Abstract
It is well-known from the elementary theory of servomechanisms that a single-input single-output linear system, whose transfer function has all zeros in the left-half complex plane can be stabilized via output feedback. If the transfer function of the system has n poles and m zeros, the feedback in question is a dynamical system of dimension \(n-m-1\), whose eigenvalues (in case \(n-m>1\)) are far away in the left-half complex plane. In this chapter, this result is reviewed using a state-space approach. This makes it possible to systematically handle the case of systems whose coefficients depend on uncertain parameters and serves as a preparation to a similar set of results that will be presented in Chap. 6 for nonlinear systems.
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Notes
- 1.
The parameter \(K^{\prime }\) is sometimes referred to as the high-frequency gain of the system.
- 2.
For a proof see, e.g., [1, pp. 142–144].
- 3.
A triplet of matrices of this kind is often referred to as a triplet in prime form.
- 4.
The formula in question is
$$ \mathrm{det}\left( \begin{matrix}S &{} P \\ Q &{} R\\ \end{matrix}\right) = \mathrm{det}(S)\,\mathrm{det}(R - QS^{-1}P). $$.
- 5.
Otherwise, T(s) could be written as strictly proper rational function in which the denominator is a polynomial of degree strictly less than n. This would imply the existence of a realization of dimension strictly less than n, contradicting the minimality of A, B, C.
- 6.
- 7.
See Theorem A.3 in Appendix A.
- 8.
The negative sign is a consequence of the standing hypothesis (2.15). If \(b(\mu )<0\), the sign must be reversed.
- 9.
See Theorem A.2 in Appendix A.
- 10.
These coefficients can be easily derived as follows. Let
$$\begin{aligned} A_{01}(\mu ) = \left( \begin{matrix}a_{01,1} &{} a_{01,2} &{} \cdots &{} a_{01,r}\\ \end{matrix}\right) . \end{aligned}$$Hence
$$ A_{01}(\mu )\xi = a_{01,1}\xi _1 + a_{01,2}\xi _2+\cdots + a_{01,r}\xi _r. $$Since
$$\begin{aligned} \xi _r = \theta - (a_0\xi _1 + a_1\xi _2 + \cdots +a_{r-2}\xi _{r-1}) \end{aligned}$$we see that
$$ A_{01}(\mu )\xi = [a_{01,1}-a_{01,r}a_0]\xi _1 + [a_{01,2}-a_{01,r}a_1]\xi _2+\cdots + [a_{01,r-1}-a_{01,r}a_{r-2}]\xi _{r-1} + a_{01,r}\theta $$The latter can be rewritten as
$$ {\tilde{a}}_{01}(\mu )\xi _1+\cdots +{\tilde{a}}_{0,r-1}(\mu )\xi _{r-1} + {\tilde{a}}_{0r}(\mu )\theta . $$A similar procedure is followed to transform \(A_{11}(\mu )\xi + a_0\dot{\xi }_1 + \cdots + a_{r-1}\dot{\xi }_{r-1}\).
- 11.
See [3, pp. 36–37].
- 12.
See (A.1) in Appendix A.
References
A. Isidori, Sistemi di Controllo, vol. II (Siderea, Roma, 1993). in Italian
H.W. Bode, Network Analysis and Feedback Amplifier Design (Van Nostrand, New York, 1945)
K.J. Astrom, R.M. Murray, Feedback Systems (Princeton University Press, Princeton, 2008)
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Isidori, A. (2017). Stabilization of Minimum-Phase 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_2
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