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Abstract

This article presents a robust stabilizing control for nonholonomic underwater systems that are affected by uncertainties. The methodology is based on adaptive integral sliding mode control. Firstly, the original underwater system is transformed in a way that the new system has uncertainties in matched form. A change of coordinates is carried out for this purpose, and the nonholonomic system is transformed into chained form system with matched uncertainties. Secondly, the chained form system with uncertainties is transformed into a special structure containing nominal part and some unknown terms through input transformation. The unknown terms are computed adaptively. Afterward, the transformed system is stabilized using integral sliding mode control. The stabilizing controller for the transformed system is constructed which consists of the nominal control plus some compensator control. The compensator controller and the adaptive laws are derived in a way that the derivative of a suitable Lyapunov function becomes strictly negative. Two different cases of perturbation are considered including the bounded uncertainty present in any single control input and the uncertainties present in the overall system model of the underwater vehicle. Finally, simulation results show the validity and correctness of the proposed controllers for both cases of nonholonomic underwater system affected by uncertainties.

Keywords

Adaptive integral sliding mode Lyapunov function nonholonomic systems autonomous underwater vehicles

Introduction

In recent years, there has been an increasing interest in design and implementation of robust control laws for underwater vehicles. The underwater vehicles are useful in performing vital roles in monitoring of coastal shallow water regions, environmental surveying, offshore oil installations, undersea cable/pipeline inspection, oil/mineral explorations, homeland security, and many other underwater tasks.^{1

–11} The survey papers^1
–3 provide a useful and relevant overview on the control of autonomous underwater vehicles (AUVs). In the study by Bi et al.,⁴ a tracking control of under-actuated AUV was designed in the presence of unknown ocean currents, whereas position-based control of underwater robotic system for maintaining position in the presence of ocean currents was presented by Kim et al.⁵ A hybrid control for dynamic positioning of an under-actuated marine system was proposed by Panagou and Kyriakopoulos,⁶ and a second order sliding mode controller was designed for AUV in the presence of unknown disturbances in the study by Joe et al.⁷ Neural network–based control technique for controlling the trajectory of AUVs was presented by Eski and Yidirim.⁸ A backstepping-based adaptive tracking control design for under-actuated AUVs has been reported in the study by Ghommam and Saad.⁹ In the study by Akcakaya and Sumer,¹⁰ a robust control of variable speed AUV was designed. Also, point stabilization for an underwater vessel in the presence of sea currents was proposed by Zaopeng et al.¹¹

Nonholonomic chained form systems

Owing to the presence of nonholonomic constraints, the kinematic model of AUVs is best described by

\dot{z} = \sum_{i = 1}^{m} g_{i} (z) u_{i}, z \in ℝ^{n}

where $u_{i}, i = 1, \dots, m$ , and m < n are the locally bounded piecewise continuous control functions defined on the interval $[0, \infty)$ and g_i are independent vector fields on ℝⁿ. An important feature of nonholonomic mechanical systems described by (1) is that the number of inputs is always fewer than the degrees of freedom. This distinguishing feature makes the stabilization of nonholonomic mechanical systems even more challenging. Although this class of nonlinear systems is controllable, however, these cannot be stabilized by continuous static state feedback as these fail the Brockett’s necessary condition for smooth stabilization.¹² As a result, the developed methods for smooth nonlinear control cannot be applied directly to the nonholonomic systems. Various approaches have been identified in the literature¹³ for resolving this problem which can be further categorized into time-varying continuous feedback control,¹⁴ discontinuous control,¹⁵ and hybrid control.¹⁶

For the purpose of analysis and control design, it is extremely helpful to first transform the original system into some canonical form via state–input transformation. One such canonical form is the chained form first introduced by Murray and Sastry.¹⁷ Many nonholonomic systems can be converted locally or globally into the chained form under coordinate/input transformation.¹³ Thus, the chained form is frequently used as a canonical form in the control and analysis of nonholonomic mechanical systems. In recent years, much effort has been put in for tracking and stabilizing control of chained form systems.^{14

–21} Detailed progress on higher order nonholonomic chained system control has been presented by Huang et al.¹⁸

Adaptive integral sliding mode

The control problem becomes further complicated whenever various uncertainties influence the nonholonomic mechanical systems.^22
–24 A simple, rather useful, approach to control nonholonomic uncertain systems is the sliding mode control (SMC). The SMC has attracted the attention of many researchers due to its simplicity, fast response, and robustness to external noise and parameter variations.^24

–29 The properties of SMC just depend upon the design of the switching surface and have nothing to do with the external interferences.²⁹ Some recent practical applications of SMC include control of steerable needles,³⁰ nonlinear control of an unmanned agricultural tractor,³¹ and impedance control of a piezoelectric microgripper.³²

The two phases of SMC are the reaching and the sliding phase. During the sliding phase, the system trajectory is forced to slide on the sliding surface. In this case, the system response solely depends on the parameters of the surface and remains insensitive to external disturbances and variations of system parameters. However, during the reaching phase, the system has no such insensitivity property. Therefore, insensitivity to external disturbances cannot be ensured for the whole system response.²⁹ Also, SMC is not useful in real-world applications because of the presence of chattering phenomenon. The integral sliding mode control (ISMC) is used to resolve the inherent chattering problem of the sliding mode. The ISMC combines the nominal control that stabilizes the nominal system and a discontinuous control that rejects the uncertainty. Various control methodologies are used in conjunction with SMC to garner the combined benefits. These include backstepping,^9,28 adaptive,²⁶ neural networks,³³ and the fuzzy control.³⁴ The neural networks have some drawbacks including the difficulty in selecting appropriate network and slow convergence to the equilibrium point. Similarly, the major drawback of fuzzy logic control is the difficulty in getting fuzzy rules and membership functions. Also, backstepping techniques are difficult to implement on nonholonomic mechanical systems, whereas the adaptive technique in combination with ISMC guarantees the robustness of system trajectory in the whole state space.^24,26

This article presents a robust method for stabilization of nonholonomic AUVs affected with uncertainties. The methodology is based on adaptive ISMC. The importance of this research is further highlighted by the fact that the proposed methodology is general and can be applied to other nonholonomic mechanical systems such as wheeled mobile robots, helicopters, vertical take off and landing aircrafts, surface vehicles, and so on. At the same time, the proposed methodology provides robustness for the whole state space and the chattering problem is significantly reduced. Firstly, the original system is transformed to a perturbed chained form system. Secondly, the perturbed chained form system is further transformed into a special structure containing nominal part and some unknown terms through input transformation. The unknown terms are computed adaptively. The transformed system is then stabilized using ISMC. The stabilizing controller for the transformed system is constructed which consists of the nominal control plus some compensator control. The compensator controller and the adaptive laws are derived in such a way that the derivative of a suitable Lyapunov function becomes negative in the strict sense. Finally, the simulations show validity of the proposed control algorithm for the original underwater vehicle model with two different cases of uncertainties.

Problem formulation

Kinematic model of underwater vehicle

Consider a model of an underwater vehicle as shown in Figure 1. The two frames of reference used for deriving the AUV model include the inertial frame (O − XYZ) and the local frame (c − xyz). The local frame is attached to the vehicle at its center c. Motion of the vehicle is described by using six coordinates, three for position (x,y,z) and three for the vehicle orientation $(ϕ, θ, ψ)$ . The Euler angle ϕ corresponds to the roll motion, while ψ and θ represent yaw and pitch motions, respectively.

Figure 1.

Model of an underwater vehicle.

The velocity of underwater vehicle is represented by v and its components along the x,y, and z axes are given by³⁵

[\begin{matrix} \dot{x} \\ \dot{y} \\ \dot{z} \end{matrix}] = [\begin{matrix} v cos ψ cos θ \\ v sin ψ cos θ \\ - v sin θ \end{matrix}]

The relationship between the angular velocity $ω = {(ω_{x}, ω_{y}, ω_{z})}^{T}$ and time rate of the Euler angles in the local frame is given as

[\begin{matrix} \dot{ϕ} \\ \dot{θ} \\ \dot{ψ} \end{matrix}] = [\begin{matrix} 1 & sin ϕ tan θ & cos ϕ tan θ \\ 0 & cos ϕ & - sin ϕ \\ 0 & sin ϕ sec θ & cos ϕ sec θ \end{matrix}] [\begin{matrix} ω_{x} \\ ω_{y} \\ ω_{z} \end{matrix}]

After combining equations (2) and (3) and using a modified set of state and control variables as $η = {(x, y, z, ϕ, θ, ψ)}^{T}$ and $(u_{1}, u_{2}, u_{3}, u_{4}) = (v, ω_{x}, ω_{y}, ω_{z})$ yields kinematic model for the underwater vehicle as

\dot{η} = [\begin{matrix} \dot{x} \\ \dot{y} \\ \dot{z} \\ \dot{ϕ} \\ \dot{θ} \\ \dot{ψ} \end{matrix}] = [\begin{matrix} cos ψ cos θ \\ sin ψ cos θ \\ - sin θ \\ 0 \\ 0 \\ 0 \end{matrix}] u_{1} + [\begin{matrix} 0 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{matrix}] u_{2} + [\begin{matrix} 0 \\ 0 \\ 0 \\ sin ϕ tan θ \\ cos ϕ \\ sin ϕ sec θ \end{matrix}] u_{3} + [\begin{matrix} 0 \\ 0 \\ 0 \\ cos ϕ tan θ \\ - sin ϕ \\ cos ϕ sec θ \end{matrix}] u_{4}

or in compact form as

\dot{η} = g_{1} (η) u_{1} + g_{2} (η) u_{2} + g_{3} (η) u_{3} + g_{4} (η) u_{4}

Case studies

Two different cases of uncertainties are considered in this article. Robust stabilizing control algorithm is developed for both cases of uncertainties. Details of the uncertainties are as follows.

Case 1

Here, it is assumed that the perturbation is present only in one control input. Consider the above system (4) and choose $u_{1} = \frac{1}{cos ψ cos θ} {\bar{u}}_{1},$ where $ψ, θ \neq \frac{π}{2}$ and ${\bar{u}}_{1}$ is the new input. Then, system (4) can be written as

\dot{η} = {\bar{g}}_{1} (η) {\bar{u}}_{1} + g_{2} (η) u_{2} + g_{3} (η) u_{3} + g_{4} (η) u_{4}

where

\begin{array}{l} {\bar{g}}_{1} (η) = [\begin{matrix} 1 & tan ψ & - sec ψ tan θ & 0 & 0 & 0 \end{matrix}]^{T} \\ g_{2} (η) = [\begin{matrix} 0 & 0 & 0 & 1 & 0 & 0 \end{matrix}]^{T} \\ g_{3} (η) = [\begin{matrix} 0 & 0 & 0 & sin ϕ tan θ & cos ϕ & sin ϕ sec θ \end{matrix}]^{T} \\ g_{4} (η) = [\begin{matrix} 0 & 0 & 0 & cos ϕ tan θ & - sin ϕ & cos ϕ sec θ \end{matrix}]^{T} \end{array}

and $η \in ℝ^{n}, u_{i}, and i = 1, \dots, 4$ are scalar control inputs.

Now, u₄ is assumed to be perturbed as $u_{4} = {\bar{u}}_{4} + Δ (η, t)$ , where $Δ (η, t)$ is an unknown scalar function bounded by time derivatives and known bound, that is, $| Δ (η, t) | < M$ .

Case 2

In this case, the perturbation is considered to be present in the overall system model. Therefore, the underwater vehicle model (5) with some nonlinear uncertainties can be written as

\dot{η} = {\bar{g}}_{1} (η) {\bar{u}}_{1} + g_{2} (η) u_{2} + g_{3} (η) u_{3} + g_{4} (η) u_{4} + p (η, t)

where $p (η, t) \in ℝ^{6}$ is an additive perturbation in the system model. Also, it is assumed that the additive uncertainties are in matched form.

The control problem

Given a desired set point $η_{des} \in ℝ^{n}$ , construct a stabilizing control law in terms of the inputs $u_{i} : ℝ^{n} \to ℝ, i = 1, \dots, 4$ so that the desired set point η_des is an attractive set for (5) and (6) in spite of the additive uncertainties and there exists an ε > 0 such that $η (t; 0, η_{0}) \to η_{des}$ as t → ∞ for any initial condition $η_{0} \in B (η_{des}; ε)$ .

Also, it is assumed without any loss of generality that $η_{des} = 0$ that can be obtained by a suitable translation of the coordinate system.

Some assumptions

For stabilizing control problem, the nonholonomic underwater systems described by (4) must satisfy the following assumptions:

Assumption 1: The vector fields $g_{i} (η), i = 1, \dots, 4$ are linearly independent.

Assumption 2: The nonholonomic system satisfies the Lie algebra rank condition (LARC) for accessibility,³⁶ that is, the Lie algebra, $L (g_{1}, \dots, g_{4}) (η)$ spans the whole state space ℝⁿ at each point $η \in ℝ^{n}$ .

The kinematic model of AUV given by (4) satisfies the above two assumptions, that is, the vector fields $g_{1}, g_{2}, g_{3}$ , and g₄ are linearly independent and the system (4) is completely controllable on the manifold $M = {η = {(x, y, z, ϕ, θ, ψ)}^{T} \in ℝ^{6} : | ϕ | < \frac{π}{2}}$ as it satisfies the LARC for controllability on M.

In order to prove the aforementioned assumption 2, it is sufficient to compute the Lie brackets given below

g_{5} (η) = [g_{1}, g_{3}] (η) = [\begin{matrix} sin θ cos ψ cos ϕ + sin ϕ sin ψ \\ sin θ sin ψ cos ϕ - sin ϕ cos ψ \\ cos ϕ cos θ \\ 0 \\ 0 \\ 0 \end{matrix}]

g_{6} (η) = [g_{1}, g_{4}] (η) = [\begin{matrix} sin ϕ sin θ cos ψ + cos ϕ sin ψ \\ - sin ϕ sin θ sin ψ - cos ϕ cos ψ \\ - sin ϕ cos θ \\ 0 \\ 0 \\ 0 \end{matrix}]

It is now straightforward to verify that if the system motion is restricted to the manifold M, then ${g_{1}, g_{2}, g_{3}, g_{4}, g_{5}, g_{6}}$ are linearly independent, thus, satisfying the LARC condition

span {g_{1} (η), \dots, g_{6} (η)} = ℝ^{6} \forall η \in M

Conversion into chained form

The original system is, first of all, transformed into a perturbed chained form system. Then, the perturbed chained form system is further transformed into a special structure containing nominal part and some unknown terms through input transformation.

Now using the following state transformation

\begin{array}{l} x_{1} = x \\ x_{2} = tan ψ \\ x_{3} = y \\ x_{4} = - tan θ sec ψ \\ x_{5} = z \\ x_{6} = ϕ \\ x = {[\begin{matrix} x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & x_{6} \end{matrix}]}^{T} \end{array}

and input transformation

[\begin{matrix} v_{1} \\ v_{2} \\ v_{3} \\ v_{4} \end{matrix}] = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 0 & a_{1} & a_{2} \\ 0 & 0 & a_{3} & a_{4} \\ 0 & 1 & a_{5} & a_{6} \end{matrix}] [\begin{matrix} {\bar{u}}_{1} \\ u_{2} \\ u_{3} \\ u_{4} \end{matrix}] i.e. v = A u

where

\begin{array}{l} a_{1} = \frac{sin θ}{cos ϕ {cos}^{2} ψ}, a_{2} = \frac{cos θ}{cos ϕ {cos}^{2} ψ}, a_{3} = \frac{- cos ϕ cos ψ - sin ϕ sin θ sin ψ}{{cos}^{2} θ {cos}^{2} ψ} \\ a_{4} = \frac{sin ϕ cos ψ - cos ϕ sin θ sin ψ}{{cos}^{2} θ {cos}^{2} ψ}, a_{5} = \frac{sin ϕ sin θ}{cos θ}, a_{6} = \frac{cos ϕ sin θ}{cos θ} \\ | A | = \frac{1}{{(cos θ cos ψ)}^{3}} \neq 0 if ψ, θ \neq \frac{π}{2} \end{array}

the system (5) can be written in chained form as

\begin{array}{l} {\dot{x}}_{1} = v_{1} \\ {\dot{x}}_{2} = v_{2} \\ {\dot{x}}_{3} = x_{2} v_{1} \\ {\dot{x}}_{4} = v_{3} \\ {\dot{x}}_{5} = x_{4} v_{1} \\ {\dot{x}}_{6} = v_{4} \end{array}

Case 1

The control problem is solved by first converting the system (5) into the following chained form system containing the uncertainty term with v₄. The system is, therefore, complicated by the presence of a bounded disturbance affecting the last equation

\begin{array}{l} {\dot{x}}_{1} = v_{1} \\ {\dot{x}}_{2} = v_{2} \\ {\dot{x}}_{3} = x_{2} v_{1} \\ {\dot{x}}_{4} = v_{3} \\ {\dot{x}}_{5} = x_{4} v_{1} \\ {\dot{x}}_{6} = v_{4} + Δ (x, t) \end{array}

where $Δ (x, t)$ is unknown scalar function bounded with first time derivatives and with known bound, that is, $| Δ (x, t) | < M$ .

Case 2

Here, consider system (6) along with model level perturbation as

\dot{η} = {\bar{g}}_{1} (η) {\bar{u}}_{1} + g_{2} (η) u_{2} + g_{3} (η) u_{3} + g_{4} (η) u_{4} + p (η, t)

where

\begin{array}{l} {\bar{g}}_{1} (η) = [\begin{matrix} 1 & tan ψ & - sec ψ tan θ & 0 & 0 & 0 \end{matrix}]^{T} \\ g_{2} (η) = [\begin{matrix} 0 & 0 & 0 & 1 & 0 & 0 \end{matrix}]^{T} \\ g_{3} (η) = [\begin{matrix} 0 & 0 & 0 & sin ϕ tan θ & cos ϕ & sin ϕ sec θ \end{matrix}]^{T} \\ g_{4} (η) = [\begin{matrix} 0 & 0 & 0 & cos ϕ tan θ & - sin ϕ & cos ϕ sec θ \end{matrix}]^{T} \end{array}

and $p (η, t) \in span {{\bar{g}}_{1} (η), g_{2} (η), g_{3} (η), g_{4} (η)}, \forall t$

According to Liang and Jianying,²⁴ if $p (η, t) \in span {{\bar{g}}_{1} (η), g_{2} (η), g_{3} (η), g_{4} (η)}$ is in matched form, then under coordinate change and state feedback (11) can be locally or globally transformed into

\begin{array}{l} {\dot{x}}_{1} = v_{1} + p_{1} (x, t) \\ {\dot{x}}_{2} = v_{2} + p_{2} (x, t) \\ {\dot{x}}_{3} = x_{2} (v_{1} + p_{1} (x, t)) \\ {\dot{x}}_{4} = v_{3} + p_{3} (x, t) \\ {\dot{x}}_{5} = x_{4} (v_{1} + p_{1} (x, t)) \\ {\dot{x}}_{6} = v_{4} + p_{4} (x, t) \end{array}

where the perturbations are $p_{1} (x, t) = x_{1} ϕ_{1} (x, t) θ_{1}$ , $p_{2} (x, t) = x_{2} ϕ_{2} (x, t) θ_{2}$ , $p_{3} (x, t) = x_{4} ϕ_{3} (x, t) θ_{3}$ , and $p_{4} (x, t) = x_{6} ϕ_{4} (x, t) θ_{4}$ with known nonlinear functions $ϕ_{i} (x, t), i = 1, \dots, 4$ and unknown constants $θ_{i}, i = 1, \dots, 4$ . Let ${\hat{θ}}_{i}$ be estimates of θ_i and let ${\tilde{θ}}_{i} = θ_{i} - {\hat{θ}}_{i}, i = 1, \dots, 4$ be errors in estimation of θ_i.

Case Study 1: Stabilizing algorithm for perturbed control input

The original underwater vehicle system (5) is, first of all, transformed into a perturbed chained form system (10). Then, the perturbed chained form system is further transformed into a special structure containing nominal part and some unknown terms through input transformation. The unknown terms are computed adaptively and the transformed system is stabilized using ISMC. The stabilizing controller for the transformed system is constructed which consists of the nominal control plus some compensator control. Step-by-step details of the stabilizing algorithm are presented in the subsection.

Control algorithm

Step 1: Choose $v_{1} = x_{2}$ , $v_{2} = x_{3}$ , $v_{3} = x_{5}$ , and $v_{4} = v$ . Then, system (10) becomes

\begin{array}{l} {\dot{x}}_{1} = x_{2} \\ {\dot{x}}_{2} = x_{3} \\ {\dot{x}}_{3} = x_{2}^{2} \\ {\dot{x}}_{4} = x_{5} \\ {\dot{x}}_{5} = x_{3} x_{2} \\ {\dot{x}}_{6} = v + Δ (t) \end{array}

Step 2: After some manipulation, the above system (13) can be written as

\begin{array}{l} {\dot{x}}_{1} = x_{2} \\ {\dot{x}}_{2} = x_{3} \\ {\dot{x}}_{3} = x_{4} + F_{3} \\ {\dot{x}}_{4} = x_{5} \\ {\dot{x}}_{5} = x_{6} + F_{5} \\ {\dot{x}}_{6} = v + Δ (t) \end{array}

where $F_{3} = - x_{4} + x_{2}^{2}, F_{5} = - x_{6} + x_{3} x_{2}$

Step 3: Now, assume that F₃ and F₅ are unknowns and can be computed adaptively. Let ${\hat{F}}_{i}$ be an estimate of F_i $, i = 3, 5$ , respectively, and ${\tilde{F}}_{i} = F_{i} - {\hat{F}}_{i}$ be the errors in estimation of F_i $, i = 3, 5$ , respectively.

Then, system (14) can be written as

\begin{array}{l} {\dot{x}}_{1} = x_{2} \\ {\dot{x}}_{2} = x_{3} \\ {\dot{x}}_{3} = x_{4} + {\hat{F}}_{3} + {\tilde{F}}_{3} \\ {\dot{x}}_{4} = x_{5} \\ {\dot{x}}_{5} = x_{6} + {\hat{F}}_{5} + {\tilde{F}}_{5} \\ {\dot{x}}_{6} = v + Δ (t) \end{array}

Step 4: The nominal system for (15) becomes

\begin{array}{l} {\dot{x}}_{1} = x_{2} \\ {\dot{x}}_{2} = x_{3} \\ {\dot{x}}_{3} = x_{4} \\ {\dot{x}}_{4} = x_{5} \\ {\dot{x}}_{5} = x_{6} \\ {\dot{x}}_{6} = v_{0} \end{array}

Now, choose a sliding surface for (16) as $σ_{0} = {(1 + \frac{d}{d t})}^{5} x_{1}$

σ_{0} = x_{1} + 5 x_{2} + 10 x_{3} + 10 x_{4} + 5 x_{5} + x_{6}

and

\begin{array}{l} {\dot{σ}}_{0} = {\dot{x}}_{1} + 5 {\dot{x}}_{2} + 10 {\dot{x}}_{3} + 10 {\dot{x}}_{4} + 5 {\dot{x}}_{5} + {\dot{x}}_{6} \\ = x_{2} + 5 x_{3} + 10 x_{4} + 10 x_{5} + 5 x_{6} + v_{0} \end{array}

Then, by taking $v_{0} = - x_{2} - 5 x_{3} - 10 x_{4} - 10 x_{5} - 5 x_{6} - k sgn (σ_{0}), k > 0$ , we get ${\dot{σ}}_{0} = - k sgn (σ_{0})$ . Thus, the nominal system (16) is asymptotically stable.

Step 5: Select the sliding surface for (15) as $σ = σ_{0} + β$ , where β is the integral term computed later. In order to avoid the reaching phase, choose $β (0)$ such that $σ (0) = 0$ . Take $v = v_{0} + v_{s}$ , where v₀ is the nominal input and v_s is compensator term to be computed later.

Then

\begin{array}{l} \dot{σ} = {\dot{σ}}_{0} + \dot{β} = {\dot{x}}_{1} + 5 {\dot{x}}_{2} + 10 {\dot{x}}_{3} + 10 {\dot{x}}_{4} + 5 {\dot{x}}_{5} + {\dot{x}}_{6} + \dot{β} \\ = x_{2} + 5 x_{3} + 10 x_{4} + 10 {\hat{F}}_{3} + 10 {\tilde{F}}_{3} + 10 x_{5} + 5 x_{6} + 5 {\hat{F}}_{5} + 5 {\tilde{F}}_{5} + v_{0} + v_{s} + Δ (t) + \dot{β} \end{array}

Step 6: Now, by choosing a Lyapunov function $V = \frac{1}{2} σ^{2} + \frac{1}{2} {\tilde{F}}_{3}^{2} + \frac{1}{2} {\tilde{F}}_{5}^{2}$ , design the adaptive laws for ${\tilde{F}}_{i}$ and ${\hat{F}}_{i}, i = 3, 5$ and compute v_s such that $\dot{V} < 0$ .

Since

\begin{array}{l} \dot{V} = σ \dot{σ} + {\tilde{F}}_{3} {\dot{\tilde{F}}}_{3} + {\tilde{F}}_{5} {\dot{\tilde{F}}}_{5} \\ = σ (x_{2} + 5 x_{3} + 10 x_{4} + 10 {\hat{F}}_{3} + 10 {\tilde{F}}_{3} + 10 x_{5} + 5 x_{6} + 5 {\hat{F}}_{5} + 5 {\tilde{F}}_{5} + v_{0} + v_{s} + \dot{β}) + {\tilde{F}}_{3} {\dot{\tilde{F}}}_{3} + {\tilde{F}}_{5} {\dot{\tilde{F}}}_{5} \\ = σ (x_{2} + 5 x_{3} + 10 x_{4} + 10 x_{5} + 5 x_{6} + 10 {\hat{F}}_{3} + 5 {\hat{F}}_{5} + v_{0} + v_{s} + Δ (t) + \dot{β}) + {\tilde{F}}_{3} ({\dot{\tilde{F}}}_{3} + 10 σ) + {\tilde{F}}_{5} ({\dot{\tilde{F}}}_{5} + 5 σ) \end{array}

Therefore, by using

\begin{array}{l} \dot{β} = - x_{2} - 5 x_{3} - 10 x_{4} - 10 x_{5} - 5 x_{6} - v_{0} \\ v_{s} = - 10 {\hat{F}}_{3} - 5 {\hat{F}}_{5} - k sgn (σ), k > M \\ {\dot{\tilde{F}}}_{5} = - 10 σ - k_{1} {\tilde{F}}_{3} \\ {\dot{\tilde{F}}}_{5} = - 5 σ - k_{2} {\tilde{F}}_{5} \\ {\dot{\hat{F}}}_{3} = - {\dot{\tilde{F}}}_{3}, {\dot{\hat{F}}}_{5} = - {\dot{\tilde{F}}}_{5} \end{array}

we have $\dot{V} = - k σ sgn (σ) - k_{1} {\tilde{F}}_{3}^{2} - k_{2} {\tilde{F}}_{5}^{2} < 0$ .

From this, we conclude that $σ, {\tilde{F}}_{3}, {\tilde{F}}_{5} \to 0$ . Since σ → 0, therefore x → 0.

Simulation results

Figures 2 and 3 show simulation results for underwater vehicle without and with uncertainty in the control input. The simulations are carried out for the same initial conditions $(x (0), y (0), z (0), ϕ (0), θ (0), ψ (0)) = (2,0, - 2, π / 6, π / 8, π / 5)$ , so that the effect of uncertainty can be observed. The applied uncertainty in the control input is $0.5 sin (2 t) + 0.4 cos (t) + x_{3} x_{2} t$ . As can be seen, the system states converge to zero for both the systems with and without the uncertainty, thus, validating the correctness of the applied algorithm. Therefore, the control objective to stabilize the underwater vehicle having any random initial condition has been achieved.

Figure 2.

System trajectory without any uncertainty. (a) System states corresponding to initial condition $(x (0), y (0), z (0), ϕ (0), θ (0), ψ (0)) = (2,0, - 2, π / 6, π / 8, π / 5)$ and (b) control effort $u = {(u_{1}, u_{2}, u_{3}, u_{4})}^{T}$ .

Figure 3.

System trajectory with uncertainty. (a) System states corresponding to initial condition $(x (0), y (0), z (0), ϕ (0), θ (0), ψ (0)) = (2,0, - 2, π / 6, π / 8, π / 5)$ and (b) control effort $u = {(u_{1}, u_{2}, u_{3}, u_{4})}^{T}$ .

Case study 2: Stabilizing algorithm for perturbed system model

Again the original system (6) is, first of all, transformed into a perturbed chained form system (12). Afterward, the perturbed chained form system is further transformed into a special structure containing nominal part and some unknown terms through input transformation. Step-by-step details of the proposed algorithm are presented next in the subsection.

Control algorithm

Step 1: After decomposing unknown constants into estimated values and errors in estimated values, the system (12) can be rewritten as

\begin{array}{l} {\dot{x}}_{1} = v_{1} + x_{1} ϕ_{1} (x, t) {\hat{θ}}_{1} + x_{1} ϕ_{1} (x, t) {\tilde{θ}}_{1} \\ {\dot{x}}_{2} = v_{2} + x_{2} ϕ_{2} (x, t) {\hat{θ}}_{2} + x_{2} ϕ_{2} (x, t) {\tilde{θ}}_{2} \\ {\dot{x}}_{3} = x_{2} (v_{1} + x_{1} ϕ_{1} (x, t) {\hat{θ}}_{1} + x_{1} ϕ_{1} (x, t) {\tilde{θ}}_{1}) \\ {\dot{x}}_{4} = v_{3} + x_{4} ϕ_{3} (x, t) {\hat{θ}}_{3} + x_{4} ϕ_{3} (x, t) {\tilde{θ}}_{3} \\ {\dot{x}}_{5} = x_{3} (v_{1} + x_{1} ϕ_{1} (x, t) {\hat{θ}}_{1} + x_{1} ϕ_{1} (x, t) {\tilde{θ}}_{1}) \\ {\dot{x}}_{6} = v_{4} + x_{6} ϕ_{4} (x, t) {\hat{θ}}_{4} + x_{6} ϕ_{4} (x, t) {\tilde{θ}}_{4} \end{array}

By choosing $v_{1} = x_{2} - x_{1} ϕ_{1} (x, t) {\hat{θ}}_{1}$ , $v_{2} = x_{3} - x_{2} ϕ_{2} (x, t) {\hat{θ}}_{2}$ , $v_{3} = x_{5} - x_{4} ϕ_{3} (x, t) {\hat{θ}}_{3}$ , and $v_{4} = v - x_{6} ϕ_{4} (x, t) {\hat{θ}}_{4}$ , system (17) becomes

\begin{array}{l} {\dot{x}}_{1} = x_{2} + x_{1} ϕ_{1} (x, t) {\tilde{θ}}_{1} \\ {\dot{x}}_{2} = x_{3} + x_{2} ϕ_{2} (x, t) {\tilde{θ}}_{2} \\ {\dot{x}}_{3} = x_{2} (x_{2} + x_{1} ϕ_{1} (x, t) {\tilde{θ}}_{1}) \\ {\dot{x}}_{4} = x_{5} + x_{4} ϕ_{3} (x, t) {\tilde{θ}}_{3} \\ {\dot{x}}_{5} = x_{3} (x_{2} + x_{1} ϕ_{1} (x, t) {\tilde{θ}}_{1}) \\ {\dot{x}}_{6} = v + x_{6} ϕ_{4} (x, t) {\tilde{θ}}_{4} \end{array}

Step 2: After some manipulation, the above system (18) can be written as

\begin{array}{l} {\dot{x}}_{1} = x_{2} + x_{1} ϕ_{1} (x, t) {\tilde{θ}}_{1} \\ {\dot{x}}_{2} = x_{3} + x_{2} ϕ_{2} (x, t) {\tilde{θ}}_{2} \\ {\dot{x}}_{3} = x_{4} + F_{3} + x_{2} x_{1} ϕ_{1} (x, t) {\tilde{θ}}_{1} \\ {\dot{x}}_{4} = x_{5} + x_{4} ϕ_{3} (x, t) {\tilde{θ}}_{3} \\ {\dot{x}}_{5} = x_{6} + F_{5} + x_{2} x_{1} ϕ_{1} (x, t) {\tilde{θ}}_{1} \\ {\dot{x}}_{6} = v + x_{6} ϕ_{4} (x, t) {\tilde{θ}}_{4} \end{array}

where $F_{3} = - x_{4} + x_{2} x_{2}, F_{5} = - x_{6} + x_{3} x_{2}$

Step 3: Now, assume that F₃ and F₅ are unknown and can be computed adaptively. Let ${\hat{F}}_{i}$ be an estimate of F_i $, i = 3, 5$ , respectively, and ${\tilde{F}}_{i} = F_{i} - {\hat{F}}_{i}$ be the errors in estimation of F_i $, i = 3, 5$ , respectively.

Then, system (19) can be written as

\begin{array}{l} {\dot{x}}_{1} = x_{2} + x_{1} ϕ_{1} (x, t) {\tilde{θ}}_{1} \\ {\dot{x}}_{2} = x_{3} + x_{2} ϕ_{2} (x, t) {\tilde{θ}}_{2} \\ {\dot{x}}_{3} = x_{4} + {\hat{F}}_{3} + {\tilde{F}}_{3} + x_{2} x_{1} ϕ_{1} (x, t) {\tilde{θ}}_{1} \\ {\dot{x}}_{4} = x_{5} + x_{4} ϕ_{3} (x, t) {\tilde{θ}}_{3} \\ {\dot{x}}_{5} = x_{6} + {\hat{F}}_{5} + {\tilde{F}}_{5} + x_{2} x_{1} ϕ_{1} (x, t) {\tilde{θ}}_{1} \\ {\dot{x}}_{6} = v + x_{6} ϕ_{4} (x, t) {\tilde{θ}}_{4} \end{array}

Step 4: The nominal system for (20) is given as

\begin{array}{l} {\dot{x}}_{1} = x_{2} \\ {\dot{x}}_{2} = x_{3} \\ {\dot{x}}_{3} = x_{4} \\ {\dot{x}}_{4} = x_{5} \\ {\dot{x}}_{5} = x_{6} \\ {\dot{x}}_{6} = v_{0} \end{array}

Select the sliding surface for (21) as $σ_{0} = {(1 + \frac{d}{d t})}^{5} x_{1}$ , that is

σ_{0} = x_{1} + 5 x_{2} + 10 x_{3} + 10 x_{4} + 5 x_{5} + x_{6}

then

\begin{array}{l} {\dot{σ}}_{0} = {\dot{x}}_{1} + 5 {\dot{x}}_{2} + 10 {\dot{x}}_{3} + 10 {\dot{x}}_{4} + 5 {\dot{x}}_{5} + {\dot{x}}_{6} \\ = x_{2} + 5 x_{3} + 10 x_{4} + 10 x_{5} + 5 x_{6} + v_{0} \end{array}

Therefore, by taking $v_{0} = - x_{2} - 5 x_{3} - 10 x_{4} - 10 x_{5} - 5 x_{6} - k sgn (σ_{0}), k > 0$ , we get ${\dot{σ}}_{0} = - k sgn (σ_{0})$ . Hence, the nominal system (21) is asymptotically stable.

Step 5: Choose the sliding surface for (20) as $σ = σ_{0} + β$ , where β is the integral term computed later. In order to avoid the reaching phase, choose β(0) such that $σ (0) = 0$ . Taking $v = v_{0} + v_{s}$ , where v₀ is the nominal input and v_s is compensator term to be computed later.

Therefore

\begin{array}{l} \dot{σ} = {\dot{σ}}_{0} + \dot{β} = {\dot{x}}_{1} + 5 {\dot{x}}_{2} + 10 {\dot{x}}_{3} + 10 {\dot{x}}_{4} + 5 {\dot{x}}_{5} + {\dot{x}}_{6} + \dot{β} \\ = x_{2} + x_{1} ϕ_{1} (x, t) {\tilde{θ}}_{1} + 5 x_{3} + 5 x_{2} ϕ_{2} (x, t) {\tilde{θ}}_{2} + 10 x_{4} + 10 {\hat{F}}_{3} + 10 {\tilde{F}}_{3} + 10 x_{2} x_{1} ϕ_{1} (x, t) {\tilde{θ}}_{1} \\ + 10 x_{5} + 10 x_{4} ϕ_{3} (x, t) {\tilde{θ}}_{3} + 5 x_{6} + 5 {\hat{F}}_{5} + 5 {\tilde{F}}_{5} + 5 x_{2} x_{1} ϕ_{1} (x, t) {\tilde{θ}}_{1} + x_{6} ϕ_{4} (x, t) {\tilde{θ}}_{4} + v_{0} + v_{s} + \dot{β} \end{array}

Step 6: By choosing a Lyapunov function $V = \frac{1}{2} σ^{2} + \frac{1}{2} {\tilde{F}}_{3}^{2} + \frac{1}{2} {\tilde{F}}_{5}^{2} + \frac{1}{2} {\tilde{θ}}_{1}^{2} + \frac{1}{2} {\tilde{θ}}_{2}^{2} + \frac{1}{2} {\tilde{θ}}_{3}^{2} + \frac{1}{2} {\tilde{θ}}_{4}^{2}$ , design the adaptive laws for ${\tilde{F}}_{i}$ and ${\hat{F}}_{i}, i = 3, 5$ , ${\tilde{θ}}_{i}$ and ${\hat{θ}}_{i}, i = 1, \dots, 4$ and compute v_s such that $\dot{V} < 0$ .

Since

\begin{array}{l} \dot{V} = σ \dot{σ} + {\tilde{F}}_{3} {\dot{\tilde{F}}}_{3} + {\tilde{F}}_{5} {\dot{\tilde{F}}}_{5} + {\tilde{θ}}_{1} {\dot{\tilde{θ}}}_{1} + {\tilde{θ}}_{2} {\dot{\tilde{θ}}}_{2} + {\tilde{θ}}_{3} {\dot{\tilde{θ}}}_{3} + {\tilde{θ}}_{4} {\dot{\tilde{θ}}}_{4} \\ = σ [x_{2} + x_{1} ϕ_{1} (x, t) {\tilde{θ}}_{1} + 5 x_{3} + 5 x_{2} ϕ_{2} (x, t) {\tilde{θ}}_{2} + 10 x_{4} + 10 {\hat{F}}_{3} + 10 {\tilde{F}}_{3} + 10 x_{2} x_{1} ϕ_{1} (x, t) {\tilde{θ}}_{1} \\ + 10 x_{5} + 10 x_{4} ϕ_{3} (x, t) {\tilde{θ}}_{3} + 5 x_{6} + 5 {\hat{F}}_{5} + 5 {\tilde{F}}_{5} + 5 x_{2} x_{1} ϕ_{1} (x, t) {\tilde{θ}}_{1} + x_{6} ϕ_{4} (x, t) {\tilde{θ}}_{4} + v_{0} + v_{s} + \dot{β}] \\ + {\tilde{F}}_{3} {\dot{\tilde{F}}}_{3} + {\tilde{F}}_{5} {\dot{\tilde{F}}}_{5} + {\tilde{θ}}_{1} {\dot{\tilde{θ}}}_{1} + {\tilde{θ}}_{2} {\dot{\tilde{θ}}}_{2} + {\tilde{θ}}_{3} {\dot{\tilde{θ}}}_{3} + {\tilde{θ}}_{4} {\dot{\tilde{θ}}}_{4} \\ = σ [x_{2} + 5 x_{3} + 10 x_{4} + 10 x_{5} + 5 x_{6} + 10 {\hat{F}}_{3} + 5 {\hat{F}}_{5} + v_{0} + v_{s} + \dot{β}] + {\tilde{F}}_{3} ({\dot{\tilde{F}}}_{3} + 10 σ) + {\tilde{F}}_{5} ({\dot{\tilde{F}}}_{5} + 5 σ) \\ + {\tilde{θ}}_{1} ({\dot{\tilde{θ}}}_{1} + σ x_{1} ϕ_{1} (x, t) + 10 σ x_{2} x_{1} ϕ_{1} (x, t) + 5 σ x_{2} x_{1} ϕ_{1} (x, t)) + {\tilde{θ}}_{2} ({\dot{\tilde{θ}}}_{2} + 5 σ x_{2} φ_{2} (x, t)) \\ + {\tilde{θ}}_{3} ({\dot{\tilde{θ}}}_{3} + 10 σ x_{4} ϕ_{3} (x, t)) + {\tilde{θ}}_{4} ({\dot{\tilde{θ}}}_{4} + σ x_{6} ϕ_{4} (x, t)) \end{array}

Therefore, by using

\begin{array}{l} \dot{β} = - x_{2} - 5 x_{3} - 10 x_{4} - 10 x_{5} - 5 x_{6} - v_{0} \\ v_{s} = - 10 {\hat{F}}_{3} - 5 {\hat{F}}_{5} - k sgn (σ) \\ {\dot{\tilde{F}}}_{3} = - 10 σ - k_{1} {\tilde{F}}_{3}, {\dot{\tilde{F}}}_{5} = - 5 σ - k_{2} {\tilde{F}}_{5}, {\dot{\hat{F}}}_{3} = - {\dot{\tilde{F}}}_{3}, {\dot{\hat{F}}}_{5} = - {\dot{\tilde{F}}}_{5}, \\ {\dot{\tilde{θ}}}_{1} = - σ x_{1} ϕ_{1} (x, t) - 10 σ x_{2} x_{1} φ_{1} (x, t) - 5 σ x_{2} x_{1} ϕ_{1} (x, t) - l_{1} {\tilde{θ}}_{1}, {\dot{\hat{θ}}}_{1} = - {\dot{\tilde{θ}}}_{1} \\ {\dot{\tilde{θ}}}_{2} = - 5 σ x_{2} ϕ_{2} (x, t) - l_{2} {\tilde{θ}}_{2}, {\dot{\hat{θ}}}_{2} = - {\dot{\tilde{θ}}}_{2} \\ {\dot{\tilde{θ}}}_{3} = - 10 σ x_{4} ϕ_{3} (x, t) - l_{3} {\tilde{θ}}_{3}, {\dot{\hat{θ}}}_{3} = - {\dot{\tilde{θ}}}_{3} \\ {\dot{\tilde{θ}}}_{4} = - σ x_{6} ϕ_{4} (x, t) - l_{4} {\tilde{θ}}_{4}, {\dot{\hat{θ}}}_{4} = - {\dot{\tilde{θ}}}_{4}, l_{i} > 0, i = 1, 2, 3, 4 \end{array}

we have $\dot{V} = - k σ sgn (σ) - k_{1} {\tilde{F}}_{3}^{2} - k_{2} {\tilde{F}}_{5}^{2} - l_{1} {\tilde{θ}}_{1}^{2} - l_{2} {\tilde{θ}}_{2}^{2} - l_{3} {\tilde{θ}}_{3}^{2} - l_{4} {\tilde{θ}}_{4}^{2} < 0$ .

From this, we conclude that $σ, {\tilde{F}}_{3}, {\tilde{F}}_{5}, {\tilde{θ}}_{i} \to 0$ . Since σ → 0, therefore $x \to 0$ .

Simulation results

Figures 4 and 5 show simulation results for underwater vehicle model affected with uncertainties. The simulations are carried out for two different initial conditions $(x (0), y (0), z (0), ϕ (0), θ (0), ψ (0))$ ,

Figure 4.

System trajectory with perturbation in system model. (a) System states corresponding to initial condition $(x (0), y (0), z (0), ϕ (0), θ (0), ψ (0)) = (0,1, - 1, π / 5, π / 8, π / 6)$ , (b) control effort $u = {(u_{1}, u_{2}, u_{3}, u_{4})}^{T}$ , and (c) estimated values of parameters $θ_{i}, i = 1, \dots, 4$ .

Figure 5.

System trajectory with perturbation in system model. (a) System states corresponding to initial condition $(x (0), y (0), z (0), ϕ (0), θ (0), ψ (0)) = (1,−1, 1, π / 6, π / 10, π / 8)$ , (b) control effort $u = {(u_{1}, u_{2}, u_{3}, u_{4})}^{T}$ , and (c) estimated values of parameters $θ_{i}, i = 1, \dots, 4$ .

(0, 1, −1, π/5, π/8, π/6)

(1, −1, 1, π/6, π/10, π/8)

Also, following sinusoidal/nonlinear functions are used for the known functions in the perturbations.

ϕ_{1} (x, t) = 0.5 sin (x_{1}) + 0.3 sin (2 x_{2})

ϕ_{2} (x, t) = x_{1} + t x_{3}

ϕ_{3} (x, t) = 0.2 cos (x_{5}) + 0.5 sin (x_{4})

ϕ_{4} (x, t) = x_{5} + x_{4} x_{5} + t x_{6}

The unknown parameters for Figure 4 are chosen as $(θ_{1}, θ_{2}, θ_{3}, θ_{4}) = (- 4, - 2, 4,3)$ , whereas the unknown parameters for Figure 5 are taken as $(θ_{1}, θ_{2}, θ_{3}, θ_{4}) = (4, - 3, - 2,1)$ . As can be seen from the simulation results, all the system states converge to zero for both systems with perturbations and the estimates of $θ_{i}, i = 1, \dots 4$ converge to their actual values, thus validating the correctness of the applied algorithm.

Conclusion and future work

An adaptive ISMC-based robust control algorithm for perturbed underwater vehicle is presented in this research. Firstly, the system is transformed into a special structure containing a nominal part and an unknown part through input transformation. The terms of the unknown part are computed adaptively. The transformed system is stabilized using ISMC. The stabilizing controller for the transformed system is constructed which consists of the nominal control plus a compensator control. The derivation of compensator controller and the adaptive laws for the unknown terms are based on the Lyapunov theory. The proposed methodology is applied on two different types of perturbed underwater vehicle.

The importance of research is further highlighted by the fact that the proposed methodology is general and can be applied to other nonholonomic systems. Also, the proposed scheme ensures robust stabilization for the whole state space along with chattering suppression. Application of other methods in conjunction with SMC has major disadvantages. The neural networks have the drawbacks of difficulty in selecting appropriate network and slow convergence to the equilibrium point. Similarly, the major drawback of fuzzy logic control is the difficulty in getting fuzzy rules and membership functions.

Future work will concern development of robust algorithm for underwater vehicle or general nonholonomic mechanical systems that are affected with unmatched uncertainties. Also, formation control of AUVs will be studied, as cooperative control of AUVs and its real-world implementation is an active area of research.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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Robust stabilizing control of nonholonomic systems with uncertainties via adaptive integral sliding mode

Abstract

Keywords

Introduction

Nonholonomic chained form systems

Adaptive integral sliding mode

Problem formulation

Kinematic model of underwater vehicle

Case studies

Case 1

Case 2

The control problem

Some assumptions

Conversion into chained form

Case 1

Case 2

Case Study 1: Stabilizing algorithm for perturbed control input

Control algorithm

Simulation results

Case study 2: Stabilizing algorithm for perturbed system model

Control algorithm

Simulation results

Conclusion and future work

Footnotes

Declaration of conflicting interests

Funding

References