Abstract
This paper proposes an AILC method for uncertain nonlinear system to solve different target tracking problems. The method uses fuzzy logic systems (FLS) to approximate every uncertain term in systems. All closed-loop signals are bounded on according to the Lyapunov theory. A time-varying boundary layer and a typical convergent series are introduced to handle initial state error, unknown bounds of errors, and nonuniform target tracking, respectively. The result is that the tracking error’s norm can converge to a small neighborhood along iteration increasing asymptotically. Finally, the simulation results of mass-spring mechanical system show the correctness of the theory and validity of the method.
1. Introduction
The research of the nonuniform trajectory is an interesting problem. The paper [1] proposed a new ILC law for first-order hybrid parametric system and the paper [2] proposed a novel AILC method for nonlinear hybrid parameter systems. Recently, AILC is presented; a nonuniform target tracking AILC method was proposed in the paper [3]. The paper [4] proposed a fault-tolerant ILC technique for mobile robots’ nonrepetitive target tracking with output constraints. From the above literature analysis, solving the nonuniform target tracking problem for uncertain mechanical nonlinear system is an important problem.
Adaptive control is used to handle system’s control problem about uncertainties which is a challenging problem. Adaptive control schemes learn uncertainties by adaptive laws. NN and FLS are used in the method as function approximators, such as the papers [5–8]. The literature [9] could complete the varying control tasks by designing an adaptive fuzzy ILC for uncertain nonlinear system. Based on RBF neural network approximation, the literature [10] proposed AILC for nonlinear pure-feedback systems to solve the nonuniform target tracking problem. The uniform AILC frame for uncertain nonlinear system was proposed in the paper [11]; by Lyapunov theory, it can prove the convergence. It should be noted that Lyapunov function-based AILC plays an important role in dealing with the time-varying parameter in the literatures [9, 11, 12]. But initial state error is a challenging one as they need to converge to zero for keeping stability. Only the papers [9, 10, 13] considered this problem recently it is an important problem for AILC.
Due to the physical limitations of actuators, in real systems, control inputs are often constrained, such as dead-zone inputs. However, these constraints may damage the performance of the system. In the papers [14–23], control performance could be changed by using different techniques recently. The question of adaptive stabilization for time-delay system had been solved in the literature [16]. The paper [17] developed an adaptive backstepping method of uncertain nonlinear systems about nonsymmetric dead-zone. As yet, there is no report from the literature for the AILC of nonlinear systems with nonsymmetric dead-zone input and initial state error. This is a problem that needs to be solved urgently.
In this paper, the nonuniform trajectory tracking issue is discussed for the uncertain nonlinear systems with nonsymmetric dead-zone input and initial state error. The contributions of the proposed control method are presented as follows:(i)The nonuniform trajectory tracking issue is studied for uncertain nonlinear systems under nonsymmetric dead-zone input and initial state error issues.(ii)The AILC method is used to uncertain nonlinear systems. FLS is introduced to learn unknown dynamic. A convergence order is introduced to solve the unknown bound and nonuniform target tracking problem.
Finally, simulation results of mass-spring mechanical system are given to verify the validity of the designed controller.
The remainder of this paper is organized as follows: in Section 2, the system description and related concepts are given in detail. The controller design process and main results are presented in Section 3. A simulation is shown in Section 4. Section 5 is the conclusion.
2. System Description
2.1. System Model
The following nonlinear systems are considered:where , is the state that is measured; represents the actuator characteristics and is the system output; , , are smooth unknown nonlinear functions.
Here, the function represents actuator output with nonsymmetric dead-zone and can be expressed aswhere and present the right and left slopes, respectively; and present the breakpoint of dead-zone. The nonsymmetric dead-zone can be rewritten as a combination of a line and a disturbance-like term [18].where
Assumption 1. Parameters , , , and are uncertain constants. There exists an unknown satisfing , which is small enough. The unknown constant is the upper bound of .
Designing an AILC law on to make the output following the target trajectory is our control objective, that is to say, , where is the small positive error. Guarantee that the closed-loop signals are bounded. is the smooth desired target. is the iteration index.
2.2. Convergent Series Sequence
The following definition and lemma can be used in the design process.
Definition 1. (see [20]). is a series convergent sequence which is shown aswhere ; and are the needed parameters, satisfing , .
Lemma 1 (see [20]). For , where , , the inequality as follows holds:
2.3. Description of Fuzzy Logic System (FLS)
The FLS has a good approximation property [24]. For a smooth can be approximated bywhere is the fuzzy approximated error, is the optimal weight vector satisfingwhere represents the set of and that is estimated , shown bywith an adjustable , and is given as follows:with being a positive constant.
Assumption 2. In this paper, the following inherent approximation error is assumed to be bounded with , where the unknown parameters denote the smallest upper bounds of with .
2.4. Time-Varying Boundary Layer
Introduce the function as follows to deal with initial state errors:where and are variables of ; is a series convergence sequence; the saturation function is given aswhere is the time-varying boundary layer. The control objective can be achieved by this. See the literature [9] for specific analysis.
Assumption 3. The initial state errors must satisfy for some positive parameters which are known; , are given later in this paper.
3. AILC Design and Convergence Analysis
The specific process about designing controller is given as follows.
Step 1. Let , , , and is a virtual controller. Introduce the error function and from Section 2.4 to deal with initial state errors asRecall thatDerivate as follows:According to Section 2.3, by FLS, is approximated as follows:where is the approximation error and is the optimal weight vector.
Define , which is needed later; , , and . The virtual controller is taken asSubstitute equations (16) and (17) into equation (15), thenwhere the estimated and are and , respectively. and are the errors of estimated parameters. The last two terms of equation (18) can be changed asEquations (18) and (19) can be rewritten asLet , then equation (20) becomes
Assumption 4. The bounded term satisfies , where is a positive parameter.
Take the following nonnegative function:where and are symmetric matrices and are positive. Derivate according to equation (21), thenIn the previous equation, is used, where .
Step j. (). Denote , , which is given later. , the same to Step 1; and from Section 2.4 are introduced asDerivate as follows:where . Denote , , then equation (25) can be rewritten asAccording to Section 2.3, by FLS can becomewhere is the approximation error and is the optimal weight vector.
The virtual controller is taken asEquations (27) and (28) are substituted into equation (26), then we havewhere and are the estimated parameters of and , respectively. and are estimated parameter errors. The last two terms of equation (30) can be changed asLet , then equation (29) becomes
Assumption j. satisfies ; here, is unknown.
Positive definite function is chosen:Derivate according to equation (31),
Step n. Define ; from Section 2.4 is introduced asDerivate as follows:where . Denote , , and by equation (3), then equation (35) can be rewritten asfrom which can be rewritten as follows:where is the approximation error and is an optimal weight vector.
Let . Take the actual controller aswhere is designed to compensate for unknown input gain .
Then, according to equations (37) and (38), equation (36) can be rewritten aswhere and are the estimated parameters of and , respectively. and are the estimated parameter errors.
The last two terms of equation (39) can be changed asThen,Take , where is the estimation of uncertain parameter and let , then equation (41) becomes
Assumption n. Because of assumptions 1 and 2, is bounded such that with being unknown.
Adaptive learning laws are taken as follows:where is the learning gain to be designed.
Positive definite function is chosen:Derivate according to equation (42) and substitute equations (43)–(45) into it, thenThe assumption below is needed.
Assumption 5. For any , as , , .
Theorem 1. Under assumptions 1, 2, 3, 4–n, and 5, design controller equation (38), and adaptive parameter estimation laws (43)–(45) for system (1), we get all signals bounded, andi.e., , as , where is a small positive parameter.
Proof. From Section 2.4, , where . By (46), we getwhere , .
Substituting equation (47) into equation (49), we getDenote , then equation (50) can be rewritten asBy equation (6), , then is bounded; by assumption 5 and equation (50), can be bounded. , soBy equation (46), for any , , by (47), thenBy equation (52), is bounded. By Definition 1, is bounded, and , so is also bounded. From above all, for any , is bounded, then we have ; and are bounded. By equation (38), is bounded. By equation (31), is bounded; is continuous uniformly; according to Barbalat lemma, when , i.e., , , then . , so , , finished.
4. Simulation
In this section, a mass-spring mechanical system is considered to show the effectiveness of the proposed controller. is a mass; assume that resistive force caused by friction is zero. The external force drives the mass, which is a control variable. is the displacement from a reference position; the motion equation of the system with nonsymmetric dead-zone input is as follows:where , is the spring’s restoring force; denotes the iteration index.
We define , , and , so equation (54) can be transformed into the state-space form
The spring’s restoring force can be modeled as
In the system, we have , , , and .
The nonsymmetric dead-zone is shown as
System objective is that the output of system (54) can follow the reference trajectory on when . In the different target case, as is even, and as is odd, where .
By Theorem 1, the AILC is chosen aswhere and the estimation laws are represented by equations (43)–(45), where , , , , , , , , and . The inputs of the FLS are , ; the membership function of every variable is given as follows:where , , , , and , and the fuzzy rules are given aswhere : as , , as , . The period of sampling is .
Here, choose the parameters and initial values as follows: , , , , , , and . Taking , Figures 1–5 are the simulation figures.
Figure 1 shows the convergence of error. Moreover, on the interval , Figures 2–5 give that , , , , , and are bounded. Figures 1–5 confirm the validity of control scheme which is developed in this article from the simulation results.
5. Conclusions
The different target tracking problem for unknown nonlinear systems with nonsymmetric dead-zone input is solved. Introduce FLS to deal with the uncertain dynamics, and the problem of approximation error and initial state error can be efficiently solved by suitable means. This paper can keep all signals being bounded on ; errors can converge to a small set along iteration increasing. Simulation study proves the correctness of AILC method of this paper.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant no. 61603296 and Natural Science Foundation Project of Shaanxi Provincial Education Department under Grant no. 16JK1541.