Nonlinear extended state observer-based adaptive higher-order sliding mode control for parallel antenna platform with input saturation

Abstract

This study introduces a robust trajectory tracking controller for a Stewart-type offshore antenna platform considering unmodeled uncertainties and external disturbances under input saturation. First, a finite-time convergent global fast sliding mode surface combining a nonsingular terminal sliding surface and an integral sliding surface is developed and assigned with an anti-windup factor. In addition, a nonlinear extended state observer is used to estimate and compensate for unknown disturbances and unmodeled uncertainties to improve system robustness. An adaptive higher-order sliding mode controller based on a super-twisting algorithm is designed to improve the robustness of the antenna platform control system while reducing sliding mode chattering by dynamically adjusting the control gain value. Finally, the Lyapunov theory is used to guarantee system stability in the presence of lumped uncertainties and input nonlinearity. The effectiveness of the proposed approach is verified by simulations. The integral of the absolute error of the proposed approach is ${78.0\%}$ lower than that of sliding mode controller based on PID sliding mode surface.

Appendix 1

Proof of Theorem 1

Let ${\mathbf{{u}}=\mathbf{{u}}_{smc}}$, and substitute Eq. (25) into (24), then, it can be written that:

$$\begin{aligned} \begin{aligned} {\dot{\textbf{S}}} =\,\,&{\mathbf{{u}}_{ST}} + {\mathbf{{z}}_3} - \mathbf{{d}}\\ = \,\,&{\mathbf{{u}}_{ST}} - {\mathbf{{e}}_3}. \end{aligned} \end{aligned}$$

(45)

According to the previous discussion, ${\mathbf{{e}}_3}$ is bounded and Lipschitz continuous, and its first derivative is also bounded, namely, $\left\| {{\mathbf{{e}}_3}} \right\| \le {\Delta _{e3}}$ and $\left\| {{{{\dot{\textbf{e}}}}_3}} \right\| \le {\Delta _{de3}}$, where ${\Delta _{e3}}$ and ${\Delta _{de3}}$ are small positive constants.

Next, substitute the control law based on the AST given by Eqs. (27) and (28), into Eq. (45):

$$\begin{aligned} \begin{aligned} {\dot{\textbf{S}}} =\,\,&- \lambda |\mathbf{{S}}{|^{\frac{1}{2}}}sign(\mathbf{{S}}) + {\varvec{\mu }} - {\mathbf{{e}}_3}\\ = \,\,&- \lambda |\mathbf{{S}}{|^{\frac{1}{2}}}sign(\mathbf{{S}}) + {{\varvec{\mu }_e}}, \end{aligned} \end{aligned}$$

(46)

where ${{\varvec{\mu }}_e} = {\varvec{\mu }} - {\mathbf{{e}}_3}$.

Then, it holds that:

$$\begin{aligned} \begin{aligned} {{{\varvec{{\dot{\mu }}}}_e}} =&{\varvec{{\dot{\mu }}}} - {{{\dot{\textbf{e}}}}_3}= - \eta sign(\mathbf{{S}}) - {{{\dot{\textbf{e}}}}_3}. \end{aligned} \end{aligned}$$

(47)

Next, we define the Lyapunov function ${V_{AST}}$ as follows:

$$\begin{aligned} {V_{AST}} = {V_{ST}} + \frac{1}{{2{\gamma _{ast1}}}}{(\lambda - {\lambda _0})^2} + \frac{1}{{2{\gamma _{ast2}}}}{(\eta - {\eta _0})^2},\nonumber \\ \end{aligned}$$

(48)

where

$$\begin{aligned} {V_{ST}}= & {} {\mathbf{{Z}}^T}{} \mathbf{{PZ}}, \end{aligned}$$

(49)

$$\begin{aligned} \mathbf{{Z}}= & {} {[\begin{array}{*{20}{c}} {{\mathbf{{Z}}_1}}&{{\mathbf{{Z}}_2}} \end{array}]^T}= {[\begin{array}{*{20}{c}} {|\mathbf{{S}}{|^{\frac{1}{2}}}sign(\mathbf{{S}})}&{{{\varvec{\mu }}_e}} \end{array}]^T}, \end{aligned}$$

(50)

$$\begin{aligned} \mathbf{{P}}= & {} \left[ {\begin{array}{*{20}{c}} {{\zeta _{ast}} + 4\varepsilon _{ast}^2}&{}{ - 2{\varepsilon _{ast}}}\\ { - 2{\varepsilon _{ast}}}&{}1 \end{array}} \right] , \end{aligned}$$

(51)

where ${\lambda _0}$, ${\eta _0}$, and ${\gamma _{ast2}}$ are positive constants.

Further, it can be written that:

$$\begin{aligned} \begin{aligned} {\dot{\textbf{Z}}} =\,\,&\frac{1}{{|{\mathbf{{Z}}_1}|}}\left[ {\begin{array}{*{20}{c}} { - \frac{1}{2}\lambda }&{}{\frac{1}{2}}\\ { - \eta }&{}0 \end{array}} \right] \mathbf{{Z}} + \left[ {\begin{array}{*{20}{c}} 0\\ { - {{{\dot{\textbf{e}}}}_3}} \end{array}} \right] \\ = \,\,&\frac{1}{{|{\mathbf{{Z}}_1}|}}{} \mathbf{{AZ}} + \mathbf{{B}}, \end{aligned} \end{aligned}$$

(52)

where $\mathbf{{A}} = \left[ {\begin{array}{*{20}{c}} { - \frac{1}{2}\lambda }&{}{\frac{1}{2}}\\ { - \eta }&{}0 \end{array}} \right]$, $\mathbf{{B}} = \left[ {\begin{array}{*{20}{c}} 0\\ { - {{{\dot{\textbf{e}}}}_3}} \end{array}} \right]$.

The derivative of ${V_{AST}}$ consists of two terms. The first term is the derivative of ${V_{ST}}$ that is given by Eq. (49), which can be expressed as follows:

(53)

where

$$\begin{aligned}{} & {} \mathbf{{W}} = \left[ {\begin{array}{*{20}{c}} 0&{}0\\ { - {\Delta _{de3}}}&{}0 \end{array}} \right] , \end{aligned}$$

(54)

$$\begin{aligned}{} & {} \mathbf{{Q}} = \left[ {\begin{array}{*{20}{c}} {\lambda ({\zeta _{ast}} + 4\varepsilon _{ast}^2) - 4{\varepsilon _{ast}}(\eta + {\Delta _{de3}})}&{}{(\eta + {\Delta _{de3}}) - \frac{1}{2}({\zeta _{ast}} + 4\varepsilon _{ast}^2) - {\varepsilon _{ast}}\lambda }\\ {(\eta + {\Delta _{de3}}) - \frac{1}{2}({\zeta _{ast}} + 4\varepsilon _{ast}^2) - {\varepsilon _{ast}}\lambda }&{}{2{\varepsilon _{ast}}} \end{array}} \right] . \end{aligned}$$

(55)

Next, substituting the second equation of (28) into (55) yields:

$$\begin{aligned} \mathbf{{Q}} = \left[ {\begin{array}{*{20}{c}} {\lambda {\zeta _{ast}} - 2{\varepsilon _{ast}}(2{\Delta _{de3}} + ({\zeta _{ast}} + 4\varepsilon _{ast}^2))}&{}{{\Delta _{de3}}}\\ {{\Delta _{de3}}}&{}{2{\varepsilon _{ast}}} \end{array}} \right] .\nonumber \\ \end{aligned}$$

(56)

Matrix $\textbf{Q}$ is positive definite when the following inequality holds:

$$\begin{aligned}{} & {} \lambda > \frac{{4\varepsilon _{ast}^2(2{\Delta _{de3}} + ({\zeta _{ast}} + 4\varepsilon _{ast}^2)) + \Delta _{de3}^2}}{{2{\varepsilon _{ast}}{\zeta _{ast}}}}. \end{aligned}$$

(57)

It is worth noting that:

$$\begin{aligned}{} & {} {\lambda _{min}}(\mathbf{{P}}){\left\| \mathbf{{Z}} \right\| ^2} \le {\mathbf{{Z}}^T}{} \mathbf{{PZ}} \le {\lambda _{\max }}(\mathbf{{P}}){\left\| \mathbf{{Z}} \right\| ^2}, \end{aligned}$$

(58)

$$\begin{aligned}{} & {} \left| {{\mathbf{{Z}}_1}} \right| \le \left\| \mathbf{{Z}} \right\| \le \frac{{{V^{\frac{1}{2}}}}}{{\lambda _{min}^{\frac{1}{2}}(\mathbf{{P}})}}. \end{aligned}$$

(59)

Accordingly, from Eq. (53),it can be deduced that:

$$\begin{aligned} \begin{aligned} {{\dot{V}}_{ST}} \le&- \frac{1}{{|{\mathbf{{Z}}_1}|}}{\mathbf{{Z}}^T}{} \mathbf{{QZ}}\\ \le&- \frac{1}{{|{\mathbf{{Z}}_1}|}}\frac{{{\lambda _{min}}(\mathbf{{Q}}){V_{ST}}}}{{{\lambda _{\max }}(\mathbf{{P}})}}\\ \le&- {c_{ST}}V_{ST}^{\frac{1}{2}}, \end{aligned} \end{aligned}$$

(60)

where ${{c_{ST}} = \frac{{{\lambda _{\min }}({\textbf{Q}})\lambda _{min}^{\frac{1}{2}}({\textbf{P}})}}{{{\lambda _{\max }}(\mathbf{{P}})}} > 0}$

Then, the derivative of ${V_{AST}}$ in Eq. (48) can be obtained by:

$$\begin{aligned} \begin{aligned} {{\dot{V}}_{AST}} \le&- {c_{ST}}V_{ST}^{\frac{1}{2}} + \frac{1}{{{\gamma _{ast1}}}}(\lambda - {\lambda _0}){\dot{\lambda }} + \frac{1}{{{\gamma _{ast2}}}}(\eta - {\eta _0}){\dot{\eta }} \\ =\,\,&- {c_{ST}}V_{ST}^{\frac{1}{2}} - \frac{{{\omega _{ast1}}}}{{\sqrt{2{\gamma _{ast1}}} }}|\lambda - {\lambda _0}| \\&- \frac{{{\omega _{ast2}}}}{{\sqrt{2{\gamma _{ast2}}} }}|\eta - {\eta _0}| + \frac{1}{{{\gamma _{ast1}}}}(\lambda - {\lambda _0})\dot{\lambda }\\&+ \frac{1}{{{\gamma _{ast2}}}}(\eta - {\eta _0}){\dot{\eta }}\\&+ \frac{{{\omega _{ast1}}}}{{\sqrt{2{\gamma _{ast1}}} }}|\lambda - {\lambda _0}| + \frac{{{\omega _{ast2}}}}{{\sqrt{2{\gamma _{ast2}}} }}|\eta - {\eta _0}|, \end{aligned} \end{aligned}$$

(61)

where ${\omega _{ast2}}$ and ${\gamma _{ast2}}$ are positive constants.

Considering the well-known inequality given by:

$$\begin{aligned} {({x^2} + {y^2} + {z^2})^{\frac{1}{2}}} \le |x| + |y| + |z|, \end{aligned}$$

(62)

parts of ${\dot{V}_{AST}}$ can be obtained as follows:

$$\begin{aligned}{} & {} - {c_{ST}}V_{ST}^{\frac{1}{2}} - \frac{{{\omega _{ast1}}}}{{\sqrt{2{\gamma _{ast1}}} }}|\lambda \nonumber \\{} & {} \quad - {\lambda _0}| - \frac{{{\omega _{ast2}}}}{{\sqrt{2{\gamma _{ast2}}} }}|\eta - {\eta _0}| \le - {c_m}{V_{AST}}, \end{aligned}$$

(63)

where ${c_m} = \min \{ {c_{ST}},{\omega _{ast1}},{\omega _{ast2}}\}$.

Then, the derivative of ${V_{AST}}$ in Eq. (61) can be rewritten as follows:

$$\begin{aligned} \begin{aligned} {{\dot{V}}_{AST}} \le&- {c_m}V_{ST}^{\frac{1}{2}} + \frac{1}{{{\gamma _{ast1}}}}(\lambda - {\lambda _0}){\dot{\lambda }} \\&+ \frac{1}{{{\gamma _{ast2}}}}(\eta - {\eta _0}){\dot{\eta }} + \frac{{{\omega _{ast1}}}}{{\sqrt{2{\gamma _{ast1}}} }}|\lambda - {\lambda _0}| \\&+ \frac{{{\omega _{ast2}}}}{{\sqrt{2{\gamma _{ast2}}} }}|\eta - {\eta _0}|. \end{aligned} \end{aligned}$$

(64)

The controller gains ${\lambda }$ and ${\eta }$ can be tuned using the adaptive law (28), so a reasonable assumption about the gains can be made, that is, both ${\lambda }$ and ${\eta }$ are bounded. The proof of this assumption is given in Sect. 4. Based on this assumption, it can inferred that there are positive constants ${\lambda _0}$ and ${\eta _0}$ satisfying the conditions of ${\lambda - \lambda _0 < 0}$ and ${\eta - \eta _0 < 0}$.

Then, Eq. (64) can be rewritten by:

$$\begin{aligned} \begin{aligned} {{\dot{V}}_{AST}} \le&- {c_m}V_{ST}^{\frac{1}{2}} - |\lambda - {\lambda _0}|(\frac{1}{{{\gamma _{ast1}}}}{\dot{\lambda }} \\&- \frac{{{\omega _{ast1}}}}{{\sqrt{2{\gamma _{ast1}}} }}) - |\eta - {\eta _0}|(\frac{1}{{{\gamma _{ast2}}}}{\dot{\eta }} \\&- \frac{{{\omega _{ast2}}}}{{\sqrt{2{\gamma _{ast2}}} }})\\ = \,\,&- {c_m}V_{ST}^{\frac{1}{2}} + \xi , \end{aligned} \end{aligned}$$

(65)

where

$$\begin{aligned} \xi= & {} - |\lambda - {\lambda _0}|(\frac{1}{{{\gamma _{ast1}}}}\dot{\lambda }- \frac{{{\omega _{ast1}}}}{{\sqrt{2{\gamma _{ast1}}} }}) - |\eta \nonumber \\{} & {} - {\eta _0}|(\frac{1}{{{\gamma _{ast2}}}}{\dot{\eta }} - \frac{{{\omega _{ast2}}}}{{\sqrt{2{\gamma _{ast2}}} }}). \end{aligned}$$

(66)

Next, the discussion can be divided into two cases.

Case 1 If $|\mathbf{{S}}| > {k_{ast}}$ and $\lambda > {\lambda _m}$, then from Eq. (28), it can obtained that:

$$\begin{aligned} {\dot{\lambda }} = {\omega _{ast1}}\sqrt{\frac{{{\gamma _{ast1}}}}{2}}. \end{aligned}$$

(67)

By selecting ${\varepsilon _{ast}} = \frac{{{\omega _{ast2}}}}{{{\omega _{ast1}}}}\sqrt{\frac{{{\gamma _{ast2}}}}{{{\gamma _{ast1}}}}}$, the derivative of the second equation in (28) can be deduced:

$$\begin{aligned} {\dot{\eta }} = {\varepsilon _{ast}}{\dot{\lambda }} = {\omega _{ast2}}\sqrt{\frac{{{\gamma _{ast2}}}}{2}}. \end{aligned}$$

(68)

After substituting Eqs. (67) and (68) into Eq. (66), term ${\xi }$ in Eq. (65) becomes ${\xi = 0}$ and is holds that:

$$\begin{aligned} {\dot{V}_{AST}} \le - {c_m}V_{ST}^{\frac{1}{2}}. \end{aligned}$$

(69)

The control gain ${\lambda }$ increases according to Eq. (67) to meet the condition defined by Eq. (57). Then, the finite-time convergence of the sliding mode surface in the domain of $|\mathbf{{S}}| \le {k_{ast}}$ can be guaranteed (Lemma 5 in [37]).

Case 2 If $|\mathbf{{S}}| < {k_{ast}}$, then from Eq. (28), it can obtained that:

$$\begin{aligned} {\dot{\lambda }} = \left\{ {\begin{array}{*{20}{c}} { - {\omega _{ast1}}\sqrt{\frac{{{\gamma _{ast1}}}}{2}} ,}&{}{\lambda > {\lambda _m}}\\ {{\rho _{ast}},}&{}{\lambda \le {\lambda _m}} \end{array}} \right. , \end{aligned}$$

(70)

and

$$\begin{aligned} \xi = \left\{ {\begin{array}{*{20}{l}} {2|\lambda - {\lambda _0}|\frac{{{\omega _{ast1}}}}{{\sqrt{2{\gamma _{ast1}}} }} + 2|\eta - {\eta _0}|\frac{{{\omega _{ast2}}}}{{\sqrt{2{\gamma _{ast2}}} }},}&{}{\lambda > {\lambda _m}}\\ { - |\lambda - {\lambda _0}|(\frac{{{\rho _{ast}}}}{{{\gamma _{ast1}}}} - \frac{{{\omega _{ast1}}}}{{\sqrt{2{\gamma _{ast1}}} }}) - |\eta - {\eta _0}|(\frac{{{\rho _{ast}}}}{{{\gamma _{ast2}}}} - \frac{{{\omega _{ast2}}}}{{\sqrt{2{\gamma _{ast2}}} }}),}&{}{\lambda \le {\lambda _m}} \end{array}} \right. .\nonumber \\ \end{aligned}$$

(71)

The first equation in Eq. (71) is positive, which guarantees the finite-time convergence of the sliding mode surface in the domain of $|\mathbf{{S}}| \le {k_{ast}}$ according to Lemma 7 in [37] when $\lambda > {\lambda _m}$. In the second equation in Eq. (71), the term ${\xi }$ is not positive definite when $\lambda \le {\lambda _m}$. However, it lasts only for a finite time, making the value of ${\lambda }$ to increase with the rate ${\rho _{ast}}$ according to the second equation in Eq. (70) until $\lambda > {\lambda _m}$. Then, the finite-time convergence of the sliding mode surface in the domain of $|\mathbf{{S}}| \le {k_{ast}}$ can be guaranteed again. In control gain adaptation, the sliding mode surface may leave the region $|\mathbf{{S}}| \le {k_{ast}}$ in a finite time. However, according to the analysis of Case 1, the sliding mode surface will converge to the region of $|\mathbf{{S}}| \le {k_{ast}}$ in a finite time. Therefore, the sliding mode surface can be guaranteed to stay within a larger domain defined by ${k_{ast}} \le |\mathbf{{S}}| \le {k_{astm}}$, where ${k_{astm}} > {k_{ast}}$.

Thus, the proof of theorem 1 is complete.