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It has been known for some time that proportional output feedback will stabilize MIMO, minimum-phase, linear time-invariant systems if the feedback gain is sufficiently large. High-gain adaptive controllers achieve stability by automatically driving up the feedback gain monotonically. More recently, it was demonstrated that sample-and-hold implementations of the high-gain adaptive controller also require adaptation of the sampling rate. In this paper, we use recent advances in the mathematical field of dynamic equations on time scales to unify and generalize the discrete and continuous versions of the high-gain adaptive controller. We prove the stability of high-gain adaptive controllers on a wide class of time scales.

The concept of high-gain adaptive feedback arose from a desire to stabilize certain classes of linear continuous systems without the need to explicitly identify the unknown system parameters. This type of adaptive controller does not identify system parameters at all, but rather adapts the feedback gain itself in order to regulate the system. A number of papers examine the details of various kinds of high-gain adaptive controllers [

In this paper, we employ results from the burgeoning new field of mathematics called

We first state two assumptions that are required in the subsequent text.

The system model and feedback law are given by the linear, time-invariant, minimum phase system

Furthermore,

Under these conditions, it has been known for some time (e.g., [

The system of (A1) can be replaced by

The design objectives are to find graininess

It is important to keep in mind the generality of the expressions above. A great deal of mathematical machinery supports the existence of delta derivatives on arbitrary times scales, as well as the existence and characteristics of solutions to (

We begin this section with a definition and theorem from the work of Pötzsche et al. [

The set of exponential stability for the time-varying scalar equation

Solutions of the scalar equation

We note here that Pötzche, Siegmund, and Wirth did not explicitly consider scenarios where

The set

Let

Defining

First, suppose

On the other hand, suppose

At this point we pause briefly to discuss Lyapunov theory on time scales. DaCunha produced two pivotal works [

Before the next lemma, we define

Given assumptions (A1) and (A2) and

We construct

We comment briefly on the intuitive implication of Lemma

We now come to the three central theorems of the paper. If

In addition to (A1) and (A2), suppose

Set

Consider the Lyapunov function

Set

Set

Set

Recalling the standard inequality

We point out that, when

If

Consider the case when

We are now in a position to state the main theorem of the paper.

In addition to (A1) and (A2), assume the prototypical update law,

For the sake of contradiction, assume

It seems possible that Theorem

We remark here that there is a great amount of freedom in the choice of the update law for

For the first example, we point out that the notation in the previous sections somewhat belies the fact that the system's time domain (its time scale) may be fully or partially discrete, and thus there is no guarantee in Theorem

We remark here that if (

For the second example, we consider a problem posed by distributed control networks (c.f. [

The example of the previous paragraph is closely modeled using the variable

In graph (A), output

In summary, the paper illustrates a new unified continuous/discrete approach to the high-gain adaptive controller. Using developments in the new field of time-scale theory, the unified results reveal that this type of feedback control works well on a much wider variety of time scales than explored in previous literature, including those that switch between continuous (or nearly continuous) and discrete domains or those without monotonically decreasing graininess. A simulation of an adaptive controller on a mixed continuous/discrete time scale is also given. It is our hope that time-scale theory may find wider application in the broad fields of signals and systems as it seems that many of the tools needed in those fields are beginning to appear in their generalized forms.

We comment on the properties of the “expc” function referenced in the main body of the paper.

The power series (

For real, scalar arguments

Parts 1–3 follow immediately from the definition. To verify 4, note

This work was supported by NSF Grants no. EHS-0410685 and CMMI no. 726996 as well as a Baylor University Research Council grant. The authors thank their colleague, Robert J. Marks II, for his very helpful suggestions throughout this project.