Tracking Controller Design for Diving Behavior of an Unmanned Underwater Vehicle

Abstract

The study has investigated the almost disturbance decoupling problem of nonlinear uncertain control systems via the fuzzy feedback linearization approach. The significant dedication of this paper is to organize a control algorithm such that the closed-loop system is active for given initial condition and bounded tracking trajectory with the input-to-state stability and almost disturbance decoupling performance. This study presents a feedback linearization controller for diving control of an unmanned underwater vehicle. Unmanned underwater vehicle proposes difficult control subject due to its nonlinear dynamics, uncertain models, and the existence of disturbances that are difficult to measure. In general, while investigating the diving dynamics of an unmanned underwater vehicle, the pitch angle is always assumed to be small. This assumption is a strong restricting constraint in many interesting practical applications and will be relaxed in this study.

1. Introduction

In the past three decades, the utilization of unmanned underwater vehicle (UUV) has rapidly developed due to the applications of them to operate in deeper and riskier environments where human divers cannot arrive as scientific inspection of the deep sea, underwater cave exploration, oceanographic mapping, exploitation of underwater resources, long range survey, and marine warfare [1, 2].

Tracking controller design for diving behavior of an UUV is difficult task in the investigation of appropriate algorithms for motion and position control because the UUV dynamics includes inherently hydrodynamics and inertial nonlinearities, modeling uncertainties, disturbances of varying drag forces, and the coupling problems between degrees of freedom. So the controller for UUV should be robust to suppress the uncertain effects from nonlinearity and error of modeling and the interferences from complicated external environment. Therefore, the traditional linear control approach cannot resolve the UUV appropriately after linearizing the UUV model for small range operation [3].

The diving behavior of an unmanned underwater vehicle has been approximated to a linear system [4–6] based on two traditional assumptions, and furthermore the traditional linear control schemes can be applied. One assumption is that the pitch angle is close to zero in the diving plane for maneuvering, and the other one is that the pitch motion dynamics can be approximated as a linear equation by using Taylor’s expansion [1]. However, these two assumptions may create severe results in some practical applications due to their large modeling inaccuracy [7]. In this study, we directly solve the nonlinear dynamics of the depth motion without any restricting condition on the pitch angle of the UUV.

Recently, variable structure approach has been applied to treat nonlinear control system. However, inherited chattering effect may result in unmodeled high-frequency and even force system to be instable for variable structure approach structure. Adaptive backstepping approach has been an important tool for nonlinear strict-feedback systems and pure-feedback systems [8]. However, if the pitch angle is not close to zero, then the UUV’s diving dynamics equation cannot be expressed in the strict-feedback form or some other specific form investigated in [9]. Therefore, it is challengeable to apply the adaptive backstepping approach directly to the depth control with large pitch angle. The output regulation control [10] is applied to the control system in which the output variables are assumed to be drived by an exosystem. However, the desired regulation problem needs to find out the troublesome solution of partial-differential equation and the problem of the transient tracking errors [11]. In general, the nonlinear 𝐻^∞ control has to address the arduous nonlinear Hamilton-Jacobi equation [12]. Only for some specific control systems we can obtain a closed-form solution. The internal-model-principle approach, addressing a first-order partial-differential equation of the center manifold [13], transfers the tracking problem to nonlinear output regulation form. Only for some particular control systems and desired trajectories, the asymptotic solutions can be obtained [14]. The 𝐻^∞ adaptive fuzzy control has been applied to systematically investigate some control systems [15]. Its shortcoming is that the complicated parameter update law makes it be impractical. During the past decade, the feedback linearization approach has been the research direction for nonlinear systems [16, 17] and has been utilized successfully to solve many practical applications including the Three-Phase Photovoltaic Inverter [18], ball and beam system [16], and holonomic constrained robotic systems [19].

Fuzzy logic approach has been utilized to many industrial processes [20, 21]. Its designing structure is summarized as follows. First express the original fuzzy system as the Takagi-Sugeno model with fuzzy defined regions where the original system is locally linearized. The desired control design of resulting augmented system is organized by a combination of linear controllers constructed for each separated local linear part of the fuzzy model based on the parallel distributed compensation technique [22, 23]. For the stability investigation of fuzzy system, many researches are addressed [24, 25]. The stability analysis of fuzzy system can be mainly focused on Tanaka-Sugeno’s theorem. However, it is troublesome to carry out the common positive definite matrix 𝑃 for linear matrix inequality approach [26] even if 𝑃 is a second-order model. To solve the shortcoming of calculating the model-matrix 𝑃 for fuzzy logic approach, we will put forward a new controller design to achieve the almost disturbance decoupling property and the overall system is stabilized. Our proposed designing process is summarized as follows. First, a tracking control is constructed in order to achieve the almost disturbance decoupling performance and the stability of the closed-loop system response within a tunable global final attractor via the feedback linearization method. Once the tracking trajectories are steered to touch the global final attractor with the given radius, the fuzzy logic control instantly is added to achieve the desired convergence rate. In order to demonstrate the interesting industrial application, this study has favorably designed the almost disturbance decoupling controller for an unmanned underwater vehicle.

2. Controller Design

2.1. Controller Design for Feedback Linearization Approach

We will investigate the following nonlinear uncertain control system with bounded time-varying noises: 𝑥̇₁𝑥̇₂⋮𝑥̇_𝑛=𝑓₁𝑥₁,𝑥₂,…,𝑥_𝑛𝑓₂𝑥₁,𝑥₂,…,𝑥_𝑛⋮𝑓_𝑛𝑥₁,𝑥₂,…,𝑥_𝑛+𝑔₁𝑥₁,𝑥₂,…,𝑥_𝑛𝑔₂𝑥₁,𝑥₂,…,𝑥_𝑛⋮𝑔_𝑛𝑥₁,𝑥₂,…,𝑥_𝑛𝑢+Δ𝑓₁Δ𝑓₂⋮Δ𝑓_𝑛+^𝑝_𝑖=1𝑞^∗_𝑖𝜃_𝑖𝑑≡𝑓(𝑋(𝑡))+𝑔(𝑋(𝑡))𝑢+Δ𝑓+^𝑝_𝑖=1𝑞^∗_𝑖𝜃_𝑖𝑑,𝑦(𝑡)=ℎ𝑥₁,𝑥₂,…,𝑥_𝑛≡ℎ(𝑋(𝑡)), (1a)(1b)where 𝑋(𝑡)≡[𝑥₁(𝑡)𝑥₂(𝑡)⋯𝑥_𝑛(𝑡)]^𝑇∈ℜ^𝑛 is the system state vector, 𝑢∈ℜ¹ is the input variable, 𝑦∈ℜ¹ is the output variable, 𝜃_𝑑≡[𝜃_1𝑑(𝑡)𝜃_2𝑑(𝑡)⋯𝜃_𝑝𝑑(𝑡)]^𝑇 is a bounded time-varying noise vector, and Δ𝑓≡[Δ𝑓₁Δ𝑓₂⋯Δ𝑓_𝑛]^𝑇∈ℜ^𝑛 is unknown function denoting as Δ𝑓=∑^𝑝_𝑖=1𝑞^∗_𝑖𝜃_𝑖𝑢, where 𝜃_𝑢≡[𝜃_1𝑢(𝑡)𝜃_2𝑢(𝑡)⋯𝜃_𝑝𝑢(𝑡)]^𝑇 is a bounded time-varying uncertainty. 𝑓,𝑔,𝑞^∗₁,…,𝑞^∗_𝑝 are smooth vector fields on ℜ^𝑛 and ℎ(𝑋(𝑡))∈ℜ¹ is a smooth function. The nominal system of original system is then described as follows: 𝑋̇(𝑡)(𝑋(𝑡(𝑋(𝑡𝑦(𝑡)(𝑋(𝑡.=𝑓))+𝑔))𝑢,=ℎ)) (2a)(2b)The relative degree 𝑟 [27] is defined 1≤𝑟<∞ such that 𝐿_𝑔𝐿^𝑘_𝑓ℎ(𝑋(𝑡𝐿))=0,𝑘<𝑟−1,_𝑔𝐿_𝑓^𝑟−1ℎ(𝑋(𝑡))≠0 (3) for all 𝑋∈ℜ^𝑛 and 𝑡∈[0,∞), where the operator 𝐿 is denoted as the Lie derivative [28]. The desired trajectory 𝑦_𝑡(𝑡) and its first 𝑟 derivatives are all uniformly bounded and 𝑦_𝑡(𝑡),𝑦_𝑡⁽¹⁾(𝑡),…,𝑦_𝑡^(𝑟)(𝑡)≤𝑂, (4) where 𝑂 is a positive real variable.

Reference [28] has exploited the fact that the transformation Φ∶ℜ^𝑛⟶ℜ^𝑛 (5) denoted as Φ_𝑖(𝑋(𝑡))≡𝜉_𝑖(𝑡)=𝐿_𝑓^𝑖−1ℎ(𝑋(𝑡Φ)),𝑖=1,2,…,𝑟,_𝑘(𝑋(𝑡))≡𝜂_𝑘(𝑡),𝑘=𝑟+1,𝑟+2,…,𝑛 (6)(7) and satisfying 𝐿_𝑔Φ_𝑘(𝑋(𝑡))=0,𝑘=𝑟+1,𝑟+2,…,𝑛 (8) is a diffeomorphism based on the assumption of well-defined relative degree. For the sake of simplification, denote the tracking error to be 𝑒_𝑖(𝑡)≡𝜉_𝑖(𝑡)−𝑦_𝑡^(𝑖−1)(𝑡)𝑒(𝑡)≡𝑒,𝑖=1,2,…,𝑟,₁(𝑡)𝑒₂(𝑡)⋯𝑒_𝑟(𝑡)^𝑇∈ℜ^𝑟 (9)(10) and the tracking error multiplied with an adjustable positive real variable 𝜎 to be 𝑒_𝑖(𝑡)≡𝜎^𝑖−1𝑒_𝑖(𝑡),𝑖=1,2,…,𝑟,𝑒(𝑡)≡𝑒₁(𝑡)𝑒₂(𝑡)⋯𝑒_𝑟(𝑡)^𝑇∈ℜ^𝑟, (11)(12)𝜉(𝑡)≡𝜉₁(𝑡)𝜉₂(𝑡)⋯𝜉_𝑟(𝑡)^𝑇∈ℜ^𝑟,𝜂(𝑡)≡𝜂_𝑟+1(𝑡)𝜂_𝑟+2(𝑡)⋯𝜂_𝑛(𝑡)^𝑇∈ℜ^𝑛−𝑟,𝑞𝜉(𝑡)(𝑡)≡𝐿,𝜂_𝑓Φ_𝑟+1(𝑡)𝐿_𝑓Φ_𝑟+2(𝑡)⋯𝐿_𝑓Φ_𝑛(𝑡)^𝑇,≡𝑞_𝑟+1𝑞_𝑟+2⋯𝑞_𝑛^𝑇. (13a)(13b)(13c)

Denote the desired phase-variable canonical matrix 𝐴_pvc to be 𝐴_pvc≡010⋯0001⋯0⋮⋮000⋯1−𝑛₁−𝑛₂−𝑛₃⋯−𝑛_{𝑟𝑟×𝑟}, (14) where 𝑛₁,𝑛₂,…,𝑛_𝑟 are adjustable real variables such that 𝐴_pvc is Hurwitz matrix and the vector 𝐵 to be 𝐵≡00⋯01^𝑇_𝑟×1. (15)

Define 𝑃_pd to be the matrix of the famous Lyapunov equation 𝐴^𝑇_pvc𝑃_pd+𝑃_pd𝐴_pvc𝜆=−𝐼,_max≡themaximumeigenvalueofthepositivedefinitematrix𝑃_pd,𝜆_min≡theminimumeigenvalueofthepositivedefinitematrix𝑃_pd. (16)

Assumption 1. For all 𝑡≥0, 𝜂∈ℜ^𝑛−𝑟 and 𝜉∈ℜ^𝑟, there exists a positive real variable 𝐿_𝑘 such that the following property holds: 𝑞₂₂𝜂,𝑒−𝑞₂₂𝜂,0≤𝐿_𝑘(‖𝑒‖), (17) where 𝑞₂₂(𝜂,𝑒)≡𝑞(𝜉,𝜂).

For the sake of simplification, denote𝑢_𝑑≡𝐿_𝑔𝐿_𝑓^𝑟−1ℎ(𝑋(𝑡,𝑢))_𝑐≡𝐿^𝑟_𝑓ℎ(𝑋(𝑡)), (18a)(18b)𝑒=𝑛₁𝑒₁+𝑛₂𝑒₂+⋯+𝑛_𝑟𝑒_𝑟. (19)

Definition 2 (see [29]). Consider the control system 𝑥̇=𝑓(𝑡,𝑥,𝜃), where 𝑓∶[0,∞)×ℜ^𝑛×ℜ^𝑛→ℜ^𝑛 is a piecewise continuous function and has the locally Lipschitz property in 𝑥 and 𝜃. This control system is described to be input-to-state stable if there exists a class 𝐾𝐿 function 𝛽, a class 𝐾 function 𝛾, and positive real variables 𝑘₁ and 𝑘₂ such that for given initial state 𝑥(𝑡₀) with ‖𝑥(𝑡₀)‖<𝑘₁ and the system input 𝜃(𝑡) with sup_𝑡≥𝑡0‖𝜃(𝑡)‖<𝑘₂, the system states satisfy‖𝑥(𝑡)‖≤𝛽𝑥𝑡₀,𝑡−𝑡₀+𝛾sup_{𝑡0≤𝜏≤𝑡}‖𝜃(𝜏)‖ (20a) for all 𝑡≥𝑡₀≥0. Now we exploit the almost disturbance decoupling property as follows.
Definition  3  (see  [30]). The tracking problem with almost disturbance decoupling property is defined to be globally solvable by the controller 𝑢 for the tracking-error system by a global diffeomorphism (5), if the controller 𝑢 possesses the following properties.(i) The stystem is input-to-state stable with respect to disturbance inputs.(ii) For any given initial value 𝑥_𝑒0∶=[𝑒(𝑡₀)𝜂(𝑡₀)]^𝑇, for any 𝑡≥𝑡₀, and for any 𝑡₀≥0, 𝑦(𝑡)−𝑦_𝑑(𝑡)≤𝛽₁₁𝑥𝑡₀,𝑡−𝑡₀+1𝛽₂₂𝛽₃₃sup_{𝑡0≤𝜏≤𝑡}‖𝜃(𝜏)‖_𝑡𝑡0,𝑦(𝜏)−𝑦_𝑑(𝜏)²≤1𝑑𝜏𝛽₄₄𝛽₅₅𝑥_𝑒0+^𝑡_𝑡0𝛽₃₃‖𝜃(𝜏)‖²𝑑𝜏, (20b)(20c)
  where 𝛽₂₂,𝛽₄₄ are some positive real variables, 𝛽₃₃,𝛽₅₅ are class 𝐾 functions, and 𝛽₁₁ is a class 𝐾𝐿 function.

Theorem 4. The tracking problem with almost disturbance decoupling performance for system (1a)-(1b) is globally solvable by the controller 𝑢 denoted by 𝐿_𝑔𝐿_𝑓^𝑟−1ℎ(𝑋(𝑡))⁻¹×−𝐿^𝑟_𝑓ℎ(𝑋)+𝑦_𝑡^(𝑟)−𝜎^−𝑟𝑛₁𝐿⁰_𝑓ℎ(𝑋)−𝑦_𝑡−𝜎^1−𝑟𝑛₂𝐿¹_𝑓ℎ(𝑋)−𝑦_𝑡⁽¹⁾−⋯−𝜎⁻¹𝑛_𝑟𝐿_𝑓^𝑟−1ℎ(𝑋)−𝑦_𝑡^(𝑟−1), (21) and the influence of disturbances on the tracking error can be arbitrarily attenuated by adjusting the real variable 𝑁₂:𝑁₂≡min𝑚₁₁,𝑚₂₂,𝑚₁₁≡𝛿−2𝛿Φ_𝜉𝑃_𝑝𝑑²𝑚−0.25,₂₂≡𝜇𝑚₃−𝜇𝐿_𝑘𝑚₄²−𝜇𝐿_𝑘𝑚₄Φ_𝜂²,Φ_𝜉(𝜎)≡𝜎𝜕𝜕𝑋ℎ𝑞^∗₁𝜕⋯𝜎𝜕𝑋ℎ𝑞^∗_𝑝𝜎⋮⋮_𝑟𝜕𝐿𝜕𝑋^𝑓𝑟−1ℎ𝑞^∗₁⋯𝜎^𝑟𝜕𝐿𝜕𝑋^𝑓𝑟−1ℎ𝑞^∗_𝑞Φ_𝜂,(𝜎)≡𝜕Φ𝜕𝑋^𝑟+1𝑞^∗₁⋯𝜕Φ𝜕𝑋^𝑟+1𝑞^∗_𝑝𝜕⋮⋮Φ𝜕𝑋^𝑛𝑞^∗₁⋯𝜕Φ𝜕𝑋^𝑛𝑞^∗_𝑞, (22a)(22b)(22c)(22d)(22e)where 𝛿 and 𝜇 are adjustable real variables, based on the fact that there exists a continuously differentiable function 𝑉₀∶ℜ^𝑛−𝑟→ℜ⁺ such that the following properties hold for all 𝜂∈ℜ^𝑛−𝑟:(𝑎)𝑚₁𝜂²≤𝑉₀𝜂≤𝑚₂𝜂²,𝑚₁,𝑚₂(𝑏)∇>0,_𝜂𝑉₀^𝑇𝑞₂₂𝜂,0≤−𝑚₃𝜂²,𝑚₃(𝑐∇>0,)_𝜂𝑉₀≤𝑚₄𝜂,𝑚₄>0. (23a)(23b)(23c) Moreover, the overall tracking error is fascinated into a ball 𝐵_𝑟,𝑟=𝑁₁/𝑁₂, with an convergence rate denoted by 12𝑁₂𝜔₂−𝑁₁𝜔₂𝑟²≡12𝛼^∗,𝑁₁≡12sup_{𝑡0≤𝜏≤𝑡}𝜃_𝑑(𝜏)+𝜃_𝑢(𝜏)². (23d)(23e)

Proof. Applying the coordinate transformation (5) results in 𝜉̇₁(𝑡)=𝜉₂(𝑡)+𝜕ℎ(𝑋)𝜕𝑋^𝑝_𝑖=1𝑞^∗_𝑖𝜃_𝑖𝑑+𝜃_𝑖𝑢⋮𝜉̇_𝑟−1(𝑡)=𝜉_𝑟(𝑡)+𝜕𝐿_𝑓^𝑟−2ℎ(𝑋(𝑡))𝜕𝑋^𝑝_𝑖=1𝑞^∗_𝑖𝜃_𝑖𝑑+𝜃_𝑖𝑢𝜉̇_𝑟(𝑡)=𝐿^𝑟_𝑓ℎ(𝑋)+𝐿_𝑔𝐿_𝑓^𝑟−1ℎ(𝑋)𝑢+^𝑝_𝑖=1𝜕𝐿_𝑓^𝑟−1ℎ(𝑋(𝑡))𝑞𝜕𝑋^∗𝑖𝜃_𝑖𝑑+𝜃_𝑖𝑢𝜂̇_𝑘(𝑡)=𝐿_𝑓Φ_𝑘+^𝑝_𝑖=1𝜕Φ_𝑘(𝑋)𝑞𝜕𝑋^∗𝑖𝜃_𝑖𝑑+𝜃_𝑖𝑢. (24) Since 𝑢_𝑐𝜉(𝑡),𝜂(𝑡)≡𝐿^𝑟_𝑓ℎ(𝑋(𝑡,𝑢))_𝑑𝜉(𝑡),𝜂(𝑡)≡𝐿_𝑔𝐿_𝑓^𝑟−1ℎ(𝑋(𝑡,𝑞))_𝑘𝜉(𝑡),𝜂(𝑡)=𝐿_𝑓Φ_𝑘(𝑋),𝑘=𝑟+1,𝑟+2,…,𝑛; (25)(26)(27) the state and ouput equations of system (1a)-(1b) can be rewritten as follows: 𝜉̇_𝑖(𝑡)=𝜉_𝑖+1(𝑡)+^𝑝_𝑖=1𝜕𝐿𝜕𝑋^𝑓𝑖−1ℎ𝑞^∗_𝑖𝜃_𝑖𝑑+𝜃_𝑖𝑢𝜉̇,𝑖=1,2,…,𝑟−1,_𝑟(𝑡)=𝑢_𝑐𝜉(𝑡),𝜂(𝑡)+𝑢_𝑑𝜉(𝑡)(𝑡)𝑢+,𝜂_𝑝𝑖=1𝜕𝐿𝜕𝑋^𝑓𝑟−1ℎ𝑞^∗_𝑖𝜃_𝑖𝑑+𝜃_𝑖𝑢,𝜂̇_𝑘(𝑡)=𝑞_𝑘𝜉(𝑡)(𝑡)+,𝜂_𝑝𝑖=1𝜕Φ𝜕𝑋^𝑘(𝑋)𝑞^∗_𝑖𝜃_𝑖𝑑+𝜃_𝑖𝑢𝑦(𝑡),𝑘=𝑟+1,…,𝑛,=𝜉₁(𝑡). (28)
Denote 𝑣≡𝑦_𝑡^(𝑟)−𝜎^−𝑟𝑛₁𝐿⁰_𝑓ℎ(𝑋)−𝑦_𝑡−𝜎^1−𝑟𝑛₂𝐿¹_𝑓ℎ(𝑋)−𝑦_𝑡⁽¹⁾−⋯−𝜎⁻¹𝑛_𝑟𝐿_𝑓^𝑟−1ℎ(𝑋)−𝑦_𝑡^(𝑟−1). (29) Utilizing (6), (9), (25), and (26) yields the controller as 𝑢=𝑢_𝑑⁻¹−𝑢_𝑐+𝑣. (30) The dynamic equations of system (1a)-(1b) can be shown as follows by substituting equation (30) into (2.35): 𝜉̇₁(𝑡)𝜉̇₂(𝑡)⋮𝜉̇_𝑟−1(𝑡)𝜉̇_𝑟(𝑡)=010⋯0𝜉0010⋯0⋮⋮000⋯1000⋯0¹(𝑡)𝜉₂(𝑡)⋮𝜉_𝑟−1(𝑡)𝜉_𝑟(𝑡)+00⋮01𝑣++∑^𝑝_𝑖=1𝜕𝜕𝑋ℎ𝑞^∗_𝑖𝜃_𝑖𝑑+𝜃_𝑖𝑢+∑^𝑝_𝑖=1𝜕𝐿𝜕𝑋^1𝑓ℎ𝑞^∗_𝑖𝜃_𝑖𝑑+𝜃_𝑖𝑢⋮+∑^𝑝_𝑖=1𝜕𝐿𝜕𝑋^𝑓𝑟−1ℎ𝑞^∗_𝑖𝜃_𝑖𝑑+𝜃_𝑖𝑢,𝜂̇_𝑟+1(𝑡)𝜂̇_𝑟+2(𝑡)⋮𝜂̇_𝑛−1(𝑡)𝜂̇_𝑛(𝑡)=𝑞_𝑟+1(𝑡)𝑞_𝑟+2(𝑡)⋮𝑞_𝑛−1(𝑡)𝑞_𝑛(𝑡)+^𝑝_𝑖∑₌₁𝜕Φ𝜕𝑋^𝑟+1𝑞^∗_𝑖𝜃_𝑖𝑑+𝜃_{𝑖𝑢𝑝𝑖}∑₌₁𝜕Φ𝜕𝑋^𝑟+2𝑞^∗_𝑖𝜃_𝑖𝑑+𝜃_𝑖𝑢𝑝⋮_𝑖∑₌₁𝜕Φ𝜕𝑋^𝑛−1𝑞^∗_𝑖𝜃_𝑖𝑑+𝜃_{𝑖𝑢𝑝𝑖}∑₌₁𝜕Φ𝜕𝑋^𝑛𝑞^∗_𝑖𝜃_𝑖𝑑+𝜃_𝑖𝑢,𝑦=10⋯00_𝑟×1𝜉₁(𝑡)𝜉₂(𝑡)⋮𝜉_𝑟−1(𝑡)𝜉_𝑟(𝑡)_𝑟×1=𝜉₁(𝑡). (31)(32)(33) Combining equations (9), (11), (14), and (29) verifies the fact that equations (31)–(33) can be rewritten into the following equations:𝜂̇(𝑡)=𝑞𝜉(𝑡),𝜂(𝑡)+Φ_𝜂𝜃_𝑑+𝜃_𝑢≡𝑞₂₂𝜂(𝑡),𝑒+Φ_𝜂𝜃_𝑑+𝜃_𝑢,𝜎𝑒^⋅(𝑡)=𝐴_pvc𝑒+Φ_𝜉𝜃_𝑑+𝜃_𝑢, (34a)(34b)𝑦(𝑡)=𝜉₁(𝑡). (35) Define 𝑉(𝑒,𝜂) by a weighted sum of 𝑉₀(𝜂) and 𝑉₁(𝑒), 𝑉𝑒,𝜂≡𝑉₁(𝑒)+𝜇𝑉₀𝜂, (36) as a composite Lyapunov function of the systems (34a) and (34b) [31], where 𝑉₁(𝑒) satisfies 𝑉₁(𝑒)≡𝜎𝛿𝑒^𝑇𝑃_pd𝑒, (37) where 𝛿 and 𝜇 are adjusting positive real variables. Utilizing (17)–(19), (21) and (23a), (23b), (23c), (23d), and (23e) yields the derivative of 𝑉(𝑒,𝜂) along the trajectories of (34a) and (34b) as 𝑉̇₁=𝛿𝜎𝑒̇^𝑇𝑃_pd𝑒+𝛿𝑒^𝑇𝑃_pd𝜎𝑒̇=𝛿𝐴_𝑐𝑒+Φ_𝜉𝜃_𝑑+𝜃_𝑢^𝑇𝑃_pd𝑒+𝛿𝑒^𝑇𝑃_pd𝐴_𝑐𝑒+Φ_𝜉𝜃_𝑑+𝜃_𝑢=𝛿𝑒^𝑇𝐴^𝑇_pvc𝑃_pd+𝑃_pd𝐴_pvc𝑒+𝛿2𝜃_𝑑+𝜃_𝑢^𝑇Φ^𝑇_𝜉𝑃_pd𝑒≤−𝛿‖𝑒‖⃗²+2𝛿𝜃_𝑑+𝜃_𝑢Φ_𝜉𝑃_pd𝑒‖⃗‖𝑒‖⃗‖≤−𝛿²+2𝛿𝜃_𝑑+𝜃_𝑢Φ_𝜉𝑃_pd𝑒‖⃗‖𝑒‖⃗‖≤−𝛿²+2𝛿Φ_𝜉𝑃_pd²‖𝑒‖⃗²+14𝜃_𝑑+𝜃_𝑢², (38) that is, 𝑉̇₁≤−‖𝑒‖⃗²𝛿−2𝛿Φ_𝜉𝑃_pd²+14𝜃_𝑑+𝜃_𝑢²𝜇𝑉̇₀=𝜇𝜕𝑉₀𝜕𝜂^𝑇𝜂̇=𝜇𝜕𝑉₀𝜕𝜂^𝑇×𝑞₂₂𝜂,𝑒+Φ_𝜂𝜃−𝑞₂₂𝜂,0+𝑞₂₂𝜂,0≤𝜇𝜕𝑉₀𝑞𝜕𝜂²²𝜂,𝑒−𝑞₂₂𝜂,0+𝜇𝜕𝑉₀𝜕𝜂^𝑇𝑞₂₂𝜂,0+𝜇𝜕𝑉₀Φ𝜕𝜂^𝜂𝜃_𝑑+𝜃_𝑢≤𝜇𝑚₄𝐿_𝑘𝜂‖𝑒‖−𝜇𝑚₃𝜂²+𝜇𝑚₄Φ_𝜂𝜂𝜃_𝑑+𝜃_𝑢≤𝜇𝑚₄𝐿_𝑘²𝜂²+14‖𝑒‖²−𝜇𝑚₃𝜂²+𝜇𝑚₄Φ_𝜂²𝜂²+14𝜃_𝑑+𝜃_𝑢², (39) that is, 𝜇𝑉̇₀≤𝜂²−𝜇𝑚₃+𝜇𝑚₄𝐿_𝑘²+𝜇𝑚₄Φ_𝜂²+14‖𝑒‖²+14𝜃_𝑑+𝜃_𝑢². (40) Therefore, 𝑉̇=𝑉̇₁+𝜇𝑉̇₀≤−‖𝑒‖⃗²𝛿−2𝛿Φ_𝜉𝑃_pd²−14−𝜂²𝜇𝑚₃−𝜇𝑚₄𝐿_𝑘²−𝜇𝑚₄Φ_𝜂²+12𝜃_𝑑+𝜃_𝑢²≤−𝑚₁₁‖𝑒‖⃗²−𝑚₂₂𝜂²+12𝜃_𝑑+𝜃_𝑢²≤−𝑁₂‖𝑒‖²+𝜂²+12𝜃_𝑑+𝜃_𝑢²≡−𝑁₂𝑦_total²+12𝜃_𝑑+𝜃_𝑢², (41) where 𝑦_total²≡‖𝑒‖²+𝜂². (42) Applying ([29], Theorem  5.2) and (41) implies the input-to-state stability property for the overall system. Furthermore, it is easy to obtain the following inequality: 𝜔₁‖𝑒‖²+𝜂≤𝑉≤𝜔₂‖𝑒‖²+𝜂², (43) that is, 𝜔₁𝑦_total²≤𝑉≤𝜔₂𝑦_total², (44) where 𝜔₁≡min{𝜎𝛿𝜆_min,𝑘₁} and 𝜔₂≡min{𝜎𝛿𝜆_max,𝜇𝑘₂}. Applying (41) and (44) gives 𝑉̇≤−𝑁₂𝜔₂1𝑉+2𝜃_𝑑+𝜃_𝑢²𝑁≤−²𝜔₂1𝑉+2sup_{𝑡0≤𝜏≤𝑡}𝜃_𝑑+𝜃_𝑢(𝜏)². (45) Hence, 𝑉(𝑡)≤𝑉𝑡₀𝑒^{−(𝑁2/𝜔2)(𝑡−𝑡0)}+𝜔₂2𝑁₂sup_{𝑡0≤𝜏≤𝑡}𝜃_𝑑+𝜃_𝑢(𝜏)², (46) which implies 𝑒₁(𝑡)≤𝑉𝑡₀𝜎𝛿𝜆_min𝑒^{−(𝑁2/2𝜔2)(𝑡−𝑡0)}+𝜔₂2𝑁₂𝜎𝛿𝜆_minsup_{𝑡0≤𝜏≤𝑡}𝜃_𝑑+𝜃_𝑢(𝜏), (47) so that the almost disturbance decoupling property (20b) is achieved. Utilizing (41) yields 𝑉̇≤−𝑁₂‖𝑒‖²+𝜂²+12𝜃_𝑑+𝜃_𝑢² (48) which easily implies ^𝑡_𝑡0𝑦(𝜏)−𝑦_𝑑(𝜏)²≤𝑉𝑡𝑑𝜏₀𝑁₂+12𝑁₂^𝑡_𝑡0𝜃_𝑑+𝜃_𝑢(𝜏)²𝑑𝜏, (49) so that the almost disturbance decoupling property (20c) is satisfied and then the tracking problem with almost disturbance decoupling property is globally solved. Finally, we will exploit the fact that the ball 𝐵_𝑟 is a global attractor of system (1a)-(1b). Applying (41) and (23e) gives 𝑉̇≤−𝑁₂𝑦_total²+𝑁₁. (50) For ‖𝑦_total‖>𝑟, we get 𝑉̇<0. Hence, any ball denoted as 𝐵_𝑟≡𝑒𝜂∶‖𝑒‖²+𝜂²≤𝑟 (51) is a global final attractor of the systems (1a)-(1b). Furthermore, for 𝑦_total∉𝐵_𝑟, we have 𝑉̇𝑉≤−𝑁₂𝜔₂+𝑁₁𝜔₂𝑦_total²≤−𝑁₂𝜔₂+𝑁₁𝜔₂𝑟²≡−𝛼^∗. (52) Utilizing the comparison theorem [32] gives 𝑉𝑦_total(𝑡)≤𝑉𝑦_total𝑡₀exp−𝛼^∗𝑡−𝑡₀. (53) Therefore, 𝜔₁𝑦_total²≤𝑉𝑦_total(𝑡)≤𝜔₂𝑦_total𝑡₀²exp−𝛼^∗𝑡−𝑡₀. (54) Consequently, we get 𝑦_total≤𝜔₂𝜔₁𝑦_total𝑡₀−1exp2𝛼^∗𝑡−𝑡₀; (55) that is, the convergence rate toward the ball 𝐵_𝑟 is equal to 𝛼^∗/2.

2.2. Controller Design for Fuzzy Control Approach

After applying feedback linearization controller as a guarantee of uniform ultimate bounded stability, the fuzzy controller can be significantly utilized to obtain the better convergence rate of tracking error dynamics. The block diagram of the fuzzy controller is given in Figure 1. In general, the tracking error 𝑒(𝑡) and its time derivative 𝑒(̇𝑡) are applied as the input fuzzy variables of the IF-THEN control rules and the output variable is the control 𝑢_fuzzy.

Figure 1 

Block diagram of fuzzy logic controller.

For the sake of simplification, the membership functions of the linguistic terms for 𝑒(𝑡), 𝑒(̇𝑡), and 𝑢_fuzzy are all expressed to be the triangular functions. We denote seven linguistic terms for each fuzzy variable as shown in Figure 2: PB (positive big), PM (positive medium), PS (positive small), ZE (zero), NS (negative small), NM (negative medium) and NB (negative big).

(a) 𝑒(𝑡)

(b) 𝑒(̇𝑡)

(c) 𝑢fuzzy

Figure 2 

Plots of membership functions for (a) 𝑒(𝑡), (b) 𝑒(̇𝑡), and (c) 𝑢fuzzy.

Rule base table of fuzzy control 𝑢_fuzzy is shown in Table 1. The rule base is constructed by the Macvicar-Whelan rule base [33] for general servo control systems. The Mamdani method [34] is utilized for fuzzy inference. The defuzzification of the output set membership value is constructed by the centroid approach. Therefore, we can integrate the feedback linearization control and fuzzy control to build the controller as follows: 𝑢_{𝑓𝑒+𝑓𝑢}≡𝑢_feedback𝑢_𝑠(𝑡)+𝑢_fuzzy𝑢_𝑠𝑡−𝑡₁=𝐿_𝑔𝐿_𝑓^𝑟−1ℎ(𝑋(𝑡))⁻¹×−𝐿^𝑟_𝑓ℎ(𝑋)+𝑦_𝑡^(𝑟)−𝜎^−𝑟𝑛₁𝐿⁰_𝑓ℎ(𝑋)−𝑦_𝑡−𝜎^1−𝑟𝑛₂𝐿¹_𝑓ℎ(𝑋)−𝑦_𝑡⁽¹⁾−⋯−𝜎⁻¹𝑛_𝑟𝐿_𝑓^𝑟−1ℎ(𝑋)−𝑦_𝑡^(𝑟−1)𝑢_𝑠(𝑡)+𝑢_fuzzy𝑢_𝑠𝑡−𝑡₁, (56) where 𝑢_𝑠 is the unit step function and 𝑡₁ is the time that the tracking error dynamics touch the final global attractor ball 𝐵_𝑟.

Table 1 

Rule base table of fuzzy control.

3. Unmanned Underwater Vehicle

The coordinate system of an unmanned underwater vehicle with one propeller, two stern planes, and two rudders is shown in Figure 3. The six degree-of-freedom nonlinear dynamics of the UUV control system can be described as follows [5]: 𝑀(𝜈)𝜈̇+𝐶_𝐷(𝜈)(𝜈)𝜂̇𝜂,𝜈+𝐷𝜈+𝑔+𝑑=𝜏,𝜂=𝐽 (57) where 𝜂=[𝑥𝑦𝑧𝜙𝜃𝜓]^𝑇 denotes the position and orientation vector in earth-fixed reference frame, 𝜈=[𝑢𝑣𝑤𝑝𝑞𝑟]^𝑇 denotes the velocity and angular rate vector in vehicle (body)-fixed reference frame, 𝑀(𝜈)∈𝑅^6×6 denotes the inertia matrix including rigid-body mass and added mass, 𝐶_𝐷(𝜈)∈𝑅^6×6 denotes the matrix of Coriolis, centripetal term, 𝐷(𝜈)∈𝑅^6×6 denotes the damping matrix, 𝑔(𝜂)∈𝑅⁶ denotes the gravitational forces and moments vector, 𝑑 denotes the exogenous disturbance term, 𝜏 is the torque input vector, and 𝐽(𝜂) denotes the transformation matrix [5]. The UUV is considered to be longitudinal and bilateral symmetry and it is reasonable to make the assumption that the origin of the body-fixed coordinate is located at the center of gravity, and the terms in 𝑔(𝜂)∈𝑅⁶ is all zero. For the dynamics of the diving plane, we can assume that the roll and yaw angular velocities are close to zeros based on appropriately adjusting the RPM of propeller and the rudder’s angles. For the discussion of the horizontal moving dynamics in this study, the forward velocity is kept constant 𝑢₀ by properly adjusting the rev of the thrusters. Under the above assumptions, the heave dynamics of UUV can be described as 𝑧̇=−𝑢sin𝜃+𝑣cos𝜃sin𝜙+𝜔cos𝜃cos𝜙≈−𝑢₀sin𝜃. (58) The roll angle 𝜙 is nearly constant, since 𝑝≈0. Without any loss of generality, we can assume 𝜙=0 and the pitch dynamics can be expressed as 𝜃̇=𝑞cos𝜙−𝑟sin𝜙≈𝑞cos𝜙≈𝑞. (59) Consequently, the diving dynamics of UUV can be certain modified as followed [5–7]:𝑧̇=−𝑢₀̇𝑚sin𝜃,𝜃=𝑞,_𝑞𝑞̇=Φ^𝑇Θ+𝐹_𝑞𝑢²₀𝛿_𝑞+𝑑_𝑞, (60a)(60b)(60c) where Φ=𝑞𝑢̇𝑢𝑢²𝜔𝑞𝑟𝑞cos𝜙sinΨ^𝑇,Θ=𝜃₁𝜃₂𝜃₃𝜃₄𝜃₅𝜃₆𝜃₇^𝑇, (60d)(60e)and 𝑚_𝑞 denotes the inertia term including the added mass, 𝐹_𝑞 denotes the fin moment coefficient, and 𝛿_𝑞 denotes the stern plane angle as depicted in Figure 4.

Figure 3 

The unmanned underwater vehicle.

Figure 4 

The stern plane angle of attack.

Substituting all the physical values 𝑚_𝑞=3.495,  𝐹_𝑞=6.51,  𝑢₀=1.54, and selecting 𝑥₁≡𝑧,  𝑥₂≡𝜃,  𝑥₃≡𝑞,  𝜃₁=1,  𝜃₇=1 into (60a), (60b), (60c), (60d), and (60e), the state equation can be rewritten as follows:𝑥̇₁=−1.54sin𝑥₂,𝑥̇₂=𝑥₃,𝑥̇₃=1𝑥3.495³+13.495sin𝑥₂+15.5126𝑢+𝑑_𝑞, (61a)(61b)(61c)where 𝑑_𝑟=0.5sin𝑡 is assumed to be the disturbance item. For a given desired trajectory 𝑦_𝑡=sin0.1𝑡, the main objective of this study is to construct a nonlinear feedback linearization controller for UUV system (61a), (61b), and (61c) with tracking performance by applying Theorem 4. Let us arbitrarily choose 𝑛₁=1.5,  𝑛₂=1.5,  𝑛₃=1.5 such that 𝐴_pvc=010001−1.5−1.5−1.5 (62) is Hurwitze and the positive definite matrix 𝑃_pd=3.52.50.252.54.751.50.251.51. (63) The UUV system is a system of relative degree three. It can be verified that the related conditions of Theorem 4 are satisfied with 𝜎=0.1,𝑚₁₁=1.75,𝑟=0.4,𝑁₁=0.125, and 𝑁₂=1.75. From (56), we obtain the desired tracking controller𝑢=𝐿_𝑔𝐿²_𝑓ℎ(𝑋(𝑡))⁻¹×−𝐿³_𝑓ℎ(𝑋)+𝑦_𝑡⁽³⁾−𝜎⁻³𝑛₁ℎ(𝑋)−𝑦_𝑡−𝜎⁻²𝑛₂𝐿¹_𝑓ℎ(𝑋)−𝑦_𝑡⁽¹⁾−𝜎⁻¹𝑛₃𝐿²_𝑓ℎ(𝑋)−𝑦_𝑡⁽²⁾𝑢_𝑠(𝑡)+𝑢_fuzzy𝑢_𝑠𝑡−𝑡₁,𝐿_𝑔𝐿²_𝑓ℎ(𝑋)=−23.889cos𝑥₂,𝐿³_𝑓ℎ(𝑋)=1.54𝑥²₃sin𝑥₂−1.54cos𝑥₂𝑘₁𝑥3.495³+𝑘₇3.495sin𝑥₂,𝑦_𝑑⁽³⁾𝑦=−0.001cos0.1𝑡,_𝑑(2)𝑦=−0.01sin0.1𝑡,_𝑑(1)𝐿=0.1cos0.1𝑡,_1𝑓ℎ(𝑋)=−1.54sin𝑥₂𝐿²_𝑓ℎ(𝑋)=−1.54𝑥₃cos𝑥₂. (64a)(64b)(64c)(64d)(64e)(64f)Hence, the tracking controller will steer the state variable 𝑥₁ to track the desired trajectory 𝑦_𝑡(𝑡)=ℎ(𝑋)=sin0.1𝑡 in view of Theorem 4. The tracking errors driven by 𝑢 for UUV system (61a), (61b), and (61c) is depicted in Figure 5.

Figure 5 

The tracking error dynamic 𝑥1−sin0.1𝑡 driven by 𝑢 for (61a), (61b), and (61c).

4. Conclusion

In this paper, we have proposed a fuzzy feedback control design which globally solves the almost disturbance decoupling problem based on the fuzzy feedback linearization approach for trajectory tracking using a rudder of a UUV. The investigation of input-output feedback linearization of nonlinear uncertain control systems by diffeomorphism has been proposed without eliminating the nonlinear terms in the UUV model, and so severe model errors will not be taken into the developing process. Moreover, this study proposes a control algorithm for diving dynamics of UUV without restricting constraint on the small pitch angle in the diving plane. Simulation results exploit the fact that our proposed design is favorably applied to input-output linearization problem and achieves the desired almost disturbance decoupling performance and better convergence rate of the closed-loop system.