Adaptive fuzzy extended state observer for a class of nonlinear systems with output constraint
-
Mahtab Delpasand
Abstract
In this study, an adaptive fuzzy extended state observer (AFESO) for single-input–single-output nonlinear affine systems in the presence of external disturbances and output constraints is proposed. In this regard, an extended state observer (ESO) was employed to estimate the unmeasured states and external disturbances simultaneously. To improve the estimation accuracy, the observer gains were adjusted using an adaptation law. To obtain a more comprehensive mathematical analysis and an accurate model for the ESO and to increase the degree of freedom, a Takagi–Sugeno fuzzy system was employed. The proposed AFESO relaxes the limitations of the ESO and improves the system performance as compared with the classical methods in the presence of time-varying disturbances. Next, a command-filtered backstepping controller is designed based on the barrier Lyapunov function method, which guarantees fast convergence of the tracking error as well as satisfying the output constraints of the system. The stability analysis showed that both the estimation error of the AFESO and the tracking error of the controller are bounded, and the tracking error converges to a small neighborhood of the origin. A simulating example of a flexible-joint manipulator shows the effectiveness of the proposed method as compared with the recently proposed method in the literature.
1 Introduction
The control of nonlinear systems in the presence of external disturbances and uncertainties is one of the crucial issues in control engineering and has been highly regarded by researchers in recent decades. Most systems are affected by external disturbances, which cause adverse effects on the system’s output and its performance [1–3]. Over the past decades, a variety of techniques have been proposed for disturbance rejection, including adaptive, intelligent, and robust control methods, all of which are considered anti-disturbance rejection control (ADRC) schemes [4]. Disturbance observer-based control [5–7], tracking differentiator [8,9], and extended state observer (ESO) [10,11] are some of the effective ADRC methods proposed in the literature. Furthermore, most controllers require full-state measurement of the dynamic systems to design controllers, which is not easy in practical applications, especially in the presence of uncertainties, disturbances, and constraints. Therefore, it is useful to design an observer such as ESOs to estimate unmeasured states and disturbances simultaneously.
ESOs are more common among other observers since the system is considered in the form of an integral-chain form. Moreover, ESOs require the least information in the design procedure, i.e., only the relative degree of the system is required [12]. This method can be applied to single-input–single-output high-order systems with matched disturbances [13–15]. However, ESOs have limitations in the face of time-varying disturbances. That is, if time-varying disturbances with a wide range of changes are applied to the system, the ESO cannot eliminate the disturbances effectively. In the study of Gandhi and Adhyaru [16], the disturbance is considered to be constant or slow time-varying in order to eliminate its effects. In previous studies [17,18], higher-order disturbance derivatives are employed in the design of the generalized system to deal with this limitation and to eliminate the effects of time-varying disturbances at the cost of more computational time, complexity of the design procedure, and multiplicity of the design parameters. In the study of Naghdi and Sadrnia [19], to overcome this issue and increase flexibility in the observer gain design, a fuzzy linear ESO is designed to adjust the observer’s bandwidth. A time-varying ESO is designed in the study of Attar et al. [20] to enhance performance in the transient response. Nevertheless, since the ESO is linear with constant gains, the bandwidth of the observer is limited, which drives the system away from optimal performance and causes the peaking phenomenon. In the study of Yang et al. [21], a nonlinear extended state observer (NESO) with time-varying gain is designed to overcome the peaking phenomenon and eliminate the effect of time-varying disturbances.
In the design of ESOs, the observer gains play a crucial role in reducing the influence of the disturbance amplitude and improving the estimation performance [22]. Adjusting observer gains using an adaptive method is one of the effective approaches. In previous studies [23,24], an adaptive extended state observer (AESO) has been proposed to solve the problem of the ESO dependency on the bandwidth of the observer. This method reduces the bandwidth dependency by introducing an adaptation law to adjust the controller parameters. However, only the effect of the external disturbances is considered in the design procedure and not the effect of the state estimation error. In previous studies [25,26], an adaptive fuzzy observer is proposed to overcome this problem. In the study of Pyrkin et al. [27], an AESO is designed without knowledge of the boundary of the disturbances.
Another issue that affects a system’s performance is the constraints of the system that are encountered in practical systems and play crucial roles in the system’s stability. Constraint control problems can be divided into three categories: (i) output constraint [28,29], (ii) state constraint [30,31], and (iii) input constraint [32,33]. To deal with the constrained problems, the barrier Lyapunov functions (BLFs) have been proposed to ensure that constraints are not violated. In the study of Tran et al. [28], time-varying output constraints for robotic manipulators are considered by employing BLF and an adaptive controller. In this approach, the ESO is designed with linear and nonlinear terms that have significantly improved its performance in the presence of output constraints. In the study of Wang et al. [34], to guarantee the tracking error and avoid “complexity explosion” caused by the standard backstepping control method, the command-filtered backstepping controller is proposed. Compared with the standard backstepping controller [35], the command-filtered backstepping is designed to avoid the computational complexity problem caused by the repeated derivatives of the virtual control signals [36,37]. In the study of Cui et al. [38], to reduce the computational problem of the standard backstepping controller, a second-order filter is designed to process the virtual control signal. In previous studies [28,29], to prevent constraint violation, the boundary of the reference signal is considered considerably lower than the time-varying output constraints. In the study of Zhu et al. [39], an adaptive fuzzy observer is designed to estimate disturbances. To guarantee that the system’s output tracks the desired signal and the state constraints are not violated, a time-varying BLF is proposed.
Motivated by the aforementioned literature reviews, the objective of this study is to design an adaptive fuzzy observer to estimate the unmeasured states and the time-varying disturbances simultaneously using an ESO. In this regard, the Takagi–Sugeno (T–S) fuzzy system is employed to model nonlinear systems. It should be mentioned that using adaptive fuzzy observers is not a new idea. However, unlike other adaptive fuzzy observers, e.g., refs [19,40–44], which employ Mamdani fuzzy systems to approximate the model, this study considers the T–S system to increase the degree of freedom in modeling and to obtain a more comprehensive mathematical analysis. In the convectional ESO, e.g., refs [20,45,46], nonlinearities, uncertainties, and unknown external disturbances are considered a lumped uncertainty that influences the accuracy of the estimation.
Compared with the convectional ESO, the proposed adaptive fuzzy extended state observer (AFESO) approximates the nonlinearities using fuzzy systems and estimates the external disturbances to improve the tracking accuracy and to obtain better performance of the closed-loop system. In traditional ESOs, e.g., refs [16,17,47,48], it is claimed that the time derivative of the disturbances converges to zero. Nevertheless, the estimation error can only converge to a compact set, whose boundaries depend on the upper bound of the disturbance derivatives. On the other hand, in this study, this dependency is relaxed in the ESO’s design and asymptotic stability of the estimation error dynamic is shown analytically in the presence of time-varying disturbances. In addition, the control objective is to design a control law to track the desired trajectory while ensuring all the closed-loop signals are bounded and the output constraint is not violated. To achieve these goals, the proposed controller is developed via the command-filtered backstepping method and the BLF strategy, which ensures that the output constraints are not violated and the estimation errors are bounded. In the constrained problems, to ensure that the output constraint is not violated, it is necessary to consider the constraints for the output estimation. Therefore, in this study, unlike refs [28,29,49], the estimation error is taken into account in the BLF to satisfy the output constraints. Moreover, as compared to previous studies [36,37], where the systems are considered in the form of strict feedback, in this study, a more general class of high-order nonlinear systems with output constraints is considered. To demonstrate the effectiveness of the proposed method, a simulation study is conducted on a single-link flexible-joint manipulator.
The main contributions of this study can be summarized as follows:
A novel AFESO with adaptive gains is proposed to estimate the system states and to estimate the time-varying disturbances simultaneously with good accuracy.
A combination of the BLF and the command-filtered backstepping controller is employed in the design procedure for the nth-order nonlinear systems to ensure less tracking errors as well as to consider the constraints of the system.
This article is organized as follows. The formulation of AFESO is introduced in Section 2. In Section 3, the proposed adaptive fuzzy observer is designed. The controller design is presented in Section 4. In Section 5, the stability analysis and disturbance rejection are presented. Section 6 presents the simulated results of a single-link flexible joint manipulator. Section 7 presents the conclusion.
2 Mathematical preliminaries
Consider the following nonlinear system with a relative degree of n:
where
where
Assumption 1
It is assumed that the external disturbance
Assumption 2
It is assumed that the desired trajectory
In order to design a fuzzy observer, the nonlinear system in (2) can be modeled using a T–S system as follows:
where
in which
where
Assumption 3
For
Remark 1
A necessary condition for
Remark 2
In order to design an observer, the fuzzy system must be observable. Therefore, in the process of the observer design, each fuzzy subsystem (and not the whole fuzzy T–S system) must be observable [50].
Proposition 1
For any constant
3 Observer design
Since it is assumed that all system states are unmeasurable, an observer must be designed. To estimate the system states and the disturbance simultaneously, a fuzzy ESO is proposed as follows:
where
Then, the estimation error dynamics for the jth subsystem can be calculated as
where
where
According to (9), it can be seen that the dynamics of the estimation error have been transformed into a time-varying system with an unknown input. Thus, the goal is to design an adaptive fuzzy ESO for the nonlinear system in (2), where the observer gains are adjusted using an adaptive law that guarantees stability of the estimation error in the presence of time-varying disturbances.
Lemma 1
[51] For all
Theorem 1
Consider system (2) under Assumptions 1 and 3, Lemma 1, and Proposition 1. If the observer gains are adjusted by the following adaptation laws:
and the following conditions are held, then the dynamics of the estimation error are asymptotically stable and the output constraint is satisfied:
where
where
Proof. By considering the estimation error as
Step
1. Consider the first state estimation error as
where
where
The adaptation law for
Eq. (14) can be written in the following compact form:
where
Hence, for the stability of the dynamic error,
where
in which
where
Step i.
where
where
The adaptation law for
Eq. (22) can be written in the following compact form:
where
Since
Step n
−1. Similarly, the following Lyapunov function is considered for
where
where
where
The adaptation law for
Eq. (27) can be written in the following compact form:
where
For the stability of the dynamic error,
where
where
where
Hence, in order to establish the stability conditions in (30), it suffices that matrices
Step n
. For
where
where
where
The adaptation law for
Eq. (36) can be written in the following compact form:
where
For the stability of the dynamic error,
Based on (37) and the Schur complement theorem, (39) can be written as
where
where
Step n + 1. In the final design step, the Lyapunov function is defined as
where
where
where
Eq. (45) can be written as
where
Using Young’s inequality and Assumption 1, (47) can be written as
where
For the stability of the dynamic error,
Based on (46) and the Schur complement theorem, (49) can be written as
where
Therefore, according to the following adaptation laws:
and satisfaction of the following conditions:
the dynamics of the estimation error are asymptotically stable (i.e.,
Remark 3
Conditions (53) are not restrictive. By properly adjusting the initial conditions of the observer gain and the scaling parameters (
4 Controller design
The objective is to design a control law (
where
Lemma 2
[52] A command filter with a unity DC gain is given by
where
Remark 4
In (55), a first-order low-pass filter is designed to simplify the analysis. However, higher-order filters can also be designed. For instance, a second-order filter is discussed in the study of Wang and Yu [36].
The compensated tracking errors are introduced as
To ensure that the error compensation system can compensate for the error, the following error compensation system with the initial condition
where
in which
where
The control law is divided into two parts:
where
where
5 Stability analysis of the closed-loop system
In this section, stability analysis is established to confirm that all signals in the closed-loop system are ultimately bounded, asymptotic output tracking is achieved, and output constraint is not violated. Therefore, to illustrate the stability of the closed-loop system, the detailed design procedures of the compensated tracking error dynamics are described in the following steps.
Step
1. Using
Since the states of the system cannot be measured, (63) is rewritten as
Substituting (55) and (57) into (64) yields
Step i
,
Substituting (55) and (57) into (66) yields
Step n
. By defining
where
Step n
+ 1. By defining
According to (65), (67). (69), and (70), the dynamics of the compensated tracking errors can be calculated as
The stability of the compensated tracking error dynamics (71) is shown in Theorem 2.
Lemma 3
[53] For all
holds, where
Theorem 2
Consider system (2) under Assumptions 1–3 and Proposition 1 with the output feedback controller given by (59), the command-filtered backstepping in (55), the compensated tracking error signals in (56), and the virtual control signal in (58). Then, all signals of the closed-loop system are bounded, the output constraint is not violated, and the system (2) is exponentially stable until
Proof. To illustrate the stability of the closed-loop system, the Lyapunov function is defined as
where
where
Considering
Based on Young’s inequality, (75) is rewritten as
Considering
Based on (72), the time derivative of
where
Using Lemma 3, (78) is written as
where
Remark 5
It is important to note that for better performance of the command-filtered backstepping, the filter bandwidth (
Corollary 1
According to Theorem 2 and the Lyapunov function in (72) , the compensated tracking error remains within the compact set
Proof. Multiplying both sides of (79) by
Using (72) yields
Adding
Based on Lemma 3, (82) becomes
Inequality (83) implies that
Therefore, it is straightforward to verify that
In addition,
Corollary 2
Consider Theorems 1 and 2. The adaptation laws (52) are stable and converge into the constant values of
Proof. According to Theorem 1 and the Lyapunov stability theory, the stability of estimation errors is guaranteed. That is,
Taking the limits of the adaptation laws in (52) yields
Based on (85) and Theorem 2, and because all signals in the closed-loop systems are exponentially stable, it yields
Additionally, due to the adaptive laws in (52), the convergence of the adaptive parameters is ensured. That is,
where
The proposed approach can be summarized as the following algorithm:
Step 1: Model the nonlinear system in (1) using the T–S linear fuzzy system in (4).
Step 2: Convert system (4) into the extended fuzzy system in (5).
Step 3: Design fuzzy ESO in (6) and adjust the observer gains using the adaptation laws in (52) for each subsystem.
Step 4: Adjust the scaling parameters and the initial conditions to satisfy conditions in (53).
Step 5: Design the controller using (59), the command-filtered backstepping in (55), the compensated tracking error signals in (56), and the virtual control signal in (58).
The block diagram of the proposed observer in the presence of external disturbance is shown in Figure 1. The controller is synthesized via BLF and the command-filtered backstepping method using output feedback.
Block diagram of the proposed AFESO method.
6 Simulating example
To verify the effectiveness of the proposed observer and controller, a single-link flexible-joint manipulator is simulated in the presence of external disturbances and measurement noise. The simulation results are compared with the NESO methods in the study of Tran et al. [28] and the ESO scheme in the study of Rsetam et al. [54], in the presence of output constraints.
The nonlinear model of a single-link flexible-joint manipulator in the presence of external disturbance can be written as [55]
where
Parameters of single-link flexile-joint manipulator
| Parameters | Value | Description |
|---|---|---|
|
|
1 kg | Mass of the link |
|
|
1 N m/rad | Spring stiffness |
|
|
1 m | Length of the link |
|
|
|
Link inertia |
|
|
|
Inertia of the motor |
Single-link flexible-joint mechanism.
The state-space equation (Eq. (89)) does not satisfy the matching conditions. Therefore, the coordination of the system (89) should be changed. By considering the state transformation as
the system will be in the normal form as follows:
The fuzzy membership functions are shown in Figure 3. Among all membership functions (triangular, z-shape, trapezoidal, s-shape, sigmoid, and Gaussian), triangular membership functions are selected due to its simplicity, i.e., calculations and modifications of parameters with triangular membership are easy with less computational time. The T–S fuzzy model is obtained by linearizing the system at the center of the membership function of
where
Membership function of
The external disturbance defined as
The adaptation laws are considered as
The stability conditions of the estimation error are indicated by
where
To find appropriate values for the design parameters and satisfaction of conditions in (92), genetic algorithm (GA) is employed here. Among all five selection techniques (tournament selection, roulette wheel selection, rank selection, uniform selection, and deterministic selection) roulette wheel selection is selected here [56]. The parameters of GA are given in Table 2. The fitness function
where
Parameters of GA
| Parameters | Value |
|---|---|
| Population size | 90 |
| Number of generations | 100 |
| Crossover rate | 0.8 |
| Mutation rate | 0.01 |
The simulation results are illustrated in Figures 4–8. Figure 4 shows that good trajectory tracking of the proposed method, the NESO in the study of Tran et al. [28], and the ESO in the study of Rsetam et al. [54] is obtained. Moreover, due to the employment of the BLF method in the proposed controller and observer, the output constraint (
![Figure 4
Output trajectory of the proposed method: the NESO in the study of Tran et al. [28], and the ESO in the study of Rsetam et al. [54].](/document/doi/10.1515/nleng-2022-0344/asset/graphic/j_nleng-2022-0344_fig_004.jpg)
Tracking error of different methods.
![Figure 6
States estimation of the proposed method: the NESO in the study of Tran et al. [28], and the ESO in the study of Rsetam et al. [54].](/document/doi/10.1515/nleng-2022-0344/asset/graphic/j_nleng-2022-0344_fig_006.jpg)
![Figure 7
Control inputs of the proposed method: the NESO in the study of Tran et al. [28] and the ESO in the study of Rsetam et al. [54].](/document/doi/10.1515/nleng-2022-0344/asset/graphic/j_nleng-2022-0344_fig_007.jpg)
Adaptive observer gains of the proposed method.
To compare quantitatively the estimation performance of three methods, the root mean square error (RMSE) is defined as follows:
where
| Method |
|
|
|
|
|
|---|---|---|---|---|---|
| AFESO | 0.0127 | 0.0553 | 0.0259 | 0.016 | 0.1073 |
| Tran et al. [28] | 0.0664 | 0.162 | 0.1346 | 0.0772 | 0.1114 |
| Restam et al. [54] | 0.0258 | 0.044 | 0.0881 | 0.0609 | 0.1216 |
In order to evaluate the efficiency of the proposed method in the presence of external disturbance and measurement noise, a white noise with a range of
Measurement noise at the output.
Output trajectory in the presence of external disturbance and measurement noise.
States estimation in the presence of external disturbance and measurement noise.
It should be mentioned that in the studies of Tran et al. [28] and Rsetam et al. [54], the gains are adjusted using the observer bandwidth, which creates susceptibility against measurement noises. On the other hand, in the proposed method, to overcome this problem, the observer gains are adjusted using the proposed adaptation laws. As indicated in Figure 12, the fluctuations in the control efforts of the proposed method are much less than those in the studies of Tran et al. [28] and Rsetam et al. [54] due to the use of a low-pass filter and an adaptive observer.
Control input in the presence of external disturbance and measurement noise.
The simulations are carried out using MATLAB® Software on a computer with Intel® Core i7, CPU 2.60 GHz, 8GB DDR3 RAM, and Windows 10 (64-bit). The computation time is 0.163 s for the proposed method, while it is 0.048 s and 0.07 for the studies of Tran et al. [28] and Rsetam et al. [54], respectively. Therefore, all methods can be applied in real-time using conventional computers. Even though the computation time of the proposed method is longer than those in the studies of Tran et al. [28] and of Rsetam et al. [54], it has better estimation accuracy and tracking performance. Moreover, it offers more flexibility in the observer gain design.
7 Conclusion
In this study, an adaptive fuzzy observer controller was designed to control and estimate the states and external disturbances simultaneously in the presence of an output constraint. The adaptive observer gains improved performance of the convectional ESO significantly. One of the advantages of the proposed observer was that it could be applied to higher-order nonlinear systems. The controller was designed based on the command-filtered backstepping scheme and the BLF method to avoid the computational complexity problem of the standard backstepping scheme and to satisfy output constraints. The efficiency of the proposed method in the presence of time-varying external disturbances and output constraints was investigated. The single-link flexible-joint manipulator was considered to demonstrate better performance of the proposed method as compared with the recently published methods in the related literature.
Acknowledgments
The authors thank the Editor-in-Chief and all those in charge of this journal.
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Funding information: The authors declare that no funds, grants, or other support were received during the preparation of this study.
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Author contributions: Mahtab Delpasand: conceptualization, methodology, software, data curation, validation, formal analysis, investigation, writing – original draft and Mohammad Farrokhi: conceptualization, methodology, validation, formal analysis, writing – review and editing, and supervision.
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Conflict of interest: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this article.
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Ethical responsibilities: The authors confirm that this work is original and has not been published elsewhere, nor is it currently under consideration for publication elsewhere. We declare that the submitted article is a personal academic research article and the authors are not submitting it as an official representative or on behalf of the government. We also declare that none of the authors of the article is employed by a government agency that has a primary function other than research and/or education.
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Consent to participate: Informed consent was obtained from all individual participants included in the study.
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Data availability statement: The authors confirm that all data generated or analyzed during this study are included in this published article.
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- The regularization of spectral methods for hyperbolic Volterra integrodifferential equations with fractional power elliptic operator
- Analytical and numerical study for the generalized q-deformed sinh-Gordon equation
- Dynamics and attitude control of space-based synthetic aperture radar
- A new optimal multistep optimal homotopy asymptotic method to solve nonlinear system of two biological species
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- Self-optimization examination system based on improved particle swarm optimization
- Overlapping grid SQLM for third-grade modified nanofluid flow deformed by porous stretchable/shrinkable Riga plate
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- Numerical simulation of Burgers’ equations via quartic HB-spline DQM
- Bond performance between recycled concrete and steel bar after high temperature
- Deformable Laplace transform and its applications
- A comparative study for the numerical approximation of 1D and 2D hyperbolic telegraph equations with UAT and UAH tension B-spline DQM
- Numerical approximations of CNLS equations via UAH tension B-spline DQM
- Nonlinear numerical simulation of bond performance between recycled concrete and corroded steel bars
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- Influence of the blade size on the dynamic characteristic damage identification of wind turbine blades
- Cilia and electroosmosis induced double diffusive transport of hybrid nanofluids through microchannel and entropy analysis
- Semi-analytical approximation of time-fractional telegraph equation via natural transform in Caputo derivative
- Analytical solutions of fractional couple stress fluid flow for an engineering problem
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- New kink-periodic and convex–concave-periodic solutions to the modified regularized long wave equation by means of modified rational trigonometric–hyperbolic functions
- Explicit Chebyshev Petrov–Galerkin scheme for time-fractional fourth-order uniform Euler–Bernoulli pinned–pinned beam equation
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- Nonlinear dynamic responses of ballasted railway tracks using concrete sleepers incorporated with reinforced fibres and pre-treated crumb rubber
- Two-component excitation governance of giant wave clusters with the partially nonlocal nonlinearity
- Bifurcation analysis and control of the valve-controlled hydraulic cylinder system
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- Electric vehicle wireless charging system for the foreign object detection with the inducted coil with magnetic field variation
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- Research on optimization of crane fault predictive control system based on data mining
- Nonlinear computer image scene and target information extraction based on big data technology
- Nonlinear analysis and processing of software development data under Internet of things monitoring system
- Nonlinear remote monitoring system of manipulator based on network communication technology
- Nonlinear bridge deflection monitoring and prediction system based on network communication
- Cross-modal multi-label image classification modeling and recognition based on nonlinear
- Application of nonlinear clustering optimization algorithm in web data mining of cloud computing
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- A review of tiger conservation studies using nonlinear trajectory: A telemetry data approach
- Multiwireless sensors for electrical measurement based on nonlinear improved data fusion algorithm
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- Research on nonlinear tracking and evaluation of sports 3D vision action
- Analysis of bridge vibration response for identification of bridge damage using BP neural network
- Numerical analysis of vibration response of elastic tube bundle of heat exchanger based on fluid structure coupling analysis
- Establishment of nonlinear network security situational awareness model based on random forest under the background of big data
- Research and implementation of non-linear management and monitoring system for classified information network
- Study of time-fractional delayed differential equations via new integral transform-based variation iteration technique
- Exhaustive study on post effect processing of 3D image based on nonlinear digital watermarking algorithm
- A versatile dynamic noise control framework based on computer simulation and modeling
- A novel hybrid ensemble convolutional neural network for face recognition by optimizing hyperparameters
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- Experimental design and data analysis and optimization of mechanical condition diagnosis for transformer sets
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- Research on railroad locomotive driving safety assistance technology based on electromechanical coupling analysis
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- The application of iterative hard threshold algorithm based on nonlinear optimal compression sensing and electronic information technology in the field of automatic control
- Equilibrium stability of dynamic duopoly Cournot game under heterogeneous strategies, asymmetric information, and one-way R&D spillovers
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