Gradient Algorithm in Subspace Predictive Control

Wang Xiao-ping; Wang Jian-hong

doi:doi:10.11648/j.com.20180602.13

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Gradient Algorithm in Subspace Predictive Control

Wang Xiao-ping, Wang Jian-hong

Published in Communications (Volume 6, Issue 2)

Received: 5 February 2018 Accepted: 20 July 2018 Published: 22 August 2018

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Abstract

In this paper, subspace predictive control strategy is applied to design predictive controller. Given the state space model, the output estimations corresponding to the predictive output is derived to be one explicit function of the measured input-output data. Then using these output estimations, the problem of designing predictive controller is formulated as one optimization problem with equality and inequality conditions. In order to solve this constrain optimization problem, we use dual decomposition idea to change the original constrain optimization problem into an unconstrain optimization problem. So the classical gradient algorithm is put forth to solve the primal dual optimization problem. The problem of designing dual decomposition controller is studied for subspace predictive control strategy under fault condition. For state space equation with fault condition, we establish one function form between fault and residual using only input-output measured data sequence, and construct one least squares optimization problem to obtain fault estimation. The statistical property about residual is analyzed based on our derived output prediction, then the Kronecker product is used to derive the detailed structure corresponding to residual vector at every time instant. After substituting our output prediction into objective function of predictive control, one quadratic programming problem with equality and inequality constraints is considered. For solving this constrained optimization problem, fast gradient method is not suited for this complex optimization problem, as one regularization term is added in our objective function. So in order to solve this complex quadratic optimization problem, we propose a dual decomposition idea so that this dual decomposition idea can convert the former constrained optimization into unconstrained optimization, then one nearest neighbor gradient algorithm is given to solve its optimal value.

Published in	Communications (Volume 6, Issue 2)
DOI	10.11648/j.com.20180602.13
Page(s)	39-44
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2018. Published by Science Publishing Group

Keywords

Subspace Predictive Control, Dual Decomposition, Gradient Algorithm

References

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[2]	A. Chiuso. On the relation between CCA and predictor based subspace identification. IEEE Transactions of Automatic Control, 2008, 52(10): 1795-1811.
[3]	Wang Jianhong. Application of subspace predictive control in active noise and vibration control. Journal of Vibration and Shock, 2011, 30(10): 129-135.
[4]	Wang Jianhong. Application of ellipsoid optimization in subspace predictive control. Journal of Applied Science, 2010, 28(4): 424-429.
[5]	Wang Jianhong. Fast gradient algorithm in subspace predictive control under fault estimation. Journal of Shanghai Jiaotong University, 2013, 47(7): 1015-1021.
[6]	Ljung, L. System identification: Theory for the user: Prentice Hall. 1999.
[7]	Boyd S, L Vandenberghe. Convex optimization: UK: Cambridge University Press, 2008.
[8]	Melanie Zeilinger. Real time suboptimal model predictive control using a combination of explicit MPC and online optimization. IEEE Transactions of Automatic Control, 2011, 56(7): 1524-1534.
[9]	S. Riverso, M. Farina, and G. Ferrani Trecate, Plug and play model predictive control based on robust control invariant sets, Automatica, 2014, 50( 8): 2179-2186.
[10]	Laurain V, R Toth. An instrumental least squares support vector machine for nonlinear system identification. Automatica, 2015, 54(4): 340-347.
[11]	Carlo Novara, Fredy Ruiz. Direct filtering: a new approach to optimal filter design for nonlinear system. IEEE Transaction on Automatic Control, 2013, 58(1): 86-99.
[12]	Carlo Novara. Direct design of discrete time LPV feedback controllers. IEEE Transaction on Automatic Control, 2015, 60 (10): 2819-2824.
[13]	Simone Formentin, Dario Piga. Direct learning of LPV controllers from data. Automatica, 2016, 65(3): 98-110.
[14]	Simone Formentin, Alirza Karimi. Optimal input design for direct data driven tuning of model reference controllers. Automatica, 2013, 49(6): 1874-1882.
[15]	Wang Jian-hong. Virtual reference feedback tuning control design for closed loop system. Journal of Control and System Engineering, 2016, 4(1): 1-9.

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APA Style

Wang Xiao-ping, Wang Jian-hong. (2018). Gradient Algorithm in Subspace Predictive Control. Communications, 6(2), 39-44. https://doi.org/10.11648/j.com.20180602.13

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ACS Style

Wang Xiao-ping; Wang Jian-hong. Gradient Algorithm in Subspace Predictive Control. Communications. 2018, 6(2), 39-44. doi: 10.11648/j.com.20180602.13

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AMA Style

Wang Xiao-ping, Wang Jian-hong. Gradient Algorithm in Subspace Predictive Control. Communications. 2018;6(2):39-44. doi: 10.11648/j.com.20180602.13

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@article{10.11648/j.com.20180602.13,
  author = {Wang Xiao-ping and Wang Jian-hong},
  title = {Gradient Algorithm in Subspace Predictive Control},
  journal = {Communications},
  volume = {6},
  number = {2},
  pages = {39-44},
  doi = {10.11648/j.com.20180602.13},
  url = {https://doi.org/10.11648/j.com.20180602.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.com.20180602.13},
  abstract = {In this paper, subspace predictive control strategy is applied to design predictive controller. Given the state space model, the output estimations corresponding to the predictive output is derived to be one explicit function of the measured input-output data. Then using these output estimations, the problem of designing predictive controller is formulated as one optimization problem with equality and inequality conditions. In order to solve this constrain optimization problem, we use dual decomposition idea to change the original constrain optimization problem into an unconstrain optimization problem. So the classical gradient algorithm is put forth to solve the primal dual optimization problem. The problem of designing dual decomposition controller is studied for subspace predictive control strategy under fault condition. For state space equation with fault condition, we establish one function form between fault and residual using only input-output measured data sequence, and construct one least squares optimization problem to obtain fault estimation. The statistical property about residual is analyzed based on our derived output prediction, then the Kronecker product is used to derive the detailed structure corresponding to residual vector at every time instant. After substituting our output prediction into objective function of predictive control, one quadratic programming problem with equality and inequality constraints is considered. For solving this constrained optimization problem, fast gradient method is not suited for this complex optimization problem, as one regularization term is added in our objective function. So in order to solve this complex quadratic optimization problem, we propose a dual decomposition idea so that this dual decomposition idea can convert the former constrained optimization into unconstrained optimization, then one nearest neighbor gradient algorithm is given to solve its optimal value.},
 year = {2018}
}

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TY  - JOUR
T1  - Gradient Algorithm in Subspace Predictive Control
AU  - Wang Xiao-ping
AU  - Wang Jian-hong
Y1  - 2018/08/22
PY  - 2018
N1  - https://doi.org/10.11648/j.com.20180602.13
DO  - 10.11648/j.com.20180602.13
T2  - Communications
JF  - Communications
JO  - Communications
SP  - 39
EP  - 44
PB  - Science Publishing Group
SN  - 2328-5923
UR  - https://doi.org/10.11648/j.com.20180602.13
AB  - In this paper, subspace predictive control strategy is applied to design predictive controller. Given the state space model, the output estimations corresponding to the predictive output is derived to be one explicit function of the measured input-output data. Then using these output estimations, the problem of designing predictive controller is formulated as one optimization problem with equality and inequality conditions. In order to solve this constrain optimization problem, we use dual decomposition idea to change the original constrain optimization problem into an unconstrain optimization problem. So the classical gradient algorithm is put forth to solve the primal dual optimization problem. The problem of designing dual decomposition controller is studied for subspace predictive control strategy under fault condition. For state space equation with fault condition, we establish one function form between fault and residual using only input-output measured data sequence, and construct one least squares optimization problem to obtain fault estimation. The statistical property about residual is analyzed based on our derived output prediction, then the Kronecker product is used to derive the detailed structure corresponding to residual vector at every time instant. After substituting our output prediction into objective function of predictive control, one quadratic programming problem with equality and inequality constraints is considered. For solving this constrained optimization problem, fast gradient method is not suited for this complex optimization problem, as one regularization term is added in our objective function. So in order to solve this complex quadratic optimization problem, we propose a dual decomposition idea so that this dual decomposition idea can convert the former constrained optimization into unconstrained optimization, then one nearest neighbor gradient algorithm is given to solve its optimal value.
VL  - 6
IS  - 2
ER  -

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Author Information

Wang Xiao-ping

School of Mechanical and Electronic Engineering, Jingdezhen Ceramic Institute, Jingdezhen, China
Wang Jian-hong

School of Electronic Engineering and Automation, Jiangxi University of Science and Technology, Ganzhou, China

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Plain Text BibTeX RIS

APA Style

Wang Xiao-ping, Wang Jian-hong. (2018). Gradient Algorithm in Subspace Predictive Control. Communications, 6(2), 39-44. https://doi.org/10.11648/j.com.20180602.13

Copy | Download

ACS Style

Wang Xiao-ping; Wang Jian-hong. Gradient Algorithm in Subspace Predictive Control. Communications. 2018, 6(2), 39-44. doi: 10.11648/j.com.20180602.13

Copy | Download

AMA Style

Wang Xiao-ping, Wang Jian-hong. Gradient Algorithm in Subspace Predictive Control. Communications. 2018;6(2):39-44. doi: 10.11648/j.com.20180602.13

Copy | Download

@article{10.11648/j.com.20180602.13,
  author = {Wang Xiao-ping and Wang Jian-hong},
  title = {Gradient Algorithm in Subspace Predictive Control},
  journal = {Communications},
  volume = {6},
  number = {2},
  pages = {39-44},
  doi = {10.11648/j.com.20180602.13},
  url = {https://doi.org/10.11648/j.com.20180602.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.com.20180602.13},
  abstract = {In this paper, subspace predictive control strategy is applied to design predictive controller. Given the state space model, the output estimations corresponding to the predictive output is derived to be one explicit function of the measured input-output data. Then using these output estimations, the problem of designing predictive controller is formulated as one optimization problem with equality and inequality conditions. In order to solve this constrain optimization problem, we use dual decomposition idea to change the original constrain optimization problem into an unconstrain optimization problem. So the classical gradient algorithm is put forth to solve the primal dual optimization problem. The problem of designing dual decomposition controller is studied for subspace predictive control strategy under fault condition. For state space equation with fault condition, we establish one function form between fault and residual using only input-output measured data sequence, and construct one least squares optimization problem to obtain fault estimation. The statistical property about residual is analyzed based on our derived output prediction, then the Kronecker product is used to derive the detailed structure corresponding to residual vector at every time instant. After substituting our output prediction into objective function of predictive control, one quadratic programming problem with equality and inequality constraints is considered. For solving this constrained optimization problem, fast gradient method is not suited for this complex optimization problem, as one regularization term is added in our objective function. So in order to solve this complex quadratic optimization problem, we propose a dual decomposition idea so that this dual decomposition idea can convert the former constrained optimization into unconstrained optimization, then one nearest neighbor gradient algorithm is given to solve its optimal value.},
 year = {2018}
}

Copy | Download

TY  - JOUR
T1  - Gradient Algorithm in Subspace Predictive Control
AU  - Wang Xiao-ping
AU  - Wang Jian-hong
Y1  - 2018/08/22
PY  - 2018
N1  - https://doi.org/10.11648/j.com.20180602.13
DO  - 10.11648/j.com.20180602.13
T2  - Communications
JF  - Communications
JO  - Communications
SP  - 39
EP  - 44
PB  - Science Publishing Group
SN  - 2328-5923
UR  - https://doi.org/10.11648/j.com.20180602.13
AB  - In this paper, subspace predictive control strategy is applied to design predictive controller. Given the state space model, the output estimations corresponding to the predictive output is derived to be one explicit function of the measured input-output data. Then using these output estimations, the problem of designing predictive controller is formulated as one optimization problem with equality and inequality conditions. In order to solve this constrain optimization problem, we use dual decomposition idea to change the original constrain optimization problem into an unconstrain optimization problem. So the classical gradient algorithm is put forth to solve the primal dual optimization problem. The problem of designing dual decomposition controller is studied for subspace predictive control strategy under fault condition. For state space equation with fault condition, we establish one function form between fault and residual using only input-output measured data sequence, and construct one least squares optimization problem to obtain fault estimation. The statistical property about residual is analyzed based on our derived output prediction, then the Kronecker product is used to derive the detailed structure corresponding to residual vector at every time instant. After substituting our output prediction into objective function of predictive control, one quadratic programming problem with equality and inequality constraints is considered. For solving this constrained optimization problem, fast gradient method is not suited for this complex optimization problem, as one regularization term is added in our objective function. So in order to solve this complex quadratic optimization problem, we propose a dual decomposition idea so that this dual decomposition idea can convert the former constrained optimization into unconstrained optimization, then one nearest neighbor gradient algorithm is given to solve its optimal value.
VL  - 6
IS  - 2
ER  -

Copy | Download