Characteristic Model-Based Adaptive Control for a Class of MIMO Uncertain Nonaffine Nonlinear Systems Governed by Differential Equations

/is paper addresses the difficulty of designing a controller for a class of multi-input multi-output uncertain nonaffine nonlinear systems governed by differential equations. We first derive the first-order characteristic model composed of a linear time-varying uncertain system for such nonaffine systems and then design an adaptive controller based on this first-order characteristic model for position tracking control. /e designed controller exhibits a simple structure that can effectively avoid the controller singularity problem. /e stability of the closed-loop system is analyzed using the Lyapunov method. /e effectiveness of our proposed method is validated with a numerical example.


Introduction
e study of complex nonlinear systems governed by differential equations has attracted considerable attention [1][2][3][4]. It is worth noting that in the real world, there are many nonaffine nonlinear systems, such as certain flight control systems [5,6], tank reactor system [7,8], biochemical processes [9,10], and carrier landing control systems for unmanned aerial vehicles [11]. e study of nonaffine systems is often much more difficult and complicated than that of affine ones because control inputs always appear implicitly in the nonlinear functions [7], and finding the explicit inversion of nonaffine functions is quite difficult, even when the existence of such nonlinear function inversion can be proved using the implicit function theorem [5,9]. On the other hand, in practical applications, uncertainties often arise owing to the presence of unknown parameter variations, modeling simplifications, unmodeled dynamics, and external disturbances, among others [12][13][14]. In certain practical situations, establishing accurate and detailed models of nonlinear systems may be difficult, costly, and time consuming. erefore, the control design for uncertain nonaffine systems is an interesting yet challenging problem. Over the past decades, nonaffine nonlinear system control has received considerable attention, and many important results have been presented on the design of such controllers [5-12, 14, 15]. Owing to the universal approximation theorem, some adaptive control schemes based on fuzzy logic systems [16][17][18][19][20][21][22] and neural networks [8,10,[23][24][25] have been developed for uncertain nonaffine nonlinear systems. Nevertheless, these approaches typically incur heavy computational costs. Apart from these approaches, other adaptive control methods have also been considered for uncertain nonaffine-in-control nonlinear systems, such as adaptive dynamic surface control [7] and generalized PI control [26].
Recently, a majority of representative adaptive control schemes focuses on the continuous-time domain for nonaffine nonlinear systems described by differential equations. However, in engineering implementations, computers are usually utilized to produce digital control signals, which require controllers to be designed in discrete time [27]. e above discussion motivates us to use a novel method, called characteristic modeling proposed by Wu [28][29][30][31][32][33][34][35], to solve the control design problem for uncertain nonaffine systems. e key idea of the characteristic modeling is to use a lower-order discrete time-varying linear system to express an original continuous system equivalently based on both the dynamic characteristics of controlled plants and performance specifications. is discrete system is called the "characteristic model of the original system." It is worth mentioning that the characteristic modeling differs from the conventional model reduction methods in that all information of the original dynamic model, including uncertainties and nonlinearities, is compressed/integrated into the time-varying coefficients of the characteristic model instead of truncating parts of the plant model [28,32,33]. e timevarying coefficients of the characteristic model are referred to as the characteristic parameters [33,35,36], and the characteristic parameters are identified online adaptively [31,33,35,37]. e characteristic model-based adaptive control might be a fresh perspective in the field of adaptive control: creating simplicity out of complexity [31], and it has already been applied successfully to more than 400 engineering systems including spacecraft, e.g., the reentry control of a manned spaceship [28,29,32], the control of servo systems with backlash and friction [30], and the control for the swing arm in a Fourier transform spectrometer [38].
However, in the existing literature, the main focus is on the second-order characteristic model; the first-order model has received scarce attention for related results on controller design based on the characteristic model, and few studies have reported on the control method for nonaffine nonlinear systems based on the characteristic model. In the controller design method based on the second-order characteristic model, almost all the existing control methods belong to the indirect method in adaptive control. In other words, the model parameters are identified by the input and output of the system, and then the control law is designed. For the second-order characteristic model, the most successful application in practical engineering is the characteristic modelbased golden section adaptive control method, and it must also be combined with the integral controller and the differential controller to improve the control performance [30,33,35]. is makes it very difficult to analyze the stability of the control system. Stability analysis, especially for MIMO systems, remains a challenging problem for adaptive control based on the second-order characteristic model despite its practical successes. Compared with the secondorder characteristic model, the first-order model has the advantages of simpler form, fewer identified parameters, and more convenient applicability for practical systems.
In this paper, inspired by [28,39], we derive the firstorder characteristic model for multi-input multi-output (MIMO) uncertain nonaffine nonlinear systems. en, we develop a characteristic model-based adaptive control law for the position tracking problem. By constructing a suitable Lyapunov functional, we investigate the stability of the closed-loop system involving the first-order characteristic model-based control law.
Compared with the existing results, the primary contributions and novelties of our study can be highlighted as follows: (1) e first-order characteristic model composed of a linear time-varying uncertain system is derived for a class of MIMO uncertain nonaffine nonlinear systems governed by differential equations. To the best of the authors' knowledge, first-order characteristic models have only been presented by Xu and Hu [39] and Sun et al. [40]. We note that SISO systems are only considered in [39,40] and linear systems are only considered in [40]. (2) An adaptive controller based on this first-order characteristic model for position tracking control (see formula (22)) is designed by adaptively estimating the time-varying coefficients of the characteristic model (see formula (23)). In the existing literature, no results have been reported on the adaptive control design based on the first-order characteristic model for our considered class of nonaffine multivariable systems. (3) e ranges of controller parameters, obtained (see Lemma 1) by utilizing some mathematical technique, guarantee the stability of the closed-loop system (see eorem 2). In [39], no intervals of controller parameters are presented. (4) e designed controller exhibits a simple structure and can effectively avoid possible controller singularity problems. e paper is organized as follows. e problem formulation and the characteristic model are given in Section 2. en, we present the design of the adaptive controller based on the characteristic model and the stability analysis by using the Lyapunov theory in Section 3. e effectiveness of the proposed method is illustrated by a numerical example in Section 4. Section 5 concludes this paper.

Problem Formulation and
Characteristic Modeling e following notation is used in the paper. ‖x‖ denotes the Euclidean norm for a vector x. ‖A‖ denotes the Frobenius norm for a matrix A. R n denotes the n-dimensional Euclidean space. I is the n-dimensional identity matrix. Let T s > 0 be the sampling period.
Consider a class of MIMO uncertain nonaffine nonlinear systems: where x � [x 1 , . . . , x n ] T ∈ R n and u � [u 1 , . . . , u n ] T ∈ R n represent the state (output) and the input vector of the controlled plant, respectively.f(x, u) � [f 1 (x, u), . . . , f n (x, u)] T is an unknown smooth nonlinear function vector.

Complexity
Let r i ∈ (0, 1) for i � 1, 2, . . . , n. For Jacobian matrix zf(x, u)/zu, u in the ith row is replaced by r i u, and the resulting matrix is denoted as f u (x, ·). We suppose that system (1) satisfies the following assumptions.
e control objective is to design the sampled data controller u i (t) � u i (kT s ), kT s ≤ t ≤ (k + 1)T s , such that the output x i (t) of the controlled system (1) tracks the preset value x ir ,i � 1, 2, . . . , n.
We first define the following set: where η i is a small positive constant.
we refer to y i,k as the unified variable. From (3), we can see that unified variables y i,k are not close to zero. (1) satisfying Assumption 1, if position keeping or position tracking control is desired, then its characteristic model can be described by the following system of first-order difference equations:

Theorem 1. Given an MIMO nonaffine nonlinear system
where θ i,k is the so-called characteristic parameter as follows: where g i,k is the ith component of the vector Proof. For a fixed i(i � 1, . . . , n), the following equation can be obtained based on the differential mean value theorem: where erefore, we have where According to Assumption 1, the inverse function f − 1 must exist. us, multiplying both sides of (8) with f − 1 yields Integrating both sides of the above equation yields Now set t � (k + 1)T s ; then, (11) can be rewritten as Setting t � kT s , (11) can also be rewritten as Subtracting (13) from (12) yields Complexity 3 Denote en, we have x((k + 1)T s ) � x(kT s ) + g k + u k , or equivalently, Next, we analyze two cases individually.
then the ith output from the controlled object is far away from zero. erefore, (17) can be rewritten as Case 2. If x i (k) ∈ Ω i , then the ith output from the controlled object is close to zero. In this case, (17) can be rewritten as Combining Cases 1 and 2 from (3), we can obtain the characteristic model described by (4). Remark 1. From Assumption 2, we can derive the range of the characteristic parameter θ i,k . In fact, from Assumption 2, there exist positive constants M f and M u such that is is not restrictive, and the supremum of M xk exists in practical engineering systems [41,42].

Controller Design and Stability Analysis
We define the tracking errors as e i,k � x i,k − x ir for i � 1, 2, . . . , n. e reference tracking signal of the unified variable y i,k can be expressed by y ir � x ir + D i d i . Note that the tracking errors of the unified variables equal the ones of the original object, which can be seen from We present the basic approach to obtain an adaptive controller that achieves our control objective. If the function f is known, then each θ i,k is also known. e control laws u * i,k � (− θ i,k y i,k + a i0 e i,k + y ir )/T s (i � 1, 2, . . . , n) applied to (4) result in y i,k+1 � a i0 e i,k + y ir , that is, e i,k+1 − a i0 e i,k � 0. Notice that if one chooses a i0 such that |a i0 | < 1, where i � 1, 2, . . . , n, then the roots of the polynomials H(q i ) � q i − a i0 related to the characteristic equation of e i,k+1 − a i0 e i,k � 0 are inside unit circles, which implies that lim k⟶+∞ e i,k � 0. is is main objective of control. Because f is unknown, the optimal control u * i,k cannot be implemented. us, our purpose is to design an adaptive controller to approximate this optimal control. Based on the above ideas, the adaptive controller is designed as where a i0 ∈ (0, 1) and θ i,k denotes the estimation of the parameter θ i,k . Note that a i0 ∈ (− 1, 0] is not considered for the convenience of theoretical analysis. e adaptive law of parameter θ i,k is designed as where λ i > 0,μ i > 0, and 0 < m i < 1 represent design constants. e adaptive controller (22) includes estimated parameters θ i,k , and the performance of asymptotic tracking with zero error or near zero can be obtained by adjusting the design parameters λ i > 0, μ i > 0, and 0 < m i < 1.
Let φ i,k � θ i,k − θ i,k be the parameter estimation error. e error equation of the closed-loop system can be expressed by substituting (22) into (4) as follows: From (23), we have us, as the basis for preparing the stability analysis of the closed-loop system, the relation between the parameter estimation errors at two adjacent instants can be obtained as follows: 4 Complexity where For subsequent use, we denote . . , n. en, we have the following lemma.
Proof. For the sake of simplicity, the subscript i for all variables is omitted in this proof. For example, a 0 written in this proof actually represents a i0 . We first need to prove that (30) is well defined. It should be noted that is well defined and clearly μ > 0.

□
We set T i � min η i , (d i − η i ) , i � 1, 2, . . . , n; T min � min 1≤i≤n T i /G i . We require the following lemma to summarize the performance of our proposed control strategy.

Theorem 2.
Consider an MIMO nonaffine nonlinear system (1) satisfying Assumptions 1 and 2. If the adaptive tracking controller is designed as (22) with update law (23), the sampling period T s < T min , and controller parameters Proof. First, we note that from (20), ε i < 1 when the sampling period T s < T min . us, δ i1 � 1 − ε i > 0. Since μ i > a i0 /δ i1 , it is easy to see that μ i > a i0 /δ i2 . From Lemma 1, there exist real numbers λ i such that λ i ∈ (λ i1 , λ i2 ) for i � 1, 2, . . . , n. Now, consider the Lyapunov function given by Part 1. Calculating the difference of the Lyapunov function (33). By utilizing (24), we can derive the first difference of the Lyapunov function (33) as Using (26), the last term − φ 2 i,k− 1 in (34) can be written as Now, substituting (27) into φ i,k w i,k in the last term of (35) and employing y i,k � e i,k + y ir , we then have Substituting (36) into (35) and then into (34) yields Denote en, (37) can be rewritten as Part 2. Enlarging the difference ΔV i,k denoted by (39). For further analysis, we derive the ranges of S i1 , S i2 , and S i3 shown in (38). Let S ij1 and S ij2 denote the minimum and maximum values of S ij (j � 1, 2, 3), respectively. Based on the expressions of S i1 and S i3 , we have provided that c i1 , c i2 , and c i3 > 0. Substituting these three inequalities into (39) yields 6 Complexity Denote en, we have this is, Part 3. Giving the properties of N i1 , N i2 , and N i3 in (46). Based on the bounds of the characteristic parameters provided by (20), we can prove that there exist c i1 , c i2 , and c i3 > 0, such that N i1 ∈ (0, 1); N i2 and N i3 > 0. To prove these, it suffices to show that Indeed, if (47) holds, then there exist c i1 that satisfy us, there exists a positive number Note that Consequently, we can obtain N i2 > 0 for any c i1 and c i2 that satisfy (48) and (49), respectively.
In what follows, we shall prove that (47) holds indeed. For simplicity, the subscript i of variables is omitted until the end of the proof of (47), except for S i11 and S i12 . For example, we use the notation μ instead of μ i .
(i) First, we prove the first formula in (47). Since )/c 2 ) > 1. Multiplying both sides of the above inequality by m 2 yields (ii) Next, we prove the second formula in (47).

Numerical Example
To verify the effectiveness of the developed control method, consider a nonaffine double-pendulum system. e dynamic equations of motion are given as [44] _     10 Complexity 2 × 5 × 5 × 5 × 5 � 1250 rules that are used in the simulation. e initial state of each variable is the same as in the above simulation.
e results of this simulation are illustrated in  e figures indicate that the adaptive fuzzy controller can also achieve better performance. However, it should be emphasized that this is the result of more than 2 min of CPU time running on a computer (Lenovo workstation P720 with dual Intel ® Xeon ® Gold 6136 Processor), whereas the results shown in Figures 1-4 only require 7 s of CPU time on the same computer.
ere are 1250 parameters that need to be updated online for the scheme in [45], whereas only 2 parameters are required in our scheme. e fewer parameters used in our scheme help to make the proposed algorithm more efficient.

Conclusions
In this study, the control method for a class of uncertain nonaffine systems is studied for the first time by using a firstorder characteristic model. Specifically, we first transformed the nonaffine uncertain system into a linear first-order characteristic model with bounded time-varying parameters. Based on this model, we designed an adaptive controller with a simple structure and proved that the tracking errors converge to the neighborhoods of the origin. e proposed approach can avoid the controller singularity because there is no inverse operator construction in the controller. As an illustration of the effectiveness of the proposed method, a numerical example is provided.

Data Availability
No data were used to support this study.