# Output Feedback Recursive Dynamic Surface Control with Antiwindup Compensation

Authors: Guofa Sun; Hui Du; Gang Wang; Hanbo Yu
Journal: Complexity
 (2021)

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## Abstract

Actuator saturation phenomenon often exists in the actual control system, which could destroy the closed-loop performance of the system and even lead to unstable behavior. Our main contribution is to provide an antiwindup recursive dynamic surface control (RDSC) for a discrete-time system with an unknown state and actuator saturation. The fuzzy compensator is added to perform as an active disturbance rejection term in the feedforward path to avoid windup caused by input saturation. To construct output feedback control, the system is transformed into the form of pure-feedback and an improved HOSM observer is designed to carry out future output prediction. Based on which RDSC is synthesized, only one fuzzy logic system (FLS) is used and the controller singularity is completely avoided. In addition, the simulation and numeral examples using a continuous stirred tank reactors (CSTRs) system with actuator saturation are provided and the results show that the strategy owns good robustness and effectively compensates for the disturbance caused by actuator saturation in the presence of a discrete-time system with an unmeasurable state.


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## Body

## 1. Introduction

Considering that actuators in the actual system usually have an inherent physical input limitation, a further increase of the control signal will not lead to a faster response of the controlled system. This phenomenon will greatly deteriorate the output performance of the system, produce unacceptable errors, and even cause instability of the system. For example, the jumping robot system is composed of a rotatable leg drive structure, which usually needs to be restrained during operation to avoid structural damage. The servomotor platform also has the same problem, which needs to constrain the actuator to ensure the safety of driving. The problem of input saturation has been studied by many researchers [1–4]. Rehman et al. [1] investigated antiwindup problems for nonlinear multiagent systems subject to actuator saturation; the sector-based approach is designed to maximize the region of stability and ensuring the operation of a control signal in the nonlinear domain. However, Zuo et al. [2] designed an antiwindup compensation for the stability and performance degradation of saturated systems and discussed a codesign technique about calculating the antiwindup gain. In order to solve the problems of actuator saturation in practical engineering, there is a huge amount of research literature that have adopted saturation compensation and saturation suppression, for example, jelastic joint robot limitation [5], oversaturation of wind turbines [6], and reactant concentration in the reactor [7]. Input saturation widely exists in the control system; extra effort should be made to avoid windup for improving output control performance.In some recently reported works, many adaptive controllers have been developed for compensating actuator nonlinearity, including deadzone, backlash, and hysteresis [8–10]. For instance, Dinh [8] introduced a mathematical model for backlash hysteresis compensation about wearable exosuit for upper limb assistance, which could accurately track the reference signal by compensating for time-varying backlash and continuously updating the model parameters. Then, Turner and Kerr [9] proposed a nonlinear modification which can be retrofitted to a dynamic antiwindup compensator. This paper will derive an adaptive dynamic surface control scheme for compensating windup caused by actuator saturation.In view of the above problems, some progress has been made in the research of dynamic surface control. Zhang [11] developed multidimensional Taylor network (MTN) to approach the unknown function, and MTN dynamic surface control (MTN-DSC) is proposed for a class of strict-feedback uncertain nonlinear systems with unmodeled dynamics and output constraint. Then, Liu et al. [12] presented a robust control design using the neural network (NN) and DSC method for path following of underactuated surface vessels with input saturation. Similarly, Xin et al. [13] investigated a dynamic surface control for nonlinear multi-input multioutput (MIMO) system based on FLS, which effectively reduced online computation in two cascaded continuous stirred tank reactors (CSTRs).Implementation of adaptive DSC obtained by backstepping requires the analytic calculation of the partial derivatives of virtual control signals. In [14], Yu et al. presented a new implementation approach for adaptive backstepping control, which used the filtering methods to produce certain command signals and their derivatives which eliminate the requirement of analytic differentiation, simplifying the controller design. In order to further improve the quality of the filter and avoid damaging the stability of the system, the position tracking controller module is developed based on the modified DSC method by using the second-order nonlinear tracking differentiator (NLTD) instead of the first-order filter at each step of the virtual inputs in [15]. In [16], Wang et al. proposed a new sliding mode control method and applied it to the motor drive system such that a prior knowledge on the bounds of uncertainties is not required in the controller design. In [17], Chen et al. included the periodic nonparametric part in the unknown expected control input and established a fully saturated repetitive learning law with continuous switching function to ensure that the estimation of the unknown expected control input is continuous and limited in a predetermined region, so as to ensure higher steady-state tracking accuracy.On the other hand, for continuous-time systems, in the aforementioned works has been carried out on DCS with antiwindup compensation. In contrast to the large amount of work on nonlinear systems with actuator saturation, only a few results are available in adaptive DCS of discrete-time systems, because discrete-time nonlinear systems are more difficult to deal with and many continuous control schemes are not suitable for discrete-time systems. For example, to solve the problem of “explosion of complexity” in the process of the backstepping method, the noncausal problem of the system should be solved in the discrete-time system first. Noncausal problems are solved mainly by changing the system model [18–20], which further increased the complexity of the controller. Therefore, Zhang et al. [19] applied dynamic surface control based on the filter to simplify the complexity of the algorithm, which resolved the problems of explosion of complexity and noncausal in discrete-time systems caused by backstepping. Then, Liu and Tong [20] presented an adaptive fuzzy controller design for uncertain nonlinear systems with the discrete-time form in a triangular structure and including backlash and external disturbance.But the control schemes mentioned above are based on the conditions with known states. In practical engineering, due to the limitation of sensors, it is impossible to observe all states of the system, so it is not completely satisfactory to design a controller under the conditions with known states. For discrete-time nonlinear systems with parametric uncertainties, there has been a lot of research on fuzzy or neural network state observers in recent years [21–25]. However, in the aforementioned works, the observer was employed to obtain unknown states while the observer performance to the closed-loop system remains undiscussed.The main contributions of this paper are summarized as follows:(1)
To mitigate the windup problem caused by input saturation nonlinearity in the discrete-time system, the whole closed-loop control system is analyzed in detail first. Then, input saturation nonlinearity is described by linear part and disturbance term, which is combined with uncertain system functions to compose the total disturbance that affect the control performance. Then, the fuzzy control object is established, where using control signals to avoid the windup of actuator winding is the key issue.(2)
By designing an improved HOSM observer, the stable chattering problem of HOSM is effectively solved and output feedback tracking control is achieved under the assumption that system states are unknown. Since most of the practical systems are states unknown, the proposed adaptive sliding surface algorithm is superior to the existed state-feedback control techniques in control nonlinear systems with input saturation in application. In this design, the improved HOSM observer is developed by a coordinate transformation.(3)
An antiwindup RDSC with fuzzy compensation is derived by the Lyapunov stability method based on filtered command and HOSM observer simplifying the complexity of the algorithm, which resolved the problems of the explosion of complexity and noncausal in discrete-time systems caused by backstepping. Compared to the traditional sliding mode (surface) controller (SMC), the chattering phenomenon is effectively avoided. Finally, the tracking errors of the closed-loop system, which can be made arbitrarily small by adjusting design parameters, are proved to be bounded by performance function.This paper is organized as follows: Section2 presents the mathematical model of saturation nonlinearity and nonlinear system. Section 3 gives the fuzzy DSC design and stability analysis of the whole closed-loop system. Section 4 provides a simulation example to validate the effectiveness of the proposed control scheme. Finally, Section 5 concludes this paper.


## 2. Problem Statement

In this Section, saturation nonlinearity is introduced first. Subsequently, a nonlinear plant in a triangular form is given for analysis and the control objective is established.


### 2.1. Saturation Nonlinearity

As shown in Figure1, an unknown saturation in the control system can be described by(1)u=satτ=mτmax,τk≥τmax,mτk,τmin<τk<τmax,mτmin,τk≤τmin,where uk and τk represent the input and output signals, respectively, τmin and τmax are the minimum and maximum value of τ, and m is the slope of the linear region.Figure 1
Saturation nonlinearity.Saturation nonlinearity (1) can be represented in the form of(2)uk=mτk+φk,with φk being a time variable disturbance signal.It is obtained from (1) that(3)φk=mτmax−τk,τk≥τmax,0,τmin<τk<τmax,mτmin−τk,τk≤τmin,where φk is defined as φk=uk−mτk.Remark 1.
It is concluded from (3) that if the control effort τ exceeds τmax, a disturbance signal will be produced to prevent τ from being further increasing. In this case, the output tracking control may be failed since no sufficient control effort is imposed to the plant. This case occurs when the tracking error is large. As a result, the key issue is to make the control effort to avoid increasing before saturation. To realize this object, an unknown nonlinear function should be effectively approximated and compensated.
In this paper, FLS will be utilized to approximate the unknown uncertain function in a nonlinear model. The considered control system is a strict-feedback nonlinear plant with unknown input saturation, which is shown in Figure2. The nonlinear plant in Figure 2 will be analyzed in detail in the following section.Figure 2
Saturation nonlinearity.


### 2.2. Nonlinear Plant in Triangular Form

For nonlinear plant, as shown in Figure2, we consider a triangular nonlinear model in the form of(4)x1k+1=f1x1k+g1x1kx2k,xik+1=fix¯ik+gix¯ikxi+1k,xnk+1=fnx¯k+gnx¯kuk,yk=x1k,where x¯k=x1k,…,xnkT is a vector of system states, x¯ik=x1k,…,xikT are vectors of system states with i=2,…,n−1, and fi and gi,i=1,…,n, are smooth nonlinear functions.For nonlinear system (4) preceded by input saturation (1) and a given bounded desired trajectory xrk with bounded derivatives up to n+1th order, the control objective of this paper is to design an adaptive recursive dynamic surface control signal τk such that the tracking error ek is as small as possible. In particular, the input control signal is antiwindup, and the whole closed-loop system is Lyapunov stable.Lemma 1 (see [26]).
A class of nonlinear systems in the pure-feedback form is(5)x1k+1=f1x1k+g1x1kx2k,xik+1=fix¯ik+gix¯ikxi+1k,xnk+1=fnx¯nk+gnx¯nkuk,yk=x1k.
Define alternative state variablesz¯=Tx¯k as(6)z1k=x1k,z2k=z1k+1,zi+1k=zik+1,i=2,…,n−1,yk=z1k=x1k.
System (5) can be represented as(7)z1k+1=z2k,zik+1=zi+1k,i=1,…,n−1,znk+1=γnx¯nk+βnx¯nkuk,yk=x1k.where (8)γnx¯nk=∑j=1n−1∂γn−1x¯n−1k∂xjkfjx¯jk+gjx¯jkxj+1k+∑j=1n−1∂βn−1x¯n−1k∂xjkxik+βn−1x¯n−1kfnx¯nk,βnx¯nk=βn−1x¯n−1kgnx¯nk=∏i=1ngix¯ik.
Furthermore, considering disturbance signalφk, (7) can be reformed as(9)z1k+1=z2k,zik+1=zi+1k,i=1,…,n−1,znk+1=anx¯nk+bnx¯nkτk,yk=x1k=z1k,where anx¯nk and bnx¯nk are unknown functions defined as(10)anx¯nk=γnx¯nk+βnx¯nkφk=γnx¯nk+dk,bnx¯nk=mβnx¯nk,with dk being a disturbance signal produced when the control effort τ exceeds τmax.Remark 2.
It should be pointed out that, by coordinate transformz¯=Tx¯k which is defined as in (6), there exists an inverse map x¯=T−1z¯k representing the relationship of x¯ and z¯k. Thus, the obtained functions anx¯nk,bnx¯nk are also unknown functions of z¯nk.
It is shown that, with the newly defined states (6) and the coordinate transform [26], the strict-feedback system (4) is now reformulated as a Brunovsky system (7) with an output signal z1k=x1k. Since yk=z1k=x1k holds, the control objective (i.e., yk tracks a reference signal xdk) can be retained by controlling system (9). However, in the newly derived system (9), the states zik are not directly measurable and the functions anx¯nk and bnx¯nk are unknown. In this sense, the state-feedback control of the strict-feedback system (4) can be viewed as output feedback control of the canonical system (9). In the following, we will consider the output feedback control design of (9) to achieve control of (4).


### 2.3. Improved Observer

To estimatezjk,j=2,…,n, a finite-time convergence observer is derived based on a high-order sliding mode (HOSM) observer (Levant’s differentiator).In the actual use of Levant's differentiator, it is found that when the function approaches stability, its small errors are amplified, and the rapid convergence rate will inevitably cause a chattering phenomenon. In short, when the error reaches the origin, it “does not stop,” thus repeatedly crossing the origin and forming chattering, which has an impact on the actual system.Theorem 1.
The structure of the improved HOSM observer in this study is given as(11)z^1k+1=ν1k,ν1k=z^2k−λ1η1k,z^ik+1=νik,νik=z^i+1k−λiηik,z^nk+1=νnk,νnk=z^n+1k−λnηnk,z^n+1k+1=−λn+1ηn+1k,where the variable gain is defined as λi=falηik,δ,κ1/ηik,i=1,2,…,n+1 and nonlinear function falηik,δ,κ is defined as(12)falηik,δ,κi=ηikκisignηik,ηik>δ,ηikδ1−κi,ηik≤δ,with κi=n+1−i/n+2−i and κ and δ are adjustable design parameters. The value of κ determines the degree of nonlinearity of the function and δ is the width of the linear interval, as shown in Figure 3:(13)η1k=z^1k−x1k=z^1k−z1k,ηjk=z^jk−νj−1k,j=2,…,n+1,satisfying (14)z˜ik≤μiεn−i+1/n+1,ηik≤ςiεn−i/n+1,i=1,…,n+1,where z˜ik=z^ik−ziki=1,…,n represent the observer errors and μiandςi are some positive constants with z˜1k≤ε.
It can be further obtained that(15)z˜ik≤μ=maxμiεn−i+1/n+1i=1,…,n.
It should be pointed out that an improved HOSM observer (11) can achieve finite-time error convergence. This means that a controller and an observer can be designed separately so that the combined controller-observer output feedback preserves the main features of the controller with a full state available.The nonlinear functionfal⋅ has the design concept of “small error amplification, large error reduction” and retains the advantage of fast convergence of high-order sliding mode in terms of convergence rate. Different from the nonlinear term of Levant's differentiator, ηik/δ1−β<ηikβsignηik exists for any δ∈0,1 and κ∈0,1 when e∈0,δ, which can effectively deal with the stable chattering problem of the system.Figure 3
Output signal of the nonlinear functionfal⋅. (a) Output of fal⋅ with different δ. (b) Output of fal⋅ with different κ.
(a)(b)Proof.
(16)z^1k+1=ν1k,ν1k=z^2k−λ1η1k,z^ik+1=νik,νik=z^i+1k−λiηik,z^nk+1=νnk,νnk=z^n+1k−λnηnk,z^n+1k+1=−λn+1ηn+1k.
The observer (16) can be represented as(17)z^1k+1=z^2k−λ1η1k,z^2k+1=z^3k−λ2η2k,⋮z^ik+1=z^i+1k−λiηik,⋮z^nk+1=z^n+1k−λnηnk,z^n+1k+1=−λn+1ηn+1k,νik=z^i+1k−λiηik,i=1,…,n+1.
The specific expressionsλi on both sides of the error boundary are different, so they are discussed separately here. When ηi≤δ, λi=1/δ1−κi,i=1,…,n+1, the observer is linear feedback. Its linear feedback form is as follows:(18)z^1k+1=z^2k−λ1z˜1k,z^2k+1=z^3k−λ2z˜2k,⋮z^ik+1=z^i+1k−λiz˜ik,⋮z^nk+1=z^n+1k−λnz˜nk,z^n+1k+1=−λn+1z˜n+1k,νik=z^i+1k−λiz˜ik,i=1,…,n+1.
Define the observation errorz˜ik=z^ik−ziki=1,…,n+1 and the time derivative of z˜ik is(19)z˜1k+1=z˜2k−λ1z^1k−z1k,z˜2k+1=z˜3k−λ2z^2k−z2k,⋮z˜ik+1=z˜i+1k−λiz^ik−zik,⋮z˜nk+1=z˜n+1k−λnz^nk−znk,z˜n+1k+1=−λn+1z^n+1k−zn+1k.
Definingz¯ik=z˜1k,…,z˜n+1k, the appropriate parameter λi is selected and (20)A0=−λ110…0−λ201…0⋮⋮⋮⋱⋮−λn00…1−λn+100…0is a strict Hurwitz matrix.
The selected Lyapunov candidate function is defined as follows:(21)V0k=z¯iTkPiz¯ik.
The derivative of the candidate function is selected as(22)V0k+1=z¯iTk+1Piz¯ik+z¯iTkPiz¯ik+1=z¯iTkA0TPiz¯ik+z¯iTkPiA0z¯ik=z¯iTA0TPiz¯ik+z¯iTkPiA0z¯ik.
GivingQiT=Qi>0. Then, there exists a positive definite matrix PiT=Pi>0 and (23)A0TPi+PiA0=−Qi.
Therefore, it is obtained that(24)V0k+1≤−qminQiz¯ik,where qminQi is a minimum eigenvector in matrix Qi.
Whenηi>δ, λi=ηikκisignηik,i=1,…,n+1, the observer is(25)z^1k+1=z^2k−z˜1kκ1signη1kz˜1k,z^2k+1=z^3k−z˜2kκ2signη2kz˜2k,⋮z^ik+1=z^i+1k−z˜ikκisignηikz˜ik,⋮z^nk+1=z^n+1k−z˜nkκnsignηnkz˜nk,z^n+1k+1=−z˜n+1kκn+1signηn+1kz˜n+1k.
Define the observation errorz˜ik=z^ik−ziki=1,…,n+1 and select appropriate parameters κi. According to the literature [27], the observation error can be stabilized in a finite time.
It is not unusual that a sliding mode is finally attained in finite time. This finite-time property of sliding mode observers is very attractive.Remark 3.
For the nonlinear system in the form of (4), lots of adaptive fuzzy control schemes have been developed based on state feedback in the past decades. In this case, one assumption is that all the system states are available when implementing these algorithms. Usually, not all the states are available in practice. Accordingly, a state observer is needed if not all states are available for feedback control. In the following, we design a recursive fuzzy DSC signal based on transform (9), where the states are obtained by HOSM observer (11).


## 2.1. Saturation Nonlinearity

As shown in Figure1, an unknown saturation in the control system can be described by(1)u=satτ=mτmax,τk≥τmax,mτk,τmin<τk<τmax,mτmin,τk≤τmin,where uk and τk represent the input and output signals, respectively, τmin and τmax are the minimum and maximum value of τ, and m is the slope of the linear region.Figure 1
Saturation nonlinearity.Saturation nonlinearity (1) can be represented in the form of(2)uk=mτk+φk,with φk being a time variable disturbance signal.It is obtained from (1) that(3)φk=mτmax−τk,τk≥τmax,0,τmin<τk<τmax,mτmin−τk,τk≤τmin,where φk is defined as φk=uk−mτk.Remark 1.
It is concluded from (3) that if the control effort τ exceeds τmax, a disturbance signal will be produced to prevent τ from being further increasing. In this case, the output tracking control may be failed since no sufficient control effort is imposed to the plant. This case occurs when the tracking error is large. As a result, the key issue is to make the control effort to avoid increasing before saturation. To realize this object, an unknown nonlinear function should be effectively approximated and compensated.
In this paper, FLS will be utilized to approximate the unknown uncertain function in a nonlinear model. The considered control system is a strict-feedback nonlinear plant with unknown input saturation, which is shown in Figure2. The nonlinear plant in Figure 2 will be analyzed in detail in the following section.Figure 2
Saturation nonlinearity.


## 2.2. Nonlinear Plant in Triangular Form

For nonlinear plant, as shown in Figure2, we consider a triangular nonlinear model in the form of(4)x1k+1=f1x1k+g1x1kx2k,xik+1=fix¯ik+gix¯ikxi+1k,xnk+1=fnx¯k+gnx¯kuk,yk=x1k,where x¯k=x1k,…,xnkT is a vector of system states, x¯ik=x1k,…,xikT are vectors of system states with i=2,…,n−1, and fi and gi,i=1,…,n, are smooth nonlinear functions.For nonlinear system (4) preceded by input saturation (1) and a given bounded desired trajectory xrk with bounded derivatives up to n+1th order, the control objective of this paper is to design an adaptive recursive dynamic surface control signal τk such that the tracking error ek is as small as possible. In particular, the input control signal is antiwindup, and the whole closed-loop system is Lyapunov stable.Lemma 1 (see [26]).
A class of nonlinear systems in the pure-feedback form is(5)x1k+1=f1x1k+g1x1kx2k,xik+1=fix¯ik+gix¯ikxi+1k,xnk+1=fnx¯nk+gnx¯nkuk,yk=x1k.
Define alternative state variablesz¯=Tx¯k as(6)z1k=x1k,z2k=z1k+1,zi+1k=zik+1,i=2,…,n−1,yk=z1k=x1k.
System (5) can be represented as(7)z1k+1=z2k,zik+1=zi+1k,i=1,…,n−1,znk+1=γnx¯nk+βnx¯nkuk,yk=x1k.where (8)γnx¯nk=∑j=1n−1∂γn−1x¯n−1k∂xjkfjx¯jk+gjx¯jkxj+1k+∑j=1n−1∂βn−1x¯n−1k∂xjkxik+βn−1x¯n−1kfnx¯nk,βnx¯nk=βn−1x¯n−1kgnx¯nk=∏i=1ngix¯ik.
Furthermore, considering disturbance signalφk, (7) can be reformed as(9)z1k+1=z2k,zik+1=zi+1k,i=1,…,n−1,znk+1=anx¯nk+bnx¯nkτk,yk=x1k=z1k,where anx¯nk and bnx¯nk are unknown functions defined as(10)anx¯nk=γnx¯nk+βnx¯nkφk=γnx¯nk+dk,bnx¯nk=mβnx¯nk,with dk being a disturbance signal produced when the control effort τ exceeds τmax.Remark 2.
It should be pointed out that, by coordinate transformz¯=Tx¯k which is defined as in (6), there exists an inverse map x¯=T−1z¯k representing the relationship of x¯ and z¯k. Thus, the obtained functions anx¯nk,bnx¯nk are also unknown functions of z¯nk.
It is shown that, with the newly defined states (6) and the coordinate transform [26], the strict-feedback system (4) is now reformulated as a Brunovsky system (7) with an output signal z1k=x1k. Since yk=z1k=x1k holds, the control objective (i.e., yk tracks a reference signal xdk) can be retained by controlling system (9). However, in the newly derived system (9), the states zik are not directly measurable and the functions anx¯nk and bnx¯nk are unknown. In this sense, the state-feedback control of the strict-feedback system (4) can be viewed as output feedback control of the canonical system (9). In the following, we will consider the output feedback control design of (9) to achieve control of (4).


## 2.3. Improved Observer

To estimatezjk,j=2,…,n, a finite-time convergence observer is derived based on a high-order sliding mode (HOSM) observer (Levant’s differentiator).In the actual use of Levant's differentiator, it is found that when the function approaches stability, its small errors are amplified, and the rapid convergence rate will inevitably cause a chattering phenomenon. In short, when the error reaches the origin, it “does not stop,” thus repeatedly crossing the origin and forming chattering, which has an impact on the actual system.Theorem 1.
The structure of the improved HOSM observer in this study is given as(11)z^1k+1=ν1k,ν1k=z^2k−λ1η1k,z^ik+1=νik,νik=z^i+1k−λiηik,z^nk+1=νnk,νnk=z^n+1k−λnηnk,z^n+1k+1=−λn+1ηn+1k,where the variable gain is defined as λi=falηik,δ,κ1/ηik,i=1,2,…,n+1 and nonlinear function falηik,δ,κ is defined as(12)falηik,δ,κi=ηikκisignηik,ηik>δ,ηikδ1−κi,ηik≤δ,with κi=n+1−i/n+2−i and κ and δ are adjustable design parameters. The value of κ determines the degree of nonlinearity of the function and δ is the width of the linear interval, as shown in Figure 3:(13)η1k=z^1k−x1k=z^1k−z1k,ηjk=z^jk−νj−1k,j=2,…,n+1,satisfying (14)z˜ik≤μiεn−i+1/n+1,ηik≤ςiεn−i/n+1,i=1,…,n+1,where z˜ik=z^ik−ziki=1,…,n represent the observer errors and μiandςi are some positive constants with z˜1k≤ε.
It can be further obtained that(15)z˜ik≤μ=maxμiεn−i+1/n+1i=1,…,n.
It should be pointed out that an improved HOSM observer (11) can achieve finite-time error convergence. This means that a controller and an observer can be designed separately so that the combined controller-observer output feedback preserves the main features of the controller with a full state available.The nonlinear functionfal⋅ has the design concept of “small error amplification, large error reduction” and retains the advantage of fast convergence of high-order sliding mode in terms of convergence rate. Different from the nonlinear term of Levant's differentiator, ηik/δ1−β<ηikβsignηik exists for any δ∈0,1 and κ∈0,1 when e∈0,δ, which can effectively deal with the stable chattering problem of the system.Figure 3
Output signal of the nonlinear functionfal⋅. (a) Output of fal⋅ with different δ. (b) Output of fal⋅ with different κ.
(a)(b)Proof.
(16)z^1k+1=ν1k,ν1k=z^2k−λ1η1k,z^ik+1=νik,νik=z^i+1k−λiηik,z^nk+1=νnk,νnk=z^n+1k−λnηnk,z^n+1k+1=−λn+1ηn+1k.
The observer (16) can be represented as(17)z^1k+1=z^2k−λ1η1k,z^2k+1=z^3k−λ2η2k,⋮z^ik+1=z^i+1k−λiηik,⋮z^nk+1=z^n+1k−λnηnk,z^n+1k+1=−λn+1ηn+1k,νik=z^i+1k−λiηik,i=1,…,n+1.
The specific expressionsλi on both sides of the error boundary are different, so they are discussed separately here. When ηi≤δ, λi=1/δ1−κi,i=1,…,n+1, the observer is linear feedback. Its linear feedback form is as follows:(18)z^1k+1=z^2k−λ1z˜1k,z^2k+1=z^3k−λ2z˜2k,⋮z^ik+1=z^i+1k−λiz˜ik,⋮z^nk+1=z^n+1k−λnz˜nk,z^n+1k+1=−λn+1z˜n+1k,νik=z^i+1k−λiz˜ik,i=1,…,n+1.
Define the observation errorz˜ik=z^ik−ziki=1,…,n+1 and the time derivative of z˜ik is(19)z˜1k+1=z˜2k−λ1z^1k−z1k,z˜2k+1=z˜3k−λ2z^2k−z2k,⋮z˜ik+1=z˜i+1k−λiz^ik−zik,⋮z˜nk+1=z˜n+1k−λnz^nk−znk,z˜n+1k+1=−λn+1z^n+1k−zn+1k.
Definingz¯ik=z˜1k,…,z˜n+1k, the appropriate parameter λi is selected and (20)A0=−λ110…0−λ201…0⋮⋮⋮⋱⋮−λn00…1−λn+100…0is a strict Hurwitz matrix.
The selected Lyapunov candidate function is defined as follows:(21)V0k=z¯iTkPiz¯ik.
The derivative of the candidate function is selected as(22)V0k+1=z¯iTk+1Piz¯ik+z¯iTkPiz¯ik+1=z¯iTkA0TPiz¯ik+z¯iTkPiA0z¯ik=z¯iTA0TPiz¯ik+z¯iTkPiA0z¯ik.
GivingQiT=Qi>0. Then, there exists a positive definite matrix PiT=Pi>0 and (23)A0TPi+PiA0=−Qi.
Therefore, it is obtained that(24)V0k+1≤−qminQiz¯ik,where qminQi is a minimum eigenvector in matrix Qi.
Whenηi>δ, λi=ηikκisignηik,i=1,…,n+1, the observer is(25)z^1k+1=z^2k−z˜1kκ1signη1kz˜1k,z^2k+1=z^3k−z˜2kκ2signη2kz˜2k,⋮z^ik+1=z^i+1k−z˜ikκisignηikz˜ik,⋮z^nk+1=z^n+1k−z˜nkκnsignηnkz˜nk,z^n+1k+1=−z˜n+1kκn+1signηn+1kz˜n+1k.
Define the observation errorz˜ik=z^ik−ziki=1,…,n+1 and select appropriate parameters κi. According to the literature [27], the observation error can be stabilized in a finite time.
It is not unusual that a sliding mode is finally attained in finite time. This finite-time property of sliding mode observers is very attractive.Remark 3.
For the nonlinear system in the form of (4), lots of adaptive fuzzy control schemes have been developed based on state feedback in the past decades. In this case, one assumption is that all the system states are available when implementing these algorithms. Usually, not all the states are available in practice. Accordingly, a state observer is needed if not all states are available for feedback control. In the following, we design a recursive fuzzy DSC signal based on transform (9), where the states are obtained by HOSM observer (11).


## 3. Recursive Dynamic Surface Controller Design

Assumption 1.
It is assumed thatfixik,i=1,…,n, are totally unknown smooth functions while gixik are known positive or negative functions.Remark 4.
In control systems, the assumption ongixk is practical since gixk is very important in control signal design. In most of the published results, assumptions should be made before controller design to make the control design possible. If gix are totally unknown, control may be failed since gixk is a time variable or even 0, significantly affecting control performance. In the worst case, instability may be occurred.
In the control design, the following dynamic sliding surfaces will be used:(26)e1k=z^1k−xrk,e2k=e1k+1+xrk+1−ϑ1k,eik=ek−1k+1+ϑi−2k+1−ϑk−1k,enk=en−1k+1+ϑn−2k+1−ϑn−1k,where i=2,…,n−1 and ϑk is the virtual control signal.
Based on the above-defined dynamic sliding surfaces, an adaptive recursive command filtered output feedback control scheme will be developed in detail with fuzzy compensation.
The process is carried out based on the coming steps:Step 1: from (26), the state tracking error is defined as(27)e1k=z^1k−xrk,wherexrk is the reference signal.Choose the Lyapunov function candidate as(28)V1=12e12k.Then, the first-order difference of (28) is computed as(29)ΔV1=12e12k+1−12e12k=12z^1k+1−xrk+12−12e12k.According to the DSC method, a virtual control signal is defined asϑ1k=xrk+1. It is very important that, in order to obtain xrk+1, let xrk pass through a tracking differentiator (TD) in the form of(30)χ1k+1=χ2k,χ2k+1=fstχ1k−xrk,χ2k,r,h0,whereχ1kandχ2k are the states of TD, h0 is a filter coefficient, and r is a velocity factor. According to [28], the nonlinear function fst⋅ is defined as(31)fst⋅=−ar,a≤d,−rsigna,a>d,with(32)a=χ2+ch0,c<d0,χ2+signca0−d2,c>d0,d=rh0,d0=dh0,c=χ1−xr+h0χ2,a0=d2+8rc,withh0 being the sample period.Then, the second error surface is(33)e2=z^2−ϑ1,withϑ1k being the virtual control signal in Step 1. Substituting (33) into (29) yields(34)ΔV1=12z^2k−xrk+12−12e12k=12e2k+ϑ1−xrk+12−12e12k=12e22k−12e12k.Step i2≤i<n: the ith error surface is defined as(35)eik=z^ik−ϑi−1k,whereϑi−1k represents the intermediate virtual control signal in the last step. The virtual control signal in this step is(36)ϑik=ϑi−1k+1,whereϑi−1k+1 is the dynamic of the virtual control signal and is very important in the DSC design spirit.In this design, similar to Step 1, letϑi−1k pass through a TD to obtain ϑi−1k+1 as(37)χi1k+1=χi2k,χi2k+1=fstχi1k−ϑi−1k,ϑi−1k,r,h0,withχi2k being tracking signal ϑi−1k+1.Choose the Lyapunov function candidate as(38)Vi=Vi−1+12ei2k.Then, the first-order difference of (38) is computed as(39)ΔVi=12z^i+1k−ϑi−1k+12−12ei2k+ΔVi−1=12ei+12k−12ei2k+ΔVi−1=12ei+12k−12e12k.Stepn: define the last error surface as(40)enk=z^nk−ϑn−1k,withϑn−1 being the last intermediate virtual control signal.Choose the Lyapunov function candidate as(41)Vn=Vn−1+12en2k.Then, the first-order difference of (41) is computed as(42)ΔVn=12en2k+1−12en2k+ΔVn−1=12z^nk+1−ϑn−1k+12−12en2k+ΔVn−1=12anz^nk+bnz^nkτk−ϑn−1k+12−12en2k+ΔVn−1.Lemma 2.
Letfx be a continuous function defined on a compact set Ω. Then, for any scalar ε>0, there exists an FLS θTσx such that(43)supx∈Ωfx−θTσx≤ε,where θ=θ1,…,θNT is the ideal constant weight vector and σx=p1x,p2x,…,pNxT∑i=1Npix is the basis function vector, with N>1 being the number of the fuzzy rules and pi being chosen as Gaussian functions, that is, for i=1,2,…,N, pix=exp−x−μiTx−μiηi2, where μi=μi1,μi2,…,μinT is the center vector and ηi is the width of the Gaussian function.
It is known that, from the approximation property of FLS, there exists a FLS satisfying the following equation:(44)Fk=−anznk+ϑn−1k+1bnznk=θTξznk+ε.
Choose the actual control signal and adaptive law to be(45)τk=W^kξz^nk,(46)W^k+1=γξz^nkenk+1−σW^k+W^k,with Wk=θT, W˜k=W^k−W∗. γ, σ are adjustable parameters.
Differentiatingenk yields(47)enk+1=anz^nk+bnz^kτk−ϑn−1k+1=W˜kξz^nk+ε.
Usingξ⋅2≤1 and according to Young's inequality, en2k+1 can be written as(48)en2k+1=W˜kξz^nk+ε2≤W˜2kξz^nk2+ε2≤W˜2k+ε2.
Substituting (48) into (42) yields(49)ΔVn≤12W˜2k+ε2−12en2k+ΔVn−1=12W˜2k+ε2−12e12k.Proof.
Consider a Lyapunov candidate function in the form of(50)Vk=∑i=1n12ei2k+1γW˜2k,where γ is a positive design parameter.
DifferentiatingVk yields(51)ΔVk=ΔVnk+1γW˜2k+1−W2k.
Using (46) obtains(52)W˜k+1=W^k+1−W∗,=W^k+γξz^nkenk+1−σW^k−W∗=1−γσW^k+γξz^nkenk+1−W∗=1−γσW˜k−γσW∗+γξz^nkenk+1.
Next,(53)W˜2k+1−W˜2k=1−γσW˜k−γσW∗+γξz^nkenk+12−W˜2k=1−γσ2W˜2k+γσW∗2+γ2ξz^nk2en2k+1−21−γσγσW∗W˜k+21−γσγξz^nkenk+1W˜k−2γ2σW∗ξz^nkenk+1−W˜2k.
Usingξ⋅2≤1 and according to Young's inequality can yield(54)21−γσγσW∗W˜k≤γσ1−γσW∗2+W˜2k,2γ2σW∗ξz^nkenk+1≤γ2σen2k+1+W∗2,21−γσγξz^nkenk+1W˜k≤γ1−γσen2k+1+W˜2k.
Substituting (54) into (53) yields(55)W˜2k+1−W˜2k≤aW˜2k+bW∗2+cen2k+1,where(56)a=2γσ2−3γσ+γ−γ2σ−1,b=2γσ2−γσ−γ2σ,c=γ2−2γ2σ+γ.
Substituting (48) into (55) can obtain(57)W˜2k+1−W˜2k≤a+cW˜2k+bW∗2+cε2.
Substituting (57) and (49) into (51) can obtain(58)ΔVk≤−12e12k+1γa+c+γ2W˜2k+bW∗2+c+γ2ε2≤−12e12k+1γa+c+γ2W˜2k+μ,where μ=1/γbW∗2+c+γ/2ε2, selecting the appropriate parameters content γ, σ content a+c+γ/2<0. Once the error variables e1k>2μ are valid, then ΔVk≤0 is obtained.
This completes the proof.Remark 5.
We employ TD (31) to eliminate the problem of “explosion of complexity” in the traditional backstepping scheme. It is noted that, in the developed RDSC scheme, TD is superior to the first-order filter for its excellent filtering performance of the high-frequency noises. As a result, the proposed DSC scheme in this research increases control performance by reducing controller complexity and filtering high-frequency noises. In the following, we will illustrate this scheme by simulation on a strict-feedback nonlinear system and a “CSTRs” nonlinear system compared to the traditional SMC method.


## 4. Simulation Study

In this section, two numerical examples are simulated to validate the effectiveness of the fuzzy DSC algorithm proposed in the above sections.Example 1.
To further validate our proposed adaptive algorithm, consider the following strict-feedback nonlinear system:(59)x1k+1=f1x1k+g1x1x2k,x2k+1=f2x¯2k+g2x¯2kτk,with input saturation parameter chosen as τmin=−3,τmax=3,m=2 and unknown nonlinear functions defined as(60)f1x1k=x12k,g1x1k=1+0.1sinx1k,f2x¯2k=0.2e−x2k+x1ksinx2k,g2x2k=1.
The aim of this example is also to make the output signaly=x1 to track a provided reference signal xr=sint.
According to the analysis inSection 2, it is obtained by coordinate transform that(61)γ2x¯2k=x1k+0.1x2kcosx2kx2+1+0.1sinx1kx2k+1+0.1sinx1k0.2e−x2k+x1ksinx2k,β2x¯2k=1+0.2sinx1k.In this simulation, values of simulation coefficients are given as in Table 1.
For comparison, the control performance by traditional sliding mode control (SMC) technique is employed as the control signal, which is in the form of(62)τk=−1bnx¯nkKs+λ1e2k−xrk+2+a^2x¯2k+ω,where ω is a robust term defined as(63)ω=εMsignse¯2k,with(64)s=λ1e1k+λ2e2k,where λ1 and λ2 are positive constants. In simulation, the values of SMC parameters are chosen to be K=5,λ1=1,λ2=1,εM=0.1.
The simulation results are depicted in Figures4–7. From the simulation results, the following advantages are obtained by using the proposed algorithm. First, a better control performance is achieved with a smaller tracking error, which can be seen by comparing the tracking curves in Figure 4(a) and Figure 6(a). Next, derivatives of reference signal xr and virtual control signal ϑ1 are both available without high-frequency noises, which are shown in Figure 4(b) and Figure 5(a). Moreover, an output feedback control is realized by applying HOSM with the state curves shown in Figure 5(b). In particular, there is no chattering phenomenon by the proposed algorithm which can be seen by comparing the curves in Figure 4(a) and Figure 6(a).Table 1
Value of coefficients in Example1.
Parameterk1k2r1r2β1β2Value10101020155Parameterκ1κ2κ3h0ΓϱValue0.752/30.50.011.5∗I250.01Figure 4
Simulation results of Example1. (a) DSC tracking performance in Example 1. (b) States of TD 1
(a)(b)Figure 5
Simulation results of Example1. (a) States of TD2. (b) States of ESO.
(a)(b)Figure 6
Simulation results in Example1. (a) SMC tracking performance. (b) Sliding surface of Example 1.
(a)(b)Figure 7
CSTRs.Example 2.
To validate the proposed DSC scheme, consider a “continuous stirred tank reactors, CSTRs” nonlinear system is described in Figure7.
The system has two identical series CSTR, and irreversible reactions take place inside the reaction tank. This system is usually used to increase the conversion rate of reactants. It is assumed that the reactions in both reactors take place under isothermal conditions, and the concentration of reactants in the reactors is controlled. The mathematical model of the two CSTR is obtained from the law of mass conservation of reactants. The specific model is shown below:(65)CA2k+1=F1CA1kV2−F2CA2kV2−K2CA2jk,CA1k+1=FiCAikV1−F1CA1kV1−K1CA1jk,y=CA2k,CAik=FCkCAC+FDkCADFCk+FDk,FDk=Fi,with K1=α1e−E/RT,K2=α2e−E/RT and variables with indices “1” and “2” refer to the first and second reactors. CA and F are, respectively, concentration and flow rate of the reactants in the reactor, and V denotes volume of reactors. CAiandFi are, respectively, adjustable concentration and invariant reactant flow stream in the input. The chosen parameters for the simulation have been identified on CSTRs [29] and are presented in Table 2.
To simplify the system expression, letx1k=CA2k,x2k=CA1k,uk=CAik; (56) can be expressed as follows:(66)x1k+1=F1V2x2k+f1x1k,x2k+1=1V1uk+f2x2k,yk=x1k,f1x1k=−F2V2x1k−k2x12k,f2x2k=−F1V1x2k−k1x22k,in which the control input is constrained by saturation nonlinearity with parameters chosen as τmin=−1.8, τmax=1.8, m=1. The reference trajectory is xr=sint. Based on the above CSTRs model, the proposed controller, differentiator, and observer are designed, and design parameter values are selected as shown in Table 3.
The simulation results to track a reference signalxr=sint are depicted in Figures 8–12. Control performance and tracking errors are shown in Figure 8. Comparing the tracking errors with and without compensation in Figure 8, it can be seen that antiwindup control performance is achieved and tracking error is significantly reduced when applying the proposed fuzzy DSC control scheme. In addition, the curve of the control signal can be found to reduce the actuator power after compensation in Figure 9.
From the state observation curve in Figures10 and 11, it can also be seen that unknown states are observed quickly and accurately by observer (11). From the analysis in Section 3, it is concluded that better observation performance than HOSM observer and high gain observer (HGOB) can be validated by curves of observation error in Figures 10 and 11. The design of observer (11) similarly makes the error convergent in finite time and effectively deals with the stable chattering problem of the HOSM observer.
In order to make a comparison effect, linear tracking differentiator (LTD) and finite-time convergent differentiator (FTCD) [30] were selected in the simulation and compared with the tracking differentiator (30) proposed in this paper. Figure 12 shows the comparison of differential signal tracking effects and errors of different differentiators. It can also be seen that nonlinear tracking differentiator used in this paper is obviously superior to linear tracking differentiator and error convergence speed is faster than that of FTCD. On the other hand, the buffeting problem is solved in the steady-state performance, so that the error curve that tends to be stable is more gentler.
All these simulation results illustrate that the proposed fuzzy RDSC method effectively achieved the control objective proposed in Section2.Table 2
CSTRs parameters.
ParameterSymbolValueInlet reactant feed streamFi10lit/minThe second reactor inlet flow streamF110lit/minThe second reactor outlet flow streamF210lit/minThe first reactor volumeV125litThe second reactor volumeV225litThe isothermal reaction coefficient 1K10.428lit/mol.minThe isothermal reaction coefficient 2K20.428lit/mol.minThe order of reactionj2Concentration ofA in diluted streamCAD0mol/litConcentration of the concentrated streamCAC20mol/litEnergy of activationE6.46×104kl/kmolThe ideal gas constantR8.314kl/kmol.CTemperature of the reactantT100CTable 3
Value of coefficients in Example2.
Parameterk1k2δϱh0Value750.250.010.01Parameterκ1κ2κ3ΓValue0.752/30.51.5∗I25Figure 8
Tracking performance with/without compensation.Figure 9
Control signal with/without compensation.Figure 10
Statez1 of the observer.Figure 11
Statez2 of the observer.Figure 12
Stateχ2 of TD.


## 5. Conclusion

In this paper, an adaptive fuzzy RDSC scheme that contains tracking differentiators and antiwindup fuzzy compensator is developed for improving the control performance of a class of nonlinear system with input saturation. Meanwhile, an improved HOSM observer is designed to achieve output feedback adaptive RDSC control. Specifically, only one FLS is employed to compensate for the total uncertainty in the last step of backstepping. Then, the stability and closed-loop performance are analyzed by Lyapunov theory. Finally, the tracking performance that guarantees the robustness of closed-loop system is validated by simulation examples.



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