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Robust control for a class of time-delay uncertain nonlinear systems based on sliding mode observer

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Abstract

In this paper, a robust control scheme is proposed for a class of time-delay uncertain nonlinear systems with unknown input using the sliding mode observer. The sliding mode state observer is given with radial basis function neural networks, and then the robust control scheme is presented based on the designed sliding mode observer. The developed observer-based control scheme consists of two parts. One term is a linear controller and the other term is a neural network controller. Using the Lyapunov method, a criterion for bounded stability of the closed-loop system is developed in terms of linear matrix inequalities. Finally, a simulation example is used to illustrate the effectiveness of the proposed robust control scheme.

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References

  1. Ge SS, Hong F, Lee TH (2003) Adaptive neural network control of nonlinear systems with unknown time delays. IEEE Trans Autom Control 48(11):2004–2010

    Article  MathSciNet  Google Scholar 

  2. Hsiao FH, Hwang JD (1996) Stabilization of nonlinear singularly perturbed multiple time-delay systems by dither. J Dyn Syst Meas Control 18(1):176–181

    Article  Google Scholar 

  3. Nguang SK (2000) Robust stabilization of a class of time-delay nonlinear systems. IEEE Trans Autom Control 45(4):756–762

    Article  MATH  MathSciNet  Google Scholar 

  4. Park JH (2004) On the design of observer-based controller of linear neutral delay-differential systems. Appl Math Comput 150:195–202

    Article  MATH  MathSciNet  Google Scholar 

  5. Mou C, Chang-sheng J, Qing-xian W (2008) Robust adaptive control of uncertain time delay systems with FLS. Int J Innov Comput Inf Control 4(8):1995–2004

    Google Scholar 

  6. Wang Z, Huang B, Unbehauen H, Robust H (2001) ∞observer design of linear time-delay systems with parametric uncertainty. Syst Control Lett 42:303–312

    Article  MATH  Google Scholar 

  7. Raghavan S, Hedrick JK (1994) Observer design for a class of nonlinear systems. Int J Control 59(2):515–528

    Article  MATH  MathSciNet  Google Scholar 

  8. Xiong Y, Saif M (2001) Sliding mode observer for nonlinear uncertain systems. IEEE Trans Autom Control 46(12):2012–2017

    Article  MATH  MathSciNet  Google Scholar 

  9. Choi HH, Ro KS (2005) LMI-based sliding-mode observer design method. IEE Proc-Control Theory Appl 152(1):113–115

    Article  Google Scholar 

  10. Koshkouei AJ, Zinober ASI (2002) Sliding mode observer for a class of nonlinear systems. In: Proceedings of the American control conference, Anchorage, AK May 8–10

  11. Tan CP, Edwards C (2002) Sliding mode observers for detection and reconstruction of sensor faults. Automatica 38:1815–1821

    Article  MATH  MathSciNet  Google Scholar 

  12. Edwards C, Spurgeon SK, Patton RJ (2002) Sliding mode observers for fault detection and isolation. Automatica 36:541–553

    Article  MathSciNet  Google Scholar 

  13. Mou C, Chang-sheng J, Qing-xian W (2008) Sensor fault diagnosis for a class of uncertain time delay nonlinear system using neural network. Int J Autom Comput 5(4):401–405

    Article  Google Scholar 

  14. Song B, Hedrick JK (1995) Observer-based dynamic surface control for a class of nonlinear systems: an lmi approach. IEEE Trans Autom Control 49(11):1995–2001

    Article  MathSciNet  Google Scholar 

  15. Wang C-H, Liu H-L, Lin T-C (2002) Direct adaptive fuzzy-neural control with state observer and supervisory controller for unknown nonlinear dynamical systems. IEEE Trans Fuzzy Syst 10(1):39–49

    Article  Google Scholar 

  16. Liu C-C, Chen F-C (1993) Adaptive control of nonlinear continuous systems using neural networks general relative degree and MIMO cases. Int J Control 58(2):317–335

    Article  MATH  Google Scholar 

  17. Noriega JR, Wang H (1998) A direct adaptive neural-network control for unknown nonlinear systems and its application. IEEE Trans Neural Netw 9(1):27–34

    Article  Google Scholar 

  18. Xu H, Ioannou PA (2000) Robust adaptive control for a class of MIMO nonlinear systems with guaranteed. IEEE Trans Autom Control 48(5):728–742

    MathSciNet  Google Scholar 

  19. Ge SS, Wang C (2004) Adaptive neural control of uncertain MIMO nonlinear systems. IEEE Trans Neural Netw 15(3):674–692

    Article  Google Scholar 

  20. Mou C, Chen W-H (2010) Sliding mode controller design for a class of uncertain nonlinear system Based on disturbance observer. Int J Adapt Control Signal Process 24(1):51–64

    MATH  Google Scholar 

  21. Mou C, Jiang C, Bing J, Qing-xian W (2009) Design of sliding mode synchronization controller for uncertain chaotic systems with neural network. Chaos, Solitons Fractals 39(4):1856–1863

    Article  Google Scholar 

Download references

Acknowledgments

The work is partially supported by NUAA Research Funding (No. NS2010060) and Jiangsu Natural Science Foundation (Granted Number: SBK2008390). The authors also gratefully acknowledge the helpful comments and suggestions of the reviews, which have improved the presentation.

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Correspondence to Mou Chen.

Appendix

Appendix

1.1 Proof of Theorem 1

Proof Consider the Lyapunov function candidate

$$ V = V_{0} + V_{1} + V_{2} + V_{3} $$
(26)

where \( V_{0} = e^{\text{T}} (t)Pe(t) + \frac{1}{2}\tilde{\theta }^{\text{T}} \Upgamma^{ - 1} \tilde{\theta } + \sum_{i = 1}^{n} {{\frac{1}{{\mu_{i} }}}} \tilde{W}_{i}^{\text{T}} \tilde{W}_{i} ,\,\,V_{1} = x^{\text{T}} (t)Rx(t),\,\,V_{2}=\beta_{1}\int_{t-\tau}^{t}x^{T}(s)x(s)\) and \( V_{3} = \beta_{2} \int_{t - \tau }^{t} {e^{\text{T}} (s)e(s)ds} \).

Considering (1), (10), (15) and (23), the time derivative V satisfies

$$ \begin{aligned} \dot{V}& = &e^{\text{T}} (PA + A^{\text{T}} P - 2C^{\text{T}} C)e + 2e^{\text{T}} PA_{1} e(t - \tau ) + 2e^{\text{T}} P[B\left( {f(x) - f(\hat{x})} \right) + d(t) - v] \\ & + \tilde{\theta }^{\text{T}} \Upgamma^{ - 1} \dot{\tilde{\theta }} + 2\sum\limits_{i = 1}^{n} {{\frac{1}{{\mu_{i} }}}} \tilde{W}_{i}^{\text{T}} \dot{\tilde{W}}_{i} + x^{\text{T}} \left( {A^{\text{T}} R + RA - 2RBB^{\text{T}} R} \right)x + 2x^{\text{T}} RA_{1} x(t - \tau ) \\ & + 2x^{\text{T}} RBB^{\text{T}} Re + 2x^{\text{T}} RBf(x) - 2x^{\text{T}} RB\hat{W}\Upphi (\hat{x}) + 2e^{\text{T}} Rd(t) + 2\hat{x}^{\text{T}} Rd(t) \\ & + \beta_{1} x^{\text{T}} x - \beta_{1} x^{\text{T}} (t - \tau )x(t - \tau ) + \beta_{2} e^{\text{T}} e - \beta_{2} e^{\text{T}} (t - \tau )e(t - \tau ) \\ & \le& e^{\text{T}} (PA + A^{\text{T}} P - 2C^{\text{T}} C)e + 2e^{\text{T}} PA_{1} e(t - \tau ) + 2e^{\text{T}} PB\tilde{W}^{\text{T}} \Upphi (\hat{x}) + 2e^{\text{T}} Pd(t) \\ & - 2e^{\text{T}} Pv + 2e^{\text{T}} PB(\Uplambda + \varepsilon ) + \tilde{\theta }^{\text{T}} \Upgamma^{ - 1} \dot{\tilde{\theta }} + 2\sum\limits_{i = 1}^{n} {{\frac{1}{{\mu_{i} }}}} \tilde{W}_{i}^{\text{T}} \dot{\tilde{W}} \\ & + x^{\text{T}} \left( {A^{\text{T}} R + RA - 2RBB^{\text{T}} R} \right)x + 2x^{\text{T}} RA_{1} x(t - \tau ) + 2x^{\text{T}} RBB^{\text{T}} Re + 2\hat{x}^{\text{T}} RB\tilde{W}^{\text{T}} \Upphi (\hat{x}) \\ & + 2e^{\text{T}} RB\tilde{W}^{\text{T}} \Upphi (\hat{x}) + 2x^{\text{T}} RB(\Uplambda + \varepsilon ) + 2e^{\text{T}} Rd + 2\hat{x}^{\text{T}} Rd \\ & + \beta_{1} x^{\text{T}} x - \beta_{1} x^{\text{T}} (t - \tau )x(t - \tau ) + \beta_{2} e^{\text{T}} e - \beta_{2} e^{\text{T}} (t - \tau )e(t - \tau ) \\ \end{aligned} $$
(27)

Considering Lemma 1, it is clear that

$$ 2e^{\text{T}} Pd \le \gamma_{1}^{ - 1} \left\| {e^{\text{T}} P} \right\|^{2} + \gamma_{1} \left\| d \right\|^{2} \le \gamma_{1}^{ - 1} \left\| {e^{\text{T}} P} \right\|^{2} + \gamma_{1} \rho^{\text{T}} \theta^{*} $$
(28)
$$ 2e^{\text{T}} Rd \le \gamma_{2}^{ - 1} \left\| {e^{\text{T}} R} \right\|^{2} + \gamma_{2} \left\| d \right\|^{2} \le \gamma_{2}^{ - 1} \left\| {e^{\text{T}} R} \right\|^{2} + \gamma_{2} \rho^{\text{T}} \theta^{*} $$
(29)
$$ 2\hat{x}^{\text{T}} Rd \le \gamma_{3}^{ - 1} \left\| {\hat{x}^{\text{T}} R} \right\|^{2} + \gamma_{3} \left\| d \right\|^{2} \le \gamma_{3}^{ - 1} \left\| {\hat{x}^{\text{T}} R} \right\|^{2} + \gamma_{3} \rho^{\text{T}} \theta^{*} $$
(30)

Considering (27), (28), (29) and (30), we have

$$ \begin{aligned} \dot{V} \le & e^{\text{T}} (PA + A^{\text{T}} P - 2C^{\text{T}} C)e + 2e^{\text{T}} PA_{1} e(t - \tau ) + 2e^{\text{T}} PB\tilde{W}^{\text{T}} \Upphi (\hat{x}) - 2e^{\text{T}} Pv \\ & + 2e^{\text{T}} PB(\Uplambda + \varepsilon ) + \tilde{\theta }^{\text{T}} \Upgamma^{ - 1} \dot{\tilde{\theta }} + 2\sum\limits_{i = 1}^{n} {{\frac{1}{{\mu_{i} }}}} \tilde{W}_{i}^{\text{T}} \dot{\tilde{W}} + x^{\text{T}} \left( {A^{\text{T}} R + RA - 2RBB^{\text{T}} R} \right)x \\ & + 2x^{\text{T}} RA_{1} x(t - \tau ) + 2x^{\text{T}} RBB^{\text{T}} Re + 2\hat{x}^{\text{T}} RB\tilde{W}^{\text{T}} \Upphi (\hat{x}) + 2e^{\text{T}} RB\tilde{W}^{\text{T}} \Upphi (\hat{x}) \\ & + 2x^{\text{T}} RB(\Uplambda + \varepsilon ) + \gamma_{1}^{ - 1} \left\| {e^{\text{T}} P} \right\|^{2} + (\gamma_{1} + \gamma_{2} + \gamma_{3} )\rho^{\text{T}} \theta^{*} + \gamma_{2}^{ - 1} \left\| {e^{\text{T}} R} \right\|^{2} + \gamma_{3}^{ - 1} \left\| {\hat{x}^{\text{T}} R} \right\|^{2} \\ & + \beta_{1} x^{\text{T}} x - \beta_{1} x^{\text{T}} (t - \tau )x(t - \tau ) + \beta_{2} e^{\text{T}} e - \beta_{2} e^{\text{T}} (t - \tau )e(t - \tau ) \\ \end{aligned} $$
(31)

Substituting (19), (20) and the expression of v into (31) yield

$$ \begin{aligned} \dot{V} \le & e^{\text{T}} (PA + A^{\text{T}} P - 2C^{\text{T}} C)e + 2e^{\text{T}} PA_{1} e(t - \tau ) + 2e^{\text{T}} PB(\Uplambda + \varepsilon ) + 2x^{\text{T}} RB(\Uplambda + \varepsilon ) \\ & + x^{\text{T}} \left( {A^{\text{T}} R + RA - 2RBB^{\text{T}} R} \right)x + 2x^{\text{T}} RA_{1} x(t - \tau ) + 2x^{\text{T}} RBB^{\text{T}} Re + (\gamma_{1} + \gamma_{2} + \gamma_{3} )\rho^{\text{T}} \theta^{*} \\ & - (\gamma_{ 1} + \gamma_{ 2} + \gamma_{3} )\rho^{\text{T}} \hat{\theta }(t) - \delta \tilde{\theta }^{\text{T}} \tilde{\theta } + (\gamma_{ 1} + \gamma_{ 2} + \gamma_{ 3} )\rho^{\text{T}} \tilde{\theta } \\ & - \delta \tilde{\theta }^{\text{T}} \theta^{*} + \beta_{1} x^{\text{T}} x - \beta_{1} x^{\text{T}} (t - \tau )x(t - \tau ) + \beta_{2} e^{\text{T}} e - \beta_{2} e^{\text{T}} (t - \tau )e(t - \tau ) \\ \end{aligned} $$
(32)

It is clear that the following fact is held:

$$ (\gamma_{1} + \gamma_{2} + \gamma_{3} )\rho^{\text{T}} \theta^{*} + (\gamma_{ 1} + \gamma_{ 2} + \gamma_{ 3} )\rho^{\text{T}} \tilde{\theta } = (\gamma_{ 1} + \gamma_{ 2} + \gamma_{ 3} )\rho^{\text{T}} \hat{\theta } $$
(33)

Thus, (32) can be rewritten as

$$ \begin{aligned} \dot{V} \le & e^{\text{T}} (PA + A^{\text{T}} P - 2C^{\text{T}} C)e + 2e^{\text{T}} PA_{1} e(t - \tau ) + 2e^{\text{T}} PB(\Uplambda + \varepsilon ) + 2x^{\text{T}} RB(\Uplambda + \varepsilon ) \\ & + x^{\text{T}} \left( {A^{\text{T}} R + RA - 2RBB^{\text{T}} R} \right)x + 2x^{\text{T}} RA_{1} x(t - \tau ) + 2x^{\text{T}} RBB^{\text{T}} Re \\ & - \delta \tilde{\theta }^{\text{T}} \tilde{\theta } - \delta \tilde{\theta }^{\text{T}} \theta^{*} + \beta_{1} x^{\text{T}} x - \beta_{1} x^{\text{T}} (t - \tau )x(t - \tau ) + \beta_{2} e^{\text{T}} e - \beta_{2} e^{\text{T}} (t - \tau )e(t - \tau ) \\ \end{aligned} $$
(34)

According to Lemma 1, we obtain

$$ e^{\text{T}} PB\Uplambda + \Uplambda^{\text{T}} B^{\text{T}} Pe \le \alpha_{1} e^{\text{T}} PBB^{\text{T}} Pe + \alpha_{1}^{ - 1} \Uplambda^{\text{T}} \Uplambda \le \alpha_{1} e^{\text{T}} PBB^{\text{T}} Pe + \alpha_{1}^{ - 1} \eta $$
(35)
$$ e^{\text{T}} PB\varepsilon + \varepsilon^{\text{T}} B^{\text{T}} Pe \le \alpha_{2} e^{\text{T}} PBB^{\text{T}} Pe + \alpha_{2}^{ - 1} \varepsilon^{\text{T}} \varepsilon \le \alpha_{2} e^{\text{T}} PBB^{\text{T}} Pe + \alpha_{2}^{ - 1} \varepsilon^{*} $$
(36)
$$ x^{\text{T}} RB\Uplambda + \Uplambda^{\text{T}} RB^{\text{T}} x \le \alpha_{3} x^{\text{T}} RBB^{\text{T}} Rx + \alpha_{3}^{ - 1} \Uplambda^{\text{T}} \Uplambda \le \alpha_{3} x^{\text{T}} RBB^{\text{T}} Rx + \alpha_{3}^{ - 1} \eta $$
(37)
$$ x^{\text{T}} RB\varepsilon + x^{\text{T}} B^{\text{T}} Re \le \alpha_{4} x^{\text{T}} RBB^{\text{T}} Rx + \alpha_{4}^{ - 1} \varepsilon^{\text{T}} \varepsilon \le \alpha_{4} x^{\text{T}} RBB^{\text{T}} Rx + \alpha_{4}^{ - 1} \varepsilon^{*} $$
(38)
$$ 2x^{\text{T}} RBB^{\text{T}} Re \le x^{\text{T}} RBB^{\text{T}} Rx + e^{\text{T}} RBB^{\text{T}} Re $$
(39)

where α i  > 0, i = 1, 2, 3, 4.

Considering the following fact

$$ - \delta \tilde{\theta }^{T} (t)\tilde{\theta }(t) + \delta \left\| {\tilde{\theta }} \right\|\left\| {\theta^{*} } \right\| \le - {\frac{\delta }{2}}\left\| {\tilde{\theta }(t)} \right\|^{2} + {\frac{\delta }{2}}\left\| {\theta^{*} } \right\|^{2} $$
(40)

and substituting (35)–(40) into (34) yield

$$ \begin{aligned} \dot{V} \le & e^{\text{T}} (PA + A^{\text{T}} P - 2C^{\text{T}} C)e + 2e^{\text{T}} PA_{1} e(t - \tau ) + \alpha_{1} e^{\text{T}} PBB^{\text{T}} Pe + \alpha_{1}^{ - 1} \eta \\ & + \alpha_{2} e^{\text{T}} PBB^{\text{T}} Pe + \alpha_{2}^{ - 1} \varepsilon^{*} + \alpha_{3} x^{\text{T}} RBB^{\text{T}} Rx + \alpha_{3}^{ - 1} \eta + \alpha_{4} x^{\text{T}} RBB^{\text{T}} Rx + \alpha_{4}^{ - 1} \varepsilon^{*} \\ & - {\frac{\delta }{2}}\left\| {\tilde{\theta }(t)} \right\|^{2} + {\frac{\delta }{2}}\left\| {\theta^{*} } \right\|^{2} + x^{\text{T}} \left( {A^{\text{T}} R + RA - 2RBB^{\text{T}} R} \right)x + 2x^{\text{T}} RA_{1} x(t - \tau ) + x^{\text{T}} RBB^{\text{T}} Rx \\ & + e^{\text{T}} RBB^{\text{T}} Re\, + \beta_{1} x^{T} x - \beta_{1} x^{\text{T}} (t - \tau )x(t - \tau ) + \beta_{2} e^{\text{T}} e - \beta_{2} e^{\text{T}} (t - \tau )e(t - \tau ) \\ \end{aligned} $$
(41)

Determining X = [xx(t − τ)]T, E = [ee(t − τ)]T, then (41) can be written as

$$ \dot{V} \le \alpha_{1}^{ - 1} \eta + \alpha_{2}^{ - 1} \varepsilon^{*} + \alpha_{3}^{ - 1} \eta + \alpha_{4}^{ - 1} \varepsilon^{*} - {\frac{\delta }{2}}\left\| {\tilde{\theta }(t)} \right\|^{2} + {\frac{\delta }{2}}\left\| {\theta^{*} } \right\|^{2} + X^{\text{T}} MX + E^{\text{T}} NE $$
(42)

where

$$ \begin{gathered} M = \left[ {\begin{array}{*{20}c} {A^{\text{T}} R + RA + \beta_{1} I - RBB^{\text{T}} R + \alpha_{3} RBB^{\text{T}} R + \alpha_{4} RBB^{\text{T}} R} & {RA_{1} } \\ {A_{1}^{T} R} & { - \beta_{1} I} \\ \end{array} } \right],\, \hfill \\ N = \left[ {\begin{array}{*{20}c} {A^{\text{T}} P + PA - 2C^{\text{T}} C + \beta_{2} I + \alpha_{1} PBB^{\text{T}} P + \alpha_{2} PBB^{\text{T}} P + RBB^{\text{T}} R} & {PA_{1} } \\ {A_{1}^{\text{T}} P} & { - \beta_{2} I} \\ \end{array} } \right] \hfill \\ \end{gathered} $$

From (24), we obtain

$$ \left[ {\begin{array}{*{20}c} {TA^{\text{T}} + AT + \beta_{1} TT - BB^{\text{T}} + \left( {\alpha_{3} + \alpha_{4} } \right)BB^{\text{T}} } & {A_{1} T} \\ {TA_{1}^{\text{T}} } & { - \beta_{1} I} \\ \end{array} } \right] < 0 $$
(43)

For (43), premultiplying and postmultiplying by diag{RI}, we have M < 0. From (25), we obtain N < 0.

When \( {\frac{\delta }{2}}\left\| {\tilde{\theta }(t)} \right\|^{2} - X^{\text{T}} MX - E^{\text{T}} NE > \alpha_{1}^{ - 1} \eta + \alpha_{2}^{ - 1} \varepsilon^{*} + \alpha_{3}^{ - 1} \eta + \alpha_{4}^{ - 1} \varepsilon^{*} + {\frac{\delta }{2}}\left\| {\theta^{*} } \right\|^{2} \), from (42), we obtain

$$ \dot{V} < 0 $$

According to (42), the semi-global uniform ultimate boundedness of estimate error of the state observer and states of the closed-loop system are guaranteed. This concludes the proof.

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Chen, M., Jiang, B., Jiang, Cs. et al. Robust control for a class of time-delay uncertain nonlinear systems based on sliding mode observer. Neural Comput & Applic 19, 945–951 (2010). https://doi.org/10.1007/s00521-010-0365-9

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