Research Article | Open Access
Improved Adaptive Sliding Mode Control for a Class of Uncertain Nonlinear Systems Subjected to Input Nonlinearity via Fuzzy Neural Networks
The paper presents an improved adaptive sliding mode control method based on fuzzy neural networks for a class of nonlinear systems subjected to input nonlinearity with unknown model dynamics. The control scheme consists of the modified adaptive and the compensation controllers. The modified adaptive controller online approximates the unknown model dynamics and input nonlinearity and then constructs the sliding mode control law, while the compensation controller takes into account the approximation errors and keeps the system robust. Based on Lyapunov stability theorem, the proposed method can guarantee the asymptotic convergence to zero of the tracking error and provide the robust stability for the closed-loop system. In addition, due to the modification in controller design, the singularity problem that usually appears in indirect adaptive control techniques based on fuzzy/neural approximations is completely eliminated. Finally, the simulation results performed on an inverted pendulum system demonstrate the advanced functions and feasibility of the proposed adaptive control approach.
Due to the wide existence of nonlinear systems in many fields of engineering, the controller design for nonlinear systems still received much attention from many researchers. The early control techniques were developed for nonlinear systems and presented their good performances [1–6]. The fundamental ideas of these control techniques are to transform a nonlinear dynamic system into a linear one through state feedback mechanism and then apply the existing methods developed for linear systems. Although good performances can be obtained with these control techniques, the major deficiency remains. The controller design largely relies on the exact cancellation of nonlinear terms or restricts to conditions in that the unknown parameters of nonlinear systems are assumed to appear linearly. This leads to awful performances of the controllers when the existing uncertainties or nonlinear terms of the nonlinear systems are completely unknown. In addition, all control methods above are carried out with an ideal assumption of linear input. Nevertheless, in practical conditions, there exist nonlinearities in the control input because of physical limitations. The existence of the nonlinear input may lead to degradation or even make the system unstable .
Nowadays, fuzzy logic and neural networks are found to be powerful tools for modeling and controlling highly uncertain, nonlinear, and complex systems due to their abilities of universal approximation [8–15]. Although fuzzy logic and neural networks have universal approximation abilities, some differences exist between them. The fuzzy logic has characteristics of linguistic information and logic control, while neural networks possess characteristics of learning, parallelism, and fault-tolerance. The combination of fuzzy logic and neural networks, known as fuzzy neural networks, which incorporate the advantages of fuzzy inference and neurolearning, was developed and has presented advanced functions in modeling and controlling nonlinear systems [15–22]. Based on universal approximation theorem, fuzzy logic and neural networks have been developed and incorporated into adaptive control techniques. In such techniques, Lyapunov approach is used to analyze the system stability and obtain the adaptive laws as well. Conceptually, there are two distinct approaches to design the adaptive controllers: direct and indirect adaptive control methods. In the direct control method, a fuzzy logic system or neural networks are employed to simulate the action of the ideal controller and the parameters are directly adjusted to meet the control objective [15, 19, 23–31]. In contrast, the indirect control method uses a fuzzy logic system or neural networks to approximate the unknown nonlinear terms of model dynamics and then synthesizes control laws based on these approximations [15, 20, 21, 32–40]. In the indirect control method [15, 21, 32–40], the authors considered the single-input single-output (SISO) nonlinear systems in the form, , where is the overall state vector, is the control input, and is the system output. and are unknown nonlinear functions. In order to meet the control objectives, they developed the indirect adaptive controllers which are in the form , where is a new input transforming the nonlinear system into the linear one. and represent the parameterized approximations of the actual nonlinear functions, and , respectively. The weighting vectors, and , which vary according to adaptive laws, are adjusted parameters of the approximations. Since the approximations and are calculated by a fuzzy logic system or neural networks, it is well known that these approximations cannot be guaranteed to be bounded away from zero for all time . In other words, may tend to zero or may be close to zero in some points in time. Such situations lead to very large control signals which may cause the controlled systems to lose their controllability or even damage the whole systems. This problem was known as a singularity problem which usually appears in indirect adaptive control method based on fuzzy/neural approximations.
Since the input nonlinearity and the singularity problem may cause negative effects on controlled systems, it is necessary to develop a new indirect adaptive control method which can bypass the singularity problem even with the existence of input nonlinearity. Therefore, by incorporating both advantages of fuzzy neural networks and sliding mode control technique, we developed a new indirect adaptive control method for a class of uncertain nonlinear systems with input nonlinearity. The proposed controller uses fuzzy neural networks to approximate the unknown nonlinear terms of model dynamics and then synthesizes the sliding mode controller. Due to the novel modifications in design of the proposed controller especially, the denominator in the adaptive control law is guaranteed to be away from zero, and, therefore, the singularity problem is completely avoided. Moreover, in order to treat the undesired effects of approximation errors, a robust compensation is added to the controller to ensure the stability of the controlled system and force the tracking errors to converge to zero as well. In contrast, many previous works dealing with indirect adaptive fuzzy control [15, 21, 32–40] still have a weakness in that the controllers may face to the singularity problem for some cases. When the controlled systems fall into the singularity problem, these controllers may produce very large control signals. These situations may lead the systems to lose their controllability or cause the serious damage to the whole systems. Moreover, these control methods are only proper when the inputs are assumed to be linear ideally. This problem induces the controllers to the restriction of applications because the control inputs may appear nonlinearly due to the physical limitations of some components in many physical systems. For this condition, the linear input may not be suitable to use or show the awful performance. Therefore, in comparison with previous methods, the proposed control method shows the improvements in controller design in that the singularity problem is completely solved. In addition, the proposed controller can show the advanced tracking performance even the controlled system is under the influence of the input nonlinearity. With the proposed controller, the output of the system is forced to follow the desired trajectory successfully and the tracking error converges to zero asymptotically. Finally, the simulations are carried out to illustrate the effectiveness and robustness of the proposed controller.
The rest of this paper is organized as follows. The conventional sliding mode control and problem statement are presented in Section 2. The design of fuzzy neural networks is addressed in Section 3, and the design of the adaptive controller is described in Section 4. In Section 5, simulation results are given to confirm the validity of the proposed method. Finally, the conclusion is given in Section 6.
2. Problem Statement and Sliding Mode Control Design
Considering the th order SISO nonlinear system and assuming that the control input is nonlinearly perturbed due to physical limitations, the dynamic equations can be expressed in normal form as follows:where is the overall state vector of the nonlinear system which is assumed to be available for measurement and , are completely unknown smooth functions. is the scalar control input, while is the scalar system output. is a continuous nonlinear function and inside the sector ; that is,where and are positive constants and . The scalar nonlinear function is illustrated in Figure 1.
Obviously according to (2), there always exists a function inside the sector satisfying and . The objective of this paper is to design the control law such that the output can successfully track a given desired trajectory , which is a known smooth function. Before designing the controller to meet the control objective, we first rewrite the dynamic equations in (1) as follows:where .
Now we define the tracking error and the sliding surface , which describes the tracking error dynamics as follows:where is a positive designed constant. The surface corresponds to a linear differential equation of which the solution denotes that the tracking error converges to zero with time . Taking the time derivative of , we can getwhere represent coefficients in the Hurwitz polynomial which expanded from (5). Here the notation denotes the derivative .
We define a new input variable as
Then in (7) can be expressed in the compact form as
If and in (9) are known, in order to meet the control objective, the control law based on ideal sliding mode control can be used aswhere is a positive designed constant, must be satisfied to make the control law in (10) proper and ensure the controllability of the system. However, in this paper, we consider that and are completely unknown, so this controllability condition is modified for stability analysis in the next sections as the following assumption.
Assumption 1. is bounded from below and above by some known positive constants and ; that is, , .
The equation in (11) implies that converges to zero exponentially fast; therefore, the tracking error converges to zero exponentially fast.
However, in fact, and are unknown, the ideal control law in (10) can no longer be used. In order to overcome this problem, we use a fuzzy neural network to approximate both and .
3. Design of Fuzzy Neural Networks
The basic configuration of a fuzzy logic system comprises four principal components: fuzzification, rule base, fuzzy inference, and defuzzification. In the fuzzification process, the inputs, state variables , are mapped to membership values in the input universes of discourse. The rule base holds a set of antecedent-consequent linguistic rules (IF-THEN rules) that quantify the knowledge that human experts have amassed about solving particular problems. Let be the number of IF-THEN rules. Then the th rule is described in the form ofwhere , and are fuzzy sets that correspond to the membership functions and , respectively. and , which stand for the approximations of and , respectively, are the outputs of the fuzzy logic system. and are fuzzy singletons, while use Gaussian functions to calculate the membership values according to the following equation:where correspond with rules and correspond with state variables. and are the parameters of Gaussian functions. The fuzzy inference engine, which uses the product inference for mapping, performs as a process of mapping membership values from the input windows through the fuzzy rule base to the output window. The engine makes successive decisions about which rules are most relevant to the current situation and applies the actions indicated by these rules. The output defuzzification is the procedure of mapping from a set of inferred fuzzy signals contained within a fuzzy output window to a crisp numeric values as control actions. Using the center-average defuzzification techniques, the outputs of the fuzzy logic system can be calculated aswhere and are weighting vectors that are online tuned according to the adaptive laws described in the next section to meet the control objective. The fuzzy singletons and reach their maximum values at points and with , respectively; that is, . The fuzzy basic vector has the element , , defined by
A fuzzy logic system, which can reason with imprecise information through the fuzzy inference, is good at explaining its actions but it cannot automatically acquire the rules it uses to make those actions. On the other hand, a neural network is good at recognizing patterns but it is not good at explaining how it reaches its decisions. These limitations have been a central driving force behind the creation of hybrid systems called fuzzy neural networks [9, 12, 13]. A fuzzy neural network can combine the human-like reasoning style of a fuzzy system with the learning and connectionist structure of a neural network. In this manner, the parameters in a fuzzy logic system can be found by a neural network through learning processes. Figure 2 shows the structure of the fuzzy neural network with four layers: input layer, membership layer, rule layer, and output layer. Nodes in the input layer are input nodes that represent input linguistic variables. In this context, the inputs are state variables and their values are directly transmitted to the membership layer. The membership layer has nodes of which each unit performs a membership function to an input and uses a Gaussian function to calculate the membership value. The rule layer has nodes of which each node stands for an element of the fuzzy basis vector and performs a fuzzy rule. Thus, all nodes of rule layer form the fuzzy rule set. The links between the rule layer and the output layer express the weighting factors, and , which are the elements of the weighting vectors, and , respectively. These factors are the parameters and are adjusted by designed adaptive laws explained in the next section. In the output layer, nodes represent the output linguistics variables. Two nodes in the output layer, as depicted in Figure 2, act for the values of and .
Therefore, the fuzzy neural network has four layers with inputs and fuzzy rules. The inputs correspond to the state variables of the system, so the number of inputs is chosen so that it equals the order (number of state variables) of the system. The parameters of the membership functions and the number of fuzzy rules are significantly relevant to the approximation accuracy of the network. These parameters must be defined so that they can appropriately cover all possible working area of state variables. For the number of fuzzy rules, in general, the more rules the network has, the more accuracy the approximation can get. However, a great number of rules lead to the complication of the designed system and the increase of the system cost. Thus, the arrangements of parameters of membership functions and the number of fuzzy rules depend on the technical experts who have much knowledge about specific systems.
4. Design of Adaptive Controller
Since and are completely unknown, the ideal sliding mode control law in (10) cannot be determined. This problem leads to the uselessness of the ideal control law in (10). In order to take care of this problem, a fuzzy neural network, as shown in Figure 2, is used to approximate and online. Then incorporating the certainty equivalent approach, the adaptive controller inspired from the ideal sliding mode control law, can be obtained aswhere and are the online approximations of and , respectively. The values of and are calculated by a fuzzy neural network as described in (14).
However, may tend to be zero or be close to zero in some point in time during the operation time, especially in initial phase. This leads to very large control signal values, which may damage the whole system. This situation is known as a singularity problem. In order to avoid the singularity problem, we modify the control law in that is replaced with , and the fundamental idea for operation of is described as follows. First we consider the value of during the operation time. With any initial value of satisfying , we consider whether reaches the lower bound . If so, may have a tendency in that its value is close to zero, and the control signal will be very large, leading to lose the controllability. Thus, when the reaches the lower bound we stop the update law for and forcedly assign to . In contrast, we assign to elsewhere. By this way, the singularity problem can be surely avoided. Therefore, the adaptive controller in (16) is replaced with the modified adaptive controller aswhere and are calculated via the fuzzy neural network as follows:
When the controller works, the values of the weighting vectors, and , are adjusted so that and reach and , respectively. The adaptive laws for and are chosen aswhere and are positive-defined weighting matrices. These matrices govern the speed of adaptation. Now we use the notation to express the -norm of a vector. If and are large, and are small, leading to low speed of adaptation. On the contrary, the small values of and imply that and are large, leading to high speed of adaptation. Nevertheless, the high adaptive speed has a drawback in that the controlled system is very sensitive to external uncertainties. This may cause the system to lose its controllability.
In the adaptive mechanism, and are adjusted so that they converge to and , respectively. In this situation, and achieve their optimal values, and , respectively. Notice that these optimal values, and , are artificial constant quantities which are introduced only for analytical purpose, and they are not used in implementation. The optimal values, and , are then defined bywhere and are sets of the acceptable values of weighting vector, and , respectively. is a compact set of the state variables . In this paper, we assume that the compact set is large enough so that the state variables remain within under closed-loop control and the designed fuzzy neural network does not violate the universal approximation property on .
In the ideal case of approximation, when and reach and , respectively, and reach and , respectively. However, the designed fuzzy neural network which has a finite number of units in the hidden layer is utilized to approximate and , so the approximation errors appear and affect the controlled system. Due to these errors, and cannot converge to and exactly even though and completely converge to and , respectively. Let and be the approximation errors, then the exact models of and can be expressed as follows:
Assumption 2. The approximation errors are bounded by some known constants and over the compact set as follows:
Now the different quantities between the approximation models and exact models can be calculated aswhere and are parameter errors.
Because the approximation errors exist and affect the controlled system, the modified adaptive controller may be difficult to ensure the stability of the closed-loop controlled system alone. In order to suppress the undesirable effects of the approximation errors, a compensation controller is developed and added to the controlled system. This controller is able to compensate the approximation error effects and keep the close-loop system robust. The compensation controller is designed according to the following equation:where the switching function is defined as
Therefore, there are two controllers working together to force the controlled system to match the control objective: the modified adaptive controller and the compensation controller . The overall scheme of the controlled system is illustrated in Figure 3. The total controller is the sum of these two controllers and its formula is given as
Theorem 3. Consider the unknown nonlinear system in (1) and suppose that Assumptions 1 and 2 are satisfied. Then the controller (27) with the designed adaptive laws (19) and (20) can guarantee that the system output tracks the desired trajectory successfully and the tracking error converges to zero asymptotically fast.
Proof. From (9) and (27), we take some basic algebraic manipulations and obtainReplacing in (28) with its expression in (17), we can rewrite (28) asThen, using (18) and (24), (29) can be rewritten asWe define a Lyapunov-like function for stability analysis as follows:Taking the time derivative of with the fact, and , and using (20), we can obtainSubstituting (30) into (32), then (32) can be expressed asApplying the adaptive laws in (19) and (20) to (33), (33) can be rewritten aswhere the switching function is defined in (26).
Replacing the compensation controller in (34) with its expression in (25), and noticing that , we can getFrom (32) and (35), we can get and . Hence, the close-loop controlled system is stable under the effect of the controller. Also, we can obtain , and . These reveal that the tracking error and the adjusted parameters are bounded.
In addition, from the inequality in (35), we have the following inequality:The inequality in (36) implies that , and incorporating leads to . On the other hand, using (5), , , , and can be determined. Then, using Barbalat’s lemma , we can get leading to . Therefore, the perfect tracking performance is achieved and the system stability is ensured, finishing the proof.
5. Illustrative Example
This section presents the simulation results of the proposed control method for a class of unknown nonlinear dynamical system to illustrate the stability and effectiveness of the control algorithm.
Let us consider the inverted pendulum system with input nonlinearity as shown in Figure 4. If is the angle of the pendulum with respect to the vertical line and denotes the angular velocity, the dynamic equations governing the inverted pendulum system are given as where is the state vector, while is the output of the system. is the mass of cart, is the mass of rod, is the half length of the rod, is the gravitational acceleration, and is the nonlinear control input. In this example, it is assumed that , kg, kg, m, and m/s2. Let ; the system in (37) can be rewritten as follows:where
With the given model parameters, it is easy to find that when . Let be the desired trajectory; then the control objective is to design a controller such that the output follows the desired trajectory as close as possible.
Since the inverted pendulum is the second order system, a fuzzy neural network with 2 inputs, and , and 2 outputs, and , is designed. Nine fuzzy rules are used to construct the fuzzy inference of the network. When the inputs are normalized in a range , the membership functions with are chosen as follows:
On the other hand, the designed parameters for modified adaptive controller are chosen as , , , where is a 9-by-9 unit matrix. The designed parameters for compensation controller are arranged as , , , and . In the numerical simulations, the fourth-order Runge-Kutta is used to solve the problem with time step size of 0.001 second. The initial condition is chosen for the simulation. Also, the initial values of adjusted parameters of the neural network are selected to be , , . Figure 5 shows the tracking performance in that the output, , follows the desired trajectory, , successfully. In Figure 6, the tracking error which asymptotically converges to zero under the effect of the proposed controller is depicted. Also, the value of the sliding surface which tends to zero is displayed in Figure 7. Figure 8 especially illustrates that the value of function is always more than or equal to during the period of simulation. This demonstrates that the singularity problem can be completely avoided with the proposed method. The control input and nonlinear control input during the simulation period are shown in Figures 9 and 10, respectively.
Furthermore, the proposed control method is compared with the observer-based adaptive fuzzy control (OAFC) developed in  to demonstrate the advantage of our control approach. The designed parameters for the controller and the observer are maintained as provided in . These parameters are given as follows:
Both simulations of the proposed method and OAFC are performed in the case of input nonlinearity. The tracking performances of the proposed method and OAFC are depicted together in Figure 11. The proposed method, denoted by a solid line, shows better tracking performance than OAFC. The proposed method especially demonstrates the absolute advantage in the settling time. In Figure 12, both tracking errors of these methods are displayed. The tracking error of the proposed control method significantly converges to zero faster than OAFC. Also, the tracking error value of the proposed method is less than OAFC even in the steady state.
In this paper, we have proposed the advanced adaptive control approach for a class of SISO nonlinear systems subjected to nonlinear inputs. The proposed control scheme has two parts where the adaptive fuzzy controller simulates the ideal feedback control law, while the compensation controller reduces the effects of the approximation errors and keeps the system robust. With the improvement in the controller design, the proposed controller not only meets the control objective even with nonlinear inputs but also surely avoids the singularity problem that may be a serious drawback in the indirect adaptive control techniques based on fuzzy/neural networks approximations. Finally, numerical simulations were executed to verify the validity of the proposed method.
Conflict of Interests
The authors, Tat-Bao-Thien Nguyen, Teh-Lu Liao, and Jun-Juh Yan, declare that there is no conflict of interests regarding the publication of this paper.
The authors gratefully acknowledge the support of National Science Council of Taiwan through Grants NSC101-2221-E-006-190-MY2 and NSC102-2221-E-366-003.
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