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Erschienen in:
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2017 | OriginalPaper | Buchkapitel

4. Regulation and Tracking in Linear Systems

verfasst von : Alberto Isidori

Erschienen in: Lectures in Feedback Design for Multivariable Systems

Verlag: Springer International Publishing

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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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Vorheriges Kapitel The Small-Gain Theorem for Linear Systems and Its Applications to Robust Stability
Nächstes Kapitel Coordination and Consensus of Linear Systems
Fußnoten
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.
 
Literatur
1.
B.A. Francis, The linear multivariable regulator problem. SIAM J. Control Optim. 14, 486–505 (1976)MathSciNet
2.
B. Francis, O.A. Sebakhy, W.M. Wonham, Synthesis of multivariable regulators: the internal model principle. Appl. Math. Optim. 1, 64–86 (1974)MathSciNetCrossRefMATH
3.
E.J. Davison, The robust control of a servomechanism problem for linear time-invariant multivariable systems. IEEE Trans. Autom. Control 21, 25–34 (1976)MathSciNetCrossRefMATH
4.
5.
A. Serrani, A. Isidori, L. Marconi, Semiglobal nonlinear output regulation with adaptive internal model. IEEE Trans. Autom. Control, AC 46, 1178–1194 (2001)MathSciNetCrossRefMATH
6.
H. Köroglu, C.W. Scherer, An LMI approach to \(H_\infty \) synthesis subject to almost asymptotic regulation constraints. Syst. Control Lett. 57, 300–308 (2008)CrossRefMATH
7.
J. Abedor, K. Nagpal, P.P. Khargonekar, K. Poolla, Robust regulation in the presence of norm-bounded uncertainty. IEEE Trans. Autom. Control 40, 147–153 (1995)MathSciNetCrossRefMATH
8.
H. Köroglu, C.W. Scherer, Scheduled control for robust attenuation of non-stationary sinusoidal disturbances with measurable frequencies. Automatica 47, 504–514 (2011)MathSciNetCrossRefMATH
Metadaten
Titel
Regulation and Tracking in Linear Systems
verfasst von
Alberto Isidori
Copyright-Jahr
2017
Buch
Lectures in Feedback Design for Multivariable Systems

Print ISBN: 978-3-319-42030-1

Electronic ISBN: 978-3-319-42031-8

Copyright-Jahr: 2017

DOI
https://doi.org/10.1007/978-3-319-42031-8_4

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