Global attitude stabilization of rigid spacecraft with unknown input delay

Abstract

Global attitude stabilization of a rigid spacecraft with unknown actuator delay time is an important problem that has rarely been studied. In this paper, first we investigate a Lyapunov-based controller for attitude regulation of a rigid spacecraft with delayed inputs. Simple conditions for global asymptotical stability are obtained by assuming that the true delay value is unknown, but approximation of its upper bound is available. It is also shown that a proper design of the controller prevents the unwinding phenomenon. Then, we extend the results for the system while taking disturbance effects into account. Based on Lyapunov–Krasovskii methodology, it is proven that the proposed controller can drive the closed-loop trajectories to a small region in the neighborhood of the origin in the presence of external disturbances and model uncertainties. Various numerical simulations are carried out to illustrate the effectiveness of the proposed control system.

Appendices

Appendix 1: Some useful inequalities

In what follows, we recall some well-known inequalities.

Jensen’s inequality: For any positive scalar $\gamma$ and a real-valued function f such that the considered integrations are well defined, we have

$$\begin{aligned} \left( \int _0^\gamma f(\eta )\, \mathrm {d}\eta \right) ^2 \le \gamma \int _0^\gamma f^2(\eta ) \, \mathrm {d}\eta . \end{aligned}$$

(26)

Cauchy–Schwarz inequality: Let $x, y \in \mathfrak {R}$. The Cauchy–Schwarz inequality says that

$$\begin{aligned} \Vert x+y\Vert ^2 \le (\Vert x\Vert + \Vert y\Vert )^2. \end{aligned}$$

(27)

Young’s inequality: Let $1 < p,q < \infty$ satisfy the constraint $1/p+1/q = 1$, and $x,y \in \mathfrak {R}_+$. Then

$$\begin{aligned} x y\le & {} \frac{1}{p} x^p + \frac{1}{q} y^q. \end{aligned}$$

An elementary case of Young’s inequality is

$$\begin{aligned} x y\le & {} \frac{1}{2\varepsilon ^2} x^2 + \frac{\varepsilon ^2}{2} y^2, \qquad \varepsilon \in \mathfrak {R}. \end{aligned}$$

From above inequalities, one can immediately follow that

$$\begin{aligned} \left( x+y\right) ^2&\le \frac{401}{400} x^2 + 401 y^2. \end{aligned}$$

(28)

$$\begin{aligned} \left( x+y\right) ^2&\le 2 x^2 + 2 y^2. \end{aligned}$$

(29)

$$\begin{aligned} \left( x+y\right) ^2&\ge \left( 1-\frac{1}{\varepsilon ^2}\right) x^2 +(1-\varepsilon ^2) y^2. \end{aligned}$$

(30)

Appendix 2: Extracting upper bound of $\Vert \varXi \Vert$

Let rewrite $\varXi$ as

$$\begin{aligned} \varXi = \varXi _1 + \varXi _2 \end{aligned}$$

(31)

where

$$\begin{aligned} \varXi _1 =&-J^{-1} S_{\alpha -P \sigma } J \left( \alpha - P \sigma \right) \end{aligned}$$

(32)

$$\begin{aligned} \varXi _2 =&P B (\alpha - P \sigma ), \qquad \varXi _1, \varXi _2 \in \mathfrak {R}^3. \end{aligned}$$

(33)

$\varXi _1$ can expand to

$$\begin{aligned} \varXi _1&= - \mathrm {diag}\left[ \frac{J_3-J_2}{J_1},\frac{J_1-J_3}{J_2},\frac{J_2-J_1}{J_3} \right] \sum _{j=1}^4 \varXi _{1_j} \end{aligned}$$

(34)

where

$$\begin{aligned} \varXi _{1_1}&= \mathrm {diag} \left[ \alpha _2, \alpha _3, \alpha _1\right] \left[ \alpha _3, \alpha _1, \alpha _2\right] ^T \\ \varXi _{1_2}&= -\mathrm {diag} \left[ P_2, P_3, P_1\right] \mathrm {diag}\left[ \sigma _2, \sigma _3, \sigma _1\right] \left[ \alpha _3, \alpha _1, \alpha _2\right] ^T \\ \varXi _{1_3}&= -\mathrm {diag} \left[ P_3, P_1, P_2\right] \mathrm {diag}\left[ \sigma _3, \sigma _1, \sigma _2\right] \left[ \alpha _2, \alpha _3, \alpha _1\right] ^T \\ \varXi _{1_4}&= \mathrm {diag} \left[ P_2, P_3, P_1\right] \mathrm {diag}\left[ P_3, P_1, P_2\right] \\&\quad \mathrm {diag}\left[ \sigma _2, \sigma _3, \sigma _1\right] \left[ \sigma _3, \sigma _1, \sigma _2\right] ^T. \end{aligned}$$

Property 2 implies $\Vert \mathrm {diag}\left[ \frac{J_3-J_2}{J_1},\frac{J_1-J_3}{J_2},\frac{J_2-J_1}{J_3} \right] \Vert \le 1$. On the other hand, Property 3 deduces $\Vert \sigma \Vert \le 1$. Hence, considering these inequalities and successive application of the triangular inequality, from (34), we obtain

$$\begin{aligned} \Vert \varXi _1\Vert \le \left( \Vert \alpha \Vert _\infty + 2 \bar{P} \right) \Vert \alpha \Vert + \bar{P}^2 \Vert \sigma \Vert . \end{aligned}$$

(35)

Also $\varXi _2$ satisfies

$$\begin{aligned} \Vert \varXi _2\Vert&\le \Vert P B \alpha \Vert +\Vert P\left( BP\sigma \right) \Vert \nonumber \\&\le \bar{P} \frac{1+\Vert \sigma \Vert ^2}{4}\left( \Vert \alpha \Vert +\bar{P}\Vert \sigma \Vert \right) \nonumber \\&\le \frac{1}{2}\bar{P} \left( \Vert \alpha \Vert +\bar{P}\Vert \sigma \Vert \right) \end{aligned}$$

(36)

where Eq. (8) in Property 1 is used to drive (36).

Finally, combining (35) and (36) yields (20).

Appendix 3: Proof of Theorem 2

Before proving Theorem 2, we introduce a function which plays a key role in the proof. Our approach lies on the application of an operator of new type that leads to develop a delay compensation control law. This algorithm may be seen as classical reduction technique [3] but as pointed in [34], there is a major difference between classical reduction approach and algorithms based on application of this operator. By this consideration, the function $\mathcal {O}(\sigma ,\alpha ):\mathfrak {R}^3\times \mathfrak {R}^3 \rightarrow \mathfrak {R}^3$ is defined by [21, 22, 34, 35]

$$\begin{aligned} \mathcal {O} = \alpha (t) + \int _{t-\tau }^t e^{\varrho \left( t-\tau -\eta \right) } J^{-1}T(\eta ) \, \mathrm {d}\eta \end{aligned}$$

(37)

and using Leibniz’s rule, its time derivative is obtained as

$$\begin{aligned} \dot{\mathcal {O}}&=\dot{\alpha } + e^{-\varrho \tau } J^{-1}T(t)-J^{-1}T(t-\tau ) \nonumber \\&\quad + \varrho \int _{t-\tau }^t e^{\varrho \left( t-\eta -\tau \right) } J^{-1}T(\eta )\, \mathrm {d}\eta . \end{aligned}$$

(38)

By substituting dynamics Eq. (18) and the control law (21) in Eq. (38), and also by adding and subtracting $\varrho \alpha$ to the referenced equation, we have

$$\begin{aligned} \dot{\mathcal {O}}&= \varXi + J^{-1} T(t-\tau ) + e^{-\varrho \tau } J^{-1}T(t)-J^{-1}T(t-\tau ) \nonumber \\&\quad + \varrho \int _{t-\tau }^t e^{\varrho \left( t-\eta -\tau \right) } J^{-1}T(\eta )\, \mathrm {d}\eta +\varrho \alpha -\varrho \alpha \nonumber \\&=\varrho \mathcal {O} - \left( 1-e^{\bar{\tau }-\tau }\right) \left[ \varrho \alpha - \varXi \right] . \end{aligned}$$

(39)

By the virtue of Krasovskii theorem (see [42, Theorem 3.7] or [16, Theorem 3.6]) and thanks to the negativity of $\varrho$ which is guaranteed by conditions (22) and (23), one can find that $\mathcal {O}$ is a solution of exponentially stable system with Lyapunov function $\mathcal {O}^T\mathcal {O}$. Thus, the existence of $\gamma _3\in \mathfrak {R}_+$ such that

$$\begin{aligned} \dot{\mathcal {O}}^T\mathcal {O} \le -\gamma _3 \Vert \mathcal {O}\Vert ^2 \end{aligned}$$

(40)

is guaranteed. This feature will be used in what follows.

Using Sepulchre–Janković–Kokotović approach [40] with the function $\mathcal {O}$, the L–K functional is constructed as

$$\begin{aligned} v(\sigma ,\alpha ) = v_0+\sum _1^5v_i \end{aligned}$$

(41)

where

$$\begin{aligned}&v_1(\sigma ) = \frac{1}{2\bar{\tau }} \int _{t-\bar{\tau }}^t\int _\lambda ^t\sigma ^T(\eta )P\sigma (\eta )\, \mathrm {d}\eta \, \mathrm {d}\lambda \nonumber \\&v_2(\alpha ) = \gamma _4 \int _{t-\bar{\tau }}^t\int _\lambda ^t \Vert \alpha (\eta )\Vert ^2\, \mathrm {d}\eta \, \mathrm {d}\lambda \nonumber \\&v_3(\sigma ) = \gamma _5 \int _{t-\bar{\tau }}^t \sigma (\eta )^TP\sigma (\eta )\, \mathrm {d}\eta \nonumber \\&v_4(\alpha ) = \gamma _5 \int _{t-\bar{\tau }}^t \Vert \alpha (\eta )\Vert ^2\, \mathrm {d}\eta \nonumber \\&v_5(\sigma , \alpha ) = \frac{1+802\bar{\tau }\gamma _4}{2\gamma _3} \Vert \mathcal {O}\Vert ^2 \nonumber \end{aligned}$$

in which $\gamma _4 = \left( 22.545 \bar{\tau }^3 \bar{P}^4/\underline{P}\right) ^{-1}$ and $\gamma _5 \in \mathfrak {R}_+$ is a constant which will be defined further ahead.

The derivatives of $v_i,\, i=0, 1, \ldots , 5$ are obtained as

$$\begin{aligned} \dot{v}_0= & {} -\sigma (t)^TP\sigma (t) + \sigma ^T(t) \alpha (t) \end{aligned}$$

(42)

$$\begin{aligned} \dot{v}_1= & {} \frac{1}{2} \sigma (t)^T P \sigma (t)- \frac{1}{2\bar{\tau }} \int _{t-\bar{\tau }}^t \sigma (\eta )^TP\sigma (\eta ) \, \mathrm {d}\eta \end{aligned}$$

(43)

$$\begin{aligned} \dot{v}_2= & {} \gamma _4\bar{\tau } \Vert \alpha \Vert ^2-\gamma _4 \int _{t-\bar{\tau }}^t \Vert \alpha (\eta )\Vert ^2\, \mathrm {d}\eta \end{aligned}$$

(44)

$$\begin{aligned} \dot{v}_3= & {} \gamma _5 \left[ \sigma (t)^TP\sigma (t) -\sigma (t-\bar{\tau })^T P\sigma (t-\bar{\tau }) \right] \end{aligned}$$

(45)

$$\begin{aligned} \dot{v}_4= & {} \gamma _5 \left[ \Vert \alpha (t)\Vert ^2 -\Vert \alpha (t-\bar{\tau })\Vert ^2 \right] \end{aligned}$$

(46)

$$\begin{aligned} \dot{v}_5= & {} \frac{1+802\bar{\tau }\gamma _4}{\gamma _3} \dot{\mathcal {O}}^T \mathcal {O}. \end{aligned}$$

(47)

To show negativity of v, we shall establish the inequalities obtain from Eqs. (42)–(44) and (47).

Since $\Vert \sigma (t)\Vert ^2\le 1/\underline{P} \sigma (t)^TP\sigma (t)$ for all $t>0$, we get

$$\begin{aligned} \dot{v}_0&\le -\sigma (t)^TP\sigma (t) + \Vert \sigma (t)\Vert \Vert \alpha (t)\Vert \nonumber \\&\le -\sigma (t)^TP\sigma (t) + \sqrt{\frac{1}{\underline{P}}\sigma (t)^TP\sigma (t)} \Vert \alpha \Vert . \end{aligned}$$

(48)

Definition of function $\mathcal {O}$ allows us to have the following inequality

$$\begin{aligned} \Vert \alpha (t)\Vert&\le \int _{t-\tau }^t e^{\varrho \left( t-\tau -\eta \right) } \Vert J^{-1}T(\eta )\Vert \, \mathrm {d}\eta +\Vert \mathcal {O}\Vert \\&\le \sup _\tau \left[ \int _{t-\tau }^t e^{\varrho \left( t-\eta +\bar{\tau }-{\tau }\right) }\Vert \left[ \varXi (\eta ) - \varrho \alpha \right] \Vert \,\mathrm {d}\eta \right] \\&\quad + \Vert \mathcal {O}\Vert . \end{aligned}$$

This inequality together with the inequality (20) and the negativity of $\varrho$ implies that

$$\begin{aligned} \Vert \alpha (t)\Vert&\le \int _{t-\bar{\tau }}^t e^{\varrho \bar{\tau }}\Vert \left[ \varXi (\eta ) - \varrho \alpha \right] \Vert \,\mathrm {d}\eta + \Vert \mathcal {O}\Vert \nonumber \\&\le \int _{t-\bar{\tau }}^t \left( \gamma _1 \Vert \sigma (\eta )\Vert + \gamma _6 \Vert \alpha (\eta )\Vert \right) \, \mathrm {d}\eta + \Vert \mathcal {O}\Vert \end{aligned}$$

(49)

where $\gamma _6$ =$\gamma _2-\varrho$.

Using Jensen’s inequality (26) in conjunction with Cauchy–Schwarz inequality (27), and Young’s inequalities which are specialized for our needs, namely (28) and (29), we conclude

$$\begin{aligned} \Vert \alpha (t)\Vert ^2&\le \left( \int _{t-\bar{\tau }}^t \left( \gamma _1 \Vert \sigma (\eta )\Vert + \gamma _6 \Vert \alpha (\eta )\Vert \right) \, \mathrm {d}\eta + \Vert \mathcal {O}\Vert \right) ^2 \nonumber \\&\le \frac{401}{400}\left( \int _{t-\bar{\tau }}^t \left( \gamma _1 \Vert \sigma (\eta )\Vert + \gamma _6 \Vert \alpha (\eta )\Vert \right) \, \mathrm {d}\eta \right) ^2 \nonumber \\&\quad + 401 \Vert \mathcal {O}\Vert ^2 \nonumber \\&\le \frac{401}{400}\bar{\tau }\int _{t-\bar{\tau }}^t \left( \gamma _1 \Vert \sigma (\eta )\Vert + \gamma _6 \Vert \alpha (\eta )\Vert \right) ^2 \, \mathrm {d}\eta \nonumber \\&\quad + 401 \Vert \mathcal {O}\Vert ^2 \nonumber \\&\le \frac{401}{200}\bar{\tau }\int _{t-\bar{\tau }}^t \left( \gamma _1^2 \Vert \sigma (\eta )\Vert ^2 + \gamma _6^2 \Vert \alpha (\eta )\Vert \right) \, \mathrm {d}\eta \nonumber \\&\quad + 401 \Vert \mathcal {O}\Vert ^2 \nonumber \\&\le \frac{401}{200}\bar{\tau }\int _{t-\bar{\tau }}^t \left( \frac{\gamma _1^2}{\underline{P}}\sigma ^T(\eta )P\sigma (\eta ) +\gamma _6^2\Vert \alpha (\eta )\Vert ^2 \right) \, \mathrm {d}\eta \nonumber \\&\quad + 401 \Vert \mathcal {O}\Vert ^2. \end{aligned}$$

(50)

Consequently, by combining (50) with (44), we deduce that the following inequality holds

$$\begin{aligned} \dot{v}_2= & {} \gamma _4 \left( -\bar{\tau } \Vert \alpha \Vert ^2- \int _{t-\bar{\tau }}^t \Vert \alpha (\eta )\Vert ^2\, \mathrm {d}\eta +2 \bar{\tau } \Vert \alpha \Vert ^2 \right) \nonumber \\\le & {} \gamma _4 \left( -\bar{\tau } \Vert \alpha \Vert ^2 + \left( 4.01\bar{\tau }^2\gamma _6^2 -1\right) \int _{t-\bar{\tau }}^t \Vert \alpha (\eta )\Vert ^2\, \mathrm {d}\eta \right. \nonumber \\&\left. + \,4.01 \bar{\tau }^2 \frac{\gamma _1^2}{\underline{P}} \int _{t-\bar{\tau }}^t \sigma ^TP\sigma \, \mathrm {d}\eta + 802 \bar{\tau } \Vert \mathcal {O}\Vert ^2\right) \!.\nonumber \\ \end{aligned}$$

(51)

The functional v is chosen such that its time derivative along the trajectories is negative definite. Hence, this implies that a value for the $\Vert \alpha (t)\Vert _\infty$ occurs over the initial condition interval, i.e., $\Vert \alpha (t)\Vert _\infty =\Vert \alpha (0)\Vert _\infty$. Bearing this point in mind, the condition (22) imposes that $4.01\bar{\tau }^2\gamma _6^2 -1 \le -0.4\bar{\tau }^2\gamma _6^2$. Combining (51) and the previous inequality, we get

$$\begin{aligned} \dot{v}_2\le & {} \gamma _4 \Big ( -\bar{\tau } \Vert \alpha \Vert ^2 -0.4 \bar{\tau }^2\gamma _6^2 \int _{t-\bar{\tau }}^t \Vert \alpha (\eta )\Vert ^2\, \mathrm {d}\eta \nonumber \\&+\,4.01 \bar{\tau }^2 \frac{\gamma _1^2}{\underline{P}} \int _{t-\bar{\tau }}^t \sigma ^TP\sigma \, \mathrm {d}\eta +802\bar{\tau } \Vert \mathcal {O}\Vert ^2 \Big ).\nonumber \\ \end{aligned}$$

(52)

Next, we need to extract an inequality for $\dot{v}_5$ which is useful for our purpose. This is immediately done by considering the inequality (40), so

$$\begin{aligned} \dot{v}_5 \le -\left( 1+802\bar{\tau }\gamma _4 \right) \Vert \mathcal {O}\Vert ^2. \end{aligned}$$

(53)

Consequently, combining the inequalities (43), (45), (46), (48), (52), and (53) together and with some standard algebraic manipulations, we conclude that, for all $t\ge 0$

$$\begin{aligned} \dot{v}\le & {} -\frac{1}{2} \sigma ^T P \sigma + \sqrt{\frac{1}{\underline{P}}\sigma ^TP\sigma } \Vert \alpha \Vert -\gamma _4 \bar{\tau } \Vert \alpha \Vert ^2 \\&\; -\, 0.4 \bar{\tau }^2\gamma _4 \gamma _6^2 \int _{t-\bar{\tau }}^t \Vert \alpha \Vert ^2\, \mathrm {d}\eta \\&\; -\, \left( \frac{1}{2\bar{\tau }}-4.01 \bar{\tau }^2 \gamma _4 \frac{\gamma _1^2}{\underline{P}}\right) \int _{t-\bar{\tau }}^t \sigma ^TP\sigma \, \mathrm {d}\eta -\Vert \mathcal {O}\Vert ^2 \\&\; +\, \dot{v}_3 + \dot{v}_4. \end{aligned}$$

The above inequality can be rewritten as

$$\begin{aligned} \dot{v}\le & {} - \left[ \begin{array}{cc} \sqrt{\sigma ^TP\sigma }&\Vert \alpha \Vert \end{array} \right] \mathcal {N} \left[ \begin{array}{cc} \sqrt{\sigma ^TP\sigma }&\Vert \alpha \Vert \end{array} \right] ^T \nonumber \\&\; -\, 0.4 \bar{\tau }^2\gamma _4 \gamma _6^2 \int _{t-\bar{\tau }}^t \Vert \alpha \Vert ^2\, \mathrm {d}\eta \nonumber \\&\; -\, \frac{50}{501\bar{\tau }} \int _{t-\bar{\tau }}^t \sigma ^TP\sigma \, \mathrm {d}\eta -\Vert \mathcal {O}\Vert ^2 \nonumber \\&\; + \,\dot{v}_3 + \dot{v}_4, \end{aligned}$$

(54)

where $\mathcal {N} = \left[ \begin{array}{ll} \frac{1}{2} &{} -\frac{1}{2\sqrt{\underline{P}}} \\ -\frac{1}{2\sqrt{\underline{P}}} &{} \gamma _4 \bar{\tau } \end{array} \right]$.

It is clear that the first term of the right-hand side of the inequality (54) is negative definite, if $\mathcal {N}$ is positive definite matrix, i.e., we have

$$\begin{aligned} \frac{1}{2}\bar{\tau }\gamma _4-\frac{1}{4\underline{P}}>0 \rightarrow \frac{\bar{P}^4}{\underline{P}^2} < \frac{1}{1.5^2 5.01 \bar{\tau }^2} \end{aligned}$$

(55)

which implies that the inequality in condition (24) must be satisfied. Furthermore, it is easy to prove that there exists a positive constat $\gamma _5$ such that $\lambda ({\mathcal {N}})> 2\gamma _5$ (for instance $4\gamma _5 = 0.5 + \bar{\tau }\gamma _4 - \sqrt{0.25\left( 1+ \underline{P}\right) +\tau ^2\gamma _4^2-\tau \gamma _4}$); so

$$\begin{aligned} -\left[ \begin{array}{cc} \sqrt{\sigma ^TP\sigma }&\Vert \alpha \Vert \end{array} \right] \mathcal {N}&\left[ \begin{array}{cc} \sqrt{\sigma ^TP\sigma }&\Vert \alpha \Vert \end{array} \right] ^T \le \\&- 2\gamma _5\left( \sigma ^TP\sigma + \Vert \alpha \Vert ^2\right) . \end{aligned}$$

Thus, by recalling the inequality (54), we conclude that for all $t\ge 0$, the following inequality holds

$$\begin{aligned} \dot{v} \le&-\gamma _5\sigma ^T P \sigma - \gamma _5 \Vert \alpha \Vert ^2 - 0.4 \bar{\tau }^2\gamma _4 \gamma _6^2 \int _{t-\bar{\tau }}^t \Vert \alpha \Vert ^2\, \mathrm {d}\eta \nonumber \\&- \frac{50}{501\bar{\tau }} \int _{t-\bar{\tau }}^t \sigma ^TP\sigma \, \mathrm {d}\eta -\Vert \mathcal {O}\Vert ^2. \end{aligned}$$

(56)

Notice that, however, we prove the negativity of $\dot{v}$ along the trajectories, but still we are not allowed to apply L–K theorem [31, Theorem2.6]. For utilizing this theorem and establishing the uniformly asymptotically stability of the closed-loop system, we also need to show that there exist $\kappa _i \in \mathcal {K}_\infty ,\, i=1,2$ such that

$$\begin{aligned}&\kappa _1(\Vert \sigma ^T,\alpha ^T\Vert )\le v\left( \sigma ,\alpha \right) \nonumber \\&\qquad \qquad \le \kappa _2 \left( \sup _{-\bar{\tau }\le s \le 0} \Vert \sigma ^T(t+s),\alpha ^T(t+s)\Vert \right) . \end{aligned}$$

(57)

To this end, since

$$\begin{aligned} v_1&\le \frac{1}{2\bar{\tau }} \bar{\tau } \sup _{\lambda \in [t-\bar{\tau }, t]}\left[ \int _\lambda ^t\sigma ^TP\sigma \, \mathrm {d}\eta \right] = \frac{1}{2}\int _{t-\bar{\tau }}^t\sigma ^TP\sigma \, \mathrm {d}\eta \end{aligned}$$

and

$$\begin{aligned} v_2&\le \gamma _4 \bar{\tau } \sup _{\lambda \in [t-\bar{\tau }, t]}\left[ \int _\lambda ^t \Vert \alpha \Vert ^2\, \mathrm {d}\eta \right] =\gamma _4 \bar{\tau } \int _{t-\bar{\tau }}^t \Vert \alpha \Vert ^2\, \mathrm {d}\eta \end{aligned}$$

from the definition of v, the following inequality is concluded

$$\begin{aligned} v \le&v_0 + \left( \frac{1}{2} + \gamma _5 \right) \int _{t-\bar{\tau }}^t\sigma ^TP\sigma \, \mathrm {d}\eta \nonumber \\ \;\quad&+ \left( \bar{\tau }\gamma _4 + \gamma _5 \right) \int _{t-\bar{\tau }}^t \Vert \alpha (\eta )\Vert ^2\, \mathrm {d}\eta \nonumber \\&+ \frac{1+802\bar{\tau }\gamma _4}{2\gamma _3} \Vert \mathcal {O}\Vert ^2 \end{aligned}$$

(58)

which guarantees the existence of $\kappa _2\in \mathcal {K}_\infty$ such that $v \le \kappa _2$.

The only thing that remains to be shown is a existence of $\kappa _1$ which satisfy (57).

The definition of the functional v implies that

$$\begin{aligned} v \ge v_0 + v_3 + v_4 +v_5. \end{aligned}$$

Considering above inequality, through lengthy but simple calculations and similar procedure which was done to obtain Eq. (50) with utilizing inequality (30), we obtain

$$\begin{aligned} v\ge & {} v_0 + v_3 + v_4 +\frac{1+802 \bar{\tau }\gamma _4}{\gamma _3}\left[ \left( 1 - \epsilon ^2\right) \Vert \alpha \Vert ^2 + \varUpsilon \right] \nonumber \\ \end{aligned}$$

(59)

where

$$\begin{aligned} \varUpsilon&= 2\tau \left( 1-1/\epsilon ^2\right) e^{2\varrho (\bar{\tau }-\tau )} \\&\,\left[ \left( 2\gamma _2^2 + \varrho ^2 \right) \int _{t-\tau }^t \Vert \alpha \Vert ^2 \, \mathrm {d}\eta + 2\gamma _1^2/\underline{P}\right. \\&\quad \left. \int _{t-\tau }^t \sigma ^T P\sigma \, \mathrm {d}\eta \right] \end{aligned}$$

and $\epsilon \in (-1, 1)$.

we observe that the $\varUpsilon$ is a negative functional which is not suitable for our purpose, but this problem is solved efficiently by proper selecting of $\epsilon$ and considering $v_3$ and $v_4$ functionals (at this point, the reason of introducing two functionals $v_3$ and $v_4$ in v is clarified). So, it follows that

$$\begin{aligned} v \ge v_0 + \frac{1+802 \bar{\tau }\gamma _4}{\gamma _3} \left( 1 - \epsilon ^2 \right) \Vert \alpha \Vert ^2 \quad \forall t\ge 0.\qquad \end{aligned}$$

(60)

Since the right-hand side of (60) is a positive definite radially unbounded function, we are able to state that there exists $\kappa _1$ such that $\kappa _1 \le v$. Therefore, according to the L–K theorem, we prove that the origin of the attitude system with control law (21) is UGAS. This completes the proof.

Appendix 4: Proof of Theorem 3

To show that the resulting closed-loop system with the proposed control law is ISS with respect to the disturbance $T_\mathrm{d}$, we just need to show that the functional (41) is an ISS-LKF for the considered system.

Recalling that what we have done in the proof of Theorem 2, from (56), we conclude that

$$\begin{aligned} \dot{v} \le&-\gamma _5 \sigma ^T P \sigma -\gamma _5 \Vert \alpha \Vert ^2 -\gamma _7 \int _{t-\bar{\tau }}^t \Vert \alpha \Vert ^2\, \mathrm {d}\eta \\ \,&- \frac{50}{501\bar{\tau }} \int _{t-\bar{\tau }}^t \sigma ^TP\sigma \, \mathrm {d}\eta -\Vert \mathcal {O}\Vert ^2 + \gamma _8 \mathcal {O}^T J^{-1} T_\mathrm{d}(t). \end{aligned}$$

where $\gamma _7 = 0.4 \bar{\tau }^2\gamma _4 \gamma _6^2$ and $\gamma _8 =\left( 1+802\bar{\tau }\gamma _4\right) /\gamma _3$.

By the inequality

$$\begin{aligned} \gamma _8 \mathcal {O}^T J^{-1} T_d(t) \le 1/2 \Vert \mathcal {O}\Vert ^2 + \gamma ^2_8/2 \Vert J^{-1}T_d\Vert ^2 \end{aligned}$$

it follows that for all $t\ge 0$, we have $\dot{v} \le - \varGamma (x,\alpha ) + \gamma ^2_8/2 \Vert J^{-1}T_d\Vert ^2$ where

$$\begin{aligned} \varGamma&= \gamma _5 \sigma ^T P \sigma + \gamma _5 \Vert \alpha \Vert ^2 + \gamma _7 \int _{t-\bar{\tau }}^t \Vert \alpha \Vert ^2\, \mathrm {d}\eta \\ \,&+ \frac{50}{501\bar{\tau }} \int _{t-\bar{\tau }}^t \sigma ^TP\sigma \, \mathrm {d}\eta +\frac{1}{2}\Vert \mathcal {O}\Vert ^2. \end{aligned}$$

The positive definiteness of $\varGamma$ implies that there exists a class $\mathcal {K}_\infty$ function $\kappa _3$ such that $\dot{v} \le -\kappa _3 + \gamma ^2_8/2 \Vert J^{-1} T_d\Vert ^2$. This inequality and the positive definiteness condition (57) imply that v is an ISS-LKF. Finally, from Definition 1, we deduce that the system is ISS with respect to $T_d$.