Abstract
For discrete-time iterative learning control systems, the discrete Fourier transform (DFT) is a powerful technique for frequency analysis, and Toeplitz matrices are a typical tool for the system input–output transmission. This paper first exploits z-transform and DFT-based frequency properties for iterative learning control systems and studies the convergence property of a Toeplitz matrix to the power of iteration index. The exploitation exhibits that for the finite-length discrete-time iterative learning control systems, the time-domain convolution theorem for the z-transform and DFT is no longer true, and the Toeplitz matrix to the power of iteration index converges if and only if the identical diagonal element lies in the unit circle. Then, by considering the DFT to a finite-length sequence as a linear transform, it is easy to equivalently reform the input–output equation of linear discrete time-invariant and time-varying ILC systems as an algebraic discrete-frequency equation. Thus the derivative-type (D-type) iterative learning control (ILC) converges in a discrete-frequency domain if and only if it converges in a discrete-time domain. Numerical simulations are carried out to exhibit the validity and effectiveness.
Highlights
1 Introduction Since the iterative learning control (ILC) has been invented three decades before, it has been acknowledged as an efficacious intelligent strategy for a robot manipulator to repetitively execute a desired trajectory tracking over a finite time interval [1,2,3]
The pursuing aim is that the generated control input may drive the system to track the desired trajectory as precise as possible as the iteration index goes on, or in other words, the ILC is convergent
It is worth noticing that the input–output transmit matrix of a linear ILC-driven system is Toeplitz, and along the iteration axis the evolution behavior of the Toeplitz matrix to the power of iteration index must convey the learning performance evolution behaviors, including the transient overshooting, asymptotical convergence, or convergence monotonicity
Summary
Since the iterative learning control (ILC) has been invented three decades before, it has been acknowledged as an efficacious intelligent strategy for a robot manipulator to repetitively execute a desired trajectory tracking over a finite time interval [1,2,3]. The frequency-domain spectrum can be precisely computed by DFT in a direct manner This motivates the paper firstly to investigate DFT properties for linear discrete time-invariant (LDTI) derivative-type (D-type) ILC systems and the convergence of the Toeplitz matrix to the power of the iteration index. The spectrum at each frequency for a finite-length sequence computed by the z-transform making use of time-domain convolution theorem is only an approximate formulation. Comparing the summation expression on the right side of Eq (11) with that of (12), we observe that usualy Y +(m) = G1(m)U(m) unless N = +∞ This means that the time-domain convolution theorem for the discrete Fourier transform applied to a finite-length sequence is not true either. Equation (16) formulates the discrete-frequency relationship of output, shift impulse responses, and input
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