Regulation and Tracking in Linear Systems

Isidori, Alberto

doi:10.1007/978-3-319-42031-8_4

Alberto Isidori⁴

Part of the book series: Advanced Textbooks in Control and Signal Processing ((C&SP))

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Abstract

In this chapter, we will study in some generality the problem of designing a feedback law to the purpose of making a controlled plant stable, and securing exact asymptotic tracking of external commands (respectively, exact asymptotic rejection of external disturbances) which belong to a fixed family of functions. The problem in question can be seen as a (broad) generalization of the classical set point control problem in the elementary theory of servomechanisms.

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Notes

1.
If some components of w are (external) commands that certain variables of interest are required to track, then some of the components of e can be seen as tracking errors, that is differences between the actual values of those variables of interest and their expected reference values. Overall, the components of e can simply be seen as variables on which the effect of w is expected to vanish asymptotically.
2.
See [1].
3.
See Appendix A.2.
4.
The arguments uses hereafter are essentially the same as those used in [2–4].
5.
It is worth observing that, since by assumption the matrix S is not affected by parameter uncertainties, so is its minimal polynomial $\psi (\lambda )$ and consequently so is the matrix $\varPhi $ defined in (4.23).
6.
Note that the filter (4.49) is an invertible system, the inverse being given by
$$\begin{array}{rcl} \dot{\eta } &{}=&{} (\varPhi -G\varGamma )\eta + G\tilde{e}\\ e &{}=&{} -\varGamma \eta + \tilde{e}.\end{array}$$
Hence, if $\varGamma $ is chosen in such that a way $\varPhi -G\varGamma $ is Hurwitz, the inverse of (4.49) is a stable system.
7.
Note that the subscript “e” has been dropped.
8.
Here and in the following we use the abbreviation “the triplet $\{A,B,C\}$” to mean the associated system (2.1).
9.
Since the parameters of the equation are $\mu $-dependent so is expected to be its solution.
10.
See Sect. 2.3.
11.
The approach in this section closely follows the approach described, in a more general context, in [5].
12.
It is seen from the construction in Example 4.1 that, in the normal form (4.11), the matrices $P_0$ and $P_1$ are found by means of transformations involving A, B, C, P and also S. Thus, if the former are functions of $\mu $ and the latter is a function of $\rho $, so are expected to be $P_0$ and $P_1$.
13.
Recall, in this respect, that both the uncertain vectors $\mu $ and $\rho $ range on compact sets.
14.
See Theorem B.6 in Appendix B.
15.
To prove (i), it suffices to observe that the positive-definite matrix
$$ Q= \left( \begin{matrix}1 &{} 0\\ 0&{} P_2\\ \end{matrix}\right) $$
satisfies
$$\begin{aligned} QF + F^\mathrm{T}Q = \left( \begin{matrix}0 &{} 0\\ 0 &{} -I\\ \end{matrix}\right) \le 0, \end{aligned}$$
and use LaSalle’s invariance principle. The proof of (ii) is achieved by direct substitution. Property (iii) is a consequence of (i) and of (ii), which says that all eigenvalues of $F+G\varGamma _\rho $ have zero real part. Property (iv) follows from Lemma 4.8.
16.
The approach in this section essentially follows the approach of [6]. See also [7, 8] for further reading.
17.
See Sect. 3.6.

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Dipartimento di Ingegneria Informatica, Automatica e Gestionale, Università degli Studi di Roma “La Sapienza”, Rome, Italy
Alberto Isidori

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Isidori, A. (2017). Regulation and Tracking in Linear Systems. In: Lectures in Feedback Design for Multivariable Systems. Advanced Textbooks in Control and Signal Processing. Springer, Cham. https://doi.org/10.1007/978-3-319-42031-8_4

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DOI: https://doi.org/10.1007/978-3-319-42031-8_4
Published: 13 August 2016
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-42030-1
Online ISBN: 978-3-319-42031-8
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