Spline Based Pseudo‐Inversion of Sampled Data Non‐Minimum Phase Systems for an Almost Exact Output Tracking

Abstract

This paper considers the problem of achieving a very accurate tracking of a pre-specified desired output trajectorỹ y(k), k ∈ ∠ Z + , for linear, multiple input multiple output, non-minimum phase and/or non hyperbolic, sampled data, and closed loop control systems. The proposed approach is situated in the general framework of model stable inversion and introduces significant novelties with the purpose of reducing some theoretical and numerical limitations inherent in the methods usually proposed. In particular, the new method does not require either a preactuation or null initial conditions of the system. The desiredỹ(k) and the corresponding sought input are partitioned in a transient component (ỹ t (k) and u t (k), respectively) and steady-state (ỹ s (k) and u s (k), respectively). The desired transient componentỹ t (k) is freely assigned without requiring it to be null over an initial time interval. This drastically reduces the total settling time. The structure of u t (k) is a priori assumed to be given by a sampled smoothing spline function. The spline coefficients are determined as the least-squares solution of the over-determined system of linear equations obtained imposing that the sampled spline function assumed as reference input yield the desired output over a properly defined transient interval. The steady-state input u s (k) is directly analytically computed exploiting the steady-state output response expressions for inputs belonging to the same set ofỹ s (k). Key Words:Model stable inversion, sampled data non-minimum phase systems, optimal transient tracking.

Using B‐spline functions (BSFs) of various degrees to obtain a powerful method for numerical solution for a special class of optimal control problems (OCPs)

“…Later, De Boor entitled to an especial class of the SFs that have minimal support with respect to a region partition and various degrees, smoothly as BSFs. The SFs and BSFs have been recently and widely used to deal with several optimal tracking problems . Any presented degree of the SFs can be denoted as a series of the BSFs of that degree.…”

“…The SFs and BSFs have been recently and widely used to deal with several optimal tracking problems. [38][39][40][41][42][43][44][45] Any presented degree of the SFs can be denoted as a series of the BSFs of that degree. The BSFs have equal distance support points, which are considered as the partition vector t = (t i , t i+1 , t i+2 , …, t i+d+1 ) and are called control points.…”

Precise Tracking for Continuous‐Time Non‐Minimum Phase Systems

“…Our goal is to solve (11) for d (t) and d (t) that are the stable inversion solutions for (3). The calculation process of (11) is similar to that of (9). For simplicity and clarity, only the calculation process of (11) is studied in detail, and the calculation process of (9) can be seen from that of (31).…”

Optimal transient performance under output set‐point reset

“…With the purpose of removing some limitations inherent in the aforementioned techniques, a new stable pseudo‐inversion approach has been recently proposed in .…”

“…In these references, the inverse problem consists in solving the convolution integral (or the convolution sum in the case of sampled data systems ) with respect to the transient input, which minimizes the L 2 norm of the transient tracking error. The reference input is ‘a priori’ assumed to belong to the space of piece‐wise continuously differentiable functions in or to the set of spline functions in . The assumed structure of the input guarantees an efficient solution of the optimization procedure and a sufficiently regular waveform of the reference input.…”