Abstract
A novel adaptive fuzzy dynamic surface control (DSC) scheme is for the first time constructed for a larger class of (multi-input multi-output) MIMO non-affine pure-feedback systems in the presence of input saturation nonlinearity. First of all, the restrictive differentiability assumption on non-affine functions has been canceled after using the piecewise functions to reconstruct the model for non-affine nonlinear functions. Then, a novel auxiliary system with bounded compensation term is firstly introduced to deal with input saturation, and the dynamic system employed in this work designs a bounded compensation term of tangent function. Thus, we successfully relax the strictly bounded assumption of the dynamic system. Additionally, the fuzzy logic systems (FLSs) are used to approximate unknown continuous systems functions, and the minimal learning parameter (MLP) technique is exploited to simplify control design and reduce the number of adaptive parameters. Finally, two simulation examples with input saturation are given to validate the effectiveness of the developed method.
1. Introduction
In the past several decades, approximation-based adaptive control of nonlinear systems has been attracting much attention, and many significant results have been achieved [1–11]. Among them, the fuzzy logic systems (FLSs) and neural networks (NNs) have been successfully employed to approximate the unknown nonlinear functions. In addition, as a breakthrough in nonlinear control, approximation-based adaptive backstepping control has been extensively introduced to achieve global stability for many classes of nonlinear systems [12–17]. For example, in [12], an adaptive fuzzy control scheme was proposed for a class of nonlinear pure-feedback systems under the framework of backstepping, which requires no priori knowledge of the systems dynamic. In [14], an adaptive fuzzy control scheme is presented for a class of pure-feedback nonlinear systems with immeasurable states by utilizing backstepping methodology. Recently, for a class of stochastic nonlinear systems with unknown control direction and unknown dead-zones, an adaptive fuzzy backstepping control method is presented in [17]. However, the problem of “explosion of complexity” caused by repeated differentiations of the virtual control law seriously limits the application of conventional backstepping technique. Thus, the dynamic surface control (DSC) technique has been creatively proposed to avoid this problem effectively by introducing a first-order low-pass filter at each step. Furthermore, compared with strict-feedback systems, pure-feedback systems have a non-affine fashion that the control inputs or variables appear nonlinearly in uncertain systems functions, which leads to the design being more difficult [18, 19]. Moreover, in contrast with SISO pure-feedback nonlinear systems, the control design of MIMO pure-feedback nonlinear systems is, as well known, more complicated due to the couplings among various inputs and outputs [20].
On the other hand, input saturation nonlinearity, as one of the most important input constraints, usually appears in many industrial control systems [21]. In many applications, the input saturation nonlinearity may severely cause degradation of system performance, instability, or even damage. Consequently, the adaptive control of nonlinear systems in the presence of input saturation nonlinearity has been an active topic and attracted increasing attention in recent years [22–30]. For example, in [22], an adaptive fuzzy controller is constructed for pure-feedback stochastic nonlinear systems to deal with input constraints based on the adjustment of commanded input signal. In [25], an adaptive neural controller is investigated for a class of pure-feedback nonlinear time-varying systems with asymmetric input saturation nonlinearity in combination with the Gaussian error function. Recently, for a class of uncertain nonlinear systems with input saturation constraint and external disturbances, a tracking control scheme is proposed by introducing an auxiliary system in [27]. However, it should be pointed out that, for all above the state-of-the-art schemes [22–30] to work for pure-feedback uncertain nonlinear systems subject to input saturation, the non-affine function is always assumed to be differentiable with respect to control variables or inputs, which is restrictive arising from the fact that non-smooth nonlinearities such as dead zone, backlash, and saturation widely exist in various kinds of practical systems [22–25], which makes the non-affine functions non-differentiable and motives us to explore new methods to overcome this limitation [22].
As a matter of fact, overcoming this limitation is challenging. This is because FLSs approximation errors will inevitably occur while adopting FLSs to approximate unknown systems functions within a compact set, this, in combination with external disturbances, may seriously degrade control performance or even give rise to closed-loop system instability. Additionally, there also exist a large number of fuzzy weights that need to be tuned online, which drastically increases the computational burden [28]. Therefore, a design technique needs to be developed that is able to guarantee that all system trajectories stay in the appropriate compact sets all the time, and the MLP technique needs to be employed to solve the explosion of learning parameters. Based on the aforementioned observations, this paper addresses the control problem for a more general class of MIMO pure-feedback nonlinear systems in the presence of input saturation nonlinearity. What is more, to the best of authors’ knowledge, the control design of this huger class of nonlinear systems has not been reported, which is still an open problem with theoretical and applicable significance. The main contributions of this paper are highlighted as follows: (1) it seems that this is the first work that considers both the MIMO non-affine nonlinear systems and input saturation even though some existing works focused on the same topic; (2) to handle input saturation, compared with the auxiliary system presented in [27, 30], the dynamic system employed in this work designs a bounded compensation term , and, thus, the assumption that is bounded is cancelled; (3) in contrast to the existing strategies [22–30], we allow the non-affine functions of MIMO input-saturated nonlinear systems to be non-differentiable via the reconstruction of non-affine functions using appropriate piecewise functions, which removes the restrictive differentiability assumption on non-affine functions.
The rest of this paper is organized as follows. Section 2 presents the problem statement and preliminaries. The adaptive controller design is given in Section 3. Section 4 is devoted to stability analysis. In Section 5 simulation results are presented to show the effectiveness of the proposed scheme, followed by the conclusion in Section 6.
2. Problem Statement and Preliminaries
Consider the following MIMO pure-feedback systems [23]:where is the state of the th subsystem, is the state vector of the whole system , where and is the order of the th subsystem. , and are the input and output of the th subsystem, respectively. are unknown non-affine continuous functions, and are the unknown external disturbances. is the plant input subject to saturation and satisfying [30]where is the bound of , is the input saturation, and .
The design objective of this work is to construct a novel dynamic surface controller such that (1) the output tracking error achieves preselected transient and steady bounds; (2) all signals of system (1) are semiglobally uniformly ultimately bounded (SGUUB); (3) the control input constraint is not violated.
Assumption 1. Define the functions . We assume that the functions satisfywhere , , , and are unknown positive constants; , , , and are unknown constants. And denote , for notation conciseness.
Remark 2. In [22–30], the non-affine functions are always assumed to satisfy and with , , , and being unknown constants. In fact, this assumption is used to ensure the controllability of system (1). However, the assumption is too restrictive due to the fact that many kinds of non-smooth nonlinearities (e.g., dead-zone, backlash, or saturation, and so on) extensively exist in control input, leading to the non-differentiability of non-affine functions, even instability of closed-loop systems [10]. Even though some existing works like [16, 19] focus on the same topic, none of them addresses the control problem for both MIMO non-affine systems and input saturation problem. In other words, in this paper, we for the first time investigate a larger class of MIMO nonlinear systems considering both non-differentiable non-affine functions and input saturation.
Remark 3. From (3), there exist functions and taking values in and satisfyingTo make the control design succinct, define the functions and asUsing (5), we can model the non-affine terms asIn view of (5), it can be known thatwhere , and . According to (6) and the definition of , system (1) can be rewritten as
Assumption 4. The reference signal is continuous and available, and there exists a positive constant such that .
Assumption 5. For , there exist unknown positive constants satisfying .
Lemma 6 (see [8]). Consider the first-order dynamical system:with , and a positive function. Then, for any given bounded initial condition , the inequality holds.
Lemma 7 (see [17]). For any and , the hyperbolic tangent function fulfillsThe fuzzy logic systems (FLSs) are employed as function approximator. Construct FLSs with the following IF-THEN rules:where and are input and output of the FLSs. Based on the singleton fuzzifier, product inference, and center average defuzzifier, the FLSs can be formulated aswhere and are the membership of and , respectively. Letwhere , , and . Then, the FLSs can be expressed as follows:
Lemma 8 (see [23]). On a compact set , if is a continuous function, for any given constant , then there exist FLSs such that
3. Fuzzy Adaptive Controller Design
In this section, an adaptive fuzzy controller is proposed for a larger class of MIMO pure-feedback nonlinear systems (1) utilizing the DSC technique. To start, consider the following change of coordinates:where is the output tracking error, is the output of the first-order filter with as the input, is a positive design parameter, and is a dynamic system defined aswhere is a design parameter.
Remark 9. It has to be noted that, compared with the existing work [27, 30], a novel auxiliary system is proposed, and the dynamic system employed in this brief designed a bounded compensation term to cope with input saturation problem. Therefore, the restrictive bounded assumption of the dynamic system has been deleted.
Since are unknown continuous functions, we use fuzzy logic systems (FLSs) to approximate them as follows:where is the approximation error and satisfies with being an unknown constant.
Definewhere are unknown constants and is the estimate of with .
Step . Differentiating along with (16) yields
Consider the following quadratic Lyapunov function candidate:
Invoking (7), (20), and Assumption 5, we have
Substituting (18) into (22) giveswhere . In view of Young’s inequality, we can further havewhere is positive constant.
Then, construct the virtual control law and parameters adaptation laws and aswhere , , , , , and are design parameters, and is the estimate of .
Remark 10. Note that (26) and (27) satisfy Lemma 6. Thus, by choosing and , one has and for . Furthermore, since the initial conditions and are selected by control law designer, we choose and .
In line with the DSC technique, introduce variable . Let pass through a first-order filter with time constant to obtain asDefine the filter error , which yields andwhere is the introduced continuous function.
By and , we haveNoting that and , and substituting (25) and (30) into (24), we can further obtainTake the following Lyapunov function candidate:where and are the estimates of and , respectively.
It follows from (31) that the time derivative of isApplying (26), (27), and Lemma 7, one has
Step . Similar to the design method in Step differentiating along with (16) yields
Consider the following quadratic Lyapunov function candidate:
In view of Young’s inequality and using (35), we can obtain the time derivative of (36) aswhere , and is positive constant.
Take the virtual control law and parameters adaptation laws and as
The design process of parameters is similar to Step . Then, let pass through a first-order filter with time constant as follows:
Define , it yields andwhere , , , and is a continuous function.
According to and , one reaches
Substituting (38) and (43) into (37) results in
Consider the following Lyapunov function candidate:where and .
Noting and following the same way as Step give rise to
Applying (39), (40), and Lemma 7 yields
Step . Similar to the former design process, we can obtain
For , there exists a continuous function such that
Consider a compact set + ≤ . It can be seen from (43) that all the variables of are included in the compact set . Thus, have maximums on . There exist unknown positive constants such that .
Choosing the quadratic function as , it giveswhere .
According to Young’s inequality, one haswhere is positive constant.
Similarly, construct the actual control law and the adaptation laws and as
The design process of parameters is also similar to Step and Step . Take the following Lyapunov function candidate:where and .
Following the same way as the former steps gives
4. Stability Analysis
The main stability results of the MIMO pure-feedback nonlinear systems (1) are presented.
Theorem 11. Supposing that Assumptions 1, 4, and 5 hold and the above proposed design procedure is employed to MIMO pure-feedback nonlinear systems described by (1), for , , and , there exist design parameters , , , , , , and such that
(1) for , and hence all of the signals in the closed-loop systems remain semiglobally uniformly ultimately bounded;
(2) the output tracking error is such that , where is a positive constant depending on the design parameters. Furthermore, the whole system output tracking error satisfies with a positive constant that relies on the design parameters;
(3) the dynamic system is bounded, and the control input constraint is not violated.
Proof. Choose the following Lyapunov function candidate for the whole systems:where = + .
According to (34), (47), and (56), we can obtain the time derivative of asUsing the following inequalitieswe can arrive atwhere = + + .
By completion of squares, one haswith and being positive constants. Then, we can further rewrite (60) asThen, it can be known from [16] that has a maximum on the compact set . Let with being a positive constant. Setting , with being any positive constant, one haswhere . Noting (32), (45), (55), and (57), it yieldswhere . Note that can be made arbitrarily small by decreasing , , and and meanwhile increasing , , , and . Hence we can have by appropriately choosing the design parameters. It follows from and (64) that on the level set . Therefore, all the signals of the closed-loop systems are SGUUB. The property (1) of Theorem 11 is proved.
Solving (64) showswith a positive constant. According to (21), (36), and (57), we have . Using the first inequality in (65), the following inequality holds:Now let us consider the Lyapunov function candidate for the whole systems as . From (65), it can be derived thatwhere and . Then, we further havewhere is a positive constant.
Similarly, we have ,which leads toNoting that the size of depends on the design parameters , , , , , , and . Thus, by appropriately online-tuning the design parameters, the tracking error can be regulated to a neighborhood of the origin as small as desired and property (2) of Theorem 11 is proved.
Furthermore, for input saturation, there exists a nonnegative scalar to satisfy with and . Choosing the Lyapunov function candidate quadratic function as , we can obtainIf , we have . Therefore, will lie in the compact set and property (3) of Theorem 11 is proved. This completes the proof.
5. Simulation Analysis
In this section, two simulation examples are given to show validity of the proposed method in this paper.
Example 1. Consider the MIMO non-affine nonlinear uncertain systems as follows:where , , and , , = , and . and are defined as follows:It can be known that the existence of input saturation nonlinearity implies that non-affine functions and are non-differentiable. In this case, the existing approaches cannot be used. However, Assumption 1 in this paper is still satisfied which means that the scheme proposed here is able to deal with the control design difficulty in spite of the input saturation nonlinearity.
According to Theorem 11, the virtual control laws and actual control laws are constructed aswith adaptive lawswhere , , and . Let the initial conditions be , and . The simulation results are provided in Figures 1–5.
From Figure 1, we can see that the outputs and track the desired trajectories and with small tracking error. Figure 2 shows that the proposed scheme works well with bounded system inputs, and the response curves of adaptive parameters , , , , , , , and are depicted in Figure 3. From Figure 4, it can be seen that the bounds for , , and are not overstepped. Finally, Figure 5 is given to explain the boundedness of states , , , and .
Example 2. Consider the following two inverted pendulums systems composed of spring and damper connections. The pendulum angle and angular velocity were controlled using the torque inputs generated by a servomotor at each base. The dynamic equations can be described as follows [6]:where and are the angular positions, and are the moments of inertia, and are the masses, , denotes the force applied by the spring and damper at the connection points, and is the distance between the connection points as follows:where , and . The relative angular position can be defined asand are assumed to be a LuGre friction model defined aswhere , , , , and .
Defining , , and , system (75) can be rewritten in the following form:where , , , , , ; , , and . Moreover, and are described by the following:It can be seen that the non-affine functions are non-differentiable with respect to and . In simulation, choose the desired reference trajectories as and ; the virtual control laws, actual control laws, and adaption laws are provided by (25), (52) and (26)-(27), (53)-(54) with design parameters , , , , , , , , , , , , , , . Let the initial conditions be , and . The simulation results are provided in Figures 6–8.
As can be seen in Figure 6, the system outputs track the desired trajectories, perfectly. Figures 7-8 illustrate the system inputs and adaptive parameters, from which, we can see that the fairly good tracking performance is obtained.
6. Conclusion
This work for the first time proposes fuzzy adaptive dynamic surface control design for a larger class of MIMO non-affine nonlinear systems in the presence of input saturation. To overcome the design difficulty of input saturation, a novel auxiliary system with bounded compensation term has been proposed, and a bounded compensation term of tangent function is designed in this paper. Thanks to this design, we successfully relax the strictly bounded assumption of the dynamic system. SGUUB stability of the closed-loop systems is rigorously proved by combining Lyapunov theory and invariant set theory.
Data Availability
The simulation data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (Grant no. 71601183).