Closed-loop identification of MIMO systems: a new look at identifiability and experiment design

Bazanella, Alexandre Sanfelice; Gevers, Michel; Mišković, Ljubiša

doi:10.23919/ecc.2007.7068792

Cited by 18 publications

(50 citation statements)

References 18 publications

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“…2.2. Because the models in

{scriptL}^{MathClass-bin*}

could have any LTI structures ,

{bold-italicW}_{bold-italicy}^{i} ((), z^{MathClass-bin- 1})

and

{bold-italicW}_{bold-italicu}^{i} ((), z^{MathClass-bin- 1})

are generally defined in impulse response forms as in the succeeding text:

{bold-italicW}_{bold-italicy}^{i} ((), z^{MathClass-bin- 1}) MathClass-rel= {falsefalse}_{MathClass-op\sum}^{j MathClass-rel= 1} {bold-italicC}_{j}^{bold-italicy MathClass-punc, i} MathClass-bin\cdot z^{MathClass-bin- j} MathClass-punc, {bold-italicW}_{bold-italicu}^{i} ((), z^{MathClass-bin- 1}) MathClass-rel= {falsefalse}_{MathClass-op\sum}^{j MathClass-rel= 1} {bold-italicC}_{j}^{bold-italicu MathClass-punc, i} MathClass-bin\cdot z^{MathClass-bin- j}

where

{C_{j}^{y, i} \in ℛ m \times m}

and

{C_{j}^{u, i} \in ℛ m \times n}

are Markov parameters of

{bold-italicW}_{bold-italicy}^{i} ((), z^{MathClass-bin- 1})

and

{bold-italicW}_{bold-italicu}^{i} ((), z^{MathClass-bin- 1})

, respectively. Definition A quasi‐stationary data set Z ∞ = { y 1 , u 1 , ⋯ , y k , u k , ⋯ } is informative if for any two pairs of models

([], bold-italicW)

…”

Section: Preliminaries and Mimo Closed‐loop System With A Switching Cmentioning

confidence: 99%

“…For the sake of generality, we do not require the model structures of G ( z − 1 ) and H ( z − 1 ) in our paper, and therefore, they are defined as the following general impulses response models

bold-italicG ((), z^{MathClass-bin- 1}) MathClass-rel= {falsefalse}_{MathClass-op\sum}^{l MathClass-rel= 1} {bold-italicG}_{l} MathClass-bin\cdot z^{MathClass-bin- l} MathClass-punc, bold H ((), z^{MathClass-bin- 1}) MathClass-rel= {falsefalse}_{MathClass-op\sum}^{i MathClass-rel= 0} {bold H}_{i} MathClass-bin\cdot z^{MathClass-bin- i}

where for

l \in {scriptZ}^{+}, G_{l} \in ℛ^{m \times n} \in ℛ^{m \times m}

, and for

i \in Z, i \geq 0, H_{i} \in ℛ^{m \times m}

. Moreover, H ( z − 1 ) is a stable, inversely stable, and monic filter (i.e., H 0 = I ) .…”

Section: Preliminaries and Mimo Closed‐loop System With A Switching Cmentioning

confidence: 99%

“…2.2. Because the models in L could have any LTI structures [4,5], W i y ´ 1 and W i u ´ 1 are generally defined in impulse response forms as in the succeeding text:…”

Section: Notationmentioning

confidence: 99%

“…where for l 2 Z C ; G l 2 R m n , and for i 2 Z; i > 0; H i 2 R m m . Moreover, H ´ 1 is a stable, inversely stable, and monic filter (i.e., H 0 D I/ [4,5]. According to [15][16][17], the switching controller is described in the state-space form ²…”

Section: Mimo Closed-loop System With a Switching Controllermentioning

confidence: 99%

“…The switching controller's impulse response model can be derived from its state-space model in (5) as the following convolution of an LTV filter and ¹y k º…”

Section: Conditions For the Switching Controller To Ensure The Informmentioning

confidence: 99%

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Informative conditions for a data set in a MIMO closed‐loop system with a switching controller

Zhang

Yang

2014

Intl J Robust & Nonlinear

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SummaryIn system identification, a data set needs to be informativeto ensure that the identification criterion has a unique global minimum asymptotically, and the parameter estimation is consistent. To guarantee a data set in a MIMO closed‐loop system with a switching controller to be informative, we develop a group of flexible conditions for the controller's switching rule and feedback laws, which provide a trade‐off between these two aspects through adjustable parameters. Compared with the results in the existing literature, our result includes all kinds of switching rules and therefore is not limited by the constraints on the switching rule.Copyright © 2014 John Wiley & Sons, Ltd.

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“…2.2. Because the models in

{scriptL}^{MathClass-bin*}

could have any LTI structures ,

{bold-italicW}_{bold-italicy}^{i} ((), z^{MathClass-bin- 1})

and

{bold-italicW}_{bold-italicu}^{i} ((), z^{MathClass-bin- 1})

are generally defined in impulse response forms as in the succeeding text:

{bold-italicW}_{bold-italicy}^{i} ((), z^{MathClass-bin- 1}) MathClass-rel= {falsefalse}_{MathClass-op\sum}^{j MathClass-rel= 1} {bold-italicC}_{j}^{bold-italicy MathClass-punc, i} MathClass-bin\cdot z^{MathClass-bin- j} MathClass-punc, {bold-italicW}_{bold-italicu}^{i} ((), z^{MathClass-bin- 1}) MathClass-rel= {falsefalse}_{MathClass-op\sum}^{j MathClass-rel= 1} {bold-italicC}_{j}^{bold-italicu MathClass-punc, i} MathClass-bin\cdot z^{MathClass-bin- j}

where

{C_{j}^{y, i} \in ℛ m \times m}

and

{C_{j}^{u, i} \in ℛ m \times n}

are Markov parameters of

{bold-italicW}_{bold-italicy}^{i} ((), z^{MathClass-bin- 1})

and

{bold-italicW}_{bold-italicu}^{i} ((), z^{MathClass-bin- 1})

, respectively. Definition A quasi‐stationary data set Z ∞ = { y 1 , u 1 , ⋯ , y k , u k , ⋯ } is informative if for any two pairs of models

([], bold-italicW)

…”

Section: Preliminaries and Mimo Closed‐loop System With A Switching Cmentioning

confidence: 99%

bold-italicG ((), z^{MathClass-bin- 1}) MathClass-rel= {falsefalse}_{MathClass-op\sum}^{l MathClass-rel= 1} {bold-italicG}_{l} MathClass-bin\cdot z^{MathClass-bin- l} MathClass-punc, bold H ((), z^{MathClass-bin- 1}) MathClass-rel= {falsefalse}_{MathClass-op\sum}^{i MathClass-rel= 0} {bold H}_{i} MathClass-bin\cdot z^{MathClass-bin- i}

where for

l \in {scriptZ}^{+}, G_{l} \in ℛ^{m \times n} \in ℛ^{m \times m}

, and for

i \in Z, i \geq 0, H_{i} \in ℛ^{m \times m}

. Moreover, H ( z − 1 ) is a stable, inversely stable, and monic filter (i.e., H 0 = I ) .…”

Section: Preliminaries and Mimo Closed‐loop System With A Switching Cmentioning

confidence: 99%

“…2.2. Because the models in L could have any LTI structures [4,5], W i y ´ 1 and W i u ´ 1 are generally defined in impulse response forms as in the succeeding text:…”

Section: Notationmentioning

confidence: 99%

Section: Mimo Closed-loop System With a Switching Controllermentioning

confidence: 99%

“…The switching controller's impulse response model can be derived from its state-space model in (5) as the following convolution of an LTV filter and ¹y k º…”

Section: Conditions For the Switching Controller To Ensure The Informmentioning

confidence: 99%

See 3 more Smart Citations

Informative conditions for a data set in a MIMO closed‐loop system with a switching controller

Zhang

Yang

2014

Intl J Robust & Nonlinear

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show abstract

Identifiability and Informative Experiments in Open and Closed-Loop Identification

Gevers

Bazanella

Mišković

Lecture Notes in Control and Information Sciences

Self Cite

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To Giorgio, who to our great surprise has not solved the rather fundamental problems raised in this chapter, we hope to offer some food for thought. * This chapter presents research results of the Belgian Programme on Interuniversity Attraction Poles, initiated by the Belgian Federal Science Policy Office. The first author is also partially supported by the Brazilian Ministry of Education through CAPES.

show abstract

Closed-loop identification of multivariable systems: With or without excitation of all references?

Mišković¹,

Karimi²,

Bonvin³

et al. 2008

Automatica

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Closed-loop identification of MIMO systems: a new look at identifiability and experiment design

Cited by 18 publications

References 18 publications

Informative conditions for a data set in a MIMO closed‐loop system with a switching controller

Informative conditions for a data set in a MIMO closed‐loop system with a switching controller

Identifiability and Informative Experiments in Open and Closed-Loop Identification

Closed-loop identification of multivariable systems: With or without excitation of all references?

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