( z u z G z y 

Models choice: I/O models

Z transform domain model explicit form Consider a single input – single output model

METTI5 Spring School – T7 – S. Malan & C. Greco

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                                                          ) ( ) ( ) ( ) ( 1 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 ) ( ) ( ) (

02 01 2 1 , 2 , 1 , , 2 2 , 2 1 , 2 , 1 2 , 1 1 , 1 2 1

z N z N z N z D z u z u z u z N z N z N z N z N z N z N z N z N z D z y z y z y

ny c nu nu ny ny ny nu nu c ny

         

) ( ) ( ) ( ) ( ) ( ) ( ) (

1 1 1 , 1 , 1 1 , ,

z u a z a z a z b z b z b z b z u z D z N z u z G z y

k n n n jk jk n n jk n n jk k c jk k jk j

           

   

 

Models choice: I/O models

The derived difference equation is The model for a multi input – single output therefore is

METTI5 Spring School – T7 – S. Malan & C. Greco

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) ( ) 1 ( ) ( ) ( ) 2 ( ) 1 ( ) (

, 1 , , 2 1

n i u b i u b i u b n i y a i y a i y a i y

k jk k n jk k n jk j j n j n j

             

  

 

) ( ) 1 ( ) ( ) ( ) 1 ( ) ( ) ( ) 1 ( ) ( ) ( ) 1 ( ) ( ) ( ) 2 ( ) 1 ( ) (

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

n i u b i u b i u b n i u b i u b i u b n i u b i u b i u b n i u b i u b i u b n i y a i y a i y a i y

u u u u u u

n jn n n jn n n jn j n j n j j n j n j j n j n j j j n j n j

                                  

     

     

EXPERIMENTAL SETUP AND DATA

METTI5 Spring School – T7 – S. Malan & C. Greco

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METTI5 Spring School – Tutorial 7 June 2011

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4 Experimental setup and data

METTI5 Spring School – T7 – S. Malan & C. Greco

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Input temperature Analog to Digital Converter Output temperature Flow meter Room temp.

Experimental setup and data

METTI5 Spring School – T7 – S. Malan & C. Greco

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  • Temperature sensor National Semiconductor LM35
  • scale 10 mV/°C, range 40 °C ÷ 110 °C, accuracy ± 0.2 °C
  • heating water input and output copper pipes (error < 0.07 °C)
  • room
  • external
  • Water flow transducer Dynasonic TFX Ultra
  • ultrasonic, not invasive
  • output current 4 ÷ 20 mA → water velocity 0 ÷ 𝑊

𝑥,𝑛𝑏𝑦

  • corresponding acquired voltage given by a 100 W resistor
  • water flow 𝑅 obtained multiplying by 49 mm2 pipe section
  • Acquisition board National Instruments USB-6211
  • range ± 10 V, conversion accuracy 16 bit, LabView interface

Experimental setup and data

METTI5 Spring School – T7 – S. Malan & C. Greco

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  • Acquisition period: lasting a few days, in the last decade
  • f March, in Torino, Piemonte, north west of Italy
  • Command: water flow regulated

by a manual valve

  • Disturbances:
  • heating water input temperature

(centrally imposed)

  • external environment temperature,
  • pening and closing windows
  • Sampling interval: 1 or 5 seconds,

subsequently decimated to 15 seconds

Experimental setup and data

METTI5 Spring School – T7 – S. Malan & C. Greco

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2000 4000 6000 8000 10000 12000 14000 16000 18000 10 20 30 40 50 Inputs: Tm, Q, Te Samples (Ts=15 s) Inputs (°C - m3/s) 2000 4000 6000 8000 10000 12000 14000 16000 18000 20 25 30 35 40 45 Outputs: Ta, Tr Samples (Ts=15 s) Outputs (°C) Tr Ta Tm Te Q * 105 Tm Q * 105 Te Ta Tr

About three days

Experimental setup and data

METTI5 Spring School – T7 – S. Malan & C. Greco

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1000 2000 3000 4000 5000 6000 7000

  • 10

10 20 30 40 50 Inputs: Tm, Q, Te Samples (Ts=15 s) Inputs (°C - m3/s) Tm Q * 105 Te 1000 2000 3000 4000 5000 6000 7000 15 20 25 30 35 40 45 Outputs: Ta, Tr Samples (Ts=15 s) Outputs (°C) Tm Te Q * 105 Ta Tr Ta Tr

About one day Window opening

ESTIMATION METHOD

Operational aspects Software tools

METTI5 Spring School – T7 – S. Malan & C. Greco

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METTI5 Spring School – Tutorial 7 June 2011

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5 Estimation method

METTI5 Spring School – T7 – S. Malan & C. Greco

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SISO system, time domain in matrix form previous input and output samples 𝑛 ≐ 2𝑜 parameters to be identified ) ( ) 2 ( ) 1 ( ) ( ) 2 ( ) 1 ( ) (

2 1 2 1

n i u b i u b i u b n i y a i y a i y a i y

n n n n

              

   

 

 

1

) (

 

i

i y

 

) ( ) 2 ( ) 1 ( ) ( ) 2 ( ) 1 (

1

n i u i u i u n i y i y i y

i

           

 

2 1 2 1

b b b a a a

n n n n

  

   

  

Estimation method

METTI5 Spring School – T7 – S. Malan & C. Greco

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Hp: time invariant parameters Rewrite 𝑧(𝑗) = 𝜒𝑗−1

′

𝜄 for 𝑧 𝑗 − 1 , … , 𝑧 𝑗 − 𝑂 + 1 and obtain 𝑂 equations in 𝑛 unknown Note: the above equation error term was skipped

           

 

i i

y M i i N i

i y i y N i y

    

                               

) ( ) 1 ( ) 1 (

1

1 2

   

i i

y M 

  1

Estimation method

METTI5 Spring School – T7 – S. Malan & C. Greco

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If 𝑂 ≫ 𝑛 solve as a least square optimization the solution 𝜄 exists if this implies particular constraints on the experimental conditions under which the I/O samples were acquired

 

 

i i i i i i

y M M M M y

1 1 1 1 2 1

ˆ min

    

       

 

m Mi 

1

rank

Estimation method: operational aspects

METTI5 Spring School – T7 – S. Malan & C. Greco

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  • a. MIMO systems: same equation 𝑁𝑗−1𝜄 = 𝑧𝑗 but different

construction of matrices; for example for a second order, 2 inputs, 2 outputs system

                                                  

) 2 ( ) 1 ( ) 2 ( ) 1 ( ) 2 ( ) 1 ( ) 1 ( ) ( ) 1 ( ) ( ) 1 ( ) ( ) 2 ( ) 1 ( ) 2 ( ) 1 ( ) 2 ( ) 1 ( ) 1 ( ) ( ) 1 ( ) ( ) 1 ( ) (

2 2 1 1 2 2 2 2 1 1 2 2 2 2 1 1 1 1 2 2 1 1 1 1 1

i u i u i u i u i y i y N i u N i u N i u N i u N i y N i y i u i u i u i u i y i y N i u N i u N i u N i u N i y N i y M i  

                         ) ( ) 1 ( ) ( ) 1 (

2 2 1 1

i y N i y i y N i y yi  

 

, 22 1 , 22 , 21 1 , 21 , 12 1 , 12 , 11 1 , 11 1

b b b b b b b b a a 

Estimation method: operational aspects

METTI5 Spring School – T7 – S. Malan & C. Greco

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  • b. SSV models: it is possible to set up the least-squares

estimation problem or use Matlab routines

  • c. Offset in the I/O variables: the system dynamics usually

develop in the neighbourhood of a constant value, then

  • 1. detrend inputs and outputs; not always possible
  • 2. use an affine model and estimate the offset 𝑍

I/O model, SISO case Y n i u b i u b i u b n i y a i y a i y a i y

n n n n

               

   

) ( ) 2 ( ) 1 ( ) ( ) 2 ( ) 1 ( ) (

2 1 2 1

 

Estimation method: operational aspects

METTI5 Spring School – T7 – S. Malan & C. Greco

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  • c. Offset in the I/O variables

Same equation 𝑧(𝑗) = 𝜒𝑗−1

′

𝜄 but different construction of vectors a constant value input equal to 1 surely makes the identification problem numerically ill conditioned it is necessary to “perturb” it with some random noise

 

1 ) ( ) 2 ( ) 1 ( ) ( ) 2 ( ) 1 (

1

n i u i u i u n i y i y i y

i

          

 

Y b b b a a a

n n n n 2 1 2 1

 

   

  

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METTI5 Spring School – Tutorial 7 June 2011

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Estimation method: operational aspects

METTI5 Spring School – T7 – S. Malan & C. Greco

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  • c. Offset in the I/O variables

SSV models 𝑌 and 𝑍 are constant to be identified in the “augmented” system           Y i Du i Cx i y X i Bu i Ax i x ) ( ) ( ) ( ) ( ) ( ) 1 (

   

                        1 ) ( ) ( ) ( 1 ) ( ) ( ) 1 ( i u Y D i Cx i y i u X B i Ax i x

Estimation method: operational aspects

METTI5 Spring School – T7 – S. Malan & C. Greco

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  • d. Acquired data absolute values / numerical ill conditioning

Numerical values given as input to the identification algorithm must be about of the same magnitude to prevent numerical ill conditioning, to satisfy algorithm requirements Often these are not satisfied not because of a requirement lack structurally related to the considered problem, but only because of a numerical ill conditioning that can be

  • vercome by a suitable numerical values scaling

For example the values of the water flow 𝑅 is multiplied by a factor 105

Estimation method: software tools

METTI5 Spring School – T7 – S. Malan & C. Greco

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  • Matlab environment plus the Identification Toolbox
  • Scilab environment
  • C language

IDENTIFICATION AND VALIDATION RESULTS

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Identification results

METTI5 Spring School – T7 – S. Malan & C. Greco

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  • Two data sets: first for identification, second for validation
  • Four typologies of models
  • IOd I/O models, without offset, identified from detrendized data
  • IOo I/O models, with offset, identified from not detrendized data
  • SSd SSV models, without offset, identified from detrendized data
  • SSo SSV models, with offset identified, from not detrendized data
  • Other combinations are possible
  • Model order identified by trial and error
  • low order
  • second order suitable to describe the system
  • same number of zeroes

Identification results

METTI5 Spring School – T7 – S. Malan & C. Greco

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  • Temperature 𝑈

𝑠

2000 4000 6000 8000 10000 12000 14000 16000 18000 20 25 30 35 40 45 Identification results Samples (Ts=15 s) Radiator output temperature (°C) Tr Tr,IOo Tr,IOd Tr,SSo Tr,SSd Measured Identified

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METTI5 Spring School – Tutorial 7 June 2011

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7 Identification results

METTI5 Spring School – T7 – S. Malan & C. Greco

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  • Temperature 𝑈

𝑏

2000 4000 6000 8000 10000 12000 14000 16000 18000 20 20.5 21 21.5 22 22.5 23 23.5 24 24.5 Identification results Samples (Ts=15 s) Room temperature (°C) Ta Ta,IOo Ta,IOd Ta,SSo Ta,SSd Measured Identified I/O models Identified SSV models

Identification results

METTI5 Spring School – T7 – S. Malan & C. Greco

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  • Numerical evaluation: standard deviation

     

l i y i y

l i s

 

1 2

Model IOo IOd SSo SSd 𝜏𝑠 for 𝑈

𝑠

0.7308 0.7324 0.6644 0.6667 𝜏a for 𝑈

𝑏

0.5341 0.5341 0.3035 0.3016

Identification results

METTI5 Spring School – T7 – S. Malan & C. Greco

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  • Transfer matrix

                     

                        z T z Q z T z z z z z z z T z T

e m a r 5 23 22 21 13 12 11

10 G G G G G G

                                  

                             z T z Q z T z D z z D z z D z z D z z D z z D z z T z T

e m C C C C C C a r 5 23 22 21 13 12 11

10 N N N N N N

Identification results

METTI5 Spring School – T7 – S. Malan & C. Greco

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  • IOo model
  • IOd model

             

) . ) (z+ . (z z D ) . ) (z . (z . z N ); . ) (z . (z . z N ) . ) (z+ . (z . z N ); . z + . + (z . z N ) . ) (z . (z . z N ); . ) (z+ . (z . z N

C

126 9845 2814 125 1 012715 3412 8036 013828 6046 793 1 0024779 162 1 566 1 0027284 8742 2 103 00018483 5303 729 1 010176

23 13 22 2 12 21 11

                   

             

) . ) (z+ . (z z D ) . ) (z . (z . z N ); . ) (z . (z . z N ) . ) (z+ . (z . z N ); . z + . + (z . z N ) . ) (z . (z . z N ); . ) (z+ . (z . z N

C

1259 9845 2815 125 1 012715 3396 8043 013841 6044 793 1 002478 173 1 558 1 0027247 8742 7 103 00018402 5253 745 1 0099935

23 13 22 2 12 21 11

                   

Identification results

METTI5 Spring School – T7 – S. Malan & C. Greco

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  • SSo model
  • SSd model

             

) . ) (z . (z z D ) . ) (z . (z . z N ); . ) (z . (z . z N ) . ) (z . (z . z N ); . ) (z . (z+ . z N ) . ) (z . (z . z N ); . ) (z . (z . z N

C

9798 9986 9759 011 1 0092313 8513 9992 015802 9221 033 1 0020483 9989 341 2 0034834 9521 9942 017656 9985 411 1 030646

23 13 22 12 21 11

                      

             

) . ) (z . (z z D ) . ) (z . (z . z N ); . ) (z . (z . z N ) . ) (z . (z . z N ); . ) (z . (z+ . z N ) . ) (z . (z . z N ); . ) (z . (z . z N

C

9798 9986 9759 011 1 0092488 8516 9992 015828 922 033 1 0020333 9989 366 2 0034558 9522 9942 017674 9985 414 1 030404

23 13 22 12 21 11

                      

Validation results

METTI5 Spring School – T7 – S. Malan & C. Greco

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  • Temperature 𝑈

𝑠

1000 2000 3000 4000 5000 6000 7000 15 20 25 30 35 40 45 50 Validation results Samples (Ts=15 s) Radiator output temperature (°C) Tr Tr,IOo Tr,IOd Tr,SSo Tr,SSd Identified I/O models Identified SSV models Measured

slide-8
SLIDE 8

METTI5 Spring School – Tutorial 7 June 2011

  • S. Malan & C. Greco - Politecnico di Torino

8 Validation results

METTI5 Spring School – T7 – S. Malan & C. Greco

43

  • Temperature 𝑈

𝑏

1000 2000 3000 4000 5000 6000 7000 18 19 20 21 22 23 24 25 26 Validation results Samples (Ts=15 s) Room temperature (°C) Ta Ta,IOo Ta,IOd Ta,SSo Ta,SSd Measured Identified IOo model Identified IOd model Identified SSo model Identified SSd model

VIRTUAL SENSORS

State space solution Input output solution

METTI5 Spring School – T7 – S. Malan & C. Greco

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Virtual sensors

METTI5 Spring School – T7 – S. Malan & C. Greco

45

  • Virtual Sensors are mathematical models able to give, in

real time, the numerical value of a physical signal, that is not directly measured by an actual sensor

  • Any previous model: open loop VS
  • Closed loop VS to correct differences between the model

and the actual system, to make the virtual measure convergent to the unknown real one

  • The correction is obtained by suitably feedback the

comparison of the simulated and measured values of the second output 𝑈

𝑏

Virtual sensors

METTI5 Spring School – T7 – S. Malan & C. Greco

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Generalized dual-input/dual-output (DIDO) model

  • Two inputs 𝑣1 and 𝑣2
  • One plant output 𝑧𝑡1 to be estimated
  • One plant output 𝑧𝑡2 measured
  • Two model outputs 𝑧1 and 𝑧2

Working assumptions

  • 𝑣1, 𝑣2 and 𝑧𝑡2 are measurable
  • 𝑧𝑡1 is not measurable, it is to be reconstructed or

estimated

Virtual sensors: state space solution

METTI5 Spring School – T7 – S. Malan & C. Greco

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VS as classical Luenberger observer gain 𝑀 are selected to obtain a closed loop asymptotically stable system with suitable time constants Estimation error gain 𝑀 are chosen in order to ensure 𝑧 2 → 𝑧𝑡2 and to guarantee also 𝑧 1 → 𝑧𝑡1

     

                 ) ( ˆ ) ( ˆ ˆ ) ( ) ( ) ( ˆ ) 1 ( ˆ

1 1 2 2 2 1

i x C i y i x C i y L i u i u B i x A i x

s

x C y y y e

s s

ˆ ˆ

2 2 2 2 2

    

Virtual sensors: state space solution

METTI5 Spring School – T7 – S. Malan & C. Greco

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Closed loop VS gain 𝑀 are chosen in order to ensure that the closed loop matrix 𝐵 − 𝑀𝐷2 is asymptotically stable Note: the measured plant output 𝑧𝑡2 becomes one of the Virtual Sensor inputs

     

                      ) ( ˆ ) ( ˆ ) ( ) ( ) ( ˆ ) 1 ( ˆ

1 1 2 2 1 2

i x C i y i y i u i u L B i x LC A i x

s

slide-9
SLIDE 9

METTI5 Spring School – Tutorial 7 June 2011

  • S. Malan & C. Greco - Politecnico di Torino

9 Virtual sensors: input output solution

METTI5 Spring School – T7 – S. Malan & C. Greco

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Input output model 𝐻(𝑨) identified from acquired data                      

22 21 12 11 2 1 2 1

where ) ( ) ( ) ( G G G G G u u U y y Y z U z G z Y

22

G

u1 u2 y1 y2

+ + + + 11

G

12

G

21

G

y2 y1 u1 u2 u1

+ +

  • +

12 22

N N

11

G

12

G D N N

12 x

d1

Virtual sensors: input output solution

METTI5 Spring School – T7 – S. Malan & C. Greco

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Input output model 𝐻(𝑨) tf’s Gij have same denominator, different numerators it can be computed where D N N N N N D G j i D N G

G ij ij

           

22 21 12 11

1 and ,

1 12 1 12 22 2

u D N N y N N y

x

 

 

2 12 1 11 1

det u G u G y N N

G x

  

Virtual sensors: input output solution

METTI5 Spring School – T7 – S. Malan & C. Greco

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Closed loop VS: filter 𝑂𝐺 𝐸𝐺 and gain 𝐿 are chosen in

  • rder to ensure 𝑧

2 → 𝑧𝑡2 and to guarantee also 𝑧 1 → 𝑧𝑡1

ys2 d1 ŷ1 ŷ2 y1 u1 u2 u1

+ +

  • +

12 22

N N

12 22

N N 12

G 11

G

eF2 ê2

+ + +

  • K

𝑂𝐺 𝐸𝐺

MATLAB SCRIPTS

METTI5 Spring School – T7 – S. Malan & C. Greco

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Matlab scripts

Identification and Validation

  • IdE_ordN_zerM_inpP_outQ

Main

  • ARXdAH

I/O Identification, detrendized data

  • ARXoAH

I/O Identification, affine model

  • ICC

Initial Conditions Computation

  • MaFiMoRe

Make Figure More Readable

  • n4sid (Identification Toolbox)

SSV Identification

Virtual Sensor Design (state space solution)

  • ViSeDe_L

linear model

  • ViSeDe_LY

affine model

Note: if interested in getting the scripts, contact the authors

METTI5 Spring School – T7 – S. Malan & C. Greco

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THANK YOU FOR YOUR KIND ATTENTION

Questions, please …

METTI5 Spring School – T7 – S. Malan & C. Greco

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