#### Abstract

A new periodic recursive least-squares (PRLS) estimator is developed with data-weighting factors for a class of linear time-varying parametric systems where the uncertain parameters are periodic with a known periodicity. The periodical time-varying parameter can be regarded as a constant in the time interval of a periodicity. Then the proposed PRLS estimates the unknown time-varying parameter from period to period in batches. By using equivalent feedback principle, the feedback control law is constructed for the adaptive control. Another distinct feature of the proposed PRLS-based adaptive control is that the controller design and analysis are done via Lyapunov technology without any linear growth conditions imposed on the nonlinearities of the control plant. Simulation results further confirm the effectiveness of the presented approach.

#### 1. Introduction

Repetitive control (RC), introduced by Inoue et al. [1, 2] originally, is an effective control scheme for tracking periodic reference and rejecting periodic disturbance signals. It is regarded as a simple learning controller and the control input is calculated using the information of the error signal in the preceding periods. The basic theory and convergence analysis were shown in the pioneering works [3–5]. The necessary and sufficient conditions for asymptotic stability of the continuous-time repetitive controller were restrictively formulated in [6]. In [7, 8], the stability of repetitive controllers is analyzed in the discrete-time domain. To enhance the robustness of these repetitive control schemes, researchers in [4, 7] modified the repetitive update rule to include the so-called Q-filter.

However, the analysis and design of repetitive control are mainly performed in the frequency domain [1–8], which makes the nonlinear study more difficult. Most of the stability results in repetitive control require that the dynamic system be assumed linear or it could be as least partially linearized with feedback control. As stated in [9], learning controllers could be synthesized and analyzed using a similar approach to the adaptive control. Adaptive control [10–15] of nonlinear systems has been an area of increasing research activity and global regulation and tracking results have been obtained for several classes of nonlinear systems. However, as indicated in [15], no adaptive control algorithms developed hitherto can solve unknown parameters with arbitrarily fast and nonvanishing variations.

It is worth pointing out that periodic variations are encountered in many real systems. These variations can exist in the system parameters [16], or as a disturbance to the system [17]. When the periodicity of system parameters is known a priori, some new adaptive controllers with periodic updating have been constructed by means of a pointwise integral mechanism [16, 18–20]. However, only a few results of discrete-time periodic adaptive control [21–23] were proposed and they have to impose linear growth conditions on the nonlinearities to provide global stability.

Note that repetitive control is closely related to iterative learning control [24], which is developed for control tasks that repeat in a finite duration with perfect tracking requirement. For example, a no-reset ILC approach was proposed in [25] because the system never actually starts, stops, resets, and then repeats, and a new nonstandard ILC algorithm was used in [26] for tracking periodic signals by defining a “trial” in terms of completion of a single “period” of the output trajectory. More recently, Chi et al. [27] proposed a discrete-time adaptive ILC (AILC), where the parameters are updated in the iteration domain. Further a nonlinear data-weighted iterative recursive least-squares algorithm [28] was developed to extend the discrete-time AILC to linear parametric systems without linear growth condition.

Motivated by the above discussion, this paper uses the formalism of discrete-time AILC [27, 28] to solve the repetitive control problem without assuming any linear growth conditions on the nonlinearities. A new periodic recursive least-squares (PRLS) algorithm is developed by using nonlinear data weighting. In the sequel, a new periodic adaptive control is proposed to overcome the sector-bounded restriction. Using the Lyapunov technology, the asymptotic convergence and global stability of the proposed periodic adaptive control are shown without requiring linear growth condition. This work is an extended version of the conference paper [23] and a more general case with multiple time-varying parameters is explored and discussed. Simulation study shows the applicability and effectiveness of the proposed approach.

The remainder of this paper is organized as follows. Section 2 gives the problem formulation and the controller design with rigorous convergence analysis. Section 3 extends the result to more general cases with multiple time-varying parameters and time-varying input gains. Some simulation results are provided in Section 4. Finally, some conclusions are given in Section 5.

#### 2. Problem Formulation and Controller Design

##### 2.1. Problem Formulation

Consider a discrete-time system with one unknown time-varying parameter where is the measurable system state; is the system control input; is an unknown time-varying parameter with a known periodicity , that is, ; and is a known nonlinear scalar function which is bounded for bounded .

It is required that the state, , follow a given reference trajectory . For the simplicity, we use denoting in the following discussion.

##### 2.2. Data-Weighting Periodic Adaptation

Defining the tracking error as , we have The new adaptive control mechanism is constructed as follows: where is used to learn the periodic time-varying parameter and updated as follows: where the initial value can be chosen according to some prior knowledge, or simple zero if no prior knowledge is available. Similarly, we can choose to be a sufficient large constant over the interval . is a nonnegative nonlinear data-weighting coefficient.

*Remark 1. *Here the weighting coefficient is used to attach more weight to those terms which are more affected by the nonlinearities. For this purpose, we allow to be a positive nonlinear function of all measured variables up to and including the time instant .

*Remark 2. *Note that the adaptation process starts only after the first cycle is completed or . The estimate for is set to be .

For the restriction of the next analysis, an assumption is exposed as follows.

*Assumption 3. *The unknown time-varying parameters and the target trajectory are uniformly bounded for all . Without loss of generality, we assume that , and , where and are some positive bounded constants.

*Remark 4. *Note that, in Assumption 3, we only assume the existence of such bounds, without requiring the exact values.

##### 2.3. Convergence Analysis

Theorem 5. *For system (1) under Assumption 3, the presented periodic adaptive control algorithm (3)–(5) can guarantee that (a) the parameter estimation value, , is bounded for all time t and that (b) the tracking error converges to zero asymptotically.*

*Proof. *There are two parts in the proof of Theorem 5, as shown in the following details.*Part (i): The Boundedness of *. Define the parametric estimation error . Substituting the control law (3) into the error dynamics (2) yields
Subtracting from both sides of (4), we have that, for any ,

Define a nonnegative function , and its difference with respect to the interval for any is

From (5), it is easy to derive

Using (9) and the error dynamics (6) leads to

Substituting (10) and (11) into (8) yields

In terms of (5), we can derive
so (12) can be rewritten as
Thus is nonincreasing, implying that is bounded. According to Assumption 3, is bounded, which directly leads to the boundedness of .*Part (ii): The Convergence of Tracking Error.* Applying (14) repeatedly for any and noticing , we have
Since , and , when , according to (15), one can derive that

Since is nonnegative and is finite in the interval of , according to the convergence theorem of the sum of series, we have
or

To show the learning convergence, we need to introduce the following lemma.

Lemma 6. *There must exist a constant such that, for all , one has
*

*Remark 7. *It is worth noting that we only need the existence of , without requiring its exact value. The proof of Lemma 6 is shown in Appendix A.

According to (9) and Lemma 6, one can derive that
that is, . So we have
Hence,
Clearly, we have
In the sequel, one can directly conclude that .

#### 3. Extension to Multiple Parameters and Time-Varying Input Gain

##### 3.1. Problem Formulation

Consider a scalar system with a specified relative degree , where are unknown periodic parameters and is a known vector-valued function. is a time-varying and uncertain gain of the system input. The prior information with regard to is that the control direction is known and invariant; that is, is either positive or negative and nonsingular for all . Without loss of generality, assume that , where is a known lower bound. Note that each unknown parameter or may have its own period or . The periodic adaptive control will still be applicable if there exists a common period , such that and can divide with an integer quotient. In such a case, can be used as the updating period. The presence of uncertain system input gain makes the controller design more complex.

Note that (24) can be incorporated as where and .

Suppose that a bounded signal represents the desired output of the system, and the value is known to the controller at time . The objective of periodic adaptive control is to generate a bounded control signal such that the state asymptotically approaches the specified bounded signal .

##### 3.2. Nonlinear Data-Weighting Periodic Adaptation

Defining the tracking error as , we have

The periodic adaptive control law is designed as

Note that the computation of requires the inverse of the system input gain estimate and may cause a singularity in the solution if is zero. To prevent the input singularity, a semisaturator is applied to the input gain estimation. The parameter updating law is where the covariance is a positive definite matrix of dimension and derived from the relationship by means of the matrix inversion lemma [10]. The initial values of , , can be chosen arbitrarily provided that . Similarly, we can choose the initial values , , with being a positive definite matrix. is a nonnegative nonlinear data-weighting coefficient.

Let denote the vector ; the semisaturator is defined as

##### 3.3. Convergence Analysis

An assumption is introduced as follows.

*Assumption 8. *The unknown time-varying parameters and the target trajectory are uniformly bounded for all . Without loss of generality, we assume that , and , where and are some bounded constants.

The validity of the above periodic adaption law is verified by the following theorem.

Theorem 9. *For system (24) under Assumption 8, the proposed period adaptive control scheme (27)–(30) has the following properties.*(a)*The parameter estimation error is bounded; that is, ,where , .*(b)*The tracking error converges to zero asymptotically as time instant t approaches to infinity.*

*Proof. *According to (27),

Substituting (31) into (26), we have
where .

Define a nonnegative function , and its difference with respect to the interval for any is

Note that, when , and . When , and the relationship holds. Thus the magnitude of the estimation error is the same or larger if no saturator is applied. As a result, we have

Furthermore, for a positive definite matrix , the following also holds:

Thus, we can further simplify (33) as

From the matrix inversion lemma [10] and (29), we have

Thus we can rearrange (36) as

Using the error dynamics (32), we can simplify (38) as

In order to evaluate the relationship between and described by (39), look at the term in (39).

From (29), one can derive

Therefore,

Note that is a positive definite matrix for any ; we can immediately obtain that
Thus is nonincreasing, and for any , , we have

From (37), we can derive

Equation (44) implies that
which establishes conclusion (a) of Theorem 9; that is,

Following the same steps that lead to (18) in Theorem 5 and by virtue of (42), one can conclude that

Since the nonlinear function is not sector-bounded, the following lemma is introduced to show the convergence performance.

Lemma 10. *There must exist a constant such that, for all , one has
*

*Proof. *See Appendix B.

Following the same steps that lead to (23) in Theorem 5 and using (48), it is easy to derive that

Hence, we can directly conclude .

#### 4. Illustrative Examples

Consider a system where and . The common periodicity of and is known as 20.

Furthermore, we can see that the nonlinear function above is not satisfied with the linear growth condition.

It is required that track a given reference

Note that the given desired trajectory is also a period function, but its periodicity known as 100 has nothing to do with the common periodicity of the unknown parameters. In the simulation, the periodicity used to update the period parameters is 20, instead of 100.

In the simulation, the initial value is set as 0. The other parameters are chosen as , , , and over the first period . By using the presented data-weighting periodic adaptive control (27)–(29), the simulation results are shown in Figures 1–3, respectively. Figure 1 is the profile of nonlinear data-weighting factor . Figure 2 is the tracking performance of system output. And the convergence of tracking error is shown in Figure 3, where is used to record the maximum absolute tracking error during the th period. Obviously, the proposed method results in good convergence in a pointwise manner.

Apparently, the effectiveness of the proposed data-weighting periodic adaptive control can be seen from Figures 1–3. Although the nonlinear system in the simulation is not sector-bounded, the tracking error converges asymptotically to zero as period number approaches to infinity. The reason is that the nonlinearities are compensated by using the data weighting factor (Figure 1).

For comparison, the following standard periodic adaptive control [22] without nonlinear data-weighting factor is applied:

By selecting the same controller parameters and the same initial value as that in the previous simulation, the profile of tracking error is shown in Figure 4. It is obvious that a finite time escape phenomenon may occur for the tracking error without using the nonlinear data-weighting factor .

#### 5. Conclusion

A new adaptive control is proposed with periodic least-squares estimate for a class of discrete-time systems to address periodic time-varying parameters. The only prior knowledge needed in the periodic adaptation is the periodicity. The periodic parameter updating law proposed here is updated in the same instance of two consecutive periods. A major distinct feature is that a nonlinear data-weighting function is introduced into the parameter updating law to address nonlinearities without requiring any growth condition. Both theoretical analysis and numerical simulations verify the effectiveness of the proposed approach.

#### Appendices

#### A. Proof of Lemma 6

*Proof. *We arbitrarily choose a positive constant and examine the following two cases.*Case 1*. When , then (19) is satisfied with .*Case 2*. When , according to the definition of and the relationship of (14), it is obvious that

Hence, for all , one has
where .

According to (1) and (3), we have

Since nonlinear function is bounded for bounded , apparently there exists a constant such that is bounded.

Apparently, we have
and there exists a constant such that .

By analogy, there exists a constant such that

Thus we can find a constant , which is bounded since is a finite periodicity, such that . Then (19) is satisfied with .

The above discussion shows that (19) is satisfied for all with .

#### B. Proof of Lemma 10

*Proof. *For any and , arbitrarily choose a positive constant and examine the following two cases.(1). Then (48) is satisfied with .(2). Since
we know that both and are bounded.

From (45), we know that for all
where .

and have been shown bounded; thus
for some constant . Clearly the control input signal is bounded.

For any , since the selected initial state values are bounded, clearly and thus are bounded according to (B.3).

For the convenience, we denote , , and in the following.

From the system (24), it is easy to get

Similarly,

As a result, is bounded for the bounded , with . According to (B.3), is bounded; thus is bounded too.

By the same steps, one can conclude that is bounded. Without loss of generality, we assume that ; then (48) is satisfied with .

The above discussion shows that (48) is satisfied for all with .

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

This work was supported by National Science Foundation of China (60974040, 61374102, and 61120106009).