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Nonlinear sampled-data ESO-based active disturbance rejection control for networked control systems with actuator saturation

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Abstract

This paper proposes a framework of anti-windup active disturbance rejection control for the networked control systems (NCSs) subjected to actuator saturation. The sensor-to-controller network is considered where only one sensor can report its measurements at each transmission instant. Both the round-robin and try-once-discard protocols are applied, respectively, to determine which sensor should be given the access to the network at a certain instant. To reflect the impact of communication constraints, a nonlinear sampled-data extended state observer (NSESO) is employed to estimate the states and ignored nonlinearities of the addressed system. Then, a composite control strategy with an anti-windup compensator is designed based on the NSESO, and the effects of actuator saturation is eliminated by the anti-windup compensator. The sufficient conditions to guarantee the convergence of the NSESO are provided, and then the input-to-state stability of the overall NCSs is given as well. Finally, a numerical example is introduced to demonstrate the effectiveness of the proposed design technique.

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References

  1. Pang, Z.-H., Liu, G.-P., Zhou, D., Sun, D.: Data-driven control with input design-based data dropout compensation for networked nonlinear systems. IEEE Trans. Contr. Syst. Technol. 25(2), 628–636 (2017)

    Article  Google Scholar 

  2. Kim, S.H.: T–S fuzzy control design for a class of nonlinear networked control systems. Nonlinear Dyn. 73(1–2), 17–27 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Yuan, Y., Yuan, H., Wang, Z., Guo, L., Yang, H.: Optimal control for networked control systems with disturbances: a delta operator approach. IET Control Theory Appl. 11(9), 1325–1332 (2017)

    Article  MathSciNet  Google Scholar 

  4. Li, H., Wu, C., Shi, P., Gao, Y.: Control of nonlinear networked systems with packet dropouts: Interval type-2 fuzzy model-based approach. IEEE Trans Cybern. 45(11), 2378–2389 (2015)

    Article  Google Scholar 

  5. Qiu, J., Gao, H., Ding, S.X.: Recent advances on fuzzy-model-based nonlinear networked control systems: a survey. IEEE Trans. Ind. Electron. 63(2), 1207–1217 (2016)

    Article  Google Scholar 

  6. Yuan, Y., Wang, Z., Zhang, P., Liu, H.: Near-optimal resilient control strategy design for state-saturated networked systems under stochastic communication protocol. IEEE Trans. Cybern. (2018). https://doi.org/10.1109/TCYB.2018.2840430

  7. Yang, H., Xu, Y., Zhang, J.: Event-driven control for networked control systems with quantization and markov packet losses. IEEE Trans. Cybern. 47(8), 2235–2243 (2017)

    Article  Google Scholar 

  8. Wang, Z., Wang, X., Liu, L.: Stochastic optimal linear control of wireless networked control systems with delays and packet losses. IET Control Theory Appl. 10(7), 742–751 (2014)

    Article  MathSciNet  Google Scholar 

  9. Postoyan, R., Nešić, D.: A framework for the observer design for networked control systems. IEEE Trans. Autom. Control 57(5), 1309–1314 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Liu, K., Fridman, E., Johansson, K.H., Xia, Y.: Quantized control under round-robin communication protocol. IEEE Trans. Ind. Electron. 63(7), 4461–4471 (2016)

    Article  Google Scholar 

  11. Dačić, D., Nešić, D.: Observer design for wired linear networked control systems using matrix inequalities. Automatica 44(11), 2840–2848 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Zou, L., Wang, Z., Gao, H., Alsaadi, F.E.: Finite-horizon \(H_\infty \) consensus Control of time-varying multiagent systems with stochastic communication protocol. IEEE Trans. Cybern. 47(8), 1830–1840 (2017)

    Article  Google Scholar 

  13. Nešić, D., Teel, A.R.: Input–output stability properties of networked control systems. IEEE Trans. Autom. Control 49(10), 1650–1667 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Walsh, G.C., Ye, H., Bushnell, L.G.: Stability analysis of networked control systems. IEEE Trans. Control Syst. Technol. 10(3), 438–446 (2002)

    Article  Google Scholar 

  15. Yuan, Y., Guo, L., Wang, Z.: Composite control of linear quadratic games in delta domain with disturbance observers. J. Frankl. Inst. 354(4), 1673–1695 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Zhang, W.A., Yu, L.: Stabilization of sampled-data control systems with control inputs missing. IEEE Trans. Autom. Control 55(2), 447–452 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Shen, B., Tan, H., Wang, Z., Huang, T.: Quantized/saturated control for sampled-data systems under noisy sampling intervals: a confluent vandermonde matrix approach. IEEE Trans. Autom. Control 62(9), 4753–4759 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. Chen, W.H., Zheng, W.X.: An improved stabilization method for sampled-data control systems with control packet loss. IEEE Trans. Autom. Control 57(9), 2378–2384 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Nesic, D., Teel, A.R.: A framework for stabilization of nonlinear sampled-data systems based on their approximate discrete-time models. IEEE Trans. Autom. Control 49(7), 1103–1122 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Yang, H., You, X., Xia, Y., Li, H.: Adaptive control for attitude synchronisation of spacecraft formation via extended state observer. IET Control Theory Appl. 8(18), 2171–2185 (2014)

    Article  MathSciNet  Google Scholar 

  21. Han, J.: From PID to active disturbance rejection control. IEEE Trans. Ind. Electron. 56(3), 900–906 (2009)

    Article  Google Scholar 

  22. Zhu, E., Pang, J., Sun, N., Gao, H., Sun, Q., Chen, Z.: Airship horizontal trajectory tracking control based on active disturbance rejection control (ADRC). Nonlinear Dyn. 75(4), 725–734 (2012)

    Article  MathSciNet  Google Scholar 

  23. Xia, Y., Dai, L., Fu, M., Li, C., Wang, C.: Application of active disturbance rejection control in tank gun control system. J. Frankl. Inst. 351(4), 2299–2314 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Zhao, L., Yang, Y., Xia, Y., Liu, Z.: Active disturbance rejection position control for a magnetic rodless pneumatic cylinder. IEEE Trans. Ind. Electron. 62(9), 5838–5846 (2015)

    Article  Google Scholar 

  25. Xia, Y., Zhu, Z., Fu, M.: Back-stepping sliding mode control for missile systems based on an extended state observer. IET Control Theory Appl. 5(1), 93–102 (2011)

    Article  MathSciNet  Google Scholar 

  26. Zhang, C., Yang, J., Li, S., Yang, N.: A generalized active disturbance rejection control method for nonlinear uncertain systems subject to additive disturbance. Nonlinear Dyn. 83(4), 2361–2372 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Hu, Q.: Robust adaptive sliding mode attitude maneuvering and vibration damping of three-axis-stabilized flexible spacecraft with actuator saturation limits. Nonlinear Dyn. 55(4), 301–321 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ran, M., Wang, Q., Dong, C.: Anti-windup design for uncertain nonlinear systems subject to actuator saturation and external disturbance. Int. J. Robust Nonlinear Control 26(15), 3421–3438 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ran, M., Wang, Q., Dong, C., Ni, M.: Multistage anti-windup design for linear systems with saturation nonlinearity: enlargement of the domain of attraction. Nonlinear Dyn. 80(3), 1543–1555 (2015)

    Article  Google Scholar 

  30. An, H., Liu, J., Wang, C., Wu, L.: Disturbance observer-based antiwindup control for air-breathing hypersonic vehicles. IEEE Trans. Ind. Electron. 63(5), 3038–3049 (2016)

    Article  Google Scholar 

  31. Tarbouriech, S., Turner, M.: Anti-windup design: an overview of some recent advances and open problems. IET Control Theory Appl. 3(1), 1–19 (2009)

    Article  MathSciNet  Google Scholar 

  32. Kamal, S., Moreno, J.A., Chalanga, A., Bandyopadhyay, B., Fridman, L.M.: Continuous terminal sliding-mode controller. Automatica 69(C), 308–314 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  33. Yuan, Y., Wang, Z., Zhang, P., Dong, H.: Nonfragile near-optimal control of stochastic time-varying multiagent systems with control-and state-dependent noises. IEEE Trans. Cybern. (2018) https://doi.org/10.1109/TCYB.2018.2829713

  34. Tarbouriech, S., Prieur, C., Silva, J.M.G.: Stability analysis and stabilization of systems presenting nested saturation. IEEE Trans. Autom. Control 51(8), 1364–1371 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  35. Jiang, Z.P., Wang, Y.: Input-to-state stability for discrete-time nonlinear systems. Automatica 37(6), 857–869 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  36. Gao, Z.: Estimation and compensation for Lipschitz nonlinear discrete-time systems subjected to unknown measurement delays. IEEE Trans. Ind. Electron. 62(9), 5950–5961 (2015)

    Article  Google Scholar 

  37. Zuo, Z., Lin, Z., Ding, Z.: Truncated prediction output feedback control of a class of Lipschitz nonlinear systems with input delay. IEEE Trans. Circuits-II 63(8), 788–792 (2016)

    Google Scholar 

  38. Wang, Q., Wang, J.: Fully distributed fault-tolerant consensus protocols for Lipschitz nonlinear multi-agent systems. IEEE Access 99(6), 17313–17325 (2018)

    Article  Google Scholar 

  39. Guo, B.Z., Zhao, Z.L.: On the convergence of an extended state observer for nonlinear systems with uncertainty. Syst. Control Lett. 60(6), 420–430 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  40. Zou, L., Wang, Z., Han, Q.L., Zhou, D.H.: Ultimate boundedness control for networked systems with try-once-discard protocol and uniform quantization effects. IEEE Trans. Autom. Control 62(12), 6582–6588 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  41. Dong, H., Wang, Z., Ding, S.X., Gao, H.: On \(H_{\infty }\) estimation of randomly occurring faults for a class of nonlinear time-varying systems with fading channels. IEEE Trans. Autom. Control 61(2), 479–484 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  42. Cao, Y., Ren, W., Casbeer, D.W., Schumacher, C.: Finite-time connectivity-preserving consensus of networked nonlinear agents with unknown Lipschitz terms. Trans. Autom. Control 61(6), 1700–1705 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  43. Khalil, H.K., Praly, L.: High-gain observers in nonlinear feedback control. Int. J. Robust Nonlinear Control 24(6), 991–992 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was funded by the National Natural Science Foundation of China (Grant Number 11572248).

Author information

Authors and Affiliations

  1. School of Astronautics, Northwestern Polytechnical University, Xi’an, 710072, China

    Yang Yu & Yuan Yuan

  2. Department of Computer Science, Brunel University London, Uxbridge, Middlesex, UB8 3PH, UK

    Yuan Yuan

  3. School of Electrical Engineering, Yanshan University, Qinhuangdao, 066004, China

    Hongjiu Yang

  4. Department of Computer Science and Technology, Tsinghua University, Beijing, 100084, China

    Huaping Liu

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Correspondence to Yuan Yuan.

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Appendix

Appendix

The positive definite function \(V_{11}(t)\) is designed by

$$\begin{aligned} V_{11}(t)=e^\mathrm{T}(t)P_{1}e(t), \end{aligned}$$
(68)

where

$$\begin{aligned} P_{1}\triangleq \left[ \begin{array}{cccc} 0.2*I_{1} &{} -\,0.05*I_{1} &{} -\,0.1*I_{1}\\ -\,0.05*I_{1} &{} 0.1*I_{1} &{} -\,0.05*I_{1}\\ -\,0.1*I_{1} &{} -\,0.05*I_{1} &{} 0.1667*I_{1} \end{array} \right] , \end{aligned}$$

is the definite solution of Lyapunov equation \(P_{1}\varLambda +\varLambda ^\mathrm{T}P_{1}=-\,0.1*I_{2}\) for

$$\begin{aligned} \varLambda \triangleq \left[ \begin{array}{cccc} -\,\beta _{1}*I_{1} &{} I_{1} &{} 0\\ -\,\beta _{2}*I_{1} &{} 0 &{} I_{1}\\ -\,\beta _{3}*I_{1} &{} 0 &{} 0 \end{array} \right] \triangleq {\left[ \begin{array}{cccc} -\,3*I_{1} &{} I_{1} &{} 0\\ -\,5*I_{1} &{} 0 &{} I_{1}\\ -\,3*I_{1} &{} 0 &{} 0 \end{array} \right] ,} \end{aligned}$$

where \(I_{1}\) and \(I_{2}\) represent the three-order and nine-order identity matrices, respectively. Moreover, MATLAB is used to calculate the eigenvalues of \(P_{1}\) and get nine eigenvalues as follows:

$$\begin{aligned}&\{0.0199,0.0199,0.0199,0.1619,0.1619,\\&0.1619,0.2849,0.2849,0.2849\}. \end{aligned}$$

From (68), the positive definite function \(V_{11}(t)\) can be rewritten as

$$\begin{aligned} V_{11}(t)= & {} 0.2e^\mathrm{T}_{1}(t)e_{1}(t)+0.1e^\mathrm{T}_{2}(t)e_{2}(t)\\&+\,0.1667e^\mathrm{T}_{3}(t)e_{3}(t)\\&-\,0.1e^\mathrm{T}_{1}(t)e_{2}(t)-0.2e^\mathrm{T}_{1}(t)e_{3}(t)\\&-\,0.1e^\mathrm{T}_{2}(t)e_{3}(t). \end{aligned}$$

Then we have

$$\begin{aligned}&\frac{\partial V_{11}}{\partial e_{1}}=0.4e_{1}(t)-0.1e_{2}(t)-0.2e_{3}(t),\\&\frac{\partial V_{11}}{\partial e_{2}}=0.2e_{2}(t)-0.1e_{1}(t)-0.1e_{3}(t),\\&\frac{\partial V_{11}}{\partial e_{3}}=0.3334e_{3}(t)-0.2e_{1}(t)-0.1e_{2}(t). \end{aligned}$$

One immediately obtains

$$\begin{aligned}&\sum ^{2}_{j=1}\frac{\partial V_{11}}{\partial e_{j}}(e_{j+1}(t)-h_{j}(e_{1}(t)))-\frac{\partial V_{11}}{\partial e_{3}}h_{3}(e_{1}(t))\nonumber \\&\quad =-\,(0.1e^\mathrm{T}_{1}(t)e_{1}(t)+0.1e^\mathrm{T}_{2}(t)e_{2}(t)+0.1e^\mathrm{T}_{3}(t)e_{3}(t)\nonumber \\&\qquad +\,0.4e^\mathrm{T}_{1}(t)g(e_{1}(t))-0.1e^\mathrm{T}_{2}(t)g(e_{1}(t))\nonumber \\&\qquad -\,0.2e^\mathrm{T}_{3}(t)g(e_{1}(t)))\nonumber \\&\quad \le -\left( 0.1e^\mathrm{T}_{1}(t)e_{1}(t)+ 0.1e^\mathrm{T}_{2}(t)e_{2}(t)\right. \nonumber \\&\qquad \left. +\,0.1e^\mathrm{T}_{3}(t)e_{3}(t)+2g^\mathrm{T}(e_{1}(t))g(e_{1}(t))\right. \nonumber \\&\qquad \left. +\left( 0.05e_{2}(t)-g(e_{1}(t))\right) ^\mathrm{T}\left( 0.05e_{2}(t)-g(e_{1}(t))\right) \right. \nonumber \\&\qquad \left. -\,0.0025e^\mathrm{T}_{2}(t)e_{2}(t)-g^\mathrm{T}(e_{1}(t))g(e_{1}(t))\right. \nonumber \\&\qquad \left. +\left( 0.1e_{3}(t)-g(e_{1}(t))\right) ^\mathrm{T}\left( 0.1e_{3}(t)-g(e_{1}(t))\right) \right. \nonumber \\&\qquad \left. -\,0.01e^\mathrm{T}_{3}(t)e_{3}(t)-g^\mathrm{T}(e_{1}(t))g(e_{1}(t))\right) \nonumber \\&\quad \le -\left( 0.1e^\mathrm{T}_{1}(t)e_{1}(t)+0.0975e^\mathrm{T}_{2}(t)e_{2}(t)+0.09e^\mathrm{T}_{3}(t)e_{3}(t)\right) \nonumber \\&\quad \triangleq -W_{11}(e_{1}(t),e_{2}(t),e_{3}(t)). \end{aligned}$$
(69)

Thus, the following inequalities are obtained as

$$\begin{aligned} \left| \frac{\partial V_{11}}{\partial e_{3}}\right|\le & {} \sqrt{0.3334^{2}+0.2^{2}+0.1^{2}}\Vert e(t)\Vert \\= & {} \sqrt{0.1612}\Vert e(t)\Vert , \end{aligned}$$

and

$$\begin{aligned} \sum ^{3}_{j=1}\left| \frac{\partial V_{11}}{\partial e_{j}}\right|\le & {} \left( \sqrt{0.4^{2}+0.1^{2}+0.2^{2}}\right. \\&\left. +\sqrt{0.2^{2}+0.1^{2}+0.1^{2}}\right. \\&\left. +\sqrt{0.3334^{2}+0.2^{2}+0.1^{2}}\right) \Vert e(t)\Vert \\= & {} \left( \sqrt{0.21}+\sqrt{0.06}+\sqrt{0.1612}\right) \Vert e(t)\Vert . \end{aligned}$$

Denote \(\lambda _{11}\triangleq \lambda _{\min }(P_{1})\triangleq 0.0199\), \(\lambda _{12}\triangleq \lambda _{\max }(P_{1})\triangleq 0.2849\), \(\lambda _{13}\triangleq 0.09\), \(\lambda _{14}\triangleq 0.1\), \(\hbar _{1}\triangleq \sqrt{0.1612}\) and \(\hbar _{2}\triangleq \sqrt{0.21}+\sqrt{0.06}+\sqrt{0.1612}=1.1047\). So the positive definite functions \(V_{11}(t)\) and \(W_{11}(t)\) are defined by (68) and (69), respectively, and the conditions in Assumption 1 are satisfied.

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Yu, Y., Yuan, Y., Yang, H. et al. Nonlinear sampled-data ESO-based active disturbance rejection control for networked control systems with actuator saturation. Nonlinear Dyn 95, 1415–1434 (2019). https://doi.org/10.1007/s11071-018-4636-9

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