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Abstract

In this paper, a new hybrid robust tracking control for an underwater vehicle in dive plane via $H^{\infty}$ method and time-variant sliding mode control is designed. The proposed controller is composed of two control signals. One control signal is designed via time-variant sliding function based on time-variant rejection parameter, and the other control signal is designed based on $H^{\infty}$ control approach. The underwater vehicle dynamics are nonlinear uncertain dynamical equation with unknown external disturbances. Therefore, in this paper, an observation of the external disturbance with a time-variant gain parameter is designed. The proposed method eliminates the effects of uncertainties and chattering phenomenon. Also, the time-variant sliding function filters all the unmodeled frequencies and external noises, such as a chain of $(n - 1)$ first-order adaptive low-pass filters. A new theorem, for facilitating the presented method, is proved. Finally, two examples for demonstrating the advantages of the proposed method are simulated.

Keywords

Sliding mode robust control time variant rejection regulator vehicle dive plane

Introduction

Autonomous underwater vehicles (AUVs) are widely used in underwater to explore in great depths. In these vehicles, control systems are required to provide signals to the actuators in order to achieve the desired position and velocity for vehicle. AUV’s dynamic is hard, nonlinear with uncertainty in nature. Therefore, the proposed controller for these systems has to indicate robust accurate tracking performance in the presence of external disturbances and noises. One of the most important techniques for control of AUV is nonlinear control technique. In literature,^1–3 many controllers designed for path following based on Lyapunov approach and back stepping technique that nonlinear feedback law yielded that the tracking error approaches to zero. Fischer et al.^4,5 developed a nonlinear control method for coupled underwater vehicle using a robust integral of the sign of the error index (RISE).

Adaptive control is a nonlinear control strategy used for uncertain or time-varying systems. This type of control is useful for AUVs due to the changing dynamics in the ocean. The controller can adapt itself to varying ocean currents or to redundant structure when the manipulator is installed. Adaptive control is applied for AUV in many literature.^6–14 In Santhakumar and Kim,¹⁵ an indirect adaptive control was developed based on extended Kalman filter (EKF) which provided a generalized framework for indirect adaptive control which considered the dynamic coupling between the vehicle and the manipulator. External disturbances were estimated using EKF. In Antonelli and Cataldi,¹⁶ an adaptive and regressive controller was developed for an arm-equipped underwater vehicle. Simplified implementation and compensation in the presence of ocean current is its advantage. For avoiding acceleration measurement, a reduced regressor is implemented.

Sliding mode control (SMC) is a powerful nonlinear control method. Pure sliding mode controller has disadvantages such as chattering phenomenon and noisy vulnerable properties.¹⁷ Sliding mode with boundary layer control (BLC) is one of the methods to attenuate chattering phenomenon.^18,19 In Elmokadem et al.,²⁰ a trajectory tracking SMC was designed for lateral motion of an AUV, in order to force the AUV to track a time-varying trajectory. For online estimation of external disturbances and uncertainties in AUV, adaptive fuzzy SMC has been used in Marzbanrad et al.²¹ AUV’s dynamic systems have uncertainties and external disturbances. The effects of these uncertainties and disturbances enter AUV from different channels to control signals. For avoiding these effects, robust control method and observation method have been implemented.^22,23 New terminologies, rejection parameter, and rejection regulator were defined, and a new time-variant sliding function based on time-variant rejection parameter was designed.²⁴ A robust wavelet SMC via time-variant sliding function and $H^{\infty}$ control method for achieving the best performance of the presented controller was proposed.²⁵ A novel integral sliding mode controller based on extended state observer has been designed in Cui et al.,²⁶ in which an adaptive gain has been used to update the algorithm. Also, the upper bound of the uncertainties has been estimated. Since there is a sign function in the control law, chattering phenomenon is inevitable.

In this paper, a hybrid robust SMC has been designed which is composed of two simultaneous control signals. The first control signal is designed based on time-variant sliding function. The time-variant sliding function filters all the unmodeled frequencies and external disturbances such as a chain of $(n - 1)$ adaptive first-order low-pass filters based on an adaptive rejection parameter. The second control signal is a robust controller designed to achieve $H^{\infty}$ performance. In order to remove the effects of external disturbances and uncertainties, a disturbance observer has been proposed to make the system behave close to the nominal model. Also, a disturbance observer with a time-variant gain parameter has been proposed to make the system behave close to the nominal model. Therefore, it can be seen that the control signal in our proposed method does not have any chattering with the less reaching time and control amplitude in comparison with the method in Cui and Zhang²⁷ and BLC method. For facilitating the present method, a theorem has been proved.

This paper is organized as follows. In section “Problem formulation,” equations and the mathematical model of an AUV in dive plane are presented. In section “Design of a new hybrid robust depth control with disturbance observer,” sliding mode with BLC for the AUV is discussed and also the proposed controller and disturbance observer are presented. In section “Simulation results,” the simulation results are analyzed in the presence of parameter uncertainties, external disturbances, and noise for the proposed controller, sliding mode with BLC, and super twisting sliding mode control (STSMC) in Ismail and Putranti²⁸ and Cui and Zhang.²⁷ Finally, section “Conclusion” concludes the paper.

Problem formulation

The underwater vehicles’ motion dynamical equations are modeled via inertial reference frame or body-fixed reference frame.²⁹ Figure 1 indicates an underwater vehicle with both inertial and body-fixed reference frames. For achieving the control objectives, the underwater vehicles’ dynamical models are designed via inertial reference frame approach based on position/attitude vector $X = [x, y, z, ϕ, θ, ψ]^{T}$ . In the particular case of remotely operated vehicles (ROVs), to keep the angle values of the roll $ϕ$ and pitch $θ$ arbitrarily small, that is, $ϕ \approx 0, θ \approx 0$ , the distance between buoyancy and gravity centers is usually large enough.²⁹ According to this approach, the vertical motion (heave) and horizontal motion are decoupled with good self-stabilizing property. Morison’s equation³⁰ is considered in the previous approach. The vertical motion along z-axis can be described by the following equation

m \overset{\cdot\cdot}{z} + c \overset{\cdot}{z} | \overset{\cdot}{z} | = u + d

(1)

where z, u, d, c, and m are the depth, control input (thrust force), disturbance caused by external forces, coefficient of the hydrodynamic quadratic damping, and vehicle’s mass plus the hydrodynamic added mass, respectively. Consider the following physically motivated assumptions.²⁹

Figure 1.

Underwater vehicle with both inertial and body-fixed reference frames.²⁹

Assumption 1

The parameter $m (t)$ is a time-varying positive bounded parameter, that is, $0 \leq m_{min} \leq m (t) \leq m_{max}$ .

Assumption 2

The parameter $c (t)$ is a time-varying bounded parameter, that is, $c_{min} \leq c (t) \leq c_{max}$ .

Design of a new hybrid robust depth control with disturbance observer

In this section, an overview of the sliding mode controller for the described system in the previous section is presented. In addition, a new sliding mode controller and the proposed disturbance observer are designed.

An overview of SMC

Consider the nth order nonlinear dynamical system as

z^{(n)} (t) = f (Z, t) + g (Z, t) u + d

(2)

where

Z = {[z, \overset{\cdot}{z}, \dots, z^{(n - 1)}]}^{T}

(3)

is the state vector, and u is the input vector. Suppose that $Z_{d} = {[z_{d}, {\overset{\cdot}{z}}_{d}, \dots, z_{d}^{(n - 1)}]}^{T}$ is the desired trajectory, then the error vector is defined as

\tilde{z} = {[{\tilde{z}}_{1}, {\tilde{z}}_{2}, \dots, {\tilde{z}}_{n}]}^{T}

(4)

Let $s (t)$ be the sliding surface manifold defined in the state space by the following equation

s (\tilde{z}, \overset{\cdot}{\tilde{z}}) = \overset{\cdot}{\tilde{z}} + λ \tilde{z}

(5)

where

\tilde{z} = z - z_{d}

(6)

here, $\tilde{z}$ , $\overset{\cdot}{\tilde{z}}$ , and $z_{d}$ are the tracking error, time derivative of $\tilde{z}$ , and desired trajectory, respectively, and $λ$ is a strictly positive constant. Consider the following assumptions for developing the control laws.

Assumption 3

The states z and $\overset{\cdot}{z}$ are available and the disturbance $d (t)$ is an unknown time-varying bounded signal, that is, $| d (t) | \leq σ (t, z, \overset{\cdot}{z})$ .

Assumption 4

Furthermore, $z_{d}$ , ${\overset{\cdot}{z}}_{d}$ , and ${\overset{\cdot\cdot}{z}}_{d}$ are desired and available with known bounds.

In most studies, remotely operated underwater vertical motion control problems are solved according to the SMC approach. In this method, the control law is composed of two components: the equivalent control term that is defined as $u_{e} = c \overset{\cdot}{z} | \overset{\cdot}{z} | + m ({\overset{\cdot\cdot}{z}}_{d} - λ \overset{\cdot}{\tilde{z}}) - \hat{d}$ and the reaching law term which is defined by $u_{reach} = - K sgn (s)$ . Also, $\hat{d}$ is an estimation of disturbance d which is discussed in the next subsection. Therefore, the conventional SMC law is defined as $u = u_{e} + u_{reach}$ that can be rewritten in the following form

u = c \overset{\cdot}{z} | \overset{\cdot}{z} | + m ({\overset{\cdot\cdot}{z}}_{d} - λ \overset{\cdot}{\tilde{z}}) - \hat{d} - K sgn (s)

(7)

where $- K sgn (s)$ is a discontinuous term and the control gain K is a positive constant. Also, $sgn (s)$ is defined as

sgn (s) = {\begin{matrix} - 1 & if x < 0 \\ 0 & if x = 0 \\ 1 & if x > 0 \end{matrix}

(8)

It can be easily verified that Equation (7) is sufficient to impose the sliding condition

\frac{1}{2} \frac{d}{dt} s^{2} \leq - η | s |

(9)

where $η$ is a strictly positive constant related to the reaching time.

Pure sliding mode controller has some undesirable properties such as chattering and being too vulnerable against noise. The first problem, chattering, causes high-frequency oscillation in the controller output and the other is the sensitivity problem. This controller is very sensitive to noise when the input signals are very close to zero.³¹ As shown in the “Simulation results,” section, in the presence of external disturbances, the controller has a good track but chattering may occur in the control input. One strategy to eliminate chattering is BLC in which the saturation function is used instead of sign function as follows

u = c \overset{\cdot}{z} | \overset{\cdot}{z} | + m ({\overset{\cdot\cdot}{z}}_{d} - λ \overset{\cdot}{\tilde{z}}) - \hat{d} - K sat (s / φ)

(10)

where $sat (x)$ is defined as

sat (x) = {\begin{matrix} sgn (x) & if | x | > 1 \\ x & if | x | \leq 1 \end{matrix}

(11)

Time-variant SMC

A sliding function is defined as $s = {(\frac{d}{dt} + λ)}^{n - 1} \tilde{z} = k^{T} \tilde{z}$ in which $k = {[k_{0}, k_{1}, \dots, k_{n - 1}]}^{T} \in R^{n}$ is the coefficient vector of the following characteristic polynomial²⁴

P (ω_{i}) = k_{0} ω^{(n - 1)} + k_{1} ω^{(n - 2)} + \dots + k_{n - 1}

(12)

and $λ$ is defined so that the following equation is satisfied

λ < ν = inf {σ_{uf} : σ_{uf} are unmodeled frequencies}

(13)

A theorem for objectively choosing the coefficients of error states in sliding equation is presented as follows.

Theorem 1

If $ω_{i}$ are the roots of the polynomial equation (12) and $ν$ is the same notation as defined in equation (13). $ω^{*} = max {| ω_{i} |, P (ω_{i}) = 0, ω_{i} < 0, i = 1, 2, \dots, n - 1}$ and $δ$ is an arbitrary real positive number (rejection parameter) such that $δ < ν$ , then by choosing $ω^{*} = ν - δ$ and

k_{j} = (\begin{matrix} n - 1 \\ j \end{matrix}) {(ω^{*})}^{j}, for j = 0, \dots, n - 1

(14)

We have $δ < ν$ , that is, the first-order low-pass filters with break frequency bandwidth $λ$ , as defined in $s = {(\frac{d}{dt} + λ)}^{n - 1}$ , reject all the unmodeled frequencies of the system.

Proof

Refer to Yarahmadi et al.²⁴

Definition 1

Define the rejection regulator in the following form²⁴

R_{j}^{n} (δ) = (\begin{matrix} n - 1 \\ j \end{matrix}) {(ν - δ)}^{j}, j = 1, \dots, n - 1

(15)

where $δ$ and $ν$ are the same notations as used in Theorem 1, and also it is assumed that $R_{0}^{1} (δ) = 1$ .

Lemma 1

If $δ$ and $R_{j}^{n} (δ)$ are rejection parameter and rejection regulator, respectively, then

\frac{\partial^{k}}{\partial δ^{k}} R_{j}^{n} (δ) = {(- 1)}^{k} \frac{(n - 1)!}{(n - k - 1)!} R_{j - k}^{n - k} (δ)

(16)

Proof

Refer to Yarahmadi et al.²⁴

If we consider the sliding equation in the following form

s = \sum_{j = 0}^{n - 1} R_{j}^{n} (δ (t)) {\tilde{z}}^{(n - j - 1)}

(17)

where $δ (t)$ is the time-variant rejection parameter, then this equation defines a time-variant sliding equation.

According to Lemma 1, we have

\sum_{j = 1}^{n - 1} \frac{\partial}{\partial δ} R_{j}^{n} (δ) {\tilde{z}}^{(n - j - 1)} = - (n - 1) \sum_{j = 1}^{n - 1} R_{j - 1}^{n - 1} (δ) {\tilde{z}}^{(n - j - 1)}

Therefore

\overset{\cdot}{s} = {\tilde{z}}^{(n)} + \sum_{j = 1}^{n - 1} R_{j}^{n} (δ) {\tilde{z}}^{(n - j)} - (n - 1) \overset{\cdot}{δ} \sum_{j = 1}^{n - 1} R_{j - 1}^{n - 1} (δ) {\tilde{z}}^{(n - j - 1)}

(18)

Substituting Equation (2) into Equation (18), we get

\begin{matrix} \overset{\cdot}{s} = f (Z, t) + g (Z, t) u + d - z_{d}^{(n)} + \sum_{j = 1}^{n - 1} R_{j}^{n} (δ) {\tilde{z}}^{(n - j)} \\ - (n - 1) \overset{\cdot}{δ} \sum_{j = 1}^{n - 1} R_{j - 1}^{n - 1} (δ) {\tilde{z}}^{(n - j - 1)} \end{matrix}

(19)

Then, we define

u = u_{eq} + u_{r}

(20)

where

u_{eq} = \frac{1}{g} (- f + z_{d}^{(n)} - \sum_{j = 1}^{n - 1} R_{j}^{n} (δ) {\tilde{z}}^{(n - j)} - \hat{d})

(21)

where $\hat{d}$ is the estimation of d. Also, ${\tilde{u}}_{r} = g u_{r}$

To estimate the unknown disturbance, Equation (1) can be written as

\hat{d} = γ + c_{1} m \overset{\cdot}{z}

(22)

Then, the disturbance observer is defined as²²

\overset{\cdot}{γ} = - c_{1} (c_{1} m \overset{\cdot}{z} + u - c \overset{\cdot}{z} | \overset{\cdot}{z} |) - c_{1} γ

(23)

where $c_{1} > 0$ ; since there is no previous knowledge about d, it is reasonable to suppose that²²

\overset{\cdot}{d} = 0

(24)

which means that the disturbance varies slowly relative to observer dynamics.

In this paper, we assume that $c_{1}$ is an exponential and time-variant term based on tracking error which is defined as follows

c_{1} = \exp (- α | \tilde{z} |)

(25)

where $α > 0$ is constant.

Now, define the estimation error as

ε = d - \hat{d}

(26)

Substituting Equations (24) and (23) in Equation (26) yields

\overset{\cdot}{ε} = \overset{\cdot}{d} - \overset{\cdot}{\hat{d}} = - \exp (- α | \tilde{z} |) ε

(27)

Therefore, the estimation error dynamic can be written as

\overset{\cdot}{ε} + \exp (- α | \tilde{z} |) ε = 0

(28)

which means that the observer is globally asymptotically stable and the exponential convergence rate is damped by $c_{1}$ variation.

Theorem 2

Consider nonlinear system described by Equation (2) and sliding surface by Equation (17); if the robust sliding mode controller is designed as Equations (20) and (21) in which an adaption law as

\overset{\cdot}{δ} = η_{1} sgn (s) \sum_{j = 1}^{n - 1} R_{j - 1}^{n - 1} (δ) {\tilde{z}}^{(n - j - 1)}

(29)

and a robust controller as

{\tilde{u}}_{r} = - \frac{s}{6} (1 + \frac{1}{ρ^{2}})

(30)

are defined, then the following properties are established:

The $H^{\infty}$ tracking performance can be achieved.

If d is square-integrable, then $lim_{n \to \infty} | s (t) | = 0$

Proof

1. Consider the Lyapunov function in the following form

V (s, t) = \frac{1}{2} s^{2} + \frac{1}{2} ε^{2}

(31)

Differentiating equation (31), with respect to time, yields

\overset{\cdot}{V} (s, t) = s \overset{\cdot}{s} + ε \overset{\cdot}{ε}

(32)

\begin{matrix} \overset{\cdot}{V} (s, t) = s ({\tilde{u}}_{r} - (n - 1) \overset{\cdot}{δ} \sum_{j = 1}^{n - 1} R_{j - 1}^{n - 1} (δ) {\tilde{z}}^{(n - j - 1)} + d - \hat{d}) + ε \overset{\cdot}{ε} \\ \overset{\cdot}{V} (s, t) = - s (n - 1) \overset{\cdot}{δ} \sum_{j = 1}^{n - 1} R_{j - 1}^{n - 1} (δ) {\tilde{z}}^{(n - j - 1)} \\ + s (ε + {\tilde{u}}_{r}) - ε^{2} (\exp (- α | \tilde{z} |)) \\ \overset{\cdot}{V} (s, t) \leq s (ε + {\tilde{u}}_{r}) \\ = s ε - \frac{s^{2}}{6} (1 + \frac{1}{ρ^{2}}) = \frac{1}{2} (2 s ε - \frac{s^{2}}{3} (1 + \frac{1}{ρ^{2}})) = \frac{1}{2} (2 s ε - \frac{s^{2}}{3} - \frac{s^{2}}{3 ρ^{2}}) \\ = \frac{1}{2} (- \frac{s^{2}}{3} - {(\frac{s}{\sqrt{3} ρ} - \sqrt{3} ρ ε)}^{2} + 3 ρ^{2} ε^{2}) \overset{\cdot}{V} (s, t) \\ \leq \frac{1}{2} (- \frac{s^{2}}{3} + 3 p^{2} ε^{2}) \end{matrix}

Then

V (τ) - V (0) \leq - \frac{1}{6} \int_{0}^{τ} s^{2} dt + \frac{3}{2} ρ^{2} \int_{0}^{τ} ε^{2} dt

(33)

As $V (τ) \geq 0$ , the following inequality simplicity can be investigated

\frac{1}{6} \int_{0}^{τ} s^{2} dt \leq \frac{1}{2} s^{2} (0) + \frac{3}{2} ρ^{2} \int_{0}^{τ} ε^{2} dt

\int_{0}^{τ} s^{2} dt \leq 3 s^{2} (0) + 9 ρ^{2} \int_{0}^{τ} ε^{2} dt

(34)

\forall τ \in [0, \infty), and ε \in L^{2} [0, τ]

where $ρ$ is the adaption gain. Thus, the robust SMC system can achieve the $H^{\infty}$ tracking performance.

2. Let $ε \in L^{2}$ , and $s (0)$ is bounded;²⁴ therefore, using non-equality (34), we conclude that $s \in L^{2}$ . Using this conclusion and Equation (17), one can infer that $\tilde{z} \in L^{2}$ . Now, Equation (19) yields $\overset{\cdot}{s} \in L^{2}$ . Using Barbalat’s lemma since $s \in L^{2} \cap L^{\infty}$ and $\overset{\cdot}{s} \in L^{2}$ , $lim_{n \to \infty} | s (t) | = 0$ and this completes the proof.

According to the proposed sliding surface and SMC laws in Theorem 2, the sliding surface, as a chain of adaptive low-pass filter, rejects all the unmolded frequencies and external disturbances. Also, the hybrid sliding mode and $H^{\infty}$ controller demonstrates an efficient performance which is simulated in next section.

Simulation results

Example 1

In this section, the performance of our proposed controller is compared with the sliding mode with BLC and STSMC in Ismail and Putranti.²⁸ Consider an AUV’s dynamical equation presented in Equation (1) with $n = 2, j = 1$ . In control performance evaluation simulation, some different initial conditions are considered for presenting controller performances via initial conditions. In this example, the controller performance was evaluated in the presence of an external disturbance. Now, let $m = 55 kg$ , $c = 250 kg / m$ , $η = 0.1$ , $φ = 0.01$ , $z_{d} = 0.5 (1 - \cos (0.1 π t))$ , and $λ = 0.6$ . The uncertain parameters $\hat{m} = m (1 + 0.1 \sin (0.1 π t))$ and $\hat{c} = c (1 + 0.1 \sin (0.1 π t))$ are considered as values for m and c in the dynamical model, respectively. For BLC method, let $k = 13$ be the gain parameter. In the proposed controller, let $η_{1} = 0.1$ , $ν = 8$ , $δ (0) = 7.5$ , and $ρ = 0.08$ . Assume that the system is under the white Gaussian noise effects. In this example, the disturbance enters in the system for $t \in [5, 30]$ .

In Figures 2 and 3, tracking performance and control signal for the proposed controller for two different initial conditions, positive initial condition (PIC) and negative initial condition (NIC), are simulated. In addition, to reach a comparison situation, tracking performances and control signals of STSMC in Ismail and Putranti²⁸ and BLC methods are illustrated in Figures 4 –7, for the two different initial conditions. Also, Figures 4 and 6 show the simulation results of tracking performance and control signal for PIC, and Figures 5 and 7 show the simulation results of NIC based on STSMC and BLC methods.

Figure 2.

(a) Tracking performance with $z (0) = 0.8$ ; (b) control signal in the proposed controller.

Figure 3.

(a) Tracking performance with $z (0) = - 0.8$ ; (b) control signal in the proposed controller.

Figure 4.

(a) Tracking performance with $z (0) = 0.8$ ; (b) control input signal in Ismail and Putranti.²⁸

Figure 5.

(a) Tracking performance with $z (0) = - 0.8$ ; (b) control input signal in Ismail and Putranti.²⁸

Figure 6.

(a) Tracking performance with $z (0) = 0.8$ ; (b) control input signal in BLC.

Figure 7.

(a) Tracking performance with $z (0) = - 0.8$ ; (b) control input signal in BLC.

In these simulations, we use integral absolute control error (IACE) as a criterion for calculating the dispersion of the control signal against noise. Integral absolute error (IAE) and IACE are defined as

IAE = \int_{0}^{t} | \tilde{z} | dt and IACE = \int_{0}^{t} | u (t) - u_{Noisy} (t) | dt

(35)

In Tables 1 and 2, the interval control effort (control amplitude), and IAE and IACE criteria are considered for comparing the control performances of BLC, STSMC,²⁸ and the proposed method without noise and in presence of noise, respectively. According to Table 1, the interval control efforts of the proposed method for two initial conditions are less than BLC and STSMC methods significantly. Also, the IAE performance index for proposed method is less than the index for STSMC and BLC methods.

Table 1.

Performance of three methods for two initial conditions without noise.

	Interval control effort		IAE
	PIC	NIC	PIC	NIC
BLC	[6.8754, –10.2859]	[65.1141, –5.8727]	3.29	1.750
STSMC	[6.4895, –6.1311]	[26.6269, –5.8726]	5.009	2.750
Proposed method	[6.1595, –6.0692]	[17.3939, –5.9687]	3.93	2.08

IAE: integral absolute error; PIC: positive initial condition; NIC: negative initial condition; BLC: boundary layer control; STSMC: super twisting sliding mode control.

Table 2.

Performance of three methods for two initial conditions in the presence of noise.

		Interval control effort	IAE	IACE
BLC	PIC	[14.4003, –12.9208]	5.815	115.6019
BLC	NIC	[65.8961, –17.0658]	5.011	217.0415
STSMC	PIC	[10.3415, –9.1683]	8.752	61.5290
STSMC	NIC	[34.0060, –9.7471]	7.263	100.7011
Proposed method	PIC	[6.5308, –7.7630]	7.279	24.2977
Proposed method	NIC	[16.6095, –7.4054]	6.057	29.7302

IAE: integral absolute error; IACE: integral absolute control error; PIC: positive initial condition; NIC: negative initial condition; BLC: boundary layer control; STSMC: super twisting sliding mode control.

Table 2 shows that the interval control efforts of the proposed method for NIC and PIC are less than BLC and STSMC methods in the presence of noise. By surveying Tables 1 and 2, it is inferred that IAE in BLC method is less than the other methods.

Finally, from Figure 8 and IACE performance index in Table 2, it can be seen that the proposed controller shows acceptable behavior against noise, but STSMC and BLC are vulnerable against noise and high chattering phenomenon occurred in STSMC and BLC methods. In addition, IACE performance index for the proposed method is very less than BLC and STSMC methods. Also, IAE performance index for the proposed method is very less than STSMC method. The proposed disturbance estimation is shown in Figure 9. It can be seen that the estimation in the proposed method is better than the other in Chen et al.²²

Figure 8.

Control signals in the presence of noise: (a) proposed method, (b) STSMC method, and (c) BLC method.

Figure 9.

External disturbance and its estimation.

Example 2

In order to show the efficiency of the proposed controller, the performance of the proposed method has been compared to the adaptive sliding mode attitude control (ASMAC) method in Cui and Zhang.²⁷ For this purpose, the presented system in this paper is simulated based on the presented method in Cui and Zhang²⁷ in the following cases:

Case 1: If the constant gains of the controller are selected as $k = 3$ , $τ = 2$ , and $k_{3} = 2$ in Cui and Zhang.²⁷ As the simulation results indicate (see Figures 10 and 11), the chattering phenomenon occurred in control signal, but the reaching time is the same as the proposed method, while in the proposed control signal, the chattering phenomenon does not occur.

Case 2: In this case, we try to select the constant gains $k = 1$ , $τ = 2$ , and $k_{3} = 0.2$ to be smaller than those of Case 1. The results show that the chattering in control signal become less than Case 1 (compare Figures 10 and 12), but the reaching time increases considerably in comparison with Case 1 (see and compare Figures 11 and 13).

However, it can be seen that the control signal in the proposed method does not have any chattering with the less reaching time in comparison with the method in Cui and Zhang²⁷ for two cases.

Figure 10.

Tracking performance and control signal using ASMAC and presented methods, for Case 1.

Figure 11.

Sliding surface using ASMAC and presented methods, for Case 1.

Figure 12.

Tracking performance and control signal using ASMAC and proposed methods, for Case 2.

Figure 13.

Sliding surface using ASMAC and proposed methods, for Case 2.

Conclusion

In this paper, a new hybrid robust tracking control for an underwater vehicle in dive plane via $H^{\infty}$ method and time-variant SMC is designed. The first part of the proposed controller is designed via time-variant sliding function based on time-variant rejection parameter. The other part of control signal is designed based on $H^{\infty}$ control approach. In order to show removing the effects of external disturbances in control signal, an observer disturbance with a time-variant gain parameter is designed. In the proposed method, the time-variant sliding function filters all the unmodeled frequencies and external disturbances, such as a chain of $(n - 1)$ first-order adaptive low-pass filters. A new theorem, for facilitating the presented method, is proved. As the simulation examples indicated, the proposed method eliminates the chattering phenomenon and reduces the reaching time significantly. Therefore, the presented method presents efficient behavior and performance, especially in the presence of external disturbance and noise.

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.

ORCID iD

Majid Yarahmadi

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A new hybrid robust sliding mode control for an underwater vehicle in dive plane

Abstract

Keywords

Introduction

Problem formulation

Assumption 1

Assumption 2

Design of a new hybrid robust depth control with disturbance observer

An overview of SMC

Assumption 3

Assumption 4

Time-variant SMC

Theorem 1

Proof

Definition 1

Lemma 1

Proof

Theorem 2

Proof

Simulation results

Example 1

Example 2

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References