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Comments on “Output feedback model predictive control for LPV systems based on quasi-min–max algorithm”
Automatica 48 (2012) 2385
Contents lists available at SciVerse ScienceDirect
Automatica journal homepage: www.elsevier.com/locate/automatica
Correspondence
Comments on ‘‘Output feedback model predictive control for LPV systems based on quasi-min–max algorithm’’✩ 1. Statement of problem
where
A state observer plus quasi-min–max state feedback MPC scheme was proposed in Park, Kim, and Sugie (2011). It claimed that the proposed MPC scheme for LPV systems can guarantee recursive feasibility and robust stability of the closed loop systems subject to input constraints. However, there exist some key issues in the proofs of feasibility and stability which will affect the results of the paper. Particularly, the recursive feasibility of the proposed algorithm was proved in Lemma 3. It states that ‘‘Then, (25) with (27) can be rewritten as zˆ (k + 1|k)T ζ (k)ˆz (k + 1|k)
− zˆ (k + 1|k)T ξ˜ (k + 1)T P (k)ξ˜ (k + 1)ˆz (k + 1|k) ≥ 0,
(E.1)
where
ζ (k) := P (k) − L − F (k)T RF (k),
(E.2)
ξ˜ (k + 1) := A(p(k + 1)) + B(p(k + 1))F (k).’’
(E.3)
zˆ (k + 2|k + 1) = A(p(k + 1))ˆz (k + 1|k + 1)
+ B(p(k + 1))u(k + 1|k + 1) + Lp (y(k + 1) − C xˆ (k + 1)).
Since Lp (y(k+1)−C xˆ (k+1)) ̸= 0 generally, the satisfaction of (E.1) does not necessarily imply (1) is satisfied, which undermines the recursive feasibility and stability. The above analysis is consistent with the fact that simple combination of a stable state observer and a stable state feedback controller might not guarantee the closed loop stability since the separation principle does not hold for nonlinear systems (Atassi & Khalil, 1999). References Atassi, A. N., & Khalil, H. K. (1999). A separation principle for the stabilization of a class of nonlinear systems. IEEE Transactions on Automatic Control, 1672–1687. Park, J.-H., Kim, T.-H., & Sugie, T. (2011). Output feedback model predictive control for LPV systems based on quasi-min–max algorithm. Automatica, 2052–2058.
Yang Su Kok Kiong Tan 1 Department of Electrical and Computer Engineering, National University of Singapore, 117576, Singapore E-mail addresses: [email protected] (Y. Su), [email protected] (K.K. Tan).
The fact that the optimization problem (24) is feasible at time k does imply that (E.1) is satisfied. However, the optimization problem (24) at time k + 1 actually requires zˆ (k + 2|k + 1)T P (k)ˆz (k + 2|k + 1) + zˆ (k + 1|k + 1)T L
× zˆ (k + 1|k + 1) + u(k + 1|k + 1)T Ru(k + 1|k + 1)
13 September 2011 Available online 18 July 2012
< zˆ (k|k)T Lzˆ (k|k) + u(k|k)T Ru(k|k) + zˆ (k + 1|k)T P (k)ˆz (k + 1|k)
(1)
✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Lalo Magni under the direction of Editor André L. Tits. The author(s) of the original paper was (were) contacted, but did not have any comment.
0005-1098/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2012.06.100
(2)
1
Tel.: +65 65162110; fax: +65 67791103.
Contents lists available at SciVerse ScienceDirect
Automatica journal homepage: www.elsevier.com/locate/automatica
Correspondence
Comments on ‘‘Output feedback model predictive control for LPV systems based on quasi-min–max algorithm’’✩ 1. Statement of problem
where
A state observer plus quasi-min–max state feedback MPC scheme was proposed in Park, Kim, and Sugie (2011). It claimed that the proposed MPC scheme for LPV systems can guarantee recursive feasibility and robust stability of the closed loop systems subject to input constraints. However, there exist some key issues in the proofs of feasibility and stability which will affect the results of the paper. Particularly, the recursive feasibility of the proposed algorithm was proved in Lemma 3. It states that ‘‘Then, (25) with (27) can be rewritten as zˆ (k + 1|k)T ζ (k)ˆz (k + 1|k)
− zˆ (k + 1|k)T ξ˜ (k + 1)T P (k)ξ˜ (k + 1)ˆz (k + 1|k) ≥ 0,
(E.1)
where
ζ (k) := P (k) − L − F (k)T RF (k),
(E.2)
ξ˜ (k + 1) := A(p(k + 1)) + B(p(k + 1))F (k).’’
(E.3)
zˆ (k + 2|k + 1) = A(p(k + 1))ˆz (k + 1|k + 1)
+ B(p(k + 1))u(k + 1|k + 1) + Lp (y(k + 1) − C xˆ (k + 1)).
Since Lp (y(k+1)−C xˆ (k+1)) ̸= 0 generally, the satisfaction of (E.1) does not necessarily imply (1) is satisfied, which undermines the recursive feasibility and stability. The above analysis is consistent with the fact that simple combination of a stable state observer and a stable state feedback controller might not guarantee the closed loop stability since the separation principle does not hold for nonlinear systems (Atassi & Khalil, 1999). References Atassi, A. N., & Khalil, H. K. (1999). A separation principle for the stabilization of a class of nonlinear systems. IEEE Transactions on Automatic Control, 1672–1687. Park, J.-H., Kim, T.-H., & Sugie, T. (2011). Output feedback model predictive control for LPV systems based on quasi-min–max algorithm. Automatica, 2052–2058.
Yang Su Kok Kiong Tan 1 Department of Electrical and Computer Engineering, National University of Singapore, 117576, Singapore E-mail addresses: [email protected] (Y. Su), [email protected] (K.K. Tan).
The fact that the optimization problem (24) is feasible at time k does imply that (E.1) is satisfied. However, the optimization problem (24) at time k + 1 actually requires zˆ (k + 2|k + 1)T P (k)ˆz (k + 2|k + 1) + zˆ (k + 1|k + 1)T L
× zˆ (k + 1|k + 1) + u(k + 1|k + 1)T Ru(k + 1|k + 1)
13 September 2011 Available online 18 July 2012
< zˆ (k|k)T Lzˆ (k|k) + u(k|k)T Ru(k|k) + zˆ (k + 1|k)T P (k)ˆz (k + 1|k)
(1)
✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Lalo Magni under the direction of Editor André L. Tits. The author(s) of the original paper was (were) contacted, but did not have any comment.
0005-1098/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2012.06.100
(2)
1
Tel.: +65 65162110; fax: +65 67791103.