September  2020, 10(3): 331-343. doi: 10.3934/naco.2020005

$ H_{\infty} $ observer-based control for large-scale systems with sparse observer communication network

Department of Automation, University of Science and Technology of China, 96 Jinzhai Road, Hefei, 230027, China

* Corresponding author: Junlin Xiong

Received  May 2019 Revised  July 2019 Published  February 2020

Fund Project: This work was supported by National Natural Science Foundation of China under Grant 61773357

This paper studies the $ H_{\infty} $ control problem for large-scale systems under a sparse observer communication network. Different from existing approaches, where the topology of the observer communication network is fixed, we aim to design the sparse observer communication network such that the closed-loop system is asymptotically stable and satisfies the $ H_{\infty} $ performance. Firstly, sufficient conditions are established to design the distributed $ H_{\infty} $ observer and controller gains in terms of LMIs. Then, the developed sufficient conditions are used to minimize the number of the links in the observer communication network. Two numerical algorithms are proposed to solve the sparse observer communication network design problem. Finally, a numerical example is given to demonstrate the effectiveness of the proposed approach.

Citation: Junlin Xiong, Wenjie Liu. $ H_{\infty} $ observer-based control for large-scale systems with sparse observer communication network. Numerical Algebra, Control & Optimization, 2020, 10 (3) : 331-343. doi: 10.3934/naco.2020005
References:
[1]

A. ArghaL. Li and S. W. Su, Design of $H_{2}$ ($H_{\infty}$)-based optimal structured and sparse static output feedback gains, Journal of the Franklin Institute, 354 (2017), 4156-4178.  doi: 10.1016/j.jfranklin.2017.03.011.  Google Scholar

[2]

P. BennerR. Lowe and M. Voigt, $L_{\infty}$-norm computation for large-scale descriptor systems using structured iterative eigensolvers, Numerical Algebra, Control & Optimization, 8 (2018), 119-133.  doi: 10.3934/naco.2018007.  Google Scholar

[3]

S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, 1994. doi: 10.1137/1.9781611970777.  Google Scholar

[4]

E. J. CandesM. B. Wakin and S. P. Boyd, Enhancing sparsity by reweighted $\ell_{1}$ minimization, Journal of Fourier Analysis and Applications, 14 (2008), 877-905.  doi: 10.1007/s00041-008-9045-x.  Google Scholar

[5]

X. Chen, J. Lam, P. Li and Z. Shu, $L_{1}$-induced performance analysis and sparse controller synthesis for interval positive systems, in Proceedings of the World Congress on Engineering, 2013. Google Scholar

[6]

R. D'Andrea and G. E. Dullerud, Distributed control design for spatially interconnected systems, IEEE Transactions on Automatic Control, 48 (2003), 1478-1495.  doi: 10.1109/TAC.2003.816954.  Google Scholar

[7]

M. FardadF. Lin and M. R. Jovanovic, Design of optimal sparse interconnection graphs for synchronization of oscillator networks, IEEE Transactions on Automatic Control, 59 (2014), 2457-2462.  doi: 10.1109/TAC.2014.2301577.  Google Scholar

[8]

P. Gahinet and P. Apkarian, A linear matrix inequality approach to $H_{\infty}$ control, International Journal of Robust and Nonlinear Control, 4 (1994), 421-448.  doi: 10.1002/rnc.4590040403.  Google Scholar

[9]

T. IshizakiH. SandbergK. KashimaJ. Imura and K. Aihara, Dissipativity-preserving model reduction for large-scale distributed control systems, IEEE Transactions on Automatic Control, 60 (2015), 1023-1037.  doi: 10.1109/TAC.2014.2370271.  Google Scholar

[10]

J. LiQ. ZhangJ. Ren and Y. Zhang, Robust decentralised stabilisation of uncertain large-scale interconnected nonlinear descriptor systems via proportional plus derivative feedback, International Journal of Systems Science, 48 (2017), 2997-3006.  doi: 10.1080/00207721.2017.1367428.  Google Scholar

[11]

F. LinM. Fardad and M. R. Jovanovic, Design of optimal sparse feedback gains via the alternating direction method of multipliers, IEEE Transactions on Automatic Control, 58 (2013), 2426-2431.  doi: 10.1109/TAC.2013.2257618.  Google Scholar

[12]

J. Lofberg, YALMIP: A toolbox for modeling and optimization in MATLAB, in Proceedings of the IEEE International Symposium on Computer Aided Control Systems Design, (2004), 284-289. Google Scholar

[13]

M. MazoA. Anta and P. Tabuada, An ISS self-triggered implementation of linear controllers, Automatica, 46 (2010), 1310-1314.  doi: 10.1016/j.automatica.2010.05.009.  Google Scholar

[14]

M. MohamedX. G. YanS. K. Spurgeon and B. Jiang, Robust sliding-mode observers for large-scale systems with application to a multimachine power system, IET Control Theory & Applications, 11 (2016), 1307-1315.  doi: 10.1049/iet-cta.2016.1204.  Google Scholar

[15]

Z. RazavinasabM. M. Farsangi and M. Barkhordari, State estimation-based distributed model predictive control of large-scale networked systems with communication delays, IET Control Theory & Applications, 11 (2017), 2497-2505.  doi: 10.1049/iet-cta.2016.1649.  Google Scholar

[16]

M. Razeghi-Jahromi and A. Seyedi, Stabilization of distributed networked control systems with minimal communications network, in Proceedings of the American Control Conference, (2011), 515-520. Google Scholar

[17]

M. Razeghi-Jahromi and A. Seyedi, Stabilization of networked control systems with sparse observer-controller networks, IEEE Transactions on Automatic Control, 60 (2015), 1686-1691.  doi: 10.1109/TAC.2014.2360310.  Google Scholar

[18]

M. Rotkowitz and S. Lall, A characterization of convex problems in decentralized control, IEEE Transactions on Automatic Control, 51 (2006), 274-286.  doi: 10.1109/TAC.2005.860365.  Google Scholar

[19]

N. SandellP. VaraiyaM. Athans and M. Safonov, Survey of decentralized control methods for large scale systems, IEEE Transactions on Automatic Control, 23 (1978), 108-128.  doi: 10.1109/tac.1978.1101704.  Google Scholar

[20]

S. SchulerP. LiJ. Lam and F. Allgöwer, Design of structured dynamic output-feedback controllers for interconnected systems, International Journal of Control, 84 (2011), 2081-2091.  doi: 10.1080/00207179.2011.634029.  Google Scholar

[21]

S. SchulerU. Münz and F. Allgöwer, Decentralized state feedback control for interconnected systems with application to power systems, Journal of Process Control, 24 (2014), 379-388.   Google Scholar

[22]

P. Shah and P. A. Parrilo, $H_{2}$-optimal decentralized control over posets: a state-space solution for state-feedback, IEEE Transactions on Automatic Control, 58 (2013), 3084-3096.  doi: 10.1109/TAC.2013.2281881.  Google Scholar

[23]

J. F. Sturm, Using SeDuMi 1.02, A MATLAB toolbox for optimization over symmetric cones, Optimization Methods and Software, 11 (1999), 625-653.  doi: 10.1080/10556789908805766.  Google Scholar

[24]

K. K. TanS. Huang and T. H. Lee, Decentralized adaptive controller design of large-scale uncertain robotic systems, Automatica, 45 (2009), 161-166.  doi: 10.1016/j.automatica.2008.06.005.  Google Scholar

[25]

X.-G. YanQ. ZhangS. K. SpurgeonQ. Zhu and L. M. Fridman, Decentralised control for complex systems - An invited survey, International Journal of Modelling, Identification and Control, 22 (2014), 285-297.   Google Scholar

[26]

J. YoneyamaM. NishikawaH. Katayama and A. Ichikawa, Output stabilization of Takagi-Sugeno fuzzy systems, Fuzzy Sets and Systems, 111 (2000), 253-266.  doi: 10.1016/S0165-0114(98)00121-3.  Google Scholar

[27]

A. I. Zecevic and D. D. Siljak, Global low-rank enhancement of decentralized control for large-scale systems, IEEE Transactions on Automatic Control, 50 (2005), 740-744.  doi: 10.1109/TAC.2005.847054.  Google Scholar

[28]

Z. ZhouB. De SchutterS. Lin and Y. Xi, Multi-agent model-based predictive control for large-scale urban traffic networks using a serial scheme, IET Control Theory & Applications, 9 (2014), 475-484.  doi: 10.1049/iet-cta.2014.0490.  Google Scholar

[29]

K. Zhou and P. P. Khargonekar, Robust stabilization of linear systems with norm-bounded time-varying uncertainty, Systems & Control Letters, 10 (1988), 17-20.  doi: 10.1016/0167-6911(88)90034-5.  Google Scholar

[30]

Y. ZhuF. Yang and Q. L. Han, Simultaneous $H_{\infty}$ stabilisation for distributed networked multimode control systems with multiple packet dropouts, IET Control Theory & Applications, 10 (2016), 625-636.  doi: 10.1049/iet-cta.2015.0563.  Google Scholar

show all references

References:
[1]

A. ArghaL. Li and S. W. Su, Design of $H_{2}$ ($H_{\infty}$)-based optimal structured and sparse static output feedback gains, Journal of the Franklin Institute, 354 (2017), 4156-4178.  doi: 10.1016/j.jfranklin.2017.03.011.  Google Scholar

[2]

P. BennerR. Lowe and M. Voigt, $L_{\infty}$-norm computation for large-scale descriptor systems using structured iterative eigensolvers, Numerical Algebra, Control & Optimization, 8 (2018), 119-133.  doi: 10.3934/naco.2018007.  Google Scholar

[3]

S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, 1994. doi: 10.1137/1.9781611970777.  Google Scholar

[4]

E. J. CandesM. B. Wakin and S. P. Boyd, Enhancing sparsity by reweighted $\ell_{1}$ minimization, Journal of Fourier Analysis and Applications, 14 (2008), 877-905.  doi: 10.1007/s00041-008-9045-x.  Google Scholar

[5]

X. Chen, J. Lam, P. Li and Z. Shu, $L_{1}$-induced performance analysis and sparse controller synthesis for interval positive systems, in Proceedings of the World Congress on Engineering, 2013. Google Scholar

[6]

R. D'Andrea and G. E. Dullerud, Distributed control design for spatially interconnected systems, IEEE Transactions on Automatic Control, 48 (2003), 1478-1495.  doi: 10.1109/TAC.2003.816954.  Google Scholar

[7]

M. FardadF. Lin and M. R. Jovanovic, Design of optimal sparse interconnection graphs for synchronization of oscillator networks, IEEE Transactions on Automatic Control, 59 (2014), 2457-2462.  doi: 10.1109/TAC.2014.2301577.  Google Scholar

[8]

P. Gahinet and P. Apkarian, A linear matrix inequality approach to $H_{\infty}$ control, International Journal of Robust and Nonlinear Control, 4 (1994), 421-448.  doi: 10.1002/rnc.4590040403.  Google Scholar

[9]

T. IshizakiH. SandbergK. KashimaJ. Imura and K. Aihara, Dissipativity-preserving model reduction for large-scale distributed control systems, IEEE Transactions on Automatic Control, 60 (2015), 1023-1037.  doi: 10.1109/TAC.2014.2370271.  Google Scholar

[10]

J. LiQ. ZhangJ. Ren and Y. Zhang, Robust decentralised stabilisation of uncertain large-scale interconnected nonlinear descriptor systems via proportional plus derivative feedback, International Journal of Systems Science, 48 (2017), 2997-3006.  doi: 10.1080/00207721.2017.1367428.  Google Scholar

[11]

F. LinM. Fardad and M. R. Jovanovic, Design of optimal sparse feedback gains via the alternating direction method of multipliers, IEEE Transactions on Automatic Control, 58 (2013), 2426-2431.  doi: 10.1109/TAC.2013.2257618.  Google Scholar

[12]

J. Lofberg, YALMIP: A toolbox for modeling and optimization in MATLAB, in Proceedings of the IEEE International Symposium on Computer Aided Control Systems Design, (2004), 284-289. Google Scholar

[13]

M. MazoA. Anta and P. Tabuada, An ISS self-triggered implementation of linear controllers, Automatica, 46 (2010), 1310-1314.  doi: 10.1016/j.automatica.2010.05.009.  Google Scholar

[14]

M. MohamedX. G. YanS. K. Spurgeon and B. Jiang, Robust sliding-mode observers for large-scale systems with application to a multimachine power system, IET Control Theory & Applications, 11 (2016), 1307-1315.  doi: 10.1049/iet-cta.2016.1204.  Google Scholar

[15]

Z. RazavinasabM. M. Farsangi and M. Barkhordari, State estimation-based distributed model predictive control of large-scale networked systems with communication delays, IET Control Theory & Applications, 11 (2017), 2497-2505.  doi: 10.1049/iet-cta.2016.1649.  Google Scholar

[16]

M. Razeghi-Jahromi and A. Seyedi, Stabilization of distributed networked control systems with minimal communications network, in Proceedings of the American Control Conference, (2011), 515-520. Google Scholar

[17]

M. Razeghi-Jahromi and A. Seyedi, Stabilization of networked control systems with sparse observer-controller networks, IEEE Transactions on Automatic Control, 60 (2015), 1686-1691.  doi: 10.1109/TAC.2014.2360310.  Google Scholar

[18]

M. Rotkowitz and S. Lall, A characterization of convex problems in decentralized control, IEEE Transactions on Automatic Control, 51 (2006), 274-286.  doi: 10.1109/TAC.2005.860365.  Google Scholar

[19]

N. SandellP. VaraiyaM. Athans and M. Safonov, Survey of decentralized control methods for large scale systems, IEEE Transactions on Automatic Control, 23 (1978), 108-128.  doi: 10.1109/tac.1978.1101704.  Google Scholar

[20]

S. SchulerP. LiJ. Lam and F. Allgöwer, Design of structured dynamic output-feedback controllers for interconnected systems, International Journal of Control, 84 (2011), 2081-2091.  doi: 10.1080/00207179.2011.634029.  Google Scholar

[21]

S. SchulerU. Münz and F. Allgöwer, Decentralized state feedback control for interconnected systems with application to power systems, Journal of Process Control, 24 (2014), 379-388.   Google Scholar

[22]

P. Shah and P. A. Parrilo, $H_{2}$-optimal decentralized control over posets: a state-space solution for state-feedback, IEEE Transactions on Automatic Control, 58 (2013), 3084-3096.  doi: 10.1109/TAC.2013.2281881.  Google Scholar

[23]

J. F. Sturm, Using SeDuMi 1.02, A MATLAB toolbox for optimization over symmetric cones, Optimization Methods and Software, 11 (1999), 625-653.  doi: 10.1080/10556789908805766.  Google Scholar

[24]

K. K. TanS. Huang and T. H. Lee, Decentralized adaptive controller design of large-scale uncertain robotic systems, Automatica, 45 (2009), 161-166.  doi: 10.1016/j.automatica.2008.06.005.  Google Scholar

[25]

X.-G. YanQ. ZhangS. K. SpurgeonQ. Zhu and L. M. Fridman, Decentralised control for complex systems - An invited survey, International Journal of Modelling, Identification and Control, 22 (2014), 285-297.   Google Scholar

[26]

J. YoneyamaM. NishikawaH. Katayama and A. Ichikawa, Output stabilization of Takagi-Sugeno fuzzy systems, Fuzzy Sets and Systems, 111 (2000), 253-266.  doi: 10.1016/S0165-0114(98)00121-3.  Google Scholar

[27]

A. I. Zecevic and D. D. Siljak, Global low-rank enhancement of decentralized control for large-scale systems, IEEE Transactions on Automatic Control, 50 (2005), 740-744.  doi: 10.1109/TAC.2005.847054.  Google Scholar

[28]

Z. ZhouB. De SchutterS. Lin and Y. Xi, Multi-agent model-based predictive control for large-scale urban traffic networks using a serial scheme, IET Control Theory & Applications, 9 (2014), 475-484.  doi: 10.1049/iet-cta.2014.0490.  Google Scholar

[29]

K. Zhou and P. P. Khargonekar, Robust stabilization of linear systems with norm-bounded time-varying uncertainty, Systems & Control Letters, 10 (1988), 17-20.  doi: 10.1016/0167-6911(88)90034-5.  Google Scholar

[30]

Y. ZhuF. Yang and Q. L. Han, Simultaneous $H_{\infty}$ stabilisation for distributed networked multimode control systems with multiple packet dropouts, IET Control Theory & Applications, 10 (2016), 625-636.  doi: 10.1049/iet-cta.2015.0563.  Google Scholar

Figure 1.  Sparse observer topology structure of large-scale systems
Figure 2.  Trajectories of system states and their estimations
Figure 3.  Number of observer network links for different $ H_{\infty} $ performance levels
Algorithm 2 Sparse Observer Communication Network Link Design
(1) Choose a small enough scalar $ \delta>0 $. Initialize $ \theta_{ij}^{(0)}=1 $ for all $ i\neq j $. Let $ k\leftarrow1 $.
(2) Solve the optimization problem (12) with $ \theta_{ij}=\theta_{ij}^{(k-1)} $ to find the optimal solution $ Z^{(k)} $.
(3) Construct the index set $ I=\left\{{(i,j) \mid \left\lVert{\mathrm{vec}(Z^{(k)}_{ij})}\right\rVert_{1}<\delta, \theta_{ij}=1, i\neq j}\right\} $. If $ I $ is an empty set, then go to step (5). Otherwise, select $ (i^{*},j^{*}) = {\rm{arg\;min}}_{(i,j)\in I} \left\lVert{\mathrm{vec}(Z^{(k)}_{ij})}\right\rVert_{1} $.
(4) Update the links $ \theta_{ij}^{(k)}=\theta_{ij}^{(k-1)} $ except that $ \theta_{i^{*}j^{*}}^{(k)}=0 $. Let $ k\leftarrow k+1 $ and go to Step (2).
(5) Let $ \theta_{ij}=\theta_{ij}^{(k-1)} $. Solve the LMI (9) and design the controller and observer gains as $ K_{i}=Y_{i}X_{i}^{-1} $, $ L_{ij}=Q_{ij}^{-1}Z_{ij} $, respectively.
Algorithm 2 Sparse Observer Communication Network Link Design
(1) Choose a small enough scalar $ \delta>0 $. Initialize $ \theta_{ij}^{(0)}=1 $ for all $ i\neq j $. Let $ k\leftarrow1 $.
(2) Solve the optimization problem (12) with $ \theta_{ij}=\theta_{ij}^{(k-1)} $ to find the optimal solution $ Z^{(k)} $.
(3) Construct the index set $ I=\left\{{(i,j) \mid \left\lVert{\mathrm{vec}(Z^{(k)}_{ij})}\right\rVert_{1}<\delta, \theta_{ij}=1, i\neq j}\right\} $. If $ I $ is an empty set, then go to step (5). Otherwise, select $ (i^{*},j^{*}) = {\rm{arg\;min}}_{(i,j)\in I} \left\lVert{\mathrm{vec}(Z^{(k)}_{ij})}\right\rVert_{1} $.
(4) Update the links $ \theta_{ij}^{(k)}=\theta_{ij}^{(k-1)} $ except that $ \theta_{i^{*}j^{*}}^{(k)}=0 $. Let $ k\leftarrow k+1 $ and go to Step (2).
(5) Let $ \theta_{ij}=\theta_{ij}^{(k-1)} $. Solve the LMI (9) and design the controller and observer gains as $ K_{i}=Y_{i}X_{i}^{-1} $, $ L_{ij}=Q_{ij}^{-1}Z_{ij} $, respectively.
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