Extended state observer-based adaptive prescribed performance control for a class of nonlinear systems with full-state constraints and uncertainties

Abstract

In this paper, an extended state observer-based adaptive prescribed performance control technique is proposed for a class of nonlinear systems with full-state constraints and uncertainties. An extraordinary feature is that not only the control problem of prescribed performance tracking and full-state constraints are solved simultaneously, but also the parametric uncertainties and disturbances are considered, which will make it difficult to design a stable controller. For this purpose, the extended state observer and adaptive technique are integrated to obtain estimations of disturbances and parameters. Then, based on the combination of prescribed performance and barrier Lyapunov function, a novel backstepping control scheme is developed with feedforward compensation of parameters and disturbances to ensure that the tracking error is kept within a specified prescribed performance bound without violation of full states at all times. Moreover, the boundedness of all signals in the closed-loop system is proved and asymptotic tracking can be realized if the disturbances are time-invariant. Finally, two simulation examples are performed to highlight the efficiency of the proposed approach.

Appendices

Appendix 1

Proof of Theorem 1. If the disturbances d_i, i = 1,..., n, are time-invariant, the following positive definite Lyapunov function is defined with x_ei = d_i in this case

$$ V_{a} = V_{n} + \sum_{i = 1}^{n} {\frac{1}{2}\varepsilon_{i}^{T} P\varepsilon_{i} } + \frac{1}{2}\tilde{\theta }^{T} \Gamma^{ - 1} \tilde{\theta } $$

(41)

Differentiating V_n, substituting (5) into it yields

$$ \begin{aligned} \dot{V}_{a} & = \dot{V}_{n} + \sum_{i = 1}^{n} {\left( {\frac{1}{2}\dot{\varepsilon }_{i}^{T} P\varepsilon_{i} + \frac{1}{2}\varepsilon_{i}^{T} P\dot{\varepsilon }_{i} } \right)} - \tilde{\theta }^{T} \Gamma^{ - 1} \dot{\hat{\theta }} \\ &= \dot{V}_{n} - \tilde{\theta }^{T} \Gamma^{ - 1} \dot{\hat{\theta }} + \sum_{i = 1}^{n} {\left( {\frac{1}{2}\left( {\omega_{i} A\varepsilon_{i} + B_{1} \tilde{\theta }^{T} \varphi_{i} } \right)^{T} P\varepsilon_{i} + \frac{1}{2}\varepsilon_{i}^{T} P\left( {\omega_{i} A\varepsilon_{i} + B_{1} \tilde{\theta }^{T} \varphi_{i} } \right)} \right)} \\ &= \dot{V}_{n} - \tilde{\theta }^{T} \Gamma^{ - 1} \dot{\hat{\theta }} + \sum_{i = 1}^{n} {\left( {\frac{1}{2}\omega_{i} \varepsilon_{i}^{T} A^{T} P\varepsilon_{i} + \frac{1}{2}\omega_{i} \varepsilon_{i}^{T} PA\varepsilon_{i} + \varepsilon_{i}^{T} PB_{1} \tilde{\theta }^{T} \varphi_{i} } \right)} \\ \end{aligned} $$

(42)

As the matrix A is Hurwitz,$A^{T} P + PA = - 2I$ is established, noting (23)–(26), we have

$$ \begin{aligned} \dot{V}_{a}& \le - \sum_{j = 2}^{n} {\frac{{k_{j} z_{j}^{2} }}{{L_{j}^{2} - z_{j}^{2} }}} - \frac{{\rho^{ - 1} k_{1} z_{1}^{2} }}{{1 - z_{1}^{2} }} + \sum_{j = 2}^{n} {\frac{{z_{j} \tilde{\theta }^{T} \varphi_{j} \left( {\overline{x}_{j} } \right)}}{{L_{j}^{2} - z_{j}^{2} }}} { + }\sum_{k = 2}^{n} {\frac{{z_{k} \left( {d_{k} \left( {\overline{x}_{k} ,t} \right) - \hat{x}_{ek} } \right)}}{{L_{k}^{2} - z_{k}^{2} }}} \hfill \\ &\quad -\, \sum_{k = 2}^{n} {\frac{{z_{k} \dot{\alpha }_{{\left( {k - 1} \right)u}} }}{{L_{k}^{2} - z_{k}^{2} }}} - \sum_{k = 2}^{n} {\frac{{z_{k}^{2} \sum_{j = 1}^{k - 1} {\left( {\omega_{j} \frac{{\partial \alpha_{k - 1} }}{{\partial x_{j} }}} \right)^{2} } }}{{2\left( {L_{k}^{2} - z_{k}^{2} } \right)^{2} }}} - \sum_{k = 2}^{n} {\frac{{\omega_{k}^{2} z_{k}^{2} }}{{2\left( {L_{k}^{2} - z_{k}^{2} } \right)^{2} }}} + \frac{{\rho^{ - 1} z_{1} }}{{1 - z_{1}^{2} }}\tilde{\theta }^{T} \varphi_{1} \left( {\overline{x}_{1} } \right) \hfill \\ &\quad +\, \frac{{\rho^{ - 1} z_{1} }}{{1 - z_{1}^{2} }}\left( {d_{1} \left( {x_{1} ,t} \right) - \hat{x}_{e1} } \right) - \frac{{\omega_{1}^{2} \rho^{ - 2} z_{1}^{2} }}{{2\left( {1 - z_{1}^{2} } \right)^{2} }} - \tilde{\theta }^{T} \Gamma^{ - 1} \dot{\hat{\theta }} - \sum_{i = 1}^{n} {\omega_{i} \left\| {\varepsilon_{i} } \right\|^{2} } + \sum_{i = 1}^{n} {\varepsilon_{i}^{T} PB_{1} \tilde{\theta }^{T} \varphi_{i} } \hfill \\ & = - \sum_{j = 2}^{n} {\frac{{k_{j} z_{j}^{2} }}{{L_{j}^{2} - z_{j}^{2} }}} - \frac{{\rho^{ - 1} k_{1} z_{1}^{2} }}{{1 - z_{1}^{2} }} + \sum_{j = 2}^{n} {\frac{{z_{j} \tilde{\theta }^{T} \varphi_{j} \left( {\overline{x}_{j} } \right)}}{{L_{j}^{2} - z_{j}^{2} }}} + \frac{{\rho^{ - 1} z_{1} }}{{1 - z_{1}^{2} }}\tilde{\theta }^{T} \varphi_{1} \left( {\overline{x}_{1} } \right){ + }\sum_{k = 2}^{n} {\frac{{z_{k} \tilde{x}_{ek} \left( {\overline{x}_{k} ,t} \right)}}{{L_{k}^{2} - z_{k}^{2} }}} \hfill \\ &\quad -\, \sum_{k = 2}^{n} {\frac{{z_{k} \sum_{j = 1}^{k - 1} {\frac{{\partial \alpha_{k - 1} }}{{\partial x_{j} }}\left( {\tilde{\theta }^{T} \varphi_{j} \left( {\overline{x}_{j} } \right) + \tilde{x}_{ej} \left( {\overline{x}_{j} ,t} \right)} \right)} }}{{L_{k}^{2} - z_{k}^{2} }}} - \sum_{k = 2}^{n} {\frac{{z_{k}^{2} \sum_{j = 1}^{k - 1} {\left( {\omega_{j} \frac{{\partial \alpha_{k - 1} }}{{\partial x_{j} }}} \right)^{2} } }}{{2\left( {L_{k}^{2} - z_{k}^{2} } \right)^{2} }}} - \sum_{k = 2}^{n} {\frac{{\omega_{k}^{2} z_{k}^{2} }}{{2\left( {L_{k}^{2} - z_{k}^{2} } \right)^{2} }}} \hfill \\ &\quad +\, \frac{{\rho^{ - 1} z_{1} }}{{1 - z_{1}^{2} }}\tilde{x}_{e1} - \frac{{\omega_{1}^{2} \rho^{ - 2} z_{1}^{2} }}{{2\left( {1 - z_{1}^{2} } \right)^{2} }} - \tilde{\theta }^{T} \Gamma^{ - 1} \dot{\hat{\theta }} - \sum_{i = 1}^{n} {\omega_{i} \left\| {\varepsilon_{i} } \right\|^{2} } + \sum_{i = 1}^{n} {\varepsilon_{i}^{T} PB_{1} \tilde{\theta }^{T} \varphi_{i} } \hfill \\ \end{aligned} $$

(43)

Utilizing the Young’s inequality, we obtain

$$ \begin{aligned} \sum_{k = 2}^{n} {\frac{{z_{k} \tilde{x}_{ek} }}{{L_{k}^{2} - z_{k}^{2} }}} &\le \sum_{k = 2}^{n} {\frac{{\omega_{k}^{2} z_{k}^{2} }}{{2\left( {L_{k}^{2} - z_{k}^{2} } \right)^{2} }}} + \sum_{k = 2}^{n} {\frac{{\varepsilon_{k2}^{2} }}{2}} \hfill \\ \sum_{k = 2}^{n} {\frac{{z_{k} \sum_{j = 1}^{k - 1} {\frac{{\partial \alpha_{k - 1} }}{{\partial x_{j} }}\tilde{x}_{ej} \left( {\overline{x}_{j} ,t} \right)} }}{{L_{k}^{2} - z_{k}^{2} }}} &\le \sum_{k = 2}^{n} {\frac{{z_{k}^{2} \sum_{j = 1}^{k - 1} {\left( {\omega_{j} \frac{{\partial \alpha_{k - 1} }}{{\partial x_{j} }}} \right)^{2} } }}{{2\left( {L_{k}^{2} - z_{k}^{2} } \right)^{2} }}} + \sum_{k = 2}^{n} {\sum_{j = 1}^{k - 1} {\frac{{\varepsilon_{j2}^{2} }}{2}} } \hfill \\ \frac{{\rho^{ - 1} }}{{1 - z_{1}^{2} }}z_{1} \tilde{x}_{e1}& \le \frac{{\omega_{1}^{2} \rho^{ - 2} z_{1}^{2} }}{{2\left( {1 - z_{1}^{2} } \right)^{2} }} + \frac{{\varepsilon_{12}^{2} }}{2} \hfill \\ \end{aligned} $$

(44)

Substituting (28) and (44) into (43), then we have

$$ \begin{aligned} \dot{V}_{a} &\le - \sum_{j = 2}^{n} {\frac{{k_{j} z_{j}^{2} }}{{L_{j}^{2} - z_{j}^{2} }}} - \frac{{\rho^{ - 1} k_{1} z_{1}^{2} }}{{1 - z_{1}^{2} }} - \sum_{i = 1}^{n} {\omega_{i} \left\| {\varepsilon_{i} } \right\|^{2} } + \sum_{k = 2}^{n} {\sum_{j = 1}^{k - 1} {\frac{{\varepsilon_{j2}^{2} }}{2}} } + \sum_{k = 1}^{n} {\frac{{\varepsilon_{k2}^{2} }}{2}} \hfill \\ &\quad -\, \tilde{\theta }^{T} \left( {\Gamma^{ - 1} \dot{\hat{\theta }} - \sum_{j = 2}^{n} {\frac{{z_{j} \varphi_{j} \left( {\overline{x}_{j} } \right)}}{{L_{j}^{2} - z_{j}^{2} }}{ + }\sum_{k = 2}^{n} {\frac{{z_{k} \sum_{j = 1}^{k - 1} {\frac{{\partial \alpha_{k - 1} }}{{\partial x_{j} }}} \varphi_{j} \left( {\overline{x}_{j} } \right)}}{{L_{k}^{2} - z_{k}^{2} }} - \sum_{i = 1}^{n} {\varepsilon_{i}^{T} PB_{1} \varphi_{i} - \frac{{\rho^{ - 1} z_{1} }}{{1 - z_{1}^{2} }}\varphi_{1} \left( {\overline{x}_{1} } \right)} } } } \right) \hfill \\ &\le - \sum_{j = 2}^{n} {\frac{{k_{j} z_{j}^{2} }}{{L_{j}^{2} - z_{j}^{2} }}} - \frac{{\rho^{ - 1} k_{1} z_{1}^{2} }}{{1 - z_{1}^{2} }} - \sum_{i = 1}^{n} {\frac{{2\omega_{i} - n}}{2}\left\| {\varepsilon_{i} } \right\|^{2} } \hfill \\ &= - W \hfill \\ \end{aligned} $$

(45)

According to Lyapunov’s theorem, V_a is uniformly ultimately bounded; thus, errors z_i, $\tilde{\theta }$, and $\tilde{\varepsilon }$ are bounded. This further guarantees the boundedness of e₁. Moreover, the adaptive parameters $\hat{\theta }$ and $\hat{x}_{ei}$ are all bounded. As x₁ = e(t) + x_1d(t), $z_{1} = e\left( t \right)/\rho \left( t \right)$, $\left| {z_{1} } \right| \le 1$ with Assumption 2 and (8), we have $\left| {x_{1} } \right| \le c_{1}$, and x₁ is bounded. α₁ in (12) is a function of x₁, z₁,$\hat{\theta }$,$\dot{x}_{1d}$ and $\hat{x}_{e1}$. Since the boundedness of x₁, z₁, $\hat{\theta }$, $\dot{x}_{1d}$ and $\hat{x}_{e1}$, α₁ is bounded. As $\left| {x_{2} } \right| \le \left| {\alpha_{1} } \right|_{\max } + \left| {z_{2} } \right|$ and $\left| {z_{2} } \right| \le L_{2}$, we obtain $x_{2} \le c_{2}$ and α₂ is bounded. Similarly,$\left| {x_{i + 1} } \right|$, α_i, i = 3, …, n-1 and the control input u are bounded. Consequently, all signals in the closed-loop system are bounded, prescribed performance tracking is obtained and full states are ensured to remain in the constrained field.

Appendix 2

Proof of Theorem 2. If the disturbances d_i, i = 1,..., n, are time-variant, x_ei = $d_{i} \left( {\overline{x}_{i} ,t} \right) + \tilde{\theta }^{T} \varphi_{i} \left( {\overline{x}_{i} } \right)$. With (6), differentiating V_b defined in (29), we have

$$ \dot{V}_{b} = \dot{V}_{n} + \sum_{i = 1}^{n} {\left( {\frac{1}{2}\omega_{i} \varepsilon_{i}^{T} A^{T} P\varepsilon_{i} + \frac{1}{2}\omega_{i} \varepsilon_{i}^{T} PA\varepsilon_{i} + \varepsilon_{i}^{T} PB_{2} \frac{{h_{i} \left( t \right)}}{{w_{i} }}} \right)} $$

(46)

As $A^{T} P + PA = - 2I$, noting (23), (25) and (28), we have

$$ \begin{aligned} \dot{V}_{b} &\le \dot{V}_{n} - \sum_{i = 1}^{n} {\omega_{i} \left\| {\varepsilon_{i} } \right\|^{2} } + \sum_{i = 1}^{n} {\varepsilon_{i}^{T} PB_{2} \frac{{h_{i} \left( t \right)}}{{\omega_{i} }}} \hfill \\ &\le - \sum_{j = 2}^{n} {\frac{{k_{j} z_{j}^{2} }}{{L_{j}^{2} - z_{j}^{2} }}} - \frac{{\rho^{ - 1} k_{1} z_{1}^{2} }}{{1 - z_{1}^{2} }}{ + }\sum_{k = 2}^{n} {\frac{{z_{k} \left( {d_{k} \left( {\overline{x}_{k} ,t} \right) + \tilde{\theta }^{T} \varphi_{k} \left( {\overline{x}_{k} } \right) - \hat{x}_{ek} } \right)}}{{L_{k}^{2} - z_{k}^{2} }}} \hfill \\ &\quad -\, \sum_{k = 2}^{n} {\frac{{z_{k} \sum_{j = 1}^{k - 1} {\frac{{\partial \alpha_{k - 1} }}{{\partial x_{j} }}\left( {\tilde{x}_{ej} \left( {\overline{x}_{j} ,t} \right)} \right)} }}{{L_{k}^{2} - z_{k}^{2} }}} - \sum_{k = 2}^{n} {\frac{{z_{k}^{2} \sum_{j = 1}^{k - 1} {\left( {\omega_{j} \frac{{\partial \alpha_{k - 1} }}{{\partial x_{j} }}} \right)^{2} } }}{{2\left( {L_{k}^{2} - z_{k}^{2} } \right)^{2} }}} - \sum_{k = 2}^{n} {\frac{{\omega_{k}^{2} z_{k}^{2} }}{{2\left( {L_{k}^{2} - z_{k}^{2} } \right)^{2} }}} \hfill \\ &\quad+\, \frac{{\rho^{ - 1} z_{1} }}{{1 - z_{1}^{2} }}\left( {d_{1} \left( {x_{1} ,t} \right) + \tilde{\theta }^{T} \varphi_{1} \left( {\overline{x}_{1} } \right) - \hat{x}_{e1} } \right) - \frac{{\omega_{1}^{2} \rho^{ - 2} z_{1}^{2} }}{{2\left( {1 - z_{1}^{2} } \right)^{2} }} - \sum_{i = 1}^{n} {\omega_{i} \left\| {\varepsilon_{i} } \right\|^{2} } + \sum_{i = 1}^{n} {\varepsilon_{i}^{T} PB_{2} \frac{{h_{i} \left( t \right)}}{{\omega_{i} }}} \hfill \\ &\le - \sum_{j = 2}^{n} {\frac{{k_{j} z_{j}^{2} }}{{L_{j}^{2} - z_{j}^{2} }}} - \frac{{\rho^{ - 1} k_{1} z_{1}^{2} }}{{1 - z_{1}^{2} }} - \sum_{i = 1}^{n} {\omega_{i} \left\| {\varepsilon_{i} } \right\|^{2} } + \sum_{j = 2}^{n} {\sum_{k = 1}^{j - 1} {\frac{{\varepsilon_{k2}^{2} }}{2}} } + \frac{1}{2}\sum_{i = 1}^{n} {\left\| {\varepsilon_{i} } \right\|^{2} } + \sum_{i = 1}^{n} {\frac{{\left\| {\varepsilon_{i} } \right\|^{2} }}{2}} + \sum_{i = 1}^{n} {\frac{{\left\| {PB_{2} } \right\|^{2} \left| {h_{i} \left( t \right)} \right|^{2} }}{{2\omega_{i}^{2} }}} \hfill \\ &\le - \sum_{j = 2}^{n} {\frac{{k_{j} z_{j}^{2} }}{{L_{j}^{2} - z_{j}^{2} }}} - \frac{{\rho^{ - 1} k_{1} z_{1}^{2} }}{{1 - z_{1}^{2} }} - \sum_{i = 1}^{n} {\frac{{2\omega_{i} - n - 1}}{2}\left\| {\varepsilon_{i} } \right\|^{2} } + \sum_{i = 1}^{n} {\frac{{\left\| {PB_{2} } \right\|^{2} \left| {h_{i} \left( t \right)} \right|^{2} }}{{2\omega_{i}^{2} }}} \hfill \\ \end{aligned} $$

(47)

As $\log \frac{{L_{j}^{2} }}{{L_{j}^{2} - z_{j}^{2} }} \le \frac{{z_{j}^{2} }}{{L_{j}^{2} - z_{j}^{2} }}$ in the interval z_j < L_j [42], then

$$ \begin{gathered} \dot{V}_{b} \le - \sum_{j = 2}^{n} {k_{j} \log \frac{{L_{j}^{2} }}{{L_{j}^{2} - z_{j}^{2} }}} - \rho^{ - 1} k_{1} \log \frac{1}{{1 - z_{1}^{2} }} - \sum_{i = 1}^{n} {\frac{{2\omega_{i} - n - 1}}{{2\lambda_{\max } \left( P \right)}}\varepsilon_{i}^{T} P\varepsilon_{i} } + \sigma \hfill \\ \le - \lambda V_{a} + \sigma \hfill \\ \end{gathered} $$

(48)

which leads to (30). Similar to the proof of Theorem 1, all signals in the closed-loop system are also bounded and prescribed performance tracking is obtained without violation of constraints of the full states.