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Abstract

In this article, an adaptive neural dynamic surface sliding mode control scheme is proposed for uncertain nonlinear systems with unknown input saturation. The non-smooth input saturation nonlinearity is firstly approximated by a smooth non-affine function, which can be further transformed into an affine form according to the mean value theorem. Then, one simple sigmoid neural network is employed to approximate the uncertain nonlinearity including the input saturation, and the approximation error is estimated using an adaptive learning law. Virtual controls are designed in each step by combing the dynamic surface control and integral sliding mode technique, and thus the problem of complexity explosion inherent in the conventional backstepping method is avoided. With the proposed control scheme, no prior knowledge is required on the bound of input saturation, and comparative simulations are given to illustrate the effectiveness and superior performance.

Keywords

Dynamic surface control integral sliding mode control neural network input saturation nonlinear system

Introduction

In many practical dynamic systems, lots of nonlinear and uncertain characteristics are encountered, such as saturation, hysteresis, dead zone, and so on.^1
–3 Input saturation is well known as one of the most common non-smooth input nonlinearities. The magnitude of control signal is always limited due to physical constraints or safety consideration of actual actuators. If the physical input saturation is ignored in the control process, unfortunately, the designed controller may severely degrade the system performance or even lead to its instability.

So far, much attention has been paid to controllers design for nonlinear systems with input saturation.^4

–8 In the study by Gao and Selmic,⁴ Chen et al.⁵ and Chen et al.,⁶ significant results have been obtained for controlling saturated nonlinear systems with the bounds of input saturation being known or estimated in prior. Recently, some research work has been investigated without the prior knowledge of saturation parameter bounds. Wen et al.⁷ uses a smooth non-affine function of the control input signal to approximate the non-smooth saturation function, and a Nussbaum function is introduced to compensate for the nonlinear term arising from the input saturation. Due to good approximation abilities of nonlinear functions, neural networks (NNs) are employed in controllers design to approximate the saturated nonlinear systems.^8,9 However, in most aforementioned works, multiple NNs are used for nonlinearity approximation in each step, which may lead to increasing complexities of the controller design.

Sliding mode control (SMC) is regarded as one of the robust control techniques against matched uncertainties and bounded disturbances. In the study by Zhu et al.,¹⁰ two adaptive SMC laws are designed to force the state variables of the closed-loop system to achieve the attitude stabilization. The backstepping method relaxes the matching condition at the expense of a high-gain feedback required for robustness, making it prone to chattering.¹¹ In the study by Taheri et al.,¹² backstepping technique is combined with the SMC to relax the matching condition in SMC design. However, a possible issue in conventional backstepping method is the problem of complexity explosion caused by the differentiation operation of virtual controls in each step. To remedy this issue, dynamic surface control (DSC) has been investigated by introducing a first-order filter in each recursive design step. In the study by Xu et al.,¹³ Wang¹⁴ and Li et al.,¹⁵ a NN-based dynamic surface technique has been proposed for nonlinear pure-feedback and strict-feedback systems, respectively. However, the effect of input saturation is not considered in the aforementioned works. Thus, it is a challenge work to develop an effective robust control scheme for uncertain nonlinear systems with unknown input saturation.

Motivated by the aforementioned discussion, this article develops a new neural dynamic surface SMC scheme for a class of uncertain nonlinear systems with unknown input saturation. The main contributions are summarized as follows.

We transform the nonlinear pure-feedback system into the canonical form using the first-order Taylor expansion and coordinate transformation. Besides, to deal with the non-smooth input saturation nonlinearity, a smooth non-affine function is used to approximate the input saturation function.

Integral sliding mode surface is combined with DSC to design the controller, in which only one simple NN is employed for approximating uncertain nonlinearities, and thus the complexity of controller design has been reduced. With the proposed control scheme, no prior knowledge is required on the bound of input saturation, and the explosion of complexity in backstepping method is avoided.

The rest of this article is organized as follows. Problem formulation and preliminaries are provided in the section “Problem formulation and preliminaries.” Controller design and stability analysis are given in sections “Controller design” and “Stability analysis,” respectively. The section “Simulations” provides comparative simulation results to validate the proposed scheme. Some conclusions are given in the section “Conclusion.”

Problem formulation and preliminaries

System description

Consider a class of nonlinear system in the following pure-feedback form

{\begin{cases} ẋ_{i} = f_{i} ({\bar{x}}_{i}, x_{i + 1}), 1 \leq i \leq n - 1 \\ ẋ_{n} = f_{n} ({\bar{x}}_{n}, v (u)) \\ y = x_{1} \end{cases}

where ${\bar{x}}_{i} = {[x_{1}, \dots, x_{i}]}^{T} \in R^{i}$ is the vector of states of the ith differential equations, and ${\bar{x}}_{n} = {[x_{1}, \dots, x_{n}]}^{T} \in R^{n}$ ; f_i, $i = 1, \dots, n - 1$ , are unknown smooth functions of $(x_{1}, \dots x_{i + 1})$ satisfying $f_{i} (0, \dots, 0) = 0; y \in R$ is the output; v(u) ∈ R is the control input subject to saturation nonlinearity described as

v (u) = sat (u) = {\begin{cases} v_{max} sgn (u), sgn | u | \geq v_{max} \\ u,, | u | \leq v_{max} \end{cases}

where v _max is a positive but unknown parameter.

Assumption 1

The state variables ${\bar{x}}_{i}$ of equation (1) are measurable, and the nonlinear functions f_i, $i = 1, \dots, n$ , are continuously differentiable to n-order with respect to the state variables ${\bar{x}}_{i}$ and the input v(u)

Since the unknown functions f_i, $i = 1, \dots, n$ , are continuously differentiable with respect to ${\bar{x}}_{i}$ , we apply the first-order Taylor expansion for f_i, $i = 1, \dots, n$ as

\begin{array}{l} f_{i} ({\bar{x}}_{i}, x_{i + 1}) = f_{i} ({\bar{x}}_{i}, x_{i + 1}^{0}) + \frac{\partial f_{i} ({\bar{x}}_{i}, x_{i + 1})}{\partial x_{i + 1}} |_{x_{i + 1} = x_{i + 1}^{α_{i}}} \times (x_{i + 1} - x_{i + 1}^{0}), 1 \leq i \leq n - 1 \\ f_{n} ({\bar{x}}_{n}, v) = f_{n} ({\bar{x}}_{n}, v^{0}) + \frac{\partial f_{n} ({\bar{x}}_{n}, v)}{\partial u} |_{v = v^{α_{n}}} \times (v - v^{0}) \end{array}

where $x_{i + 1}^{α_{i}} = α_{i} x_{i + 1} + (1 - α_{i}) x_{i + 1}^{0}$ , with $0 < α_{i} < 1, 1 \leq i \leq n - 1$ , and $v^{α_{n}} = α_{n} v + (1 - α_{n}) v^{0}$ , with $0 < α_{n} < 1$ . By choosing $x_{i + 1}^{0} = 0$ and v⁰ = 0, equation (3) can be rewritten as

\begin{array}{l} f_{i} ({\bar{x}}_{i}, x_{i + 1}) = f_{i} ({\bar{x}}_{i}, 0) + \frac{\partial f_{i} ({\bar{x}}_{i}, x_{i + 1})}{\partial x_{i + 1}} |_{x_{i + 1} = x_{i + 1}^{α_{i}}} \times x_{i + 1}, 1 \leq i \leq n - 1 \\ f_{n} ({\bar{x}}_{n}, v) = f_{n} ({\bar{x}}_{n}, 0) + \frac{\partial f_{n} ({\bar{x}}_{n}, v)}{\partial v} |_{v = v^{α_{n}}} \times v \end{array}

For the analysis convenience, it is defined that

\begin{array}{l} g_{i} ({\bar{x}}_{i}, x_{i + 1}^{α_{i}}) = \frac{\partial f_{i} ({\bar{x}}_{i}, x_{i + 1})}{\partial x_{i + 1}} |_{x_{i + 1} = x_{i + 1}^{α_{i}}}, 1 \leq i \leq n - 1 \\ g_{n} ({\bar{x}}_{n}, v^{α_{n}}) = \frac{\partial f_{n} ({\bar{x}}_{n}, v)}{\partial v} |_{v = v^{α_{n}}} \end{array}

which are unknown nonlinear functions. From equations (4) and (5), equation (1) can be re-expressed as

{\begin{cases} ẋ_{i} = f_{i} ({\bar{x}}_{i}, 0) + g_{i} ({\bar{x}}_{i}, x_{i + 1}^{α_{i}}) \times x_{i + 1}, 1 \leq i \leq n - 1 \\ ẋ_{n} = f_{n} ({\bar{x}}_{n}, 0) + g_{n} ({\bar{x}}_{n}, v^{α_{i}}) \times v \\ y = x_{1} \end{cases}

System coordinate transformation

In the following, it will be shown that the original system (1) can be transformed into the canonical form with respect to the newly defined state variables.¹⁶

Let

\begin{array}{l} z_{1} = y = x_{1} \\ z_{2} = {\dot{z}}_{1} = f_{1} (x_{1}) + g_{1} (x_{1}, x_{2}^{α_{1}}) x_{2} \end{array}

The time derivative of z₂ is derived as

\begin{array}{l} ż_{2} = \frac{\partial f_{1} (x_{1})}{\partial x_{1}} ẋ_{1} + (\frac{\partial g_{1} (x_{1}, x_{2}^{α_{1}})}{\partial x_{1}} ẋ_{1} + \frac{\partial g_{1} (x_{1}, x_{2}^{α_{1}})}{\partial x_{2}} ẋ_{2}) x_{2} + g_{1} (x_{1}, x_{2}^{α_{1}}) ẋ_{2} \\ = (\frac{\partial f_{1}}{\partial x_{1}} + \frac{\partial g_{1}}{\partial x_{1}} x_{2}) (f_{1} + g_{1} x_{2}) + (\frac{\partial g_{1}}{\partial x_{2}} x_{2} + g_{1}) (f_{2} + g_{2} x_{3}) \\ \overset{Δ}{=} a_{2} ({\bar{x}}_{2}) + b_{2} ({\bar{x}}_{2}, x_{3}^{α_{2}}) x_{3} \end{array}

where $a_{2} ({\bar{x}}_{2}) = (\frac{\partial f_{1}}{\partial x_{1}} + \frac{\partial g_{1}}{\partial x_{1}} x_{2}) (f_{1} + g_{1} x_{2}) + (\frac{\partial g_{1}}{\partial x_{2}} x_{2} + g_{1}) f_{2}$ and $b_{2} ({\bar{x}}_{2}, x_{3}^{α_{2}}) = (\frac{\partial g_{1}}{\partial x_{2}} x_{2} + g_{1}) g_{2}$ . Again, let $z_{3} = a_{2} + b_{2} x_{3}$ , and its time derivative is

\begin{array}{l} ż_{3} = \sum_{j = 1}^{2} \frac{\partial a_{2}}{\partial x_{j}} ẋ_{j} + \sum_{j = 1}^{3} \frac{\partial b_{2}}{\partial x_{j}} {\dot{x}}_{j} x_{3} + b_{2} ẋ_{3} \\ = \sum_{j = 1}^{2} (\frac{\partial a_{2}}{\partial x_{j}} + \frac{\partial b_{2}}{\partial x_{j}}) (f_{j} + g_{j} x_{j + 1}) + (\frac{\partial b_{2}}{\partial x_{3}} x_{3} + b_{2}) (f_{3} + g_{3} x_{4}) \\ \overset{Δ}{=} a_{3} ({\bar{x}}_{3}) + b_{3} ({\bar{x}}_{3}, x_{4}^{α_{3}}) x_{4} \end{array}

where $a_{3} ({\bar{x}}_{3}) = \sum_{j = 1}^{2} (\frac{\partial a_{2}}{\partial x_{j}} + \frac{\partial b_{2}}{\partial x_{j}}) (f_{j} + g_{j} x_{j + 1}) + (\frac{\partial b_{2}}{\partial x_{3}} x_{3} + b_{2}) f_{3}$ and $b_{3} ({\bar{x}}_{3}, x_{4}^{α_{3}}) = (\frac{\partial b_{2}}{\partial x_{3}} x_{3} + b_{2}) g_{3}$ . When defining a_i−1 and $b_{i - 1}, i = 2, \dots, n$ , we can obtain

\begin{array}{l} z_{i} \overset{Δ}{=} a_{i - 1} ({\bar{x}}_{i - 1}) + b_{i - 1} ({\bar{x}}_{i - 1}, x_{i}^{α_{i - 1}}) x_{i} \\ ż_{i} = a_{i} ({\bar{x}}_{i}) + b_{i} ({\bar{x}}_{i}) x_{i + 1} \end{array}

where

\begin{array}{l} a_{i} ({\bar{x}}_{i}) \overset{Δ}{=} \sum_{j = 1}^{i - 1} (\frac{\partial a_{i - 1}}{\partial x_{j}} + \frac{\partial b_{i - 1}}{\partial x_{j}} x_{i}) (f_{j} + g_{j} x_{j + 1}) + (\frac{\partial b_{i - 1}}{\partial x_{i}} x_{i} + b_{i - 1}) f_{i} \\ b_{i} ({\bar{x}}_{i}, x_{i + 1}^{α_{i}}) \overset{Δ}{=} (\frac{\partial b_{i - 1}}{\partial x_{i}} x_{i} + b_{i - 1}) g_{i} \end{array}

Thus, the pure feedback system (6) can be rewritten in the canonical form with respect to the newly defined state variables as

{\begin{cases} ż_{i} = z_{i + 1}, i = 1, \dots, n - 1 \\ ż_{n} = a_{n} ({\bar{x}}_{n}) + b_{n} ({\bar{x}}_{n}, v^{α_{n}}) v \\ y = z_{1} \end{cases}

To proceed the design procedure, the control function $b_{n} ({\bar{x}}_{n}, v^{α_{n}})$ in equation (12) is assumed to be positive and bounded satisfying $0 < b_{1} < b_{n} ({\bar{x}}_{n}, v^{α_{n}}) < b_{2}$ , where b₁ and b₂ are positive constants. It is pointed out that this condition has been widely used in the literature^17

–20 as a necessary condition for the controllability of equation (1).

The control objective of this article is to design a dynamic surface sliding-mode controller v(t) for the system (12), such that the system output y can track the desired reference signal y_d and all signals in the closed-loop system are bounded.

Nonlinear saturation model

As shown in Figure 1, the control input v(t) ∈ R is the output of the following nonlinear input saturation and u(t) ∈ R is the input of the saturation (practical control signal). The saturation is approximated by a smooth non-affine function defined as

\begin{array}{l} g (u) = v_{max} \times tanh (\frac{u}{v_{max}}) \\ = v_{max} \times \frac{e^{u / v_{max}} - e^{- u / v_{max}}}{e^{u / v_{max}} + e^{- u / v_{max}}} \end{array}

Figure 1.

Saturation sat(u) (solid line) and smooth function g(u) (dot line).

Then, $v (u) = sat (u)$ in equation (2) can be expressed as

v (u) = sat (u) = g (u) + d (u)

where $d (u) = sat (u) - g (u)$ is a bounded function and its bound is

| d (u) | = | sat (u) - g (u) | \leq v_{max} (1 - tanh (1)) = D

where D is the upper bound of |d(u)|.

According to the mean value theorem,⁸ there exists a constant ξ with 0 < ξ < 1, such that

g (u) = g (u_{0}) + g_{u_{ξ}} (u - u_{0})

where $g_{u_{ξ}} = \frac{\partial g (u)}{\partial u} |_{u = u_{ξ}} > 0, u_{ξ} = ξ u + (1 - ξ) u_{0}$ and $u_{0} \in [0, u]$ . By choosing u₀ = 0, equation (16) can be rewritten in the following affine form

g (u) = g_{u_{ξ}} u

Substituting equations (17) and (14) into equation (12), we can obtain

{\begin{cases} ż_{i} = z_{i + 1}, i = 1, \dots, n - 1 \\ ż_{n} = a ({\bar{x}}_{n}) + b ({\bar{x}}_{n}, v^{α_{n}}) u \\ y = z_{1} \end{cases}

where $a ({\bar{x}}_{n}) = a_{n} ({\bar{x}}_{n}) + d, b ({\bar{x}}_{n}, v^{α_{n}}) = b_{n} ({\bar{x}}_{n}, v^{α_{n}}) g_{u_{ξ}}$ .

NN approximation

Due to good capabilities in function approximation, NNs are usually used for the approximation of nonlinear functions.^8,9 The following NN will be used to approximate the continuous function

h (X) = W^{* T} ϕ (X) + ∊^{*}

where $W^{*} \in R^{n_{1} \times n_{2}}$ is the ideal weight matrix, $ϕ (X) \in R^{n_{1} \times 1}$ is the basis function of the NN,∊* is the NN approximation error satisfying $| ∊^{*} | \leq ∊_{N}$ , and ϕ(X) can be chosen as the commonly used sigmoid function, which is in the following form

ϕ (X) = \frac{r_{1}}{r_{2} + \exp (- X / r_{3})} + r_{4}

where r ₁, r ₂, r ₃, and r ₄ are appropriate parameters, and exp is an exponential function.

Remark 1

The employed NN with sigmoid function represents a class of linearly parameterized approximation methods and can be replaced by any other approximation approaches such as spline functions, RBF functions, or fuzzy systems. However, the structure of the employed NN in this article is simpler than the other NNs that are commonly used in other works. There is no hidden layer in the employed NN, in which five inputs and one output are included and the corresponding weight matrix is 5 × 1.

Controller design

In this section, we will incorporate the DSC and integral sliding mode techniques into a NN-based adaptive control design scheme for the nth-order system described by equation (12). Similar to the traditional backstepping design, the recursive design procedure contains n steps. From step 1 to step n − 1, virtual control $z_{i + 1}, i = 1, \dots n - 1$ , is designed at each step, and the integral sliding mode surface is proposed in the first step. Finally, the control law u is obtained at step n.

Step 1

In this step, we consider the first equation of equation (12), that is, $ż_{1} = z_{2}$ .

Define the tracking error and its sliding surface as

${\begin{cases} e = y - y_{d} \\ s_{1} = e + λ \int e d t \end{cases}$
21

where y_d is the desired reference signal and λ is a positive constant.

The derivatives of e and s ₁ are

${\begin{cases} \dot{e} = \dot{y} - ẏ_{d} = z_{2} - ẏ_{d} \\ {\dot{s}}_{1} = \dot{e} + λ e = z_{2} - ẏ_{d} + λ e \end{cases}$
22

Choose a virtual control ${\bar{z}}_{2}$ as

${\bar{z}}_{2} = - k_{1} s_{1} + ẏ_{d} - λ e$
23

where k₁ is a positive constant.

Introduce a new state variable ²₂ and let ${\bar{z}}_{2}$ pass through a first-order filter with time constant τ₂ > 0, and we have

$τ_{2} {\dot{β}}_{2} + β_{2} = {\bar{z}}_{2}, β_{2} (0) = {\bar{z}}_{2} (0)$
24

Define

$y_{2} = β_{2} - {\bar{z}}_{2}$
25

Substituting equation (25) into equation (24), we can obtain

${\dot{β}}_{2} = \frac{{\bar{z}}_{2} - β_{2}}{τ_{2}} = - \frac{y_{2}}{τ_{2}}$
26

Step 2

Consider

$ż_{2} = z_{3}$
27

Let

$s_{2} = z_{2} - β_{2}$
28

which is called the second error surface. Then, we have

${\dot{s}}_{2} = z_{3} - {\dot{β}}_{2}$
29

Choose a virtual control ${\bar{z}}_{3}$ as

${\bar{z}}_{3} = - k_{2} s_{2} - s_{1} + {\dot{β}}_{2}$
30

where k₂ is a positive constant.

Again, introducing a new state variable β₃ and let ${\bar{z}}_{3}$ pass through a first-order filter with time constant τ₃ > 0, we have

$τ_{3} {\dot{β}}_{3} + β_{3} = {\bar{z}}_{3}, β_{3} (0) = {\bar{z}}_{3} (0)$
31

Define

$y_{3} = β_{3} - {\bar{z}}_{3}$
32

Substituting equation (32) into equation (31), we can obtain

${\dot{β}}_{3} = \frac{{\bar{z}}_{3} - β_{3}}{τ_{3}} = - \frac{y_{3}}{τ_{3}}$
33

Step i

Consider

$ż_{i} = z_{i + 1}$
34

Let

$s_{i} = z_{i} - β_{i}$
35

which is called the ith error surface. Then, we have

${\dot{s}}_{i} = z_{i + 1} - {\dot{β}}_{i}$
36

Choose a virtual control ${\bar{z}}_{i + 1}$ as

${\bar{z}}_{i + 1} = - k_{i} s_{i} - s_{i - 1} + {\dot{β}}_{i}$
37

where k_i is a positive constant.

Introduce a new state variable β_i+1 and let ${\bar{z}}_{i + 1}$ pass through a first-order filter with time constant τ_i+1 > 0, and we have

$τ_{i + 1} {\dot{β}}_{i + 1} + β_{i + 1} = {\bar{z}}_{i + 1}, β_{i + 1} (0) = {\bar{z}}_{i + 1} (0)$
38

Define

$y_{i + 1} = β_{i + 1} - {\bar{z}}_{i + 1}$
39

Substituting equation (39) into equation (38), we can obtain

${\dot{β}}_{i + 1} = \frac{{\bar{z}}_{i + 1} - β_{i + 1}}{τ_{i +!}} = - \frac{y_{i + 1}}{τ_{i + 1}}$
40

Step n. The final control law will be derived in this step. Consider

${\dot{z}}_{n} = a ({\bar{x}}_{n}) + b ({\bar{x}}_{n}, v^{α_{n}}) u$
41

Given a compact set $Ω_{z n} \in R^{n}$ , let W* and ∊* be such that for any $(z_{1}, \dots, z_{n}) \in Ω_{z n}$

$H = \frac{a ({\bar{x}}_{n}) - {\dot{β}}_{n}}{b ({\bar{x}}_{n}, v^{α_{n}})} = W^{* T} φ ({\bar{z}}_{n}) + ∊^{}$
42

with $| ∊^{} | \leq ∊_{N}$ . Let

$s_{n} = z_{n} - β_{n}$
43

which is called the nth error surface. From equations (41) and (43), we have

${\dot{s}}_{n} = a ({\bar{x}}_{n}) + b ({\bar{x}}_{n}, v^{α_{n}}) u - {\dot{β}}_{n}$
44

Finally, design the final law u as

$u = - k_{n} s_{n} - s_{n - 1} - {\hat{W}}^{T} φ ({\bar{z}}_{n}) - {\hat{∊}}_{N} tanh (\frac{s_{n}}{δ})$
45

where $\hat{W}$ is the estimation of W* and ${\hat{∊}}_{N}$ is the estimation of the upper limit for |∊|. The adaptive learning laws of $\hat{W}$ and ${\hat{∊}}_{N}$ are given by

${\begin{cases} \hat{W} = \tilde{W} = Γ [ϕ ({\bar{z}}_{n}) s_{n} - σ \hat{W}] \\ {\hat{∊}}_{N} = {\tilde{∊}}_{N} = v_{∊ N} [s_{n} tanh (\frac{s_{n}}{δ})] \end{cases}$
46

where σ and δ are positive small constants and Γ = Γ^T* > 0 is a constant matrix.

Stability analysis

In this section, a theorem is provided to show the boundedness of all signals in system (12) and convergence of tracking error e as well as sliding surface s.

Theorem 1

Consider the nonlinear system (12) with unknown input saturation (14), the integral sliding mode surface (21), control law (45), and adaptive learning laws (46). Given any positive number, for all initial conditions satisfying $(\sum_{j = 1}^{n - 1} (s_{i}^{2} + y_{i + 1}^{2}) + \frac{1}{b} s_{n}^{2} + {\tilde{W}}^{T} Γ^{- 1} \tilde{W} + \frac{1}{v_{∊ N}} {\tilde{∊}}_{N}^{2}) \leq 2 p$ , all the closed-loop signals are semi-global uniformly ultimately bounded, and the tracking error can be made arbitrarily small by properly choosing the design parameters.

Proof

Firstly, define the estimation error as

${\begin{cases} \tilde{W} = \hat{W} - W^{} \\ \tilde{∊} = \hat{∊} - ∊^{} \end{cases}$
47

Then, the closed-loop system in the new coordinates, s_i, β_i, and ${\tilde{W}}_{i}$ , can be expressed as follows

$\begin{array}{l} {\dot{s}}_{1} = s_{2} + β_{2} + λ e - ẏ_{d} \\ {\dot{s}}_{2} = s_{3} + β_{3} - {\dot{β}}_{2} \\ ⋮ \\ {\dot{s}}_{i} = s_{i + 1} + β_{i + 1} - {\dot{β}}_{i}, i = 3, \dots n - 1 \\ ⋮ \\ \frac{{\dot{s}}_{n}}{b} = - k_{n} s_{n} - s_{n - 1} - \tilde{W} φ ({\bar{z}}_{n}) + ∊_{N}^{} - {\hat{∊}}_{N} tanh (\frac{s_{n}}{δ}) \end{array}$
48

From equation (48), we have $β_{i + 1} = y_{i + 1} + {\bar{z}}_{i + 1}, i = 1, \dots n - 1$ . Then, we can obtain

$\begin{array}{l} {\dot{s}}_{1} = s_{2} + y_{2} - k_{1} s_{1} \\ {\dot{s}}_{2} = s_{3} + y_{3} - k_{2} s_{2} - s_{1} - {\dot{β}}_{2} \\ ⋮ \\ {\dot{s}}_{i} = s_{i + 1} + y_{i + 1} - k_{i} s_{i} - s_{i - 1} - {\dot{β}}_{i}, i = 3, \dots n - 1 \\ ⋮ \\ \frac{{\dot{s}}_{n}}{b} = - k_{n} s_{n} - s_{n - 1} - \tilde{W} φ ({\bar{z}}_{n}) + ∊_{N}^{} - {\hat{∊}}_{N} tanh (s_{n} / δ) . \end{array}$
49

Besides, we know the fact that

$\begin{array}{l} ẏ_{2} = β_{2} - {\dot{\bar{z}}}_{2} = - \frac{y_{2}}{τ_{2}} - (- k_{1} {\dot{s}}_{1} + ÿ_{d} - λ \dot{e}) \\ = - \frac{y_{2}}{τ_{2}} + B_{2} (s_{1}, s_{2}, y_{2}, y_{d}, {\dot{y}}_{d}, ÿ_{d}) \end{array}$
50

where $B_{2} (s_{1}, s_{2}, y_{2}, y_{d}, {\dot{y}}_{d}, ÿ_{d}) = - (- k_{1} {\dot{s}}_{1} + ÿ_{d} - λ \dot{e})$ , which is a continuous function.

Similarly, for $i = 2, \dots n - 1$ , we have

$\begin{matrix} ẏ_{i + 1} = - \frac{y_{i + 1}}{τ_{i + 1}} - k_{i} {\dot{s}}_{i} - {\dot{s}}_{i - 1} = - \frac{y_{i + 1}}{τ_{i + 1}} \\ + B_{i + 1} (s_{1}, \dots, s_{i + 1}, y_{2}, \dots y_{i}, y_{d}, {\dot{y}}_{d}, ÿ_{d}) . \end{matrix}$
51

Consider the Lyapunov function candidate

$V = \frac{1}{2} \sum_{i = 1}^{n - 1} (s_{i}^{2} + y_{i + 1}^{2}) + \frac{1}{2 b} s_{n}^{2} + \frac{1}{2} {\tilde{W}}^{T} Γ^{- 1} \tilde{W} + \frac{1}{2 v_{∊ N}} {\tilde{∊}}_{N}^{2}$
52

The derivative of the Lyapunov function is

$\dot{V} = \sum_{i = 1}^{n - 1} (s_{i} {\dot{s}}_{i} + y_{i + 1} ẏ_{i + 1}) + \frac{1}{b} s_{n} {\dot{s}}_{n} + {\tilde{W}}^{T} Γ^{- 1} \hat{W} . + \frac{1}{v_{∊ N}} {\tilde{∊}}_{N} {\hat{∊} .}_{N} = \sum_{i = 1}^{n} (- k_{i} s_{i}^{2}) + \sum_{i = 1}^{n - 1} (s_{i} y_{i + 1} - \frac{y_{i + 1}^{2}}{τ_{i + 1}} + B_{i + 1} y_{i + 1}) + s_{n} (- \tilde{W} φ ({\bar{z}}_{n}) + ∊_{N}^{} - {\hat{∊}}_{N} tanh (\frac{s_{n}}{δ})) + {\tilde{W}}^{T} Γ^{- 1} \hat{W} . + \frac{1}{v_{∊ N}} {\tilde{∊}}_{N} {\hat{∊} .}_{N} = \sum_{i = 1}^{n} (- k_{i} s_{i}^{2}) + \sum_{i = 1}^{n - 1} (s_{i} y_{i + 1} - \frac{y_{i + 1}^{2}}{τ_{i + 1}} + B_{i + 1} y_{i + 1}) + s_{n} (∊_{N}^{} - {\hat{∊}}_{N} tanh (\frac{s_{n}}{δ})) - σ {\tilde{W}}^{T} \hat{W} + \frac{1}{v_{∊ N}} {\tilde{∊}}_{N} {\hat{∊} .}_{N} \leq \sum_{i = 1}^{n} (- k_{i} s_{i}^{2}) + \sum_{i = 1}^{n - 1} (s_{i} y_{i + 1} - \frac{y_{i + 1}^{2}}{τ_{i + 1}} + B_{i + 1} y_{i + 1}) + ∊_{N}^{} (| s_{n} | - s_{n} tanh (\frac{s_{n}}{δ})) + ∊_{N}^{} s_{n} tanh (\frac{s_{n}}{δ}) - s_{n} {\hat{∊}}_{N} tanh (\frac{s_{n}}{δ}) - σ {\tilde{W}}^{T} \hat{W} + \frac{1}{v_{∊ N}} {\tilde{∊}}_{N} {\hat{∊} .}_{N} \leq \sum_{i = 1}^{n} (- k_{i} s_{i}^{2}) + \sum_{i = 1}^{n - 1} (s_{i} y_{i + 1} - \frac{y_{i + 1}^{2}}{τ_{i + 1}} + B_{i + 1} y_{i + 1}) + ∊_{N}^{} (| s_{n} | - s_{n} tanh (\frac{s_{n}}{δ})) - σ {\tilde{W}}^{T} \hat{W}$
53

Using the following property with regard to function tanh(.), we have

$0 \leq | x | - x tanh (\frac{x}{δ}) \leq 0.2785 δ$
54

Using the fact

$\begin{array}{l} - σ {\tilde{W}}^{T} \hat{W} \leq - σ {\tilde{W}}^{T} (\tilde{W} + W^{}) \\ \leq - σ ‖ \tilde{W} ‖^{2} + σ ‖ \tilde{W} ‖ ‖ W^{} ‖ \\ \leq - σ ‖ \tilde{W} ‖^{2} + \frac{σ}{2} ‖ \tilde{W} ‖^{2} + \frac{σ}{2} ‖ W^{} ‖^{2} \\ \leq - \frac{σ}{2} ‖ \tilde{W} ‖^{2} + \frac{σ}{2} ‖ W^{} ‖^{2} \\ \leq \frac{σ}{2} ‖ W^{} ‖^{2} \end{array}$
55

and substituting equations (54) and (55) into equation (53), we can obtain

$\dot{V} \leq \sum_{i = 1}^{n} (- k_{i} s_{i}^{2}) + \sum_{i = 1}^{n - 1} (s_{i} y_{i + 1} - \frac{y_{i + 1}^{2}}{τ_{i + 1}} + B_{i + 1} y_{i + 1}) + 0.2785 ∊_{N}^{} δ - \frac{σ}{2} {‖ W^{} ‖}^{2}$
56

Using the fact

$s_{i}^{2} + \frac{1}{4} y_{i + 1}^{2} \geq s_{i} y_{i + 1}$
57

we have

$\dot{V} \leq \sum_{i = 1}^{n} (- k_{i} s_{i}^{2}) + \sum_{i = 1}^{n - 1} (s_{i}^{2} + \frac{1}{4} y_{i + 1}^{2} - \frac{y_{i + 1}^{2}}{τ_{i + 1}} + B_{i + 1} y_{i + 1}) + 0.2785 ∊_{N}^{} δ - \frac{σ}{2} {‖ W^{} ‖}^{2}$
58

Choose

$\begin{array}{l} k_{i} = 1 + α_{0}, i = 1, \dots, n - 1 \\ k_{n} = α_{0} \\ \frac{1}{τ_{i + 1}} = \frac{1}{4} + \frac{M_{i + 1}^{2}}{2 η} + α_{0} \end{array}$
59

where α₀ and η are positive constants and $| B_{i + 1} | \leq M_{i + 1}$ .

Remark 2

As pointed out in the study by Li et al.¹⁵ for any B₀ > 0 and p > 0, the sets $\prod : = {(y_{d}, {y .}_{d}, ÿ_{d}) ÿ_{d}^{2} + {y .}_{d}^{2} + ÿ_{d}^{2} \leq B_{0}}$ and $\prod_{i} : = (\sum_{j = 1}^{i - 1} (s_{i}^{2} + y_{i + 1}^{2}) + \frac{1}{b} s_{n}^{2} + {\tilde{W}}^{T} Γ^{- 1} \tilde{W} + \frac{1}{v_{∊ N}} {\tilde{∊}}_{N}^{2}) \leq 2 p, i = 2, \dots, n$ , are compact in R₃ and $R^{(\sum_{j = 1}^{i} i + 2)}$ , respectively. ∏ × ∏_i is also compact in $R^{(\sum_{j = 1}^{i} i + 5)}$ . Therefore, |B_i+1| has a maximum M_i+1 on ∏ × ∏_i.

Noting that, for any positive number η

$\frac{y_{i + 1}^{2} B_{i + 1}^{2}}{2 η} + \frac{η}{2} \geq | B_{i + 1} y_{i + 1} |$
60

we can obtain

$\begin{array}{l} \dot{V} \leq \sum_{i = 1}^{n} (- α_{0} s_{i}^{2}) + \sum_{i = 1}^{n - 1} (\frac{1}{4} y_{i + 1}^{2} - (\frac{1}{4} + \frac{M_{i + 1}^{2}}{2 η} + α_{0}) y_{i + 1}^{2} + \frac{M_{i + 1}^{2} y_{i + 1}^{2} B_{i + 1}^{2}}{2 η M_{i + 1}^{2}} + \frac{η}{2}) + 0.2785 ∊_{N}^{} δ - \frac{σ}{2} {‖ W^{} ‖}^{2} \\ \leq \sum_{i = 1}^{n} (- α_{0} s_{i}^{2}) + \sum_{i = 1}^{n - 1} (- α_{0} y_{i + 1}^{2} - (1 - \frac{B_{i + 1}^{2}}{M_{i + 1}^{2}}) \frac{M_{i + 1}^{2} y_{i + 1}^{2}}{2 η}) + 0.2785 ∊_{N}^{} δ - \frac{σ}{2} {‖ W^{} ‖}^{2} \\ \leq \sum_{i = 1}^{n} (- α_{0} s_{i}^{2}) + 0.2785 ∊_{N}^{} δ - \frac{σ}{2} {‖ W^{} ‖}^{2} \end{array}$
61

Hence, we can conclude $\dot{V} \leq 0$ if

$| s_{i} | \geq \sqrt{\frac{0.2785 ∊_{N}^{} δ + σ ‖ \tilde{W} ‖ ‖ W^{} ‖}{α_{0}}}$
62

Then, the ultimate boundedness of s_i is guaranteed, and s_i will converge to the following positively invariant set

$Ω = {| s_{i} | \leq γ_{s}}$
63

with $γ_{s} \geq \sqrt{\frac{0.2785 ∊_{N}^{} δ + σ ‖ \tilde{W} ‖ ‖ W^{} ‖}{α_{0}}}$ .

When s₁ reaches the positively invariant set γ_s, it remains inside thereafter. From equation (12), the error dynamics inside γ_s is

$\dot{e} + λ e = {\dot{s}}_{1}$
64

Due to the boundedness of s₁ and ${\dot{s}}_{1}$ , the ultimate boundedness of the tracking error e can be directly concluded using the Lyapunov function $W = (1 / 2) \times e^{2}$ .²¹

Simulations

In order to show the superior tracking performance of the proposed scheme, we consider three different control schemes for comparison: (S1) neural dynamic SMC with saturation compensation; (S2) neural dynamic SMC without saturation compensation; and (S3) neural DSC without saturation compensation.¹⁵

The following four indices are adopted to compare the tracking performance of each control algorithm.
$IAE = \int | e (t) | d t$ , which is the integrated absolute error to measure the system tracking performance.

$ISDE = {\int e (t) - e_{0}}^{2} d t$ , where e ₀ is the average error of whole process. ISDE is the integrated square error and used to demonstrate the smoothness of the profile.

$IAV = \int | v (t) | d t$ , which is the integrated absolute control and taken as a measurement of the overall amount of control effort.

$ISDV = {\int v (t) - v_{0}}^{2} d t$ , where v₀ is the average control input of whole process. ISDV is the integrated square control and used as a measurement of the fluctuations of control signal around its mean value.

In the following, two simulation examples are adopted for the fair comparison of different control schemes.

Spring mass and damper system

The considered spring mass and damper system represents a class of widely used second-order electromechanical servo systems, such as hydraulic systems, rigid robots, or turntable systems.^22
–24 In those systems, the number of freedom degrees is always equal to the number of control inputs, and thus the controller design is relatively easier.

As shown in Figure 2, a second-order system is described as⁷

$\begin{array}{l} {x .}_{1} = x_{2} \\ {x .}_{2} = - \frac{k}{m} x_{1} - \frac{c}{m} x_{2} + \frac{1}{m} v (u) + E (t) \end{array}$
65

where y = x₁, x₁ and x₂ are the position and velocity, respectively; m is the mass of the object; k is the stiffness constant of the spring; and c is the damping.

Figure 2.
Spring mass and damper system.

According to equation (8), equation (65) can be transformed into

$\begin{array}{l} {x .}_{1} = x_{2} \\ {x .}_{2} = a ({\bar{x}}_{2}) + b ({\bar{x}}_{2}, v) \times v (u) \end{array}$
66

where ${\bar{x}}_{2} = {[x_{1}, x_{2}]}^{T} a ({\bar{x}}_{2}, v) = - \frac{k}{m} x_{1} - \frac{c}{m} x_{2} + E (t), b ({\bar{x}}_{2}, v) = 1 / m$ .

In the simulation, two different signal waves are adopted as the desired reference signals, and the system parameters are fixed for various reference signals. The initial states of the system are ${[x_{1}, x_{2}]}^{T} = {[0, 0]}^{T}$ . The constants of adaptive leaning laws $δ = 0.5, σ = 0.01, {\hat{∊}}_{N} = 0.01, v_{∊ N} = 1$ . The time constant is τ = 0.01. The NN parameters are $r_{1} = 1, r_{2} = 5, r_{3} = 5, r_{4} = - 0.1$ , and Γ = diag{5}. The system parameters are set as $m = 1 k g, c = 2 N \cdot s / m$ , and k = 8 N/ m, which are not needed to be known in our controller design. The external disturbance is $E (t) = 0.1 sin (2 π t)$ . The control parameters are $k_{1} = 10, k_{2} = 8$ , and λ = 5.

Case 1

$y_{d} = - 0.2 \cos (3 π t) + 0.2$ is employed as the reference signal.

The input saturation bound is v_max = 14 N. Comparative tracking performance, tracking errors, and control input are shown in Figures 3, 4, and 5, respectively. From Figures 3 and 4, we can see that compared with the proposed S1 method, S2 has the larger overshoot, and S2 and S3 have larger tracking errors. From Figure 5(a), compared with S2 and S3, the control input of S1 is more smooth, and the compensation effect of input saturation is shown in Figure 5(b). From the figures, we can clearly observe the significantly improved performances with the S1.

Figure 3.
Tracking performance of $y_{d} = - 0.2 \cos (3 π t) + 0.2$ .

Figure 4.
Tracking errors of $y_{d} = - 0.2 \cos (3 π t) + 0.2$ . (a) Control inputs of three methods. (b) The saturated control v(t) and the practical control u(t) in S2.

Figure 5.
Control inputs of $y_{d} = - 0.2 \cos (3 π t) + 0.2$ .

In order to compare the control performance, four indices are given in Table 1. From Table 1, we can obtain that S3 controller gives the largest IAV and ISDV, while S2 controller gives the largest IAE and ISDE. The proposed S1 has the smallest IAE, ISDE, IAV, and ISDV, which means it performs best among three controllers. The comparative result from Table 1 is consistent with the Figures 3 to 5.

Table 1.
Comparison for $y_{d} = - 0.2 \cos (3 π t) + 0.2$ .

S1 S2 S3

IAE 0.0311 0.1190 0.1436

ISDE 0.0002 0.0037 0.0013

IAV 43.3104 55.4190 51.0590

ISDV 452.2407 688.8545 609.4313

IAE: integrated absolute error; ISDE: integrated square error; ISDV: integrated square control; IAV: integrated absolute control.

Case 2

$y_{d} = 0.5 sin (t) + sin (0.5 t)$ is employed as the reference signal.

The first reference trajectory is sinusoidal signal, and all parameters are tuned based on this signal. In order to show the high robustness of the proposed method, we give the second reference trajectory (sinusoidal signal with harmonics). The control parameters are set the same as case 1, and the input saturation bound becomes v_max = 6 N, which is more stringent than that of case 1. Comparative tracking performance, tracking errors, and control inputs are shown in Figures 6, 7, and 8, respectively. From Figures 6 and 7, we can see that S2 and S3 have larger tracking errors, while the proposed S1 method achieves the smallest tracking errors and fastest convergence speed. From Figure 8(a), compared with S2 and S3, the control input of S1 is more smooth, and the compensation effect of input saturation is shown in Figure 8(b). In conclusion, S1 has the best performance when tracking the sinusoidal signal with harmonics.

Figure 6.
Tracking performance of $y_{d} = 0.5 (sin (t) + sin (0.5 t))$ .

Figure 7.
Tracking errors of $y_{d} = 0.5 (sin (t) + sin (0.5 t))$ . (a) Control inputs of three methods. (b) The saturated control v(t) and the practical control u(t) in S2.

Figure 8.
Control inputs of $y_{d} = 0.5 (sin (t) + sin (0.5 t))$ .

In addition, the comparative results of the IAE are shown in Table 2. From Table 2, we can see that the proposed S1 method has the smallest IAE among all the three control schemes. Besides, other three indices (i.e. ISDE, IAV, and ISDV) of the proposed S1 method are also smallest, which means S1 has the smoothness of tracking error and control signal. Therefore, S1 has the best tracking preference, which is consistent with the results given by Figures 6 to 8.

Table 2.
Comparison for $y_{d} = 0.5 (sin (t) + sin (0.5 t))$ .

S1 S2 S3

IAE 0.0821 0.4360 0.5413

ISDE 0.0005 0.0303 0.0246

IAV 50.9398 58.3204 53.8104

ISDV 192.0874 263.9002 223.3383

IAE: integrated absolute error; ISDE: integrated square error; ISDV: integrated square control; IAV: integrated absolute control.

Furthermore, the comparative results of IAE for different input saturation values of case 1 and case 2 are shown in Tables 3 and 4, respectively. From Table 3, we can see that the proposed S1 method has the smallest IAE in case 1 compared with S2 and S3, although the input saturation values are changed from 10 to 18. Table 4 shows that for case 2, the proposed S1 scheme with the fixed parameters can still achieve the best tracking performance in the case of various input saturation values. It should be noted that with the input saturation becoming more stringent, the IAE of S2 and S3 will be changed much larger than that of S1. In conclusion, the proposed S1 scheme can achieve a satisfactory tracking performance for different input saturation values.

Table 3.
Comparison of IAE for case 1.

S1 S2 S3

v_max = 10 0.0248 0.6036 0.3756

v_max = 14 0.0311 0.1190 0.1436

v_max = 18 0.0257 0.0289 0.0853

IAE: integrated absolute error.

Table 4.
Comparison of IAE for case 2.

S1 S2 S3

v_max = 4 0.0915 3.8849 1.7634

v_max = 6 0.0821 0.0430 0.5413

v_max = 8 0.0850 0.0924 0.5194

IAE: integrated absolute error.

A single-link flexible-joint robotic manipulator system

In the following, we give the second example, that is, a single-link flexible-joint robotic manipulator system.²⁵ Due to the introduction of joint flexibility in the robot model, the motion equations become more complicated. In particular, the order of the related dynamics becomes twice that of the rigid robots, and the number of freedom degrees is larger than the number of control inputs, which may lead the control task more difficult.

As shown in Figure 9 (figure 1 in the study by Talole²⁵), the mechanical dynamics of the robotic manipulator system can be described as
${\begin{cases} I \overset{..}{q} + K (q - θ) + M g L sin (q) = 0 \\ J \overset{..}{t} - K (q - θ) = v (u) \end{cases}$
67
where q and θ are the position of the link and motor angles, respectively, I is the link inertia, J is the inertia of the motor, K is the spring stiffness, M and L are the mass and length of link, respectively, and v(u) ∈ R is the plant input subject to saturation nonlinearity.

Figure 9.
Schematic of flexible-joint manipulator.

For convenience of the controller design, defining $x_{1} = q, x_{2} = \dot{q} = {\dot{x}}_{1}, x_{3} = θ$ , and $x_{4} = \dot{θ} = {\dot{x}}_{3}$ , equation (67) is transformed into

${\begin{cases} {x .}_{1} = x_{2} \\ {x .}_{2} = - \frac{M g L}{I} sin x_{1} - \frac{K}{I} (x_{1} - x_{3}) \\ {x .}_{3} = x_{4} \\ {x .}_{4} = \frac{1}{J} v (u) + \frac{K}{J} (x_{1} - x_{3}) \end{cases}$
68

Let $z_{1} = x_{1}, z_{2} = x_{2}, z_{3} = - \frac{M g L}{I} sin x_{1} - \frac{K}{I} (x_{1} - x_{3})$ and $z_{4} = - x_{2} \frac{M g L}{I} \cos x_{1} - \frac{K}{I} (x_{2} - x_{4})$ , and equation (68) can be rewritten in terms of the new coordinates as

${\begin{cases} {z .}_{1} = z_{2} \\ {z .}_{2} = z_{3} \\ {z .}_{3} = z_{4} \\ {z .}_{4} = a_{1} (\bar{z}) + b_{1} (\bar{z}, v) \times v (u) \\ y = z_{1} \end{cases}$
69

where $\bar{z} = {[z_{1}, z_{2}, z_{3}]}^{T}, a_{1} (\bar{z}) = \frac{M g L}{I} sin z_{1} (z_{2}^{2} - \frac{K}{J}) - (\frac{M g L}{I} \cos (z_{1}) + \frac{K}{J} + \frac{K}{I}) z_{3}$ , and $b_{1} (\bar{z}, v) = \frac{K}{I J}$ .

According to equation (8), equation (69) can be rewritten as

${\begin{cases} {z .}_{1} = z_{2} \\ {z .}_{2} = z_{3} \\ {z .}_{3} = z_{4} \\ {z .}_{4} = a_{2} (\bar{z}) + b_{2} (\bar{z}, v) \times u \\ y = z_{1} \end{cases}$
70

where $\bar{z} = {[z_{1}, z_{2}, z_{3}]}^{T}, a_{2} (\bar{z}) = \frac{M g L}{I} sin z_{1} (z_{2}^{2} - \frac{K}{J}) - (\frac{M g L}{I} \cos z_{1} + \frac{K}{J} + \frac{K}{I}) z_{3} + d$ , and $b_{2} (\bar{z}, v) = \frac{K}{I J} g_{u_{ξ}}$ .

In the simulation, two different signal waves are adopted as the desired reference signals, and the system parameters are fixed for various reference signals. The initial states of the system are ${[z_{1}, z_{2}, z_{3}, z_{4}]}^{T} = {[0, 0, 0, 0]}^{T}$ . The constants of adaptive leaning laws are $δ = 0.1, σ = 0.01$ , and ${\hat{∊}}_{N} = 0.01$ . The time constants are $τ_{2} = τ_{3} = τ_{4} = 0.01$ . The NN parameters are $r_{1} = 10, r_{2} = 10, r_{3} = 1, r_{4} = - 1, Γ = d i a g {5}$ , and $v_{∊ N} = 0.01$ . The system parameters are $M g l = 5, I = 1, J = 1$ , and K = 40, which are not known in prior for the proposed controller design. The control parameters are $k_{1} = 0.5, k_{2} = 8, k_{3} = 8, k_{4} = 2$ , and λ = 7.

Case 3

Step signal y_d = 1 is employed as the reference signal.

The input saturation is $v_{max} = 30 (N \cdot m)$ , and the results are shown in Figures 10 to 12. From the simulation, we can discover that S1 has a smaller overshoot than S2. S1 and S2 have a faster convergence speed than S3 because of the impact of integral SMC. From Figure 12(a), the effect of saturation compensation in the proposed S1 is apparent when the robotic manipulator has the step response characteristic. From Figure 12(b), we can conclude that if the input saturation becomes smaller, S2 may not be convergent or even be seriously divergent. Consequently, S1 can achieve the best performance among all three control schemes.

Figure 10.
Tracking performance of step signal y_d = 1.

Figure 11.
Tracking errors of step signal y_d = 1. (a) Control inputs of three methods. (b) The saturated control v(t) and the practical control u(t) in S1.

Figure 12.
Control inputs of step signal y_d = 1.

Case 4

Trapezoidal wave is employed as the reference signal.

This reference signal is expressed by

$y_{d} = {\begin{cases} 0, 0 \leq t < 2 \\ 5 (t - 2), 2 \leq t < 4 \\ 10, 4 \leq t < 6 \\ - 5 (t - 8), 6 \leq t < 10 \\ - 10, 10 \leq t < 12 \\ 5 (t - 14), 12 \leq t < 16 \\ 10, 16 \leq t < 18 \\ - 5 (t - 20), 18 \leq t \leq 20 \end{cases}$
71

The control parameters are set the same as case 3. Comparative tracking performance, tracking errors, and control inputs are shown in Figures 13, 14 and 15, respectively. The input saturation of S2 or S3 becomes $v_{max} = 294 (N \dot{s} m)$ , while the S1 is $v_{max} = 75 (N \dot{s} m)$ . From Figures 13 and 14, we can see that S1 has the smallest tracking errors and fastest convergence speed. S2 becomes divergent although its tracking error is small at first, and S3 has the largest static error among all three methods. It means that control method with integral SMC can decrease the system static errors and improve the convergence speed, which can also lead to system divergence because of its integral term when time goes into infinity. From Figures 13 to 15, we have that S1 can achieve the better tracking performance with much less saturated control cost.

Figure 13.
Tracking performance of trapezoidal wave (equation (71)).

Figure 14.
Tracking errors of trapezoidal wave (equation (71)). (a) The saturated control v(t) and the practical control u(t) in S1. (b) Control inputs of three methods.

Figure 15.
Control input of trapezoidal wave (equation (71)).

From all the simulation results, we can conclude that compared with S2 and S3, the proposed S1 scheme has the better tracking performance with respect to tracking errors, convergence speed, and control cost.

Conclusion

In this article, an adaptive neural dynamic surface SMC scheme is proposed for uncertain nonlinear systems with unknown input saturation. The non-smooth saturation is transformed into an affine form by defining a non-affine function and using the mean value theorem. One simple NN is employed for nonlinearity approximation, and the approximation error is estimated by an adaptive learning law. By combing the DSC and the integral sliding mode technique, the controller is designed to improve the system robustness, and comparative simulations are given to illustrate the effectiveness of the proposed method.

	S1	S2	S3
IAE	0.0311	0.1190	0.1436
ISDE	0.0002	0.0037	0.0013
IAV	43.3104	55.4190	51.0590
ISDV	452.2407	688.8545	609.4313

	S1	S2	S3
IAE	0.0821	0.4360	0.5413
ISDE	0.0005	0.0303	0.0246
IAV	50.9398	58.3204	53.8104
ISDV	192.0874	263.9002	223.3383

	S1	S2	S3
v_max = 10	0.0248	0.6036	0.3756
v_max = 14	0.0311	0.1190	0.1436
v_max = 18	0.0257	0.0289	0.0853

	S1	S2	S3
v_max = 4	0.0915	3.8849	1.7634
v_max = 6	0.0821	0.0430	0.5413
v_max = 8	0.0850	0.0924	0.5194

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors would like to thank the support from the National Natural Science Foundation of China under grant numbers 61403343 and 61433003 and the China Postdoctoral Science Foundation funded project under grant number 2015M580521.

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Adaptive neural dynamic surface sliding mode control for uncertain nonlinear systems with unknown input saturation

Abstract

Keywords

Introduction

Problem formulation and preliminaries

System description

Assumption 1

System coordinate transformation

Nonlinear saturation model

NN approximation

Remark 1

Controller design

Step 1

Step 2

Step i

Step n. The final control law will be derived in this step. Consider

Stability analysis

Theorem 1

Proof

Remark 2

Simulations

Spring mass and damper system

Case 1

Case 2

A single-link flexible-joint robotic manipulator system

Case 3

Case 4

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References