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

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.

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 = [x, x(t − τ)]^T, E = [e, e(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{R, I}, 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.