# Signum-Function-Based Multi-Input Multi-Output Adaptive Homogeneous Finite-Time Control for a Rigid-Body Attitude Tracking

Authors: Sen Jiang; Zhong Yang; Hao Xu
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This study is devoted to investigating the robust adaptive finite-time attitude tracking control problem for a rigid body subject to unknown uncertainties. Considering about the nonlinear attitude dynamics for a rigid body, a homogeneous sliding variable is designed by employing the signum-function technique. For the presented homogeneous sliding variable, a real sliding manifold, on which the singularity problem is avoided, could be achieved. Subsequently, a novel signum-function-based MIMO adaptive homogeneous finite-time control (SMAHFTC) algorithm is proposed. Finite-time convergence of the tracking errors to a region around the origin is rigorously proved through the Lyapunov approach. The control gains will not be overestimated, which implies that minimal control gains will be obtained by the adaptation law. Moreover, the controls’ signals are continuous for the SMAHFTC law. Then, by means of the proposed design method, an attitude tracking controller is designed for a rigid body described by the modified Rodrigues parameters’ (MRPs) representation. A comparison simulation is carried out to show the effectiveness and superiority of the proposed method.


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## Body

## 1. Introduction

A large amount of research efforts have been focused on the attitude control of a rigid body due to both its fundamentality to other senior control tasks and its widespread applications on the realm of aeronautics [1–4], astronautics [5–9], and robotics [10–12]. On those situations, resulting from various uncertainties including parameter perturbations, measurement noises, discrete implementations on digit processing unit (DPU), and external disturbances, robustness of an attitude controller is a crucial performance. Meanwhile, control accuracy and transient time (i.e., the time interval of system states change from the initial state to the steady state) are also the key requirements of an attitude controller.However, in many cases, a lumped uncertainty consisting of internal perturbations and external disturbances will degrade the performance of control precision and even destroy the system stability. Therefore, developing an attitude controller for a rigid body, which simultaneously satisfies the performance requirements of robustness, control precision, and short transient time, is a design challenge. To solve this, lots of works have been conducted. For examples, based on a new family of Lie algebra-valued functions on the special orthogonal group, a novel class of attitude controllers was proposed by [13]. The induced geometric attitude controllers are coordinate-free and almost-global, but the robust problem is not the major issue that the authors care about. A nonlinear robust attitude controller was designed for robotic quadrotors by [12]. The presented controller, which consists of a nominal controller and a robust compensator, could be applied for an uncertain nonlinear plant. Furthermore, the influence of disturbances could be drastically attenuated. A new nonlinear disturbance observer was developed as a stepping stone by [14]. Subsequently, a special integral sliding mode attitude stabilizing control law was proposed for a rigid-body spacecraft. Robustness against various uncertainties is obtained, which includes dead-zone effect of the actuators. However, only asymptotic convergence of the system states is achieved, and the control signal is discontinuous, which limits applications of the presented control method. In [15], a novel fixed-time attitude stabilizing control law was proposed for a rigid-body spacecraft, which is described by the modified Rodrigues parameters (MRPs). Robustness against lumped uncertainties is guaranteed. Meanwhile, a fixed-time convergence, which is independent of the initial states, is obtained by means of a new modified backstepping technique. Among the control methods having already been used to design attitude controller for rigid bodies, sliding mode control (SMC) has received more attentions [3] resulting from its favorable robustness. Chattering effect of the control signals is the main drawback of traditional SMC, which could be improved by high-order sliding mode control (HOSMC) [16] or finite-time control (FTC) [17, 18]. However, considering about HOSMC or FTC, boundary information of the uncertainties must be known in prior, which may not be met in many practical engineering. To overcome this drawback, adaptive SMC/HOSM (ASMC) or adaptive FTC (AFTC) was proposed.Many investigations have focused on ASMC or AFTC. By means of barrier Lyapunov functions, an AFTC law was proposed in [19] for strict-feedback nonlinear systems. Taking into account dead-zone input and full-state constraints, the closed-loop system states could be steered into a bounded region including the origin in a finite time. By [20], based on fuzzy logic technique, an adaptive fuzzy sliding mode control law was proposed. The control signal is continuous, but only asymptotic convergence of the closed-loop system states is achieved. By [21], a novel adaptive super-twisting control law was proposed. Boundary information of the uncertainties is not needed, except for the boundedness on the uncertainties. Furthermore, control gains will not be overestimated, which guarantees the minimal value of the control gains that could still maintain the sliding mode. However, those abovementioned literature could only be applied on single-input single-output (SISO) systems. In [22], a new coupled form super-twisting algorithm (STA) was developed for a class of MIMO systems.This study considers the robust adaptive finite-time attitude tracking control problem for a rigid body in the presence of unknown uncertainties. To accomplish this target, a SMAHFTC strategy, which combines a coupled form STA and an adaptive SMC with the signum-function technique, is proposed. Compared with some existing results, the main contributions of this study are as follows:(i)
A homogeneous sliding variable is proposed, and a real sliding manifold (RSM) can be reached in a finite time. The control gains, which are dynamically governed by the presented adaptation law, will not be overestimated. Furthermore, the control signals are continuous, and hence, chattering effects could drastically be mitigated.(ii)
When the closed-loop system reaches the RSM, finite-time convergence of the tracking errors to a bounded set is achieved. By means of the signum-function technique, the singularity problem is removed. The stability analysis of the closed-loop system is rigorously presented via a modified backstepping technique [23].The rest of this study is arranged as follows. Notations, lemmas, problem statements, and designing objective are given in Section2. A signum-function-based adaptive homogeneous finite-time tracking control for MIMO cascade nonlinear systems is proposed. Meanwhile, rigorous analysis of the finite-time convergence is given by using the Lyapunov method in Section 3. Then, in Section 4, based on the proposed method, an adaptive homogeneous finite-time attitude tracking controller is designed for a rigid body. Simulations are carried out to verify effectiveness of the proposed method in Section 5, which is followed by the conclusion in Section 6.


## 2. Preliminaries and Problem Statements

### 2.1. Notations and Some Lemmas

Some notations are given in the beginning. The set of all positive real numbers is represented byR+. ⋅ refers to the absolute value function, and ⋅ represents the Euclidean norm of a vector. sign⋅ denotes the signum function. For a given real number a∈R+ and any column vector ∀x∈Rn, define sgnax=sgnax1,…,sgnaxnT and xa=x1a,…,xnaT, where sgnaxi=signxixia,i=1,…n. x× describes a skew-symmetric matrix x×∈R3×3 as(1)x×=0−x3x2x30−x1−x2x10.For any three-dimensional column vector,∀x=x1,x2,x3T∈R3.Some lemmas, which will be used hereinafter, are given as follows.Lemma 1 (see [5]).
Given a real numberα∈0,1, for ∀x∈R and ∀y∈R, the following inequality(2)sgnαx−sgnαy≤21−αx−yαholds.Lemma 2 (see [24]).
Provided some constantsc>0, d>0, and α∈0,1, for ∀x∈R, ∀y∈R, and γ>0, inequality(3)xcyd≤cc+dγxc+d+dc+dγ−cdyc+dholds.Lemma 3 (see [25]).
If a constant0<p<1, for ∀xi∈R,i=1,…m, the following inequality holds:(4)x1+⋯+xmp≤x1p+⋯+xmp.


### 2.2. Problem Statements

A cascade MIMO nonlinear system is considered, which is described as(5)x˙1=x2,x˙2=ρx1,x2,t+u,where x1,x2∈Rn are the state variables, u∈Rn is the control input vector, and ρ⋅∈Rn is an unknown vector function, whose first time derivative is bounded but with unknown boundary.The control objective is to steer the system states follow some given command profiles subjected to a lumped uncertainties including internal perturbations and external disturbances. The proposed control law must ensure that the tracking errors and their first time derivatives are enforced to a vicinity of the origin in a finite time and kept on it thereafter by means of a state-feedback control schema. In addition, the control signals must be continuous to mitigate the chattering effects.


## 2.1. Notations and Some Lemmas

Some notations are given in the beginning. The set of all positive real numbers is represented byR+. ⋅ refers to the absolute value function, and ⋅ represents the Euclidean norm of a vector. sign⋅ denotes the signum function. For a given real number a∈R+ and any column vector ∀x∈Rn, define sgnax=sgnax1,…,sgnaxnT and xa=x1a,…,xnaT, where sgnaxi=signxixia,i=1,…n. x× describes a skew-symmetric matrix x×∈R3×3 as(1)x×=0−x3x2x30−x1−x2x10.For any three-dimensional column vector,∀x=x1,x2,x3T∈R3.Some lemmas, which will be used hereinafter, are given as follows.Lemma 1 (see [5]).
Given a real numberα∈0,1, for ∀x∈R and ∀y∈R, the following inequality(2)sgnαx−sgnαy≤21−αx−yαholds.Lemma 2 (see [24]).
Provided some constantsc>0, d>0, and α∈0,1, for ∀x∈R, ∀y∈R, and γ>0, inequality(3)xcyd≤cc+dγxc+d+dc+dγ−cdyc+dholds.Lemma 3 (see [25]).
If a constant0<p<1, for ∀xi∈R,i=1,…m, the following inequality holds:(4)x1+⋯+xmp≤x1p+⋯+xmp.


## 2.2. Problem Statements

A cascade MIMO nonlinear system is considered, which is described as(5)x˙1=x2,x˙2=ρx1,x2,t+u,where x1,x2∈Rn are the state variables, u∈Rn is the control input vector, and ρ⋅∈Rn is an unknown vector function, whose first time derivative is bounded but with unknown boundary.The control objective is to steer the system states follow some given command profiles subjected to a lumped uncertainties including internal perturbations and external disturbances. The proposed control law must ensure that the tracking errors and their first time derivatives are enforced to a vicinity of the origin in a finite time and kept on it thereafter by means of a state-feedback control schema. In addition, the control signals must be continuous to mitigate the chattering effects.


## 3. Adaptive Homogeneous Finite-Time Tracking Control for MIMO Cascade Systems

Define the tracking errors and their derivatives as(6)e1=x1−xd,e2=e˙1=x2−x˙d,where xd,x˙d are the command profiles and their first time derivatives. Differentiating (6) and using (5), dynamic equations of the tracking errors are described as(7)e˙1=e2,e˙2=ρ˜x1,x2,t+u,ρ˜x1,x2,t=ρx1,x2,t−x¨d.The following definition and assumptions will be exploited in the subsequent development.Definition 1.
(see [26]). Given a sliding variable se, where e=e1T,e2TT, a real sliding manifold (RSM) is defined as(8)ℜ=e∈R2n:se≤ε¯,with ε¯∈R+.Assumption 1.
The vector functionρ˜⋅∈Rn must be differentiable. Moreover, its first time derivative could be discontinuous but must be bounded as(9)ρ˜˙x1,x2,t≤δ,where the finite boundary δ>0 exists but is not necessary to be known.
Using signum-function technique and homogeneous theory [27], a nonlinear sliding variable is constructed as(10)s=e2+∫0tc2sgn2a−1sgn1/ae2ι+c11/ae1ιdι.
Seeingρ˜⋅ as an unknown lumped disturbance, inspired by [22], the control law is designed as(11)u=−c2sgn2a−1sgn1/ae2+c11/ae1−k1ts1/2s−∫0tk2ι2sιsιdι,along with a gain adaptation law, which is described as(12)k˙1t=ω1γ12signs−μ,k1>k1,t,ϖ,k1≤k1,t,k2t=2εk1t,where a∈1/2,1, c1∈R+, c2∈R+, ω1∈R+, γ1∈R+, ε∈R+, and ϖ∈R+ are design parameters. The parameters μ and k1,t are some small positive constants.
Then, it is the time ready to state the first main result of this paper as following.Theorem 1.
Consider the nonlinear closed-loop system (7) and (11) with the adaptive scheme (12). Provided that the lumped disturbance ρ˜⋅ satisfies Assumption 1 with unknown constant δ, for any initial conditions s0, a bounded compact set (a RSM) including the origin,(13)ℑ=s,s˙∈R2n:s≤r1,s˙≤r2,r1>μ,will be reached in a finite time, with arbitrary positive constants ω1∈R+ and γ1∈R+. The corresponding parameter ε∈R+ is depending on ω1 and γ1. In addition, inequality γ1>μ is satisfied.Proof.
Differentiating sliding variables with respect to time and using the second line of (7) lead to(14)s˙=−k1ts1/2s−∫0tk2ι2sιsιdι+ρ˜⋅.
A new state vector is introduced(15)θ1=s,θ2=−∫0tk22sιsιdι+ρ˜x1,x2,t,θ1∈Rn,θ2∈Rn,where, for conciseness and clarity, parameters k1t and k2t sometimes briefly expressed as k1 and k2, hereinafter. Differentiating (15), a canonical form of the MIMO super-twisting system is obtained as(16)θ˙1=−k1θ11/2θ1+θ2,θ˙2=−k22θ1θ1+ρ˜˙x1,x2,t.
Due to the super-twisting algorithm [26], solution of (16) is understood in the Filippov sense [28].
Choose a Lyapunov candidate as(17)V1θ1,θ2,k1,k2=V0θ1,θ2+12γ1k1−k1∗2+12γ2k2−k2∗2,where(18)V0θ1,θ2=λ+4ε2θ1−4εθ1Tθ2θ11/2+θ2Tθ2,and λ,ε∈R+ and γ1,γ2∈R+ are some constants.
Define a subspace as(19)ℵ=θ1,θ2∈R2n:θ1=0.
Then,V0θ1,θ2 in (18) is everywhere continuous, and differentiable everywhere except on the subspace ℵ. Furthermore, it is easy to verify that V0⋅ is positive definite and radially unbounded. Differentiating (18) and using (16) yield(20)V˙0=−k1λ+4ε2−2k2εθ13/2θ1Tθ1+λ+4ε2+2k1ε−2k2θ1θ1Tθ2−2εθ11/2θ2Tθ2−4εθ11/2θ1Tρ˜˙+θ2Tρ˜˙,for all θ1,θ2∉ℵ. In order to derive the convergency condition of system (16), set(21)k2=2εk1.
Then, equation,(22)V˙0=−k1λθ13/2θ1Tθ1+λ+4ε2θ1θ1Tθ2−2εθ11/2θ2Tθ2−4εθ11/2θ1Tρ˜˙+θ2Tρ˜˙,is obtained. Taking into account Assumption 1 and using Cauchy–Schwarz inequality in (22) result in(23)V˙0≤−12θ11/2θ11/2θ2Wθ11/2θ2,W=2k1λ−4εδ−λ+4ε2+2δ−λ+4ε2+2δ4ε.
One can easily get the positive definite condition ofW with a minimal eigenvalue λminW≥2ε:(24)k1>λ+4ε2+2δ24ελ+4εδ+ελ.
Definingζ=θ11/2,θ2T, (23) becomes(25)V˙0≤−12θ11/2λminWζ2≤−12θ11/2εζ2.
Applying the inequalityθ11/2≤ζ induces inequality(26)V˙0≤−1ζεζ2=−εζ.
Meanwhile, taking into account,(27)V0θ1,θ2≤λ+4ε2θ1+4εθ11/2θ2+θ22=ζTPζ,≤λmaxPζ2,P=λ+4ε22ε2ε1.
Provided that (21) and (24) hold, inserting (27) into (26) yields(28)V˙0≤−εζ≤−cυ0V01/2,cυ0=ελ1/2/maxP,for all θ1,θ2∉ℵ. Note that the trajectories of the Filippov solution of (16) cannot stay on the set ℵ/0. Because if θ1,θ2∈ℵ/0 at a time instant ts, θ1ts=0∈Rn and θ2ts≠0∈Rn; according to the first line of (16), one can easily know that θ˙1ts=θ2ts≠0∈Rn. Then, trajectories of (16) will cross the set ℵ/0 immediately. That is to say V0⋅ is a continuous function on R2n/0. In addition, if (24) is satisfied, along the solution of (16), inequality (28) holds almost everywhere.
Taking time derivative ofV1 and applying inequality (28) lead to(29)V˙1θ1,θ2,k1,k2≤−cυ0V01/2−ω12γ1k1−k1∗−ω22γ2k2−k2∗+1γ1k1−k1∗k˙1+ω12γ1k1−k1∗+1γ2k2−k2∗k˙2+ω22γ2k2−k2∗,where ω2∈R+ is an arbitrary constant. Assuming that k1<k1∗ and k2<k2∗ (the existence of k1∗ and k2∗ will be proven in Proposition 1) and considering an important inequality,(30)cυ02V0+ω122γ1k1−k1∗2+ω222γ2k2−k2∗21/2≤cυ0V01/2+ω12γ1k1−k1∗+ω22γ2k2−k2∗.
Inequality (29) becomes(31)V˙1θ1,θ2,k1,k2≤−cυ1V01/2+κ,κ=−1γ1k˙1−ω12γ1k1−k1∗−1γ2k˙2−ω22γ2k2−k2∗,where cυ1=mincυ0,ω1,ω2. Then, the following two cases are discussed:(1)
Whens>μ, k1>k1,t choose an adaptation law as(32)k˙1=ω1γ12.According to (21), taking time derivative of k2, and setting ε=ω2/2ω1γ2/γ1 yield(33)k˙2=ω2γ22.In view of (32) and (33), the term κ is equal to zero. Hence, (31) becomes(34)V˙1θ1,θ2,k1,k2≤−cυ1V01/2,which satisfies the conditions of finite-time convergency. For a detailed analysis of inequality (34), the readers could refer to [17]. Therefore, as long as condition (24) be met by the adaptive scheme (32), trajectories of system (16) will converge to a region around the origin in a finite time, in which the domain s≤μ is included.(2)
Whens≤μ, the adaptive scheme could be chosen as(35)k˙1=−ω1γ12,k1>k1,t,k˙1=ϖ,k1≤k1,t.
In accordance withε=ω2/2ω1γ2/γ1, the term κ becomes(36)κ=2ω12γ1k1−k1∗+2ω22γ2k2−k2∗,k1>k1,t,κ=−ϖγ1−ω12γ1ϖt−t0+k1,t−k1∗−2εϖγ2−ω22γ22εϖt−t0+2εk1,t−k2∗,k1≤k1,t,where t0 is the time instant when k1 from k1>k1,t becomes to k1=k1,t. Note that the second line of (36) takes effect only for a small time interval. Indeed, as soon as k1 becomes less or equal to k1,t, its value immediately begins to increase. Quickly, k1 is again greater than k1,t, and the first line of (36) is operated.
In view of (36), time derivative of the Lyapunov function V1, as described in (31), may become positive. Consequently, s may become larger than μ due to the decreasing of the control gains k1 and k2. As soon as s becomes larger than μ, the condition described in case (1) holds. Then, s will be less than μ again in a finite time, so forth again and again. Ultimately, it could be summarized that the sliding variable s reaches the domain s<μ in a finite time during the adaptation process, and then, it may leave this domain but impossibly leaves a larger domain thereafter. That domain could be expressed as s<r1,r1>μ.
When in the domains<μ, one can also estimate the boundary of s˙. Indeed, using (16) induces that(37)s˙≤r¯2,r¯2=k1t1μ1/2+εk1t1+δt2−t1,where t1 and t2 are the time instants when s enters into the domain s<μ and deviates from this domain, respectively. When in the domain μ<s≤r1,(38)s˙≤r˜2,r˜2=r112+εt3−t2k1t2+ω1γ12t3−t2+δt3−t2,where t2 and t3 are the time instants when s deviates from the domain s<μ and re-enters into this domain, respectively. Entrances of s into the two domains s<μ and μ<s≤r1 are repeatedly switched thenceforward. In addition, (37) and (38) can be combined into(39)s˙≤r2,r2=maxr¯2,r˜2.
The combination form of adaptive schemes (32) and (35) is exactly as expressed in (12). It is worth stressing that if parameter μ=0, adaptation law (12) could be changed into(40)k˙1t=ω1γ12,s>0,0,s=0,k2t=2εk1t.
Then, the setℑ is replaced by a second-order sliding manifold s=0,s˙=0. In other words, if with μ⟶0, the condition r1⟶0,r2⟶0 is achieved. Indeed, in accordance with Definition 1, the set ℑ is a real sliding manifold. For more statements about real sliding mode, the reader could refer to [26].
Furthermore, integrating (31) with (34), (39) yields that trajectories of (14) will reach the set ℑ in a finite time Tf. The time interval Tf could be estimated as(41)Tf=ta+tf,where ta is the first time when (24) is satisfied under the adaptation law (12), and tf is described as(42)tf≤2V11/2tacυ1,cυ1=mincυ0,ω1,ω2.
Proof of Theorem1 is complete.Remark 1.
It is worth noting that estimation ofr1 and r2 in Theorem 1 is conservative, but existence of the RSM ℑ is rigorously proved.Proposition 1.
In view of the adaptation law (12), the adaptive gains k1t and k2t are bounded.Proof.
The proof is carried out in the same vein as Proposition1 presented by [21]. When s is out of the domain s<μ, solution of (12) can be expressed as(43)k1t=k10+ω1γ12t,0<t<ta+tf,k2t=2εk1t.
Considering the right-hand side of inequality (24), it is in essence a second-order polynomial w.r.t δ. Due to the boundness of δ and the monotonously increasing property of the function k1t w.r.t time, it is easily known that inequality (24) will be firstly satisfied in a finite time, which is denoted as ta. According to the proof of Theorem 1, we can also know tf is a finite-time interval, so as to ta+tf. Therefore, k1t and k2t are also bounded. This guarantees the existence of k1∗ and k2∗, which are used by the Lyapunov function V1 defined in (17).
Whens is in the domain s<μ, according to the adaptation law (12), k1t and k2t start to decrease. Obviously, k1t and k2t still meet the inequalities k1,t≤k1t<k1∗ and 2εk1,t≤k2t<k2∗.
As pointed out in Theorem1, trajectories of (7) will be forced to the RSM ℑ by the control law (11) and will remain on it subsequently, when t>ta+tf holds. It can be easily seen that when system trajectories are on the RSM ℑ, states of system (7) are governed by(44)e˙1=e2,e˙2=−c2sgn2a−1sgn1/ae2+c11/ae1+s˙.
From now, stability analysis of (44), which is the second main result of this study, will be given in Theorem 2.Theorem 2.
Consider the dynamic equations (44), by which system (7) is governed on the RSM ℑ. If the parameters c1 and c2 are properly chosen, the tracking errors e=e1T,e2TT will be steered to a region Ω=e:e≤Δ2 in a finite time, where Δ2=Δ1+21−a+c1n1−a/2Δ1a and Δ1=4r2/c01/2a−1. The origin is contained in the region Ω.Proof.
Choosing a Lyapunov candidate,(45)V1,se1=12e1Te1.
Differentiating (45) yields(46)V˙1,s=e1Te˙1=e1Te2=e1Tu1,υ+e1Te2−u1,υ.
Choose a virtual controlu1,υ=−c1sgnaξ1, where ξ1=e1, a∈1/2,1, and c1∈R+; then, (46) becomes(47)V˙1,s=e1Te2=−c1sgnaξ1+ξ1Te2−u1,υ.
Considering the termξ1Te2−u1,υ and defining ξ2=sgn1/ae2−sgn1/au1,υ, it is easy to show that(48)ξ1Te2−u1,υ≤∑i=1n21−a1+aξ1i1+a+ξ2i1+a=21−a1+aξ1Tsgnaξ1+aξ2Tsgnaξ2,where Lemma 1 and Lemma 2 are applied successively. Inserting (48) into (46) leads to(49)V˙1,s≤−c1−c^11ξ1Tsgnaξ1+c^12ξ2Tsgnaξ2,c^11=21−a1+a,c^12=a21−a1+a.
Parameterc1 is chosen as c1=c0+c^11, where c0∈R+ is a designing parameter. Then, (49) becomes(50)V˙1,s≤−c0ξ1Tsgnaξ1+c^12ξ2Tsgnaξ2.
Define(51)Vse1,e2=V1,se1+V2,se1,e2,V2,se1,e2=∑i=1nV2i,se1i,e2i,=∑i=1n∫u1i,υe2isgn2−asgn1/aι−sgn1/au1i,υdι,where e11,…,e1nT=e1, e21,…,e2nT=e2, and u11,υ,…,u1n,υT=u1,υ. In view of Propositions B.1 and B.2 by [24], it can be easily known that V2i,s is positive definite and once differentiable. Furthermore, it is easy to obtain the inequality:(52)Vs≤12ξ1Tξ1+21−aξ2Tξ2≤2ξ1Tξ1+ξ2Tξ2.
Taking time derivative ofV2i,s and using u1,a,…,un,aT=ua, s˙1,…,s˙nT=s˙ yield(53)V˙2i,s=s˙isgn2−aξ2i−ui,asgn2−aξ2i+2−ac11/ae2i∫u1i,υe2isgn1/aι−sgn1/au1i,υ1−adι≤s˙isgn2−aξ2i−ui,asgn2−aξ2i+21−a2−ac11/ae2iξ2i.
Noting thate2i=sgnaξ2i−c11/aξ1i, using Lemma 3 yields e2i≤ξ2ia+c1ξ1ia, and inserting it into (53) induce that(54)V˙2i,s=s˙isgn2−aξ2i−ui,asgn2−aξ2i+c02ξ1i1+a+c^22ξ2i1+a,c^22=1a+12ac0a+1a21−a2−ac11+1/aa+1+21−a2−ac11/a.
In accordance with (51), taking time derivative of Vs and substituting (49) and (54) into it result in(55)V˙s=V˙1,s+V˙2,s≤−c02ξ1Tsgnaξ1+c^12+c^22ξ2Tsgnaξ2+s˙Tsgn2−aξ2−uaTsgn2−aξ2.
Choosing controlua as ua=−c2sgn2a−aξ2, c2=c^12+c^22+c0/2, which is identical to the first line of (11), and continuing (55) yield(56)V˙s≤−c02ξ1Tsgnaξ1−c02ξ2Tsgnaξ2+s˙Tsgn2−aξ2.
In proof of Theorem1, it is shown that s˙≤r2 if trajectories of (7) are on the RSM ℑ, provided that parameters of c1 and c2 are chosen as(57)c1=c0+c^11,c2=c02+c^12+c^22,c0>0,where c^11, c^12, and c^22 are defined in (49) and (54). Considering the inequality,(58)sgn2−aξ2≤ξ22−a≤ξ1Tξ1+ξ2Tξ22−a/2.
Inserting (57) and s˙≤r2 into (56) results in(59)V˙s≤−c02−5+a/2Vs1+a/2−c04ξ1Tξ1+ξ2Tξ22a−1/2−4r2c0ξ1Tξ1+ξ2Tξ22−a/2,in which (52) is also used. When ξ1Tξ1+ξ2Tξ22a−1/2≥4r2/c0, (59) becomes(60)V˙s≤−c02−5+a/2Vs1+a/2,which implies that e1 and e2 will converge to the region ξ1Tξ1+ξ2Tξ2≤4r2/c01/2a−1=Δ1 in a finite time. Furthermore, e1 and e2 will be upper bounded by(61)e≤e1+e2≤e1+e2−u1,υ+u1,υ,≤ξ1+21−an1−a/2ξ2a+c1n1−a/2ξ1a,≤Δ1+21−a+c1n1−a/2Δ1a=Δ2.
In other words, if the parametersc1 and c2 are chosen according to (57), the tracking error e=e1T,e2TT will converge to the region Ω=e:e≤Δ2 in a finite time.
Proof of Theorem2 is complete.Remark 2.
As similar as the method used in [29], the underlying idea of the proposed nonlinear sliding variable s in (10) is homogeneous. Considering a weighted homogeneous dilation,(62)Dς:e1,e2↦ςe1,ςae2,∀ς>0.
With system (44), it is easily to verified that system (44) is a homogeneous system with respect to the dilation Dς, when s˙=0. Besides, the homogeneous degree of system (44) is a−1, which is a negative real number. However, proof of Theorem 2 is carried out by means of a modified backstepping method, as shown above. Compared with the geometric homogeneous method [16], advantages of this modified backstepping method are more specific proof details and more precisely parameter scope as pointed out in (49), (54), and (57). Furthermore, it is worthy to point out that, with the help of the similar construction method used in (10) and the modified backstepping method used in the proof of Theorem 2, the proposed SMAHFTC law could be extended for higher order system, which has the vector relative degree more than two.
To further illustrate thinking of the proposed SMAHFTC method, a block diagram is shown in Figure1.Figure 1
Principle diagram of the proposed SMAHFTC law.Remark 3.
It is worthy to point out that non-overestimation of the control gains need some additional expense, which is some precision degradations. In other words, a real sliding motion is established, which takes the replace of an ideal sliding motion.


## 4. A Rigid-Body Attitude Tracking Control Design

By means of the MRPs, attitude of a rigid body could be described asη=η1,η2,η3T. The directional cosine matrix from the body-fixed frame B to the inertial frame I (reference frame) is given as(63)Cbr=I3+41−ηn21+ηn22η×+811+ηn22η×η×,ηn2=ηTη.Furthermore, the relation between the Euler eigenangleϕ, the Euler eigenaxis, and the MRPs η is(64)η=tanϕ4ς,ϕ∈0,2π,where ς is the unit vector of the corresponding Euler eigenaxis.The kinematic equation of the rigid body is presented as(65)η˙=Rηωb,Rη=141+ηn2I3+2η×+2η×η×,where ωb=ω1,ω2,ω3T is the angular velocity of the rigid body with respect to the inertial frame I. Furthermore, ωb is expressed in the body-fixed frame B. The rotational dynamic equation is given as(66)ω˙b=J−1τb−ωb×Jωb+du⋅,where J is the moment of inertial matrix and du⋅ denotes the lumped uncertainties.The attitude command profile could be expressed by MRPs asηc=ηc,1,ηc,2,ηc,3T. The error MRPs between η and ηc is computed according to following equation:(67)ηe=1−ηc,n2η−1−ηn2ηc+2η×ηc1+ηn2ηc,n2+2ηcTη,ηn2=ηTη,ηc,n2=ηcTηc.Like the process presented in [5], dynamic equations of the error MRPs ηe could be derived as below:(68)η¨e=τ˜b+Dηe,η˙e,ωc,ω˙c,τ˜b=RηeJ−1τb,d˜u⋅=RηeJ−1du⋅,D⋅=−RηeJ−1ωb×Jωb+d˜u⋅−RηeCcbω˙c,+RηeR−1ηeη˙eCcbωc−RηeddtR−1ηeη˙e.In view of (7), see ηe and η˙e as the states e1 and e2 and see τ˜b and D⋅ as the control input u and the lumped uncertainty ρ˜⋅, respectively. An attitude tracking controller is derived from (11):(69)s=η˙e+c2∫0tsgn2a−1sgn1/aη˙eι+c11/aηeιdι,τ˜b=−c2sgn2a−1sgn1/aη˙e+c11/aηe,−k1ts1/2s−∫0tk2ι2sιsιdι,where k1t and k2t are dynamically determined by the adaptation law (12).


## 5. Simulations

In this section, a rigid body is considered to demonstrate the effectiveness of the proposed control method. The moment of inertial matrix is(70)J=500050001.To show the advantages of the proposed SMAHFTC method, a comparison study is carried out between the proposed SMAHFTC and the adaptive nonsingular fast terminal sliding mode control (ANFTSMC). The compared algorithm was proposed by the authors of [30], which has several properties similar to the proposed SMAHFTC, e.g., robustness against uncertainties and time-varying control gains by an adaptation law. Results of SMAHFTC and ANFTSMC are depicted by blue lines and red lines in Figures 2 and 3, respectively. The controller parameters are given in Table 1.Figure 2
Evolution of the attitude tracking and its errors: (a) tracking errorηe,1, (b) η1 tracking, (c) tracking error ηe,2, (d) η2 tracking, (e) tracking error ηe,3, and (f)η3 tracking.
(a)(b)(c)(d)(e)(f)Figure 3
Control signals, disturbances, and control gains. (a)−τ˜1 tracking error τ˜1,d (unit: N⋅m). (b) Zoom-in on −τ˜1 tracking error τ˜1,d. (c) −τ˜3 tracking error τ˜3,d (unit: N⋅m). (d) Zoom-in on −τ˜3 tracking error τ˜3,d. (e) Control gains of SMAHFTC. (f) Control gains of ANFTSMC.
(a)(b)(c)(d)(e)(f)Table 1
Control parameters.
ControllerParametersSMAHFTCc1c2aω1γ1εμk1,tϖ0.963.190.7310.60.050.54.2ANFTSMCαβk1k2kημ0μ1μ21.61.21352111The attitude command profiles are set toηc=0.2sin0.2t,cos0.2t,3T. The lumped disturbances are assumed to be Dex⋅=0.52sin0.3t,0.52cos0.3t,0.68cos0.3tT, rad/s2. Performances of attitude tracking are demonstrated in Figure 2. Subsequently, control inputs, disturbances, and control gains are illustrated in Figure 3.As shown in Figure2. The attitudes of SMAHFTC and ANFTSMC both could follow the corresponding command profiles precisely in a finite time, regardless of the presence of a certain disturbance. Convergence rate of the proposed SMAHFTC is faster than ANFTSMC.As demonstrated in Figure3(e), control gains of SMAHFTC could dynamically be increased or decreased in accordance with the time-varying disturbances. Therefore, minimum control gains could be obtained. However, as shown in Figure 3(f), control gains of ANFTSMC cannot be decreased with the time-varying disturbances. This implies that control gains of ANFTSMC may be overestimated, which maybe deteriorate the chattering effect. This phenomenon could also be seen in Figures 3(a)–3(d).


## 6. Conclusions

With the help of an innovative MIMO homogeneous finite-time sliding variable based on the signum-function technique, a SMAHFTC algorithm has been proposed for a class of MIMO cascade systems with vector relative degree two. The control gains, which are dynamically changed according to the uncertainties by the presented adaptation law, will not be overestimated. Meanwhile, a real sliding motion will be established in a finite time. Additionally, the control signals are continuous due to the adoption of STC. If overestimation of the control gains is permitted, a second-order ideal sliding mode will be achieved. Consequently, the control precision could be improved further. This trade-off is determined according to the specific practical situation by the designer. Stability analysis of the closed-loop system has rigorously been given, and the proposed method has been applied to the attitude tracking control for a rigid body. A comparison simulation has been conducted to verify the effectiveness of the proposed method. As pointed out in Remark2, though the result is presented for the second-order MIMO system, the proposed design method can easily be generalized to higher order system by means of the signum-function-based technique and the modified backstepping technique.



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*Source: 3665537-2022-04-28.xml*