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.
Repetitive control (RC), introduced by Inoue et al. [
However, the analysis and design of repetitive control are mainly performed in the frequency domain [
It is worth pointing out that periodic variations are encountered in many real systems. These variations can exist in the system parameters [
Note that repetitive control is closely related to iterative learning control [
Motivated by the above discussion, this paper uses the formalism of discrete-time AILC [
The remainder of this paper is organized as follows. Section
Consider a discrete-time system with one unknown time-varying parameter
It is required that the state,
Defining the tracking error as
Here the weighting coefficient
Note that the adaptation process starts only after the first cycle is completed or
For the restriction of the next analysis, an assumption is exposed as follows.
The unknown time-varying parameters
Note that, in Assumption
For system (
There are two parts in the proof of Theorem
Define a nonnegative function
From (
Using (
Substituting (
In terms of (
Since
To show the learning convergence, we need to introduce the following lemma.
There must exist a constant
It is worth noting that we only need the existence of
According to (
Consider a scalar system with a specified relative degree
Note that (
Suppose that a bounded signal
Defining the tracking error as
The periodic adaptive control law is designed as
Note that the computation of
Let
An assumption is introduced as follows.
The unknown time-varying parameters
The validity of the above periodic adaption law is verified by the following theorem.
For system ( The parameter estimation error is bounded; that is, The tracking error converges to zero asymptotically as time instant
According to (
Substituting (
Define a nonnegative function
Note that, when
Furthermore, for a positive definite matrix
Thus, we can further simplify (
From the matrix inversion lemma [
Thus we can rearrange (
Using the error dynamics (
In order to evaluate the relationship between
From (
Therefore,
Note that
From (
Equation (
Following the same steps that lead to (
Since the nonlinear function is not sector-bounded, the following lemma is introduced to show the convergence performance.
There must exist a constant
See Appendix
Following the same steps that lead to (
Hence, we can directly conclude
Consider a system
Furthermore, we can see that the nonlinear function above is not satisfied with the linear growth condition.
It is required that
Note that the given desired trajectory
In the simulation, the initial value
The profile of nonlinear data-weighting factor
The tracking performance of system output.
The convergence property of tracking error with period.
Apparently, the effectiveness of the proposed data-weighting periodic adaptive control can be seen from Figures
For comparison, the following standard periodic adaptive control [
By selecting the same controller parameters and the same initial value
Finite time escape phenomenon of tracking error with standard periodic adaptive control.
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.
We arbitrarily choose a positive constant
Hence, for all
According to (
Since nonlinear function
Apparently, we have
By analogy, there exists a constant
Thus we can find a constant
The above discussion shows that (
For any
we know that both
From (
For any
For the convenience, we denote
From the system (
Similarly,
As a result,
By the same steps, one can conclude that
The above discussion shows that (
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by National Science Foundation of China (60974040, 61374102, and 61120106009).