# Observer-based robust control of uncertain systems via an integral quadratic constraint approach

• Cong Cong
Article

## Abstract

The focus of the present paper is to design an observer-based robust control for an uncertain linear systems where admissible uncertainty satisfies integral quadratic constraints. The design methodology involves astute utilization of S-procedure and auxiliary matrix inequality or equality. A exist condition for the observer and observer-based robust control is established given from linear matrix inequality optimal solution, which is numerically efficient owing to recent advances in convex optimization. This control law guarantees the stability of the closed-loop system with a specified level of disturbance attenuation when the uncertainty exists. Active structural control for a wind turbine is performed to verify the proposed control law.

## Keywords

Observer-based control Robust control Integral quadratic constraints (IQC) LMI

## List of symbols

$${{A}^{T}}({{x}^{T}})$$

Transpose of matrix A(resp., vector x)

$$diag(\cdot )$$

Diagonal matrix with diagonal elements

I

Unit matrix

$$P>0$$

Positive definite symmetric matrix

$${{L}_{2}}[0,\infty )$$

The Hilbert space of square integrable

$$\left\| \cdot \right\|$$

The standard Euclidean norm

$$\left\| \cdot \right\| ^2_2$$

$$\int _{0}^{t}{{{\left\| \cdot \right\| }^{2}}dt}$$

$${\Bigg [ } \begin{matrix} A &{} \quad * \\ B &{} \quad C \\ \end{matrix} {\Bigg ]}$$

Stand for $${\Bigg [} \begin{matrix} A &{}\quad {{B}^{T}} \\ B &{}\quad C \\ \end{matrix} {\Bigg ]}$$

## References

1. 1.
Yin S, Yang H, Kaynak O (2017) Sliding mode observer-based FTC for Markovian jump systems with actuator and sensor faults. IEEE Trans Autom Control 62(7):3551–3558
2. 2.
Derakhshan SF, Fatehi A (2015) Non-monotonic robust H2 fuzzy observer-based control for discrete time nonlinear systems with parametric uncertainties. Int J Syst Sci 46(12):2134–2149
3. 3.
Chen CLP, Ren CE, Du T (2016) Fuzzy observed-based adaptive consensus tracking control for second-order multiagent systems with heterogeneous nonlinear dynamics. IEEE Trans Fuzzy Syst 24(4):906–915
4. 4.
Wu TS, Karkoub M, Chen HS et al (2015) Robust tracking observer-based adaptive fuzzy control design for uncertain nonlinear MIMO systems with time delayed states. Inf Sci 290:86–105
5. 5.
Boulkroune A, M’Saad M (2012) On the design of observer-based fuzzy adaptive controller for nonlinear systems with unknown control gain sign. Fuzzy Sets Syst 201(201):71–85
6. 6.
Tlili AS, Dhbaibi S, Braiek NB (2012) Robust decentralised observer based guaranteed cost control for nonlinear uncertain interconnected systems. Application to multi-machine power systems. Int J Syst Sci 43(9):1713–1727
7. 7.
Li JG, Yuan JQ, Lu JG (2010) Observer-based $$H\infty$$ control for networked nonlinear systems with random packet losses. Isa Trans 49(1):39–46
8. 8.
Yin Y, Shi P, Liu F et al (2014) Observer-based $$H\infty$$ control on nonhomogeneous Markov jump systems with nonlinear input. Int J Robust Nonlinear Control 24(13):1903–1924
9. 9.
Zhang D, Han Q, Jia X (2015) Observer-based $$H\infty$$ output tracking control for networked control systems. Int J Robust Nonlinear Control 24(17):2741–2760
10. 10.
Abbaszadeh M, Marquez HJ (2009) LMI optimization approach to robust $$H\infty$$, observer design and static output feedback stabilization for discrete-time nonlinear uncertain systems. Int J Robust Nonlinear Control 19(3):313–340
11. 11.
Lien CH (2005) $$H\infty$$ Observer-based control for a class of uncertain neutral time-delay systems via LMI optimization approach. J Optim Theory Appl 127(1):129–144
12. 12.
Mobayen S, Tchier F (2017) Nonsingular fast terminal sliding-mode stabilizer for a class of uncertain nonlinear systems based on disturbance observer. Sci Iran 24(3):1410–1418Google Scholar
13. 13.
Zhou B, Li ZY, Lin Z (2013) Observer based output feedback control of linear systems with input and output delays. Automatica 49(7):2039–2052
14. 14.
Mobayen S, Tchier F (2016) An LMI approach to adaptive robust tracker design for uncertain nonlinear systems with time-delays and input nonlinearities. Nonlinear Dyn 85(3):1965–1978
15. 15.
Mobayen S, Tchier F (2015) A new LMI-based robust finite-time sliding mode control strategy for a class of uncertain nonlinear systems. Kybernetika 51(6):1035–1048
16. 16.
Lien CH (2004) Robust observer-based control of systems with state perturbations via LMI approach. IEEE Trans Autom Control 49(8):1365–1370
17. 17.
Ahmad S, Rehan M, Hong KS (2016) Observer-based robust control of one-sided Lipschitz nonlinear systems. Isa Trans 65:230–240
18. 18.
Liu Q, Zhou B (2017) Extended observer based feedback control of linear systems with both state and input delays. J Frankl Inst 354:8232–8255
19. 19.
Wang Y, Rajamani R, Zemouche A (2018) Sequential LMI approach for the design of a BMI-based robust observer state feedback controller with nonlinear uncertainties. Int J Robust Nonlinear Control 28(4):1–15
20. 20.
Tlili AS, Braiek NB (2014) $$H\infty$$ optimization-based decentralized control of linear interconnected systems with nonlinear interconnections. J Frankl Inst 351(6):3286–3304
21. 21.
Savkin AV, Petersen IR (1996) Robust $$H\infty$$ control of uncertain systems with structured uncertainty. J Math Syst Estim Control 6(4):339–342
22. 22.
Petersen IR (2009) Robust $$H\infty$$ control of an uncertain system via a stable output feedback controller. IEEE Trans Autom Control 54(6):1418–1423
23. 23.
Li L, Ugrinovskii VA (2007) On necessary and sufficient conditions for $$H\infty$$ output feedback control of Markov jump linear systems. IEEE Trans Autom Control 52(7):1287–1292
24. 24.
Ma S, Xiong J, Ugrinovskii VA et al (2013) Robust decentralized stabilization of Markovian jump large-scale systems: a neighboring mode dependent control approach. Automatica 49(10):3105–3111
25. 25.
Shaiju AJ, Petersen IR (2013) Discrete-time robust $$H\infty$$ control of a class of nonlinear uncertain systems. Int J Robust Nonlinear Control 23(14):1629–1641
26. 26.
Ugrinovskii VA, Petersen IR (2002) Robust output feedback stabilization via risk-sensitive control. Automatica 38:945–955
27. 27.
Li H, Fu M (1997) A linear matrix inequality approach to robust $$H\infty$$, filtering. IEEE Trans Signal Process 45(9):2338–2350
28. 28.
Petersen IR (2009) Robust guaranteed cost state estimation for nonlinear stochastic uncertain systems via an IQC approach. Syst Control Lett 58:865–870
29. 29.
Kallapur AG, Petersen IR, Anavatti SG (2009) A discrete-time robust extended Kalman filter for uncertain systems with sum quadratic constraints. IEEE Trans Autom Control 54(4):850–854
30. 30.
Megretski A, Rantzer A (1997) System analysis via integral quadratic constraints. IEEE Trans Autom Control 42(6):819–830
31. 31.
Kheloufi H, Zemouche A, Bedouhene F et al (2013) On LMI conditions to design observer-based controllers for linear systems with parameter uncertainties. Automatica 49(12):3700–3704
32. 32.
Staino A, Basu B (2013) Dynamics and control of vibrations in wind turbines with variable rotor speed. J Eng Struct 56:58–67
33. 33.
Cong C (2018) Stochastic vibrations control of wind turbine blades based on wireless sensor. Wirel Pers Commun 102(5):3503–3515
34. 34.
Jonkman JM, Butterfield S, Musial W, et al (2009) Definition of a 5-MW reference wind turbine for offshore system development. National Renewable Energy Laboratory, Technical Report NREL/TP-500-38060, Golden, ColoradoGoogle Scholar
35. 35.
Wu R, Zhang W, Song F et al (2015) Observer-based stabilization of one-sided Lipschitz systems with application to flexible link manipulator. Adv Mech Eng 7(12):1–8