Robust Stabilization of a Class of Underactuated Mechanical Systems of 2 DOF via Continuous Higher-Order Sliding-Modes

Abstract

In this chapter we design robust controllers for a Class of underactuated mechanical systems of two DOF, using a continuous Higher Order Sliding-Mode strategy. Two kinds of controller designs are presented: One generates a fifth-order sliding-mode and achieves Local Finite-Time Stability (LFTS). The other one is a robust controller that provides Global Asymptotic Stability (GAS). These controllers compensate matched Lipschitz disturbances and/or uncertainties, cope with an uncertain control coefficient and generate a continuous control signal, possibly reducing the chattering effect. We provide evidence of the performance of the controllers using simulations for the Reaction Wheel Pendulum (RWP) and the Translational Oscillator with Rotational Actuator (TORA) systems, and by means of experiments on the RWP system.

Appendices

Appendix 1: Homogeneity

Let vector $x \in \mathbb {R}^n$, its dilation operator is defined $\varDelta _\epsilon ^\mathbf{r }: = (\epsilon ^{r_1} x_1, \ldots , \epsilon ^{r_n} x_n)$, $\forall \epsilon > 0$, where $r_i > 0$ are the weights of the coordinates and $\mathbf{r}$ is the vector of weights. A function $V: \mathbb {R}^n \rightarrow \mathbb {R}$ (respectively, a vector field $f: \mathbb {R}^n \rightarrow \mathbb {R}^n$, or vector-set $F(x) \subset \mathbb {R}^n$) is called $\mathbf{r}$-homogeneous of degree $m \in \mathbb {R}$ if the identity $V(\varDelta _\epsilon ^\mathbf{r }) = \epsilon ^{m}V(x)$ holds (or $f(\varDelta _\epsilon ^\mathbf{r } x) = \epsilon ^{m} \varDelta _\epsilon ^\mathbf{r } f(x)$, $F(\varDelta _\epsilon ^\mathbf{r } x) = \epsilon ^{m} \varDelta _\epsilon ^\mathbf{r } F(x)$) [5, 33].

Suppose that the vector $\mathbf{r}$ is fixed. The homogeneous norm is defined by $\left| \left| x \right| \right| _\mathbf{r , p} := \Big (\sum _{i = 1}^{n} \left| x_i \right| ^{\frac{p}{r_i}} \Big )^{\frac{1}{p}}$, $\forall x \in \mathbb {R}^n$, for any $p \ge 1$ and the set $S = \{x \in \mathbb {R}^n: \left| \left| x \right| \right| _\mathbf{r ,p} = 1\}$ is the homogeneous unit sphere [5].

1.1 Some Properties of Homogeneous Functions

Consider $V_1$ and $V_2$ two $\mathbf{r}$-homogeneous functions (respectively, a vector field $f_1$) of degree $m_1$, $m_2$ (and $l_1$), then: (i) $V_1 V_2$ is homogeneous of degree $m_1 + m_2$, (ii) there exist a constant $c_1 > 0$, such that $V_1 \le c_1 \left| \left| x \right| \right| _\mathbf{r ,p}^{m_1}$, moreover, if $V_1$ is positive definite, there exists $c_2$ such that $V_1 \ge c_2 \left| \left| x \right| \right| _\mathbf{r , p}^{m_1}$, (iii) $\partial V_1(x)/ \partial x_i$ is homogeneous of degree $m_1 - r_i$, (iv) $L_f V_1(x)$ is homogeneous of degree $m_1 + l_1$ [5].

The following result is well known for continuous homogeneous functions (see [16, Theorem 4.4] or [2, 33]), and can be extended to semi-continuous functions [9].

Lemma 1

Let $\eta : \mathbb {R}^n \rightarrow \mathbb {R}$ and $\gamma : \mathbb {R}^n \rightarrow \mathbb {R}$ be two $\mathbf{r}$-homogeneous and upper semi-continuous single-valued functions, with the same weights $\mathbf{r} = (r_1, \ldots , r_n)$ and homogeneity degree $m>0$. Suppose that $\gamma (x) \le 0$ in $\mathbb {R}^n$. If

$$\begin{aligned} \{x \in \mathbb {R}^n \setminus \{0\}: \gamma (x) = 0 \} \subseteq \{x \in \mathbb {R}^n \setminus \{0\}: \eta (x) < 0 \}, \end{aligned}$$

then there exists a real number $\lambda ^*$ and a constant $c > 0$ so that, for all $\lambda \ge \lambda ^*$ and for all $x \in \mathbb {R}^n \setminus \{0\}$ the following inequality is satisfied:

$$\eta (x) + \lambda \gamma (x) \le -c \left| \left| x \right| \right| _\mathbf{r , p}^m.$$

The following Lemma is simple (just monotonicity) but useful

Lemma 2

Consider the real variables x, y, it is always true that

$$\text {sign} \big (\left\lceil x + y \right\rfloor ^\beta - \left\lceil y \right\rfloor ^\beta \big ) = \text {sign}(x), \quad \beta > 0.$$

Appendix 2: Proof of Theorem 1

Using the back-stepping-like procedure to obtain the state-feedback controller and its Lyapunov Function used in [9], and introducing an additional term and a modification, we obtain the following homogeneous candidate Lyapunov function of degree m (with $m \ge 9$ to render it differentiable)

$$\begin{aligned} \begin{aligned} V_4&= \gamma _3V_3 + \frac{2}{m} \left| z_4 \right| ^{\frac{m}{2}} + k_3^{\frac{m - 2}{2}} \left\lceil \left\lceil z_3 \right\rfloor ^{\frac{5}{3}} + k_2^{\frac{5}{3}} \left\lceil z_2 \right\rfloor ^{\frac{5}{4}} + k_2^{\frac{5}{3}} k_1^{\frac{5}{4}} \xi _1 \right\rfloor ^{\frac{m - 2}{5}} z_4 + \\&\quad \,\,\left( 1 - \frac{2}{m} \right) k_3^{\frac{m}{2}} \left| \left\lceil z_3 \right\rfloor ^{\frac{5}{3}} + k_2^{\frac{5}{3}} \left\lceil z_2 \right\rfloor ^{\frac{5}{4}} + k_2^{\frac{5}{3}} k_1^{\frac{5}{4}} \xi _1 \right| ^{\frac{m}{5}} + \frac{1}{m} \left| z_5 \right| ^{m}, \end{aligned} \end{aligned}$$

where $\gamma _1,\gamma _2,\gamma _3>0$ are arbitrary and

$$\begin{aligned} \xi _1&= z_1 - k_{\xi } \left\lceil z_5 \right\rfloor ^{5}, \quad \dot{\xi }_1 = \alpha _1 z_2 - 5 k_{\xi } \left| z_5 \right| ^{4} \dot{z}_5, \quad k_\xi = k_4^{-5} k_{3}^{-\frac{5}{2}} k_{2}^{-\frac{5}{3}} k_{1}^{-\frac{5}{4}}\,, \\ \end{aligned}$$

$$\begin{aligned} V_3&= \gamma _2 V_2 + \frac{3}{m} \left| z_3 \right| ^{\frac{m}{3}} + k_2^{\frac{m - 3}{3}} \left\lceil \left\lceil z_2 \right\rfloor ^{\frac{5}{4}} + k_1^{\frac{5}{4}} \xi _1 \right\rfloor ^{\frac{m - 3}{5}} z_3 + \left( 1 - \frac{3}{m} \right) k_2^{\frac{m}{3}} \left| \left\lceil z_2 \right\rfloor ^{\frac{5}{4}} + k_1^{\frac{5}{4}} \xi _1 \right| ^{\frac{m}{5}},\\ \end{aligned}$$

$$\begin{aligned} V_2&= \frac{5}{m} \gamma _1 \left| \xi _1 \right| ^{\frac{m}{5}} + \frac{4}{m} \left| z_2 \right| ^{\frac{m}{4}} + k_1^{\frac{m - 4}{4}} \left\lceil \xi _1 \right\rfloor ^{\frac{m - 4}{5}} z_2 + \left( 1 - \frac{4}{m} \right) k_1^{\frac{m}{4}} \left| \xi _1 \right| ^{\frac{m}{5}}\,. \end{aligned}$$

Using Young’s inequality, it is possible to show that the function $V_4$ is positive definite. Its derivative along the trajectories of (22) is given by

$$\begin{aligned} \begin{aligned} \dot{V}_4&= \gamma _3 \left[ \gamma _2 F_{k_{2,3}} + F_{k_{3,4}} \right] + \beta (t,\,z)k_4 G_{k_4} + F_{k_4} + \left\lceil z_5 \right\rfloor ^{m - 1} \dot{z}_5, \end{aligned} \end{aligned}$$

(37)

where

$$\begin{aligned} \begin{aligned} G_{k_4}&= - \left[ \left\lceil z_4 \right\rfloor ^{\frac{m-2}{2}} + k_3^{\frac{m - 2}{2}} \left\lceil \left\lceil z_3 \right\rfloor ^{\frac{5}{3}} + k_2^{\frac{5}{3}} \left\lceil z_2 \right\rfloor ^{\frac{5}{4}} + k_2^{\frac{5}{3}} k_1^{\frac{5}{4}} \xi _1 \right\rfloor ^{\frac{m - 2}{5}} \right] \times \\&\quad \,\,\Bigg [- k_4^{-1} z_{5} + \left\lceil \left\lceil z_4 \right\rfloor ^{\frac{5}{2}} + k_{3}^{\frac{5}{2}} \left\lceil z_{3} \right\rfloor ^{\frac{5}{3}} + k_{3}^{\frac{5}{2}} k_{2}^{\frac{5}{3}} \left\lceil z_2 \right\rfloor ^{\frac{5}{4}} + k_{3}^{\frac{5}{2}} k_{2}^{\frac{5}{3}} k_{1}^{\frac{5}{4}} \xi _1 + k_4^{-5} \left\lceil z_5 \right\rfloor ^5 \right\rfloor ^{\frac{1}{5}} \Bigg ],\\ \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} F_{k_4}&= \left( \frac{m - 2}{5} \right) k_3^{\frac{m - 2}{2}} \left| \left\lceil z_3 \right\rfloor ^{\frac{5}{3}} + k_2^{\frac{5}{3}} \left\lceil z_2 \right\rfloor ^{\frac{5}{4}} + k_2^{\frac{5}{3}} k_1^{\frac{5}{4}} \xi _1 \right| ^{\frac{m - 7}{5}} \times \\&\quad \,\, \left[ \frac{5}{3} \left| z_3 \right| ^{\frac{2}{3}} z_4 + \frac{5}{4} k_2^{\frac{5}{3}} \alpha _2 \left| z_2 \right| ^{\frac{1}{4}} z_3 + k_2^{\frac{5}{3}} k_1^{\frac{5}{4}} \left[ \alpha _1 z_2 - 5 k_{\xi } \left| z_5 \right| ^{4} \dot{z}_5 \right] \right] \times \\&\quad \,\, \left[ z_4 + k_3 \left\lceil \left\lceil z_3 \right\rfloor ^{\frac{5}{3}} + k_2^{\frac{5}{3}} \left\lceil z_2 \right\rfloor ^{\frac{5}{4}} + k_2^{\frac{5}{3}} k_1^{\frac{5}{4}} \xi _1 \right\rfloor ^{\frac{2}{5}} \right] ,\\ \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} F_{k_{3,4}}&= \left[ \left\lceil z_3 \right\rfloor ^{\frac{m - 3}{3}} + k_2^{\frac{m - 3}{3}} \left\lceil \left\lceil z_2 \right\rfloor ^{\frac{5}{4}} + k_1^{\frac{5}{4}} \xi _1 \right\rfloor ^{\frac{m - 3}{5}} \right] z_4 + \left( \frac{m - 3}{5} \right) k_2^{\frac{m - 3}{3}} \left| \left\lceil z_2 \right\rfloor ^{\frac{5}{4}} + k_1^{\frac{5}{4}} \xi _1 \right| ^{\frac{m - 8}{5}} \times \\&\quad \,\, \left[ z_3 + k_2 \left\lceil \left\lceil z_2 \right\rfloor ^{\frac{5}{4}} + k_1^{\frac{5}{4}} \xi _1 \right\rfloor ^{\frac{3}{5}} \right] \left[ \frac{5}{4} \alpha _2 \left| z_2 \right| ^{\frac{1}{4}} z_3 + k_1^{\frac{5}{4}} \left[ \alpha _1 z_2 - 5 k_{\xi } \left| z_5 \right| ^{4} \dot{z}_5 \right] \right] ,\\ \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} F_{k_{2,3}}&= \gamma _1 \left\lceil \xi _1 \right\rfloor ^{\frac{m - 5}{5}} \left[ \alpha _1 z_2 - 5 k_{\xi } \left| z_5 \right| ^{4} \dot{z}_5 \right] + \alpha _2 \left[ \left\lceil z_2 \right\rfloor ^{\frac{m - 4}{4}} + k_1^{\frac{m - 4}{4}} \left\lceil \xi _1 \right\rfloor ^{\frac{m - 4}{5}} \right] z_3 \\&\quad + \left( \frac{m - 4}{5} \right) k_1^{\frac{m - 4}{4}} \left| \xi _1 \right| ^{\frac{m - 9}{5}} \left[ z_2 + k_1 \left\lceil \xi _1 \right\rfloor ^{\frac{4}{5}} \right] \left[ \alpha _1 z_2 - 5 k_{\xi } \left| z_5 \right| ^{4} \dot{z}_5 \right] . \end{aligned} \end{aligned}$$

Using Lemma 2 we conclude that the term $G_{k_4}$ is negative semi-definite and it vanishes only on the set

$$S_1 = \left\{ z_4 = - k_3 \left\lceil \left\lceil z_3 \right\rfloor ^{\frac{5}{3}} + k_2^{\frac{5}{3}} \left\lceil z_2 \right\rfloor ^{\frac{5}{4}} + k_2^{\frac{5}{3}} k_1^{\frac{5}{4}} \xi _1 \right\rfloor ^{\frac{2}{5}} \right\} . $$

Evaluating $\dot{V}_4$ on this set, and noting that $\left. F_{k_4} \right| _{S_1}=0$ and $\left. F_{k_{3,4}} \right| _{S_1}=k_3 G_{k_3} + F_{k_3}$, we obtain

$$\begin{aligned} \left. \dot{V}_4 \right| _{S_1} = \gamma _3 \left[ \gamma _2 F_{k_{2,3}} + k_3 G_{k_3} + F_{k_3} \right] + {z_5}^{m - 1} \left. \dot{z}_5 \right| _{S_{1}} \,, \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} G_{k_3}&= - \left[ \left\lceil z_3 \right\rfloor ^{\frac{m - 3}{3}} + k_2^{\frac{m - 3}{3}} \left\lceil \left\lceil z_2 \right\rfloor ^{\frac{5}{4}} + k_1^{\frac{5}{4}} \xi _1 \right\rfloor ^{\frac{m - 3}{5}} \right] \left\lceil \left\lceil z_3 \right\rfloor ^{\frac{5}{3}} + k_2^{\frac{5}{3}} \left\lceil z_2 \right\rfloor ^{\frac{5}{4}} + k_2^{\frac{5}{3}} k_1^{\frac{5}{4}} \xi _1 \right\rfloor ^{\frac{2}{5}},\\ F_{k_3}&= \left( \frac{m - 3}{5} \right) k_2^{\frac{m - 3}{3}} \left| \left\lceil z_2 \right\rfloor ^{\frac{5}{4}} + k_1^{\frac{5}{4}} \xi _1 \right| ^{\frac{m - 8}{5}} \left[ z_3 + k_2 \left\lceil \left\lceil z_2 \right\rfloor ^{\frac{5}{4}} + k_1^{\frac{5}{4}} \xi _1 \right\rfloor ^{\frac{3}{5}} \right] \times \\&\quad \,\, \left[ \frac{5}{4} \alpha _2 \left| z_2 \right| ^{\frac{1}{4}} z_3 + k_1^{\frac{5}{4}} \left[ \alpha _1 z_2 - 5 k_{\xi } \left| z_5 \right| ^{4} \dot{z}_5|_{S_1} \right] \right] . \end{aligned} \end{aligned}$$

We note that the term $G_{k_3}$ is negative semi-definite and it is zero only on the set

$$ S_2 = \left\{ z_3 = - k_2 \left\lceil \left\lceil z_2 \right\rfloor ^{\frac{5}{4}} + k_1^{\frac{5}{4}} \xi _1 \right\rfloor ^{\frac{3}{5}} \right\} \,. $$

Evaluating $\left. \dot{V}_4 \right| _{S_1}$ on this set, and noting that $\left. F_{k_3} \right| _{S_2}=0$ and $\left. F_{k_{2,3}} \right| _{S_{2}}=F_{k_2} + k_2 G_{k_2}$, we get

$$\begin{aligned} \left. \dot{V}_4 \right| _{S_{1} \cap S_2} = \gamma _3 \gamma _2 \left[ F_{k_2} + k_2 G_{k_2} \right] + \left\lceil z_5 \right\rfloor ^{m - 1} \left. \dot{z}_5 \right| _{S_{1} \cap S_2}\,, \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} G_{k_2}&= - \alpha _2 \left[ \left\lceil z_2 \right\rfloor ^{\frac{m - 4}{4}} + k_1^{\frac{m - 4}{4}} \left\lceil \xi _1 \right\rfloor ^{\frac{m - 4}{5}} \right] \left\lceil \left\lceil z_2 \right\rfloor ^{\frac{5}{4}} + k_1^{\frac{5}{4}} \xi _1 \right\rfloor ^{\frac{3}{5}}, \\ F_{k_2}&= \gamma _1 \left\lceil \xi _1 \right\rfloor ^{\frac{m - 5}{5}} \left[ \alpha _1 z_2 - 5 k_{\xi } \left| z_5 \right| ^{4} \dot{z}_5|_{S_{1} \cap S_2} \right] + \left( \frac{m - 4}{5} \right) k_1^{\frac{m - 4}{4}} \left| \xi _1 \right| ^{\frac{m - 9}{5}} \times \\&\quad \left[ z_2 + k_1 \left\lceil \xi _1 \right\rfloor ^{\frac{4}{5}} \right] \left[ \alpha _1 z_2 - 5 k_{\xi } \left| z_5 \right| ^{4} \dot{z}_5|_{S_{1} \cap S_2} \right] . \end{aligned} \end{aligned}$$

$G_{k_2}$ is negative semi-definite and it vanishes only on the set

$$ S_3 = \left\{ z_2 = - k_1 \left\lceil \xi _1 \right\rfloor ^{\frac{4}{5}} \right\} \,. $$

Evaluating $\left. \dot{V}_4 \right| _{S_1 \cap S_2}$ on this set we get

$$\begin{aligned} \begin{aligned} \left. \dot{V}_4 \right| _{S_1 \cap S_2 \cap S_3}&\in \gamma _2 \gamma _3 \left. F_{k_2} \right| _{S_3} + \left. \left\lceil z_5 \right\rfloor ^{m - 1} \dot{z}_5 \right| _{S_{1} \cap S_2 \cap S_3}\\&\in - \alpha _1 k_1 \gamma _3 \gamma _2 \gamma _1 \left| \xi _1 \right| ^{\frac{m - 1}{5}} - k_{I1} \left[ \left\lceil z_5 \right\rfloor ^{m - 1} - \right. 5 k_{\xi } \gamma _3 \gamma _2 \gamma _1 \left. \left\lceil \xi _1 \right\rfloor ^{\frac{m - 5}{5}} \left| z_5 \right| ^{4} \right] \times \\&\left[ \left\lceil \left[ 1 - k_{I2} k_1^{\frac{m}{4}} \right] \xi _1 + k_{\xi } \left\lceil z_5 \right\rfloor ^{5} \right\rfloor ^{0} + \left[ -\bar{L}, \bar{L} \right] \right] . \end{aligned} \end{aligned}$$

Since $k_1>0$ the first term in the latter expression is non-positive and it is zero only on the set

$$ S_{4} = \left\{ \xi _1 = 0 \right\} \,. $$

Evaluating $\left. \dot{V}_4 \right| _{S_1 \cap S_2 \cap S_3}$ on this set we get

$$\begin{aligned} \left. \dot{V}_4 \right| _{S_1 \cap S_2 \cap S_3 \cap S_4} \in - k_{I1} \left\lceil z_5 \right\rfloor ^{m - 1} \left[ \left\lceil z_5 \right\rfloor ^{0} + \left[ -\bar{L}, \bar{L} \right] \right] \,. \end{aligned}$$

The latter expression is negative if $\bar{L} =\frac{L}{k_{I1}} <1$, that is, for

$$\begin{aligned} L < k_{I1}\,, \end{aligned}$$

(38)

since $\left\lceil z_5 \right\rfloor ^{0}$ is a multivalued function, defined as

$$ \left\lceil z_5 \right\rfloor ^{0} = \left\{ \begin{array}{ll} +1 &{} \text{ if } z_5 > 0\,, \\ \left[ -1,\,+1\right] &{} \text{ if } z_5 = 0\,, \\ -1 &{} \text{ if } z_5 < 0\,. \\ \end{array} \right. $$

Lemma 1 implies that it is possible to render $\left. \dot{V}_4 \right| _{S_1 \cap S_2 \cap S_3}<0$ selecting $k_{I1}>0$ small. Applying Lemma 1 once more, we conclude that $\left. \dot{V}_4 \right| _{S_1 \cap S_2}<0$ selecting $k_2>0$ sufficiently large. Again, Lemma 1 shows that we can get $\left. \dot{V}_4 \right| _{S_1}<0$ selecting $k_3>0$ sufficiently large. Finally, Lemma 1 implies that $\left. \dot{V}_4 \right| <0$ if $k_4>0$ is sufficiently large.

Since $\dot{V}_4$ is negative definite by appropriate selection of the gains (what is always feasible), then the origin of system (22) is asymptotically stable. Moreover, since the system is homogeneous of negative degree, the origin is finite-time stable (see, e.g., [24]). Since (22) is a local homogeneous approximation of system (11), we conclude that it is locally finite-time stable.    $\square$

Appendix 3: Gain Selection

From the previous proof we derive conditions for the gains to render $\dot{V}_4<0$. The following inequalities are obtained for the gain selection:

$$\begin{aligned} k_{I1}^{-1} > \underset{ z\in { \varOmega }_{ 1 } }{ \max } \left\{ \frac{- k_1^{\frac{m}{4}} \gamma _3 \gamma _2 \gamma _1 \left| \xi _1 \right| ^{\frac{m - 1}{5}} }{\left[ \left\lceil z_5 \right\rfloor ^{m - 1} - \right. 5 k_{\xi } \gamma _3 \gamma _2 \gamma _1 \left. \left\lceil \xi _1 \right\rfloor ^{\frac{m - 5}{5}} \left| z_5 \right| ^{4} \right] \dot{ z }_5|_{S_{1} \cap S_2 \cap S_3} } \right\} \,, \end{aligned}$$

(39)

$$\begin{aligned} k_2 > \underset{ z\in { \varOmega }_{ 2 } }{ \max } \left\{ \frac{F_{k_2} + \gamma _3^{-1} \gamma _2^{-1} \left. \left\lceil z_5 \right\rfloor ^{m - 1} \dot{z}_5 \right| _{S_{1} \cap S_2} }{G_{k_2} } \right\} , \end{aligned}$$

(40)

$$\begin{aligned} k_3 > \underset{ z\in { \varOmega }_{ 3 } }{ \max } \left\{ \frac{\gamma _2 F_{k_{2,3}} + F_{k_3} + \gamma _3^{-1} \left. \left\lceil z_5 \right\rfloor ^{m - 1} \dot{z}_5 \right| _{S_1} }{G_{k_3} } \right\} , \end{aligned}$$

(41)

$$\begin{aligned} k_4 > \frac{1}{b_m} \underset{ z\in { \varOmega }_{ 4 } }{ \max } \left\{ \frac{\gamma _3 \left[ \gamma _2 F_{k_{2,3}} + F_{k_{3,4}} \right] + F_{k_4} + \left\lceil z_5 \right\rfloor ^{m - 1} \dot{z}_5 }{G_{k_4} } \right\} , \end{aligned}$$

(42)

where the homogeneous unit spheres $\varOmega _i$ are given by $\varOmega _1 = \left\{ \left| \xi _{ 1 } \right| ^{ \frac{1}{5} } + \left| z_{ 5 } \right| = 1 \right\}$, $\varOmega _2 = \left\{ \left| \xi _{ 1 } \right| ^{ \frac{1}{5} } + \left| z_{ 2 } \right| ^{ \frac{1}{4} } + \left| z_{ 5 } \right| = 1 \right\}$, $\varOmega _3 = \left\{ \left| \xi _{ 1 } \right| ^{ \frac{1}{5} } + \left| z_{ 2 } \right| ^{ \frac{1}{4} } + \left| z_{ 3 } \right| ^{ \frac{1}{3} } + \left| z_{ 5 } \right| = 1 \right\}$, and $\varOmega _4 = \left\{ \left| \xi _{ 1 } \right| ^{ \frac{1}{5} } + \left| z_{ 2 } \right| ^{ \frac{1}{4} } + \left| z_{ 3 } \right| ^{ \frac{1}{3} } + \left| z_{ 4 } \right| ^{ \frac{1}{2} } + \left| z_{ 5 } \right| = 1 \right\}$. Functions (39)–(42) are shown to have a maximum value. Since they are homogeneous of degree $d=0$ the maximum can be found on the homogeneous unit sphere $\varOmega _ i$.