Sliding mode learning control of uncertain nonlinear systems with Lyapunov stability analysis

Abstract

This paper addresses the Sliding Mode Learning Control (SMLC) of uncertain nonlinear systems with Lyapunov stability analysis. In the control scheme, a conventional control term is used to provide the system stability in compact space while a type-2 neuro-fuzzy controller (T2NFC) learns system behaviour so that the T2NFC completely takes over overall control of the system in a very short time period. The stability of the sliding mode learning algorithm has been proven in the literature; however, it is restrictive for systems without overall system stability. To address this shortcoming, a novel control structure with a novel sliding surface is proposed in this paper, and the stability of the overall system is proven for nth-order uncertain nonlinear systems. To investigate the capability and effectiveness of the proposed learning and control algorithms, the simulation studies have been carried out under noisy conditions. The simulation results confirm that the developed SMLC algorithm can learn the system behaviour in the absence of any mathematical model knowledge and exhibit robust control performance against external disturbances.

Keywords

adaptive control neuro-fuzzy control online learning algorithm sliding mode control theory uncertain nonlinear systems

Introduction

Control of uncertain nonlinear systems is one of the most crucial topics in modern control engineering (Chen et al., 2016; Kayacan and Fossen, 2019; Kayacan and Peschel, 2016; Kayacan et al., 2012, 2013b, 2017). Robust controllers aim to ensure the best control performance in the presence of uncertainties, and high controller gain is a common method to overcome the uncertainity problem in nonlinear control theory (Sun et al., 2016). However, these methods bring about large control actions, and require very powerful actuators (Ren et al., 2017). Moreover, robust control performance against uncertainties is mostly obtained at the cost of sacrificing nominal control performance of the system (Liu et al., 2017a). Therefore, a control method is required to exhibit robust control performance in the presence of uncertainties while either maintaining or improving nominal control performance (Huang et al., 2015). As a model-free method, fuzzy logic control is an alternative solution to the model-based control approaches in the presence of uncertainties and has been applied to different real-time systems, such as mobile robots (Castillo et al., 2012), spherical rolling robots (Kayacan et al., 2013a) and agricultural ground vehicles (Kayacan et al., 2015b, 2018). Even though type-1 fuzzy logic controllers are the most well known and widely used types of fuzzy logic controllers, researchers have recently focused on type-2 fuzzy logic controllers, which can deal with high levels of uncertainty in real-time applications (Castillo et al., 2014; Lee et al., 2015; Liu et al., 2017c; Maldonado et al., 2013; Muhuri and Shukla, 2017; Rubio-Solis and Panoutsos, 2015; Zaheer et al., 2015).

As a learning controller, a feedback–error learning control structure was proposed by Gomi and Kawato (1993) and first implemented for the control of robot manipulators. This relied on the parallel structure of a conventional controller and a neuro-fuzzy based controller. The generated control signal by the conventional controller provides a learning error signal to train learning algorithms. The latter learns the system behaviour online and thus eliminates the former from the control of the system (Kalanovic et al., 2000; Ruan et al., 2007). Since the gradient-based methods in training algorithms are complex and computationally more expensive, sliding mode control (SMC) theory-based learning algorithms have been proposed for feedback-error learning control structures (Efe et al., 2000). These can ensure faster convergence compared to conventional learning techniques for online adjustment in neural networks and type-1 and type-2 neuro-fuzzy structures (Kayacan et al., 2015a). Moreover, the stability and robustness of neuro-fuzzy controllers (NFCs) based on SMC theory-based learning algorithms can also be examined (Topalov et al., 2007a). The novelty of SMC theory-based learning algorithms is that the parameters are adjusted by the designed algorithm in order to fulfil a stable equation by imposing the error instead of minimizing an error function. On the other hand, SMC is inherently robust to uncertainties (Yin et al., 2017a,b).

Despite their practicality and easiness to design, model-free control methods – such as fuzzy control and neuro-fuzzy control – are mostly criticized by the model-based control community due to the fact that there is generally no convincing analytical proof of system stability (Feng, 2006). These criticisms are reasonable since trial-and-error methods cannot be tolerated for sophisticated and expensive robotic systems such as unmanned air vehicles and spacecraft systems. Therefore, analytical stability analysis is an overriding requirement so researchers tend to design model-based controllers (Kayacan et al., 2014, 2016; Lam, 2018; Mayne et al., 2000). With the aim of overcoming this limitation of model-free controllers, the overall system stability proven in this paper lightens the major disadvantage of model-free control methods. In previous studies in the literature, since the overall system stability of the sliding mode learning algorithm could not be proven, stability analysis was accomplished and a proportional-derivative controller was used to ensure the stability in compact space (Topalov et al., 2007b). Recently, it has been shown that if the system is a second-order system, it is possible to guarantee the stability of the overall system by adding a robust term to the control scheme (Khanesar et al., 2015). However, this stability analysis is too restrictive.

The contributions of this paper are as follows.

The stability of the overall system is proven for an nth-order uncertain nonlinear system. The overall system stability analysis removes the most significant shortcoming of model-free control methods.

A novel sliding surface is proposed to train the sliding mode learning algorithm for the type-2 neuro-fuzzy control (T2NFC) algorithm.

The learning rate of the online sliding mode learning algorithm of T2NFC and the controller gain in the conventional control term are adaptive. Therefore, it is possible to control the system without foreknowledge of the upper bounds of the system states and their derivatives.

The developed algorithm adjusts the proportion of the lower and upper membership functions (MFs) in the T2NFC, which allows us to handle non-uniform uncertainties in type-2 fuzzy logic systems (T2FLSs).

This paper consists of four sections. The SMLC structure consisting of the T2NFC and the conventional control term, and the sliding mode control theory-based learning algorithm are given in the next section. Following this, the overall system stability is discussed. Simulation results are presented to discuss control performance and noise analysis, and finally some conclusions are drawn.

Sliding model learning control structure

Control scheme

The whole control structure consists of a conventional controller and a type-2 neuro-fuzzy controller (T2NFC) as illustrated in Figure 1. The T2NFC controls the system by learning the system behaviour owing to the fact its parameters are updated by a sliding mode learning algorithm. The conventional control action is added to guarantee the stability of the overall system in compact space while the T2NFC learns the system behaviour. In this control structure, the aim is that the output of the conventional controller converges to zero in finite time while T2NFC takes the overall control signal. The total control action applied to the system u is determined as follows:

u = u_{c} + u_{n}

(1)

where $u_{c}$ and $u_{n}$ denote respectively the conventional control signal and the control signal generated by the T2NFC. The conventional control law is formulated as follows:

u_{c} = k sgn (s)

(2)

where s and k denote the sliding surface and the controller gain, respectively. The controller gain k is positive (i.e. $k > 0$ ), adaptive and updated with the following rule:

\overset{\cdot}{k} = \frac{γ_{k} | s |}{2},

(3)

where $γ_{k}$ is the coefficient of the adaptation for the controller gain and positive (i.e. $γ_{k} > 0$ ).

Figure 1.

Block diagram of the control scheme.

Type-2 neuro-fuzzy controller

An interval type-2 Takagi–Sugeno–Kang fuzzy if–then rule $R_{ij}$ is employed in this study (Minjie et al., 2018; Wang et al., 2017). The premise parts are type-2 fuzzy functions while the consequent parts are crisp numbers. The rule $R_{ij}$ is defined as below:

R_{ij} : If e is {\tilde{1}}_{i} and \overset{\cdot}{e} is {\tilde{2}}_{j}, then f_{ij} = d_{ij},

(4)

where e and $\overset{\cdot}{e}$ denote respectively the error and the error rate while ${\tilde{1}}_{i}$ and ${\tilde{2}}_{j}$ denote respectively type-2 fuzzy sets for input 1 and input 2. The function $f_{ij}$ is the output of the rules and the total number of rules is equal to $K = I \times J$ in which I and J denote respectively the numbers of membership functions used for input 1 and input 2. The detail of the proposed type-2 neuro-fuzzy system (T2NFS) is shown in Figure 2.

Figure 2.

Structure of the proposed T2NFS for two inputs.

The lower and upper Gaussian MFs for T2FLSs are written as follows:

{\underline{μ}}_{1 i} (e) = \exp (- {(\frac{e - {\underline{c}}_{1 i}}{{\underline{σ}}_{1 i}})}^{2})

(5)

{\bar{μ}}_{1 i} (e) = \exp (- {(\frac{e - {\bar{c}}_{1 i}}{{\bar{σ}}_{1 i}})}^{2})

(6)

{\underline{μ}}_{2 j} (\overset{\cdot}{e}) = \exp (- {(\frac{\overset{\cdot}{e} - {\underline{c}}_{2 j}}{{\underline{σ}}_{2 j}})}^{2})

(7)

{\bar{μ}}_{2 j} (\overset{\cdot}{e}) = \exp (- {(\frac{\overset{\cdot}{e} - {\bar{c}}_{2 j}}{{\bar{σ}}_{2 j}})}^{2})

(8)

where $\underline{c}$ and $\bar{c}$ denote resp ectively the lower and upper means the membership functions, and $\underline{σ}$ and $\bar{σ}$ the lower and upper standard deviations of the membership functions. They are adjusted for the T2NFC method throughout the simulations.

The lower and upper memberships $\underline{μ}$ and $\bar{μ}$ of fuzzy system are written for each input signal being fed to the system. The firing strengths of rules are formulated as follows:

{\underline{w}}_{ij} = {\underline{μ}}_{1 i} (e) {\underline{μ}}_{2 j} (\overset{\cdot}{e}) and {\bar{w}}_{ij} = {\bar{μ}}_{1 i} (e) {\bar{μ}}_{2 j} (\overset{\cdot}{e}) .

(9)

The control signal generated by the network is formulated below (Biglarbegian et al., 2010):

u_{n} = q \sum_{i = 1}^{I} \sum_{j = 1}^{J} f_{ij} {\tilde{\underline{w}}}_{ij} + (1 - q) \sum_{i = 1}^{I} \sum_{j = 1}^{J} f_{ij} {\tilde{\bar{w}}}_{ij},

(10)

where ${\tilde{\underline{w}}}_{ij}$ and ${\tilde{\bar{w}}}_{ij}$ are the normalized firing strengths of the lower and upper output signals of the neuron ij, and are written as follows:

{\tilde{\underline{w}}}_{ij} = \frac{{\underline{w}}_{ij}}{\sum_{i = 1}^{I} \sum_{j = 1}^{J} {\underline{w}}_{ij}} and {\tilde{\bar{w}}}_{ij} = \frac{{\bar{w}}_{ij}}{\sum_{i = 1}^{I} \sum_{j = 1}^{J} {\bar{w}}_{ij}} .

(11)

The parameter q determines the contributions of the lower and upper firing levels to the control signal and the adaptation rule for the parameter q is given in the next section.

Sliding mode learning algorithm

The sliding surface utilized in the fundamental basis of SMC theory (Utkin, 1992) is written as follows:

s = {(\frac{d}{d t} + λ)}^{n - 1} e,

(12)

where $λ$ , e and n denote respectively the slope of the sliding surface, the system error and the order of the system to be controlled. The slope of the sliding surface is a strictly positive constant (i.e. $λ > 0$ ). The error is the difference between a desired setpoint $x_{d}$ and a measured state x (i.e. $e = x_{d} - x$ ). The sliding surface is utilized to train the sliding mode learning algorithm. The T2NFC parameters are updated with the following equations:

{\underline{\overset{\cdot}{c}}}_{1 i} = \overset{\cdot}{e} + (e - {\underline{c}}_{1 i}) α sgn (s)

(13)

{\overset{\cdot}{\bar{c}}}_{1 i} = \overset{\cdot}{e} + (e - {\bar{c}}_{1 i}) α sgn (s)

(14)

{\underline{\overset{\cdot}{c}}}_{2 j} = \overset{\cdot\cdot}{e} + (\overset{\cdot}{e} - {\underline{c}}_{2 j}) α sgn (s)

(15)

{\overset{\cdot}{\bar{c}}}_{2 j} = \overset{\cdot\cdot}{e} + (\overset{\cdot}{e} - {\bar{c}}_{2 j}) α sgn (s)

(16)

{\underline{\overset{\cdot}{σ}}}_{1 i} = - {\underline{σ}}_{1 i} (1 + (\frac{{\underline{σ}}_{1 i}}{e - {\underline{c}}_{1 i}})^{2}) α sgn (s)

(17)

{\overset{\cdot}{\bar{σ}}}_{1 i} = - {\bar{σ}}_{1 i} (1 + (\frac{{\bar{σ}}_{1 i}}{e - {\bar{c}}_{1 i}})^{2}) α sgn (s)

(18)

{\underline{\overset{\cdot}{σ}}}_{2 j} = - {\underline{σ}}_{2 j} (1 + (\frac{{\underline{σ}}_{2 j}}{\overset{\cdot}{e} - {\underline{c}}_{2 j}})^{2}) α sgn (s)

(19)

{\overset{\cdot}{\bar{σ}}}_{2 j} = - {\bar{σ}}_{2 j} (1 + (\frac{{\bar{σ}}_{2 j}}{\overset{\cdot}{e} - {\bar{c}}_{2 j}})^{2}) α sgn (s)

(20)

{\overset{\cdot}{f}}_{ij} = \frac{q {\tilde{\underline{w}}}_{ij} + (1 - q) {\tilde{\bar{w}}}_{ij}}{{(q \tilde{\underline{W}} + (1 - q) \tilde{\bar{W}})}^{T} (q \tilde{\underline{W}} + (1 - q) \tilde{\bar{W}})} α sgn (s)

(21)

\overset{\cdot}{q} = \frac{1}{F {(\tilde{\underline{W}} - \tilde{\bar{W}})}^{T}} α sgn (s)

(22)

\overset{\cdot}{α} = γ_{α} | s |

(23)

where $α$ is the learning rate while $γ_{α}$ is the coefficient to update the learning rate, and they are positive (i.e. $α, γ_{α} > 0$ ). Normalized firing strength vectors $\tilde{\underline{W}} (t)$ and $\tilde{\bar{W}} (t)$ , and the consequent part vector F are defined as follows:

\begin{matrix} \tilde{\underline{W}} = [{\tilde{\underline{w}}}_{11} {\tilde{\underline{w}}}_{12} \dots {\tilde{\underline{w}}}_{21} \dots {\tilde{\underline{w}}}_{ij} \dots {\tilde{\underline{w}}}_{IJ}]^{T} \\ \tilde{\bar{W}} = [{\tilde{\bar{w}}}_{11} {\tilde{\bar{w}}}_{12} \dots {\tilde{\bar{w}}}_{21} \dots {\tilde{\bar{w}}}_{ij} \dots {\tilde{\bar{w}}}_{IJ}]^{T} \\ F = [f_{11} f_{12} \dots f_{21} \dots f_{ij} \dots f_{IJ}] \end{matrix}

where $0 < {\tilde{\underline{w}}}_{ij} \leq 1$ and $0 < {\tilde{\bar{w}}}_{ij} \leq 1$ . Moreover $\sum_{i = 1}^{I} \sum_{j = 1}^{J} {\tilde{\underline{w}}}_{ij} = 1$ and $\sum_{i = 1}^{I} \sum_{j = 1}^{J} {\tilde{\bar{w}}}_{ij} = 1$ .

Assumption 1. The input signal and its time-derivative are assumed to be bounded in a compact set:

| u | < B_{u}, | \overset{\cdot}{u} | < B_{\overset{\cdot}{u}}

(24)

where $B_{u}$ and $B_{\overset{\cdot}{u}}$ are considered as positive constants.

Theorem 2. Stability of sliding mode learning algorithm. If adaptation rules are proposed as in equations (13)–(23), the final value of the learning rate $α^{*}$ is large enough (i.e. $α^{*} > B_{\overset{\cdot}{u}}$ ) and the term, which is the product of the controller gain k and its final value $k^{*}$ , is larger than the absolute value of the sliding surface $| s |$ (i.e. $k^{*} k > | s |$ ), this ensures that the output of the conventional controller $u_{c}$ will converge to zero in finite time.

Proof. The stability of the learning algorithm is checked by using the following Lyapunov candidate function:

V = \frac{1}{2 k^{*}} u_{c}^{2} + \frac{{(k^{*})}^{2}}{2 γ_{α}} {(\frac{α}{{(k^{*})}^{2}} - α^{*})}^{2} .

(25)

The time-derivative of the aforementioned Lyapunov candidate function is calculated as follows:

\overset{\cdot}{V} = \frac{u_{c} {\overset{\cdot}{u}}_{c}}{k^{*}} + \frac{\overset{\cdot}{α}}{γ_{α}} (\frac{α}{{(k^{*})}^{2}} - α^{*}) .

(26)

If equations (1) and (23) are inserted into (26), it is obtained as:

\overset{\cdot}{V} = \frac{u_{c}}{k^{*}} (- {\overset{\cdot}{u}}_{n} + \overset{\cdot}{u}) + | s | (\frac{α}{{(k^{*})}^{2}} - α^{*}) .

(27)

${\overset{\cdot}{u}}_{n}$ in equation (57) is inserted into (27), and the time-derivative of the Lyapunov function is written as:

\overset{\cdot}{V} = \frac{u_{c}}{k^{*}} (- 2 α sgn (s) + \overset{\cdot}{u}) + | s | (\frac{α}{{(k^{*})}^{2}} - α^{*}) .

(28)

As it is stated in Assumption 1, the total input rate $\overset{\cdot}{u}$ is upper bounded by $B_{\overset{\cdot}{u}}$ , and the conventional control action in equation (2) is inserted into equation (28):

\begin{matrix} \overset{\cdot}{V} < \frac{- 2 α k}{k^{*}} + \frac{k}{k^{*}} | s | B_{\overset{\cdot}{u}} + \frac{α | s |}{{(k^{*})}^{2}} - α^{*} | s | \\ \overset{\cdot}{V} < α \underset{< 0}{\underset{︸}{(\frac{- 2 k k^{*} + | s |}{{(k^{*})}^{2}})}} + | s | \underset{< 0}{\underset{︸}{(\frac{k}{k^{*}} B_{\overset{\cdot}{u}} - α^{*})}}, \end{matrix}

(29)

where $α^{*}$ and $k^{*}$ are the final values of the learning rate and controller gain, and considered to be unknown parameters that are determined throughout the adaptations of learning rate and controller gain. If the final value of the learning rate $α^{*}$ is large enough (i.e. $α^{*} > B_{\overset{\cdot}{u}}$ ), and the term, which is the product of the controller gain k and its final value $k^{*}$ , is larger than the absolute value of the sliding surface $| s |$ (i.e. $k^{*} k > | s |$ ), then the time-derivative of the Lyapunov candidate function is negative (i.e. $\overset{\cdot}{V} < 0$ ). This purports that the sliding mode learning algorithm is stable. In other words, the output of the conventional controller $u_{c}$ converges to zero after the T2NFC algorithm learns the system.

Remark 3. As can be seen in equation (3), the controller gain only increases throughout simulations. Therefore, the final value of the controller $k^{*}$ is equal to or bigger than k. This results in $\frac{k}{k^{*}} \leq 1$ .

Remark 4. The adaptation rules for the conventional controller gain in equation (3) and learning rate in equation (23) are enforced; therefore, the final values of the controller gain and learning rate are unknown and are determined throughout the adaptations of the controller gain and learning rate, and they can reach large values to make the system and sliding mode learning algorithm stable.

Stability analysis of the overall system

An nth-order uncertain nonlinear system is described in the following form:

\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ ⋮ ⋮ ⋮ \\ {\overset{\cdot}{x}}_{n - 1} = x_{n} \\ {\overset{\cdot}{x}}_{n} = f (x) + gu + d, \end{matrix}

(30)

where $f (x)$ is the nonlinear function, g is the coefficient of the control input and positive, u is the control input, d is the disturbance and $x = [x_{1}, x_{2},, x_{n}]^{T}$ is the state vector. The error e is defined as $e = x_{d} - x_{1}$ where $x_{d}$ is the desired state.

Assumption 5. The nth-order time-derivative of the desired state, the nonlinear function, the disturbance, the output of the T2NFC and the term E, which is equal to $\overset{\cdot}{s} - \frac{d^{n} e}{d t^{n}}$ , are assumed to be bounded:

| \frac{d^{n} x_{d}}{d t^{n}} | + | f (x) | + | d | + g | u_{n} | + | E | < B,

(31)

where B is considered as a positive constant.

Theorem 6. Stability of the overall system. The system to be controlled is in the form of equation (30) and the controller in equation (1) is applied to the system. Then, if the final value of the controller gain $k^{*}$ is large enough (i.e. $k^{*} > 2 B$ ), the sliding surface s converges to zero for an arbitrary initial condition.

Proof. The stability of the overall system is checked for an nth-order uncertain nonlinear system by following the Lyapunov candidate function:

V = \frac{1}{2} s^{2} + \frac{g}{2 γ_{k}} (k - k^{*})^{2} .

(32)

The time-derivative of V is calculated as:

\overset{\cdot}{V} = s \overset{\cdot}{s} + \frac{g \overset{\cdot}{k}}{γ_{k}} (k - k^{*}) .

(33)

The sliding surface rate $\overset{\cdot}{s}$ is obtained by taking the time-derivative of the sliding surface defined in equation (12). Then, it is inserted into equation (33) as follows:

\overset{\cdot}{V} = s (\frac{d^{n} e}{d t^{n}} \underset{E}{\underset{︸}{+ \dots + λ^{n - 1} \overset{\cdot}{e})}} + \frac{g \overset{\cdot}{k}}{γ_{k}} (k - k^{*}) .

(34)

Since $E = \overset{\cdot}{s} - \frac{d^{n} e}{d t^{n}}$ , as stated in Assumption 5, and $\frac{d^{n} e}{d t^{n}} = \frac{d^{n} x_{d}}{d t^{n}} - \frac{d^{n} x}{d t^{n}}$ , it is obtained as

\overset{\cdot}{V} = s (\frac{d^{n} x_{d}}{d t^{n}} - \frac{d^{n} x}{d t^{n}} + E) + \frac{g \overset{\cdot}{k}}{γ_{k}} (k - k^{*}) .

(35)

Then, it is rewritten by taking the nonlinear system in equation (30) into consideration as follows:

\overset{\cdot}{V} = s (\frac{d^{n} x_{d}}{d t^{n}} - f (x) - d - gu + E) + \frac{g \overset{\cdot}{k}}{γ_{k}} (k - k^{*}) .

(36)

The control law in equation (1) and the adaptation for the controller gain in equation (3) are inserted into equation (36).

\overset{\cdot}{V} = s (\frac{d^{n} x_{d}}{d t^{n}} - f (x) - d - g u_{n} - g u_{c} + E) + \frac{g | s |}{2} (k - k^{*}) .

(37)

The conventional control action in equation (2) is inserted into (37):

\overset{\cdot}{V} = s (\frac{d^{n} x_{d}}{d t^{n}} - f (x) - d - g u_{n} + E - gk sgn (s)) + \frac{g | s |}{2} (k - k^{*}) .

(38)

As stated in Assumption 5, the term E, the nth-order time-derivative of the reference $\frac{d^{n} x_{d}}{d t^{n}}$ , the function $f (x)$ , the disturbance d and the output of the T2NFC $u_{n}$ are assumed to be upper bounded; the aforementioned equation (38) is obtained as follows:

\overset{\cdot}{V} < | s | (B - gk) + \frac{g | s |}{2} (k - k^{*}),

(39)

where $k^{*}$ is the final value of the controller gain and considered to be an unknown parameter that is determined throughout the adaptation of the controller gain. If the final value of the controller gain $k^{*}$ is large enough (i.e. $k^{*} > 2 B$ ), $\overset{\cdot}{V}$ yields

\overset{\cdot}{V} < - \frac{gk | s |}{2} .

(40)

Since $g, k > 0$ , this guarantees the stability of the overall system and purports that the sliding surface converges to zero in finite time.

Remark 7. Since the controller gain can reach a large value to make the system stable, it is not required to know the upper bound. This is one of the advantages of the SMLC algorithm.

Remark 8. A sliding motion appears on the sliding manifold $s = 0$ after finite time $t_{h}$ if the condition $s \overset{\cdot}{s} < 0$ is satisfied for all t in some nontrivial semi-open sub-interval of time of the form $[t, t_{h}) \subset (- \infty, t_{h})$ . Therefore, a sliding motion will be maintained on $s = 0$ for all $t > t_{h}$ .

Simulation results

The performance of the developed SMLC algorithm against an external disturbance is first shown in Scenario 1 and then analysed under noisy conditions in Scenario 2. Throughout the simulation studies, the number of membership functions of the T2NFC is set to 3 (i.e. $I = J = 3$ ) and the sampling time is set to 0.01 s. Since SMC theory suffers from high-frequency oscillations, the following function in equation (41) is used as the sign function to remove the chattering effect:

sgn (s) : = \frac{s}{| s | + χ},

(41)

with $χ = 0.05$ .

To avoid division by zero in the adaptation laws of equations (13)–(22), an instruction is included in the algorithm to make the denominator equal to 0.001 when its calculated value is smaller than this threshold. Moreover, the controller gain of the conventional controller and learning rate of the learning algorithm must not have infinite values under noisy conditions; therefore, a dead zone is employed. If the sliding surface is within the bounds of the dead-zone parameter $ε = 0.001$ (i.e. $| s | < ε$ ), then these parameters are not updated. Furthermore, the reference must be changing smoothly and there must exist no discontinuous disturbance to avoid infinite values for the controller gain and learning rate.

Scenario 1: control performance against external disturbance

To evaluate control performance against disturbances, the developed SMLC algorithm is applied to an adaptive cruise control system, which is illustrated in Figure 3; the control structure is shown in Figure 4. The equations of motion for the longitudinal vehicle dynamics are written as follows (Kayacan, 2017; Stankovic et al., 2000):

\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = x_{3} \\ {\overset{\cdot}{x}}_{3} = - 2 \frac{k_{a}}{m} x_{2} x_{3} - \frac{1}{τ} [x_{3} + \frac{k_{a}}{m} x_{2}^{2}] + \frac{u}{m τ} + d, \end{matrix}

(42)

where $x_{1}$ , $x_{2}$ , $x_{3}$ , m, $k_{a}$ , $τ$ , u and d represent respectively the position, velocity and acceleration, the mass of the vehicle, the aerodynamic drag coefficient, the engine time lag, the throttle command input and the external disturbance. The numerical values for the parameters in this study are $m = 9$ kg, $k_{a} = 0.26$ , $τ = 0.1$ (Mehra et al., 2015).

Figure 3.

Schematic diagram of the adaptive cruise control system.

Figure 4.

Control structure of the adaptive cruise control system.

The time–headway spacing policy to follow the preceding vehicle with the desired relative distance is formulated as follows:

z = h x_{2},

(43)

where z is the desired relative inter-vehicle distance and h is the desired time headway. The spacing policy is called the constant time–headway time policy, which aims for a constant inter-vehicle time gap. The position reference in this study is considered as the position of the preceding vehicle. The relative spacing error is defined by taking the spacing policy in equation (43) into account as follows:

e = x_{d} - x_{1} - h x_{2} .

(44)

The sliding surface for the adaptive cruise control system is second-order, since the adaptive cruise control system is a third-order system. The slope of the sliding surface $λ$ to update the T2NFC is set to 1; therefore, the sliding surface becomes $s = \overset{\cdot\cdot}{e} + 2 \overset{\cdot}{e} + e$ as formulated in equation (12). The coefficients $γ_{k}$ and $γ_{α}$ for the adaptations of the controller gain in equation (3) and the learning rate in equation (23) are set to 0.1. The initial conditions on the states of the system are considered as $[x_{1}, x_{2},, x_{3}] = [0, 0, 0]^{T}$ . An external disturbance is imposed in the vehicle dynamics as follows:

d (t) = 1 + 0.25 \sin (t) .

(45)

The test trajectory is defined by the position reference $x_{d} (t)$ , which consists of ramp signals as follows:

x_{d} (t) = {\begin{matrix} 0 \leq t < 20 & x_{d} = t \\ 20 \leq t < 40 & x_{d} = t + \frac{0.05}{2} {(t - 20)}^{2} \\ 40 \leq t < 40 & x_{d} = t + \frac{0.05}{2} {(t - 20)}^{2} - \frac{0.05}{2} {(t - 40)}^{2} . \end{matrix}

(46)

The position, speed and acceleration responses of the vehicle are respectively shown in Figures 5(a)–5(c). The developed SMLC algorithm can learn the system behaviour online in finite time so that it exhibits robust control performance against the external disturbance. As the error is shown in Figure 5(d), it controls the system without any steady-state error.

Figure 5.

Control performance on an adaptive cruise control system: (a) position; (b) speed; (c) acceleration; (d) error; (e) control signals; (f) adaptation of the controller gain; (g) adaptation of the learning rate; (h) adaptation of parameter q.

The control signals are shown in Figure 5(e). The generated control signal by the conventional control action $u_{c}$ converges to zero while the generated control signal by the T2NFC $u_{n}$ takes the overall control signal. Thus, the T2NFC becomes the leading controller after a short time period. The output of the conventional controller $u_{c}$ becomes non-zero only at the beginning of the simulations while the T2NFC is learning system behaviour.

The adaptations of the controller gain k, the learning rate $α$ and the parameter q are presented in Figures 5(f)–5(h). The parameter q is generally set to 0.5 in T2NFCs, while the controller gain and learning rate are chosen arbitrarily in this study. Therefore, the initial conditions of these parameters are set to $k (0) = 1$ , $α (0) = 3$ and $q (0) = 0.5$ . Thanks to these adaptations, the controller gain, the learning rate and the proportions of the upper and lower membership functions are updated throughout the simulations. The adaptations of the controller gain k and learning rate $α$ converge to a certain value since (1) the reference is changing smoothly, (2) there is no discontinuous disturbance on the system and (3) the dead-zone parameter is not lower than the noise level in real-time applications.

In the literature, disturbance observer-based control approaches have been generally implemented for uncertain systems (Liu et al., 2017b). In these approaches, unmodelled dynamics are considered as an external disturbance, and a control law including the disturbance estimate is derived. In our work, there is no need for such an observer due to the fact that the proposed algorithm learns system behaviour. This is one of the advantages of the developed algorithm in this paper.

Scenario 2: noise analysis

To evaluate control performance under noisy conditions, the developed SMLC algorithm is applied to the following numerical system (Yang et al., 2013):

\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = - 2 x_{1} - x_{2} + e^{x_{1}} + u . \end{matrix}

(47)

The initial conditions of the system states are set to $x (0) = [1, - 1]^{T}$ . The slope of the sliding surface $λ$ is set to 2. The coefficients $γ_{k}$ and $γ_{α}$ for the adaptations of the controller gain in equation (3) and the learning rate in equation (23) are respectively set to 1 and 0.1.

In order to test the robustness of this control approach, the states $x_{1}$ and $x_{2}$ are measured with a noise level signal-to-noise ratio (SNR) = 50 dB. The states’ responses are shown in Figures 6(a) and 6(b). The T2NFC can learn system behaviour online and thus regulates the system without any steady-state error. Thanks to the proposed T2NFC structure, our control algorithm ensures robust control performance against noisy measurements.

The control signals are shown in Figure 6(c). As seen in this figure, the generated control signal by the robust control action $u_{c}$ converges to zero while the generated control signal by the T2NFC $u_{n}$ is different from zero. It is concluded that the T2NFC can take over the overall control of the system in a very short time period.

Figure 6.

Control performance under noisy measurements: (a) state $x_{1}$ ; (b) state $x_{2}$ ; (c) control signals; (d) adaptation of the controller gain; (e) adaptation of the learning rate; (f) adaptation of parameter q.

The adaptations of the controller gain k, the learning rate $α$ and the parameter q are presented in Figures 6(d)–6(f). The initial conditions of these parameters are considered as $k (0) = 1$ , $α (0) = 0.03$ and $q (0) = 0.5$ . Thanks to these adaptations, the controller gain, the learning rate and the proportions of the upper and lower membership functions are updated throughout the simulations. Moreover, the controller gain converges to a certain value under noisy conditions owing to the dead zone.

Conclusion

The developed SMLC algorithm for uncertain nonlinear systems has been investigated in this paper. In addition to the stability of the training algorithm, the analytical proof of the overall system stability has been achieved for nth-order uncertain nonlinear systems. The simulation results show that the developed control structure exhibits robust control performance in the presence of external disturbances and noisy measurements. The parameters of the T2NFC controller are automatically regulated through the sliding mode learning algorithm so that the T2NFC can learn the system behaviour and take overall control of systems in a very short time period. The proposed sliding mode learning algorithm is valid for only single-input systems. As a future work, the extension of the proposed learning algorithm for multi-input systems is an interesting topic to investigate.

Footnotes

Appendix: the time-derivative of the output of the T2NFC u · n

The following equations are obtained by taking the time-derivative of equations (5)–(8):

(48)

{\underline{\overset{\cdot}{μ}}}_{1 i} (x_{1}) = - 2 {\underline{N}}_{1 i} {\underline{\overset{\cdot}{N}}}_{1 i} {\underline{μ}}_{1 i} (x_{1}), {\overset{\cdot}{\bar{μ}}}_{1 i} (x_{1}) = - 2 {\bar{N}}_{1 i} {\overset{\cdot}{\bar{N}}}_{1 i} {\bar{μ}}_{1 i} (x_{1})

(49)

{\underline{\overset{\cdot}{μ}}}_{2 j} (x_{2}) = - 2 {\underline{N}}_{2 j} {\underline{\overset{\cdot}{N}}}_{2 j} {\underline{μ}}_{2 j} (x_{2}), {\overset{\cdot}{\bar{μ}}}_{2 j} (x_{2}) = - 2 {\bar{N}}_{2 j} {\overset{\cdot}{\bar{N}}}_{2 j} {\bar{μ}}_{2 j} (x_{2}),

where ${\underline{N}}_{1 i} = (\frac{x_{1} - {\underline{c}}_{1 i}}{{\underline{σ}}_{1 i}})$ , ${\bar{N}}_{1 i} = (\frac{x_{1} - {\bar{c}}_{1 i}}{{\bar{σ}}_{1 i}})$ , ${\underline{N}}_{2 j} = (\frac{x_{2} - {\underline{c}}_{2 j}}{{\underline{σ}}_{2 j}})$ , and ${\bar{N}}_{2 j} = (\frac{x_{2} - {\bar{c}}_{2 j}}{{\bar{σ}}_{2 j}})$ .

By taking the time-derivative of equation (11), the following equations are obtained as follows:

(50)

\begin{matrix} {\underline{\overset{\cdot}{\tilde{w}}}}_{ij} = \frac{({\underline{μ}}_{1 i} (x_{1}) {\underline{μ}}_{2 j} (x_{2}))' (\sum_{i = 1}^{I} \sum_{j = 1}^{J} {\underline{w}}_{ij})}{(\sum_{i = 1}^{I} \sum_{j = 1}^{J} {\underline{w}}_{ij})^{2}} - \frac{({\underline{w}}_{ij}) (\sum_{i = 1}^{I} \sum_{j = 1}^{J} {\underline{μ}}_{1 i} (x_{1}) {\underline{μ}}_{2 j} (x_{2}))'}{(\sum_{i = 1}^{I} \sum_{j = 1}^{J} {\underline{w}}_{ij})^{2}} \\ {\overset{\cdot}{\tilde{\bar{w}}}}_{ij} = \frac{({\bar{μ}}_{1 i} (x_{1}) {\bar{μ}}_{2 j} (x_{2}))' (\sum_{i = 1}^{I} \sum_{j = 1}^{J} {\underline{w}}_{ij})}{(\sum_{i = 1}^{I} \sum_{j = 1}^{J} {\bar{w}}_{ij})^{2}} - \frac{({\bar{w}}_{ij}) (\sum_{i = 1}^{I} \sum_{j = 1}^{J} {\bar{μ}}_{1 i} (x_{1}) {\bar{μ}}_{2 j} (x_{2}))'}{(\sum_{i = 1}^{I} \sum_{j = 1}^{J} {\bar{w}}_{ij})^{2}} . \end{matrix}

Since ${\tilde{\underline{w}}}_{ij} = \frac{{\underline{w}}_{ij}}{\sum_{i = 1}^{I} \sum_{j = 1}^{J} {\underline{w}}_{ij}}$ ,

(51)

\begin{matrix} {\underline{\overset{\cdot}{\tilde{w}}}}_{ij} = \frac{({\underline{\overset{\cdot}{μ}}}_{1 i} (x_{1}) {\underline{μ}}_{2 j} (x_{2}) + {\underline{μ}}_{1 i} (x_{1}) {\underline{\overset{\cdot}{μ}}}_{2 j} (x_{2}))}{(\sum_{i = 1}^{I} \sum_{j = 1}^{J} {\underline{w}}_{ij})} - \frac{({\tilde{\underline{w}}}_{ij}) (\sum_{i = 1}^{I} \sum_{j = 1}^{J} ({\underline{\overset{\cdot}{μ}}}_{1 i} (x_{1}) {\underline{μ}}_{2 j} (x_{2}) + {\underline{μ}}_{1 i} (x_{1}) {\underline{\overset{\cdot}{μ}}}_{2 j} (x_{2}))}{(\sum_{i = 1}^{I} \sum_{j = 1}^{J} {\underline{w}}_{ij})} \\ = \frac{(- 2 {\underline{N}}_{1 i} {\underline{\overset{\cdot}{N}}}_{1 i} {\underline{μ}}_{1 i} (x_{1}) {\underline{μ}}_{2 j} (x_{2}) - 2 {\underline{N}}_{2 j} {\underline{\overset{\cdot}{N}}}_{2 j} {\underline{μ}}_{1 i} (x_{1}) {\underline{μ}}_{2 j} (x_{2}))}{(\sum_{i = 1}^{I} \sum_{j = 1}^{J} {\underline{w}}_{ij})} - \frac{({\tilde{\underline{w}}}_{ij}) \sum_{i = 1}^{I} \sum_{j = 1}^{J} (- 2 {\underline{N}}_{1 i} {\underline{\overset{\cdot}{N}}}_{1 i} {\underline{μ}}_{1 i} (x_{1}) {\underline{μ}}_{2 j} (x_{2}))}{(\sum_{i = 1}^{I} \sum_{j = 1}^{J} {\underline{w}}_{ij})} \\ - \frac{({\tilde{\underline{w}}}_{ij}) \sum_{i = 1}^{I} \sum_{j = 1}^{J} (- 2 {\underline{N}}_{2 j} {\underline{\overset{\cdot}{N}}}_{2 j} {\underline{μ}}_{1 i} (x_{1}) {\underline{μ}}_{2 j} (x_{2}))}{(\sum_{i = 1}^{I} \sum_{j = 1}^{J} {\underline{w}}_{ij})} \\ = - {\tilde{\underline{w}}}_{ij} {\underline{\overset{\cdot}{K}}}_{ij} + {\tilde{\underline{w}}}_{ij} \sum_{i = 1}^{I} \sum_{j = 1}^{J} {\tilde{\underline{w}}}_{ij} {\underline{\overset{\cdot}{K}}}_{ij}, \end{matrix}

where

{\underline{\overset{\cdot}{K}}}_{ij} = 2 ({\underline{N}}_{1 i} {\underline{\overset{\cdot}{N}}}_{1 i} + {\underline{N}}_{2 j} {\underline{\overset{\cdot}{N}}}_{2 j}) .

In a similar way, ${\overset{\cdot}{\tilde{\bar{w}}}}_{ij}$ obtained:

(52)

{\overset{\cdot}{\tilde{\bar{w}}}}_{ij} = - {\tilde{\bar{w}}}_{ij} {\overset{\cdot}{\bar{K}}}_{ij} + {\tilde{\bar{w}}}_{ij} \sum_{i = 1}^{I} \sum_{j = 1}^{J} {\tilde{\bar{w}}}_{ij} {\overset{\cdot}{\bar{K}}}_{ij}

where

{\overset{\cdot}{\bar{K}}}_{ij} = 2 ({\bar{N}}_{1 i} {\overset{\cdot}{\bar{N}}}_{1 i} + {\bar{N}}_{2 j} {\overset{\cdot}{\bar{N}}}_{2 j})

The time-derivative of the output of the T2NFC algorithm is obtained as:

(53)

{\overset{\cdot}{u}}_{n} = q \sum_{i = 1}^{I} \sum_{j = 1}^{J} ({\overset{\cdot}{f}}_{ij} {\tilde{\underline{w}}}_{ij} + f_{ij} {\underline{\overset{\cdot}{\tilde{w}}}}_{ij}) + (1 - q) \sum_{i = 1}^{I} \sum_{j = 1}^{J} ({\overset{\cdot}{f}}_{ij} {\tilde{\bar{w}}}_{ij} + f_{ij} {\overset{\cdot}{\tilde{\bar{w}}}}_{ij}) + \overset{\cdot}{q} \sum_{i = 1}^{I} \sum_{j = 1}^{J} f_{ij} {\tilde{\underline{w}}}_{ij} - \overset{\cdot}{q} \sum_{i = 1}^{I} \sum_{j = 1}^{J} f_{ij} {\tilde{\bar{w}}}_{ij} .

If equations (51) and (52) are inserted into the previous equation:

(54)

\begin{matrix} {\overset{\cdot}{u}}_{n} = q \sum_{i = 1}^{I} \sum_{j = 1}^{J} ((- {\tilde{\underline{w}}}_{ij} {\underline{\overset{\cdot}{K}}}_{ij} + {\tilde{\underline{w}}}_{ij} \sum_{i = 1}^{I} \sum_{j = 1}^{J} {\tilde{\underline{w}}}_{ij} {\underline{\overset{\cdot}{K}}}_{ij}) f_{ij} + {\tilde{\underline{w}}}_{ij} {\overset{\cdot}{f}}_{ij}) \\ + (1 - q) \sum_{i = 1}^{I} \sum_{j = 1}^{J} ((- {\tilde{\bar{w}}}_{ij} {\overset{\cdot}{\bar{K}}}_{ij} + {\tilde{\bar{w}}}_{ij} \sum_{i = 1}^{I} \sum_{j = 1}^{J} {\tilde{\bar{w}}}_{ij} {\overset{\cdot}{\bar{K}}}_{ij}) f_{ij} + {\tilde{\bar{w}}}_{ij} {\overset{\cdot}{f}}_{ij}) + \overset{\cdot}{q} \sum_{i = 1}^{I} \sum_{j = 1}^{J} f_{ij} ({\tilde{\underline{w}}}_{ij} - {\tilde{\bar{w}}}_{ij}) \\ = q \sum_{i = 1}^{I} \sum_{j = 1}^{J} ((- {\tilde{\underline{w}}}_{ij} (2 {\underline{N}}_{1 i} {\underline{\overset{\cdot}{N}}}_{1 i} + 2 {\underline{N}}_{2 j} {\underline{\overset{\cdot}{N}}}_{2 j}) + {\tilde{\underline{w}}}_{ij} \sum_{i = 1}^{I} \sum_{j = 1}^{J} {\tilde{\underline{w}}}_{ij} (2 {\underline{N}}_{1 i} {\underline{\overset{\cdot}{N}}}_{1 i} + 2 {\underline{N}}_{2 j} {\underline{\overset{\cdot}{N}}}_{2 j})) f_{ij} + {\tilde{\underline{w}}}_{ij} {\overset{\cdot}{f}}_{ij}) \\ + (1 - q) \sum_{i = 1}^{I} \sum_{j = 1}^{J} ((- {\tilde{\bar{w}}}_{ij} (2 {\bar{N}}_{1 i} {\overset{\cdot}{\bar{N}}}_{1 i} + 2 {\bar{N}}_{2 j} {\overset{\cdot}{\bar{N}}}_{2 j}) + {\tilde{\bar{w}}}_{ij} \sum_{i = 1}^{I} \sum_{j = 1}^{J} {\tilde{\bar{w}}}_{ij} (2 {\bar{N}}_{1 i} {\overset{\cdot}{\bar{N}}}_{1 i} + 2 {\bar{N}}_{2 j} {\overset{\cdot}{\bar{N}}}_{2 j})) f_{ij} + {\tilde{\bar{w}}}_{ij} {\overset{\cdot}{f}}_{ij}) \\ + \overset{\cdot}{q} \sum_{i = 1}^{I} \sum_{j = 1}^{J} f_{ij} ({\tilde{\underline{w}}}_{ij} - {\tilde{\bar{w}}}_{ij}), \end{matrix}

where

\begin{matrix} {\underline{\overset{\cdot}{N}}}_{1 i} = \frac{({\overset{\cdot}{x}}_{1} - {\underline{\overset{\cdot}{c}}}_{1 i}) {\underline{σ}}_{1 i} - (x_{1} - {\underline{c}}_{1 i}) {\underline{\overset{\cdot}{σ}}}_{1 i}}{{\underline{σ}}_{1 i}^{2}}, {\underline{\overset{\cdot}{N}}}_{2 i} = \frac{({\overset{\cdot}{x}}_{2} - {\underline{\overset{\cdot}{c}}}_{2 i}) {\underline{σ}}_{2 i} - (x_{2} - {\underline{c}}_{2 i}) {\underline{\overset{\cdot}{σ}}}_{2 i}}{{\underline{σ}}_{2 i}^{2}} \\ {\overset{\cdot}{\bar{N}}}_{1 i} = \frac{({\overset{\cdot}{x}}_{1} - {\overset{\cdot}{\bar{c}}}_{1 i}) {\bar{σ}}_{1 i} - (x_{1} - {\bar{c}}_{1 i}) {\overset{\cdot}{\bar{σ}}}_{1 i}}{{\bar{σ}}_{1 i}^{2}}, {\overset{\cdot}{\bar{N}}}_{2 i} = \frac{({\overset{\cdot}{x}}_{2} - {\overset{\cdot}{\bar{c}}}_{2 i}) {\bar{σ}}_{2 i} - (x_{2} - {\bar{c}}_{2 i}) {\overset{\cdot}{\bar{σ}}}_{2 i}}{{\bar{σ}}_{2 i}^{2}} . \end{matrix}

Equation (55) can be obtained as follows:

(55)

{\underline{N}}_{1 i} {\underline{\overset{\cdot}{N}}}_{1 i} = {\underline{N}}_{2 i} {\underline{\overset{\cdot}{N}}}_{2 i} = {\bar{N}}_{1 i} {\overset{\cdot}{\bar{N}}}_{1 i} = {\bar{N}}_{2 i} {\overset{\cdot}{\bar{N}}}_{2 i} = α sgn (s) .

Equation (55) is inserted into (54):

(56)

\begin{matrix} {\overset{\cdot}{u}}_{n} = q \sum_{i = 1}^{I} \sum_{j = 1}^{J} (2 (- {\tilde{\underline{w}}}_{ij} f_{ij} 2 α sgn (s) + {\tilde{\underline{w}}}_{ij} f_{ij} \sum_{i = 1}^{I} \sum_{j = 1}^{J} {\tilde{\underline{w}}}_{ij} 2 α sgn (s)) + {\tilde{\underline{w}}}_{ij} {\overset{\cdot}{f}}_{ij}) \\ + (1 - q) \sum_{i = 1}^{I} \sum_{j = 1}^{J} (2 (- {\tilde{\bar{w}}}_{ij} f_{ij} 2 α sgn (s) + {\tilde{\bar{w}}}_{ij} f_{ij} \sum_{i = 1}^{I} \sum_{j = 1}^{J} {\tilde{\bar{w}}}_{ij} 2 α sgn (s)) {\tilde{\bar{w}}}_{ij} {\overset{\cdot}{f}}_{ij}) + \overset{\cdot}{q} \sum_{i = 1}^{I} \sum_{j = 1}^{J} f_{ij} ({\tilde{\underline{w}}}_{ij} - {\tilde{\bar{w}}}_{ij}) \\ = - 4 q α sgn (s) \sum_{i = 1}^{I} \sum_{j = 1}^{J} ({\tilde{\underline{w}}}_{ij} f_{ij} - {\tilde{\underline{w}}}_{ij} f_{ij} \sum_{i = 1}^{I} \sum_{j = 1}^{J} {\tilde{\underline{w}}}_{ij}) - 4 (1 - q) α sgn (s) \sum_{i = 1}^{I} \sum_{j = 1}^{J} ({\tilde{\bar{w}}}_{ij} f_{ij} - {\tilde{\bar{w}}}_{ij} f_{ij} \sum_{i = 1}^{I} \sum_{j = 1}^{J} {\tilde{\bar{w}}}_{ij}) \\ + q \sum_{i = 1}^{I} \sum_{j = 1}^{J} {\tilde{\underline{w}}}_{ij} {\overset{\cdot}{f}}_{ij} + (1 - q) \sum_{i = 1}^{I} \sum_{j = 1}^{J} {\tilde{\bar{w}}}_{ij} {\overset{\cdot}{f}}_{ij} + \overset{\cdot}{q} \sum_{i = 1}^{I} \sum_{j = 1}^{J} f_{ij} ({\tilde{\underline{w}}}_{ij} - {\tilde{\bar{w}}}_{ij}) \end{matrix}

Since $\sum_{i = 1}^{I} \sum_{j = 1}^{J} {\tilde{\bar{w}}}_{ij} = 1$ and $\sum_{i = 1}^{I} \sum_{j = 1}^{J} {\tilde{\underline{w}}}_{ij} = 1$ , ${\overset{\cdot}{u}}_{n}$ becomes, by using equations (21) and (22), as below:

(57)

\begin{matrix} {\overset{\cdot}{u}}_{n} = q \sum_{i = 1}^{I} \sum_{j = 1}^{J} {\tilde{\underline{w}}}_{ij} {\overset{\cdot}{f}}_{ij} + (1 - q) \sum_{i = 1}^{I} \sum_{j = 1}^{J} {\tilde{\bar{w}}}_{ij} {\overset{\cdot}{f}}_{ij} + \overset{\cdot}{q} \sum_{i = 1}^{I} \sum_{j = 1}^{J} f_{ij} ({\tilde{\underline{w}}}_{ij} - {\tilde{\bar{w}}}_{ij}) \\ = 2 α sgn (s) \end{matrix}

References

Biglarbegian

Melek

Mendel

(2010) On the stability of interval type-2 TSK fuzzy logic control systems. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics) 40(3): 798–818.

Castillo

Castro

Melin

et al . (2014) Application of interval type-2 fuzzy neural networks in non-linear identification and time series prediction. Soft Computing 18(6): 1213–1224.

Castillo

Martínez-Marroquín

Melin

et al . (2012) Comparative study of bio-inspired algorithms applied to the optimization of type-1 and type-2 fuzzy controllers for an autonomous mobile robot. Information Sciences 192: 19–38.

Chen

Yang

Guo

et al . (2016) Disturbance-observer-based control and related methods: an overview. IEEE Transactions on Industrial Electronics 63(2): 1083–1095.

Efe

Kaynak

(2000) Sliding mode control of a three degrees of freedom anthropoid robot by driving the controller parameters to an equivalent regime. ASME Journal of Dynamic Systems, Measurement, and Control 122(4): 632–640.

Feng

(2006) A survey on analysis and design of model-based fuzzy control systems. IEEE Transactions on Fuzzy Systems 14(5): 676–697.

Gomi

Kawato

(1993) Neural network control for a closed-loop system using feedback-error-learning. Neural Networks 6(7): 933–946.

Huang

Liu

et al . (2015) Nonlinear disturbance observer-based dynamic surface control of mobile wheeled inverted pendulum. IEEE Transactions on Control Systems Technology 23(6): 2400–2407.

Kalanovic

Popovic

Skaug

(2000) Feedback error learning neural network for trans-femoral prosthesis. IEEE Transactions on Rehabilitation Engineering 8(1): 71–80.

10.

Kayacan

(2017) Multiobjective h_∞ control for string stability of cooperative adaptive cruise control systems. IEEE Transactions on Intelligent Vehicles 2(1): 52–61.

11.

Kayacan

Fossen

(2019) Feedback linearization control for systems with mismatched uncertainties via disturbance observers. Asian Journal of Control 21(4): 1–13.

12.

Kayacan

Peschel

(2016) Robust model predictive control of systems by modeling mismatched uncertainty. IFAC-PapersOnLine 49(18): 265–269.

13.

Kayacan

Chen

et al . (2018) On the Comparison of Model-Based and Model-Free Controllers in Guidance, Navigation and Control of Agricultural Vehicles. Cham: Springer.

14.

Kayacan

Khanesar

(2015a) Identification of nonlinear dynamic systems using type-2 fuzzy neural networks: a novel learning algorithm and a comparative study. IEEE Transactions on Industrial Electronics 62(3): 1716–1724.

15.

Kayacan

Ramon

et al . (2012) Velocity control of a spherical rolling robot using a grey-PID type fuzzy controller with an adaptive step size. IFAC Proceedings Volumes 45(22): 863–868.

16.

Kayacan

Ramon

et al . (2013a) Adaptive neuro-fuzzy control of a spherical rolling robot using sliding-mode-control-theory-based online learning algorithm. IEEE Transactions on Cybernetics 43(1): 170–179.

17.

Kayacan

Ramon

et al . (2013b) Modeling and identification of the yaw dynamics of an autonomous tractor. In: 2013 9th Asian Control Conference (ASCC), Istanbul, Turkey, 23–26 June 2013, pp. 1–6.

18.

Kayacan

Ramon

et al . (2014) Nonlinear modeling and identification of an autonomous tractor-trailer system. Computers and Electronics in Agriculture 106(Supplement C): 1–10.

19.

Kayacan

Ramon

et al . (2015b) Towards agrobots: trajectory control of an autonomous tractor using type-2 fuzzy logic controllers. IEEE/ASME Transactions on Mechatronics 20(1): 287–298.

20.

Kayacan

Peschel

Chowdhary

(2017) A self-learning disturbance observer for nonlinear systems in feedback-error learning scheme. Engineering Applications of Artificial Intelligence 62(Supplement C): 276–285.

21.

Kayacan

Peschel

Kayacan

(2016) Centralized, decentralized and distributed nonlinear model predictive control of a tractor-trailer system: a comparative study. In: 2016 American control conference, Boston, MA, 6–8 July 2016, pp. 4403–4408.

22.

Khanesar

Kayacan

Reyhanoglu

et al . (2015) Feedback error learning control of magnetic satellites using type-2 fuzzy neural networks with elliptic membership functions. IEEE Transactions on Cybernetics 45(4): 858–868.

23.

Lam

(2018) A review on stability analysis of continuous-time fuzzy-model-based control systems: from membership-function-independent to membership-function-dependent analysis. Engineering Applications of Artificial Intelligence 67(Supplement C): 390–408.

24.

Lee

Wang

et al . (2015) T2fs-based adaptive linguistic assessment system for semantic analysis and human performance evaluation on game of Go. IEEE Transactions on Fuzzy Systems 23(2): 400–420.

25.

Liu

Zhao

Zuo

et al . (2017a) Robust control for quadrotors with multiple time-varying uncertainties and delays. IEEE Transactions on Industrial Electronics 64(2): 1303–1312.

26.

Liu

Vazquez

et al . (2017b) Extended state observer-based sliding-mode control for three-phase power converters. IEEE Transactions on Industrial Electronics 64(1): 22–31.

27.

Liu

Wang

et al . (2017c) Reliable filter design for sensor networks using type-2 fuzzy framework. IEEE Transactions on Industrial Informatics 13(4): 1742–1752.

28.

Maldonado

Castillo

Melin

(2013) Particle swarm optimization of interval type-2 fuzzy systems for FPGA applications. Applied Soft Computing 13(1): 496–508.

29.

Mayne

Rawlings

Rao

et al . (2000) Constrained model predictive control: stability and optimality. Automatica 36(6): 789–814.

30.

Mehra

Berg

et al . (2015) Adaptive cruise control: experimental validation of advanced controllers on scale-model cars. In: 2015 American control conference, Chicago, IL, 1–3 July 2015, pp. 1411–1418.

31.

Minjie

Yujie

Shenhua

et al . (2018) Sampled-data control for nonlinear singular systems based on a Takagi–Sugeno fuzzy model. Transactions of the Institute of Measurement and Control. Epub before print 24 January 2018. DOI: 10.1177/0142331217740420.

32.

Muhuri

Shukla

(2017) Semi-elliptic membership function: representation, generation, operations, defuzzification, ranking and its application to the real-time task scheduling problem. Engineering Applications of Artificial Intelligence 60(Supplement C): 71–82.

33.

Ren

Zhong

Dai

(2017) Asymptotic reference tracking and disturbance rejection of UDE-based robust control. IEEE Transactions on Industrial Electronics 64(4): 3166–3176.

34.

Ruan

Ding

Gong

et al . (2007) On-line adaptive control for inverted pendulum balancing based on feedback-error-learning. Neurocomputing 70(4–6): 770–776.

35.

Rubio-Solis

Panoutsos

(2015) Interval type-2 radial basis function neural network: a modeling framework. IEEE Transactions on Fuzzy Systems 23(2): 457–473.

36.

Stankovic

Stanojevic

Siljak

(2000) Decentralized overlapping control of a platoon of vehicles. IEEE Transactions on Control Systems Technology 8(5): 816–832.

37.

Sun

Zheng

Man

et al . (2016) Robust control of a vehicle steer-by-wire system using adaptive sliding mode. IEEE Transactions on Industrial Electronics 63(4): 2251–2262.

38.

Topalov

Cascella

Giordano

et al . (2007a) Sliding mode neuro-adaptive control of electric drives. IEEE Transactions on Industrial Electronics 54(1): 671–679.

39.

Topalov

Kaynak

Aydin

(2007b) Neuro-adaptive sliding-mode tracking control of robot manipulators. International Journal of Adaptive Control and Signal Processing 21(8–9): 674–691.

40.

Utkin

(1992) Sliding Modes in Control Optimization. Berlin: Springer-Verlag.

41.

Wang

Chen

Zhang

et al . (2017) Application of Takagi–Sugeno fuzzy model optimized with an improved free search algorithm to industrial polypropylene melt index prediction. Transactions of the Institute of Measurement and Control 39(11): 1613–1622.

42.

Yang

(2013) Sliding-mode control for systems with mismatched uncertainties via a disturbance observer. IEEE Transactions on Industrial Electronics 60(1): 160–169.

43.

Yin

Gao

Qiu

et al . (2017a) Descriptor reduced-order sliding mode observers design for switched systems with sensor and actuator faults. Automatica 76: 282–292.

44.

Yin

Yang

Kaynak

(2017b) Sliding mode observer-based FTC for Markovian jump systems with actuator and sensor faults. IEEE Transactions on Automatic Control 62(7): 3551–3558.

45.

Zaheer

Choi

Jung

et al . (2015) A modular implementation scheme for nonsingleton type-2 fuzzy logic systems with input uncertainties. IEEE/ASME Transactions on Mechatronics 20(6): 3182–3193.