Improvement on stability bounds for singularly perturbed systems via state feedback. (English) Zbl 0891.34055

Consider the linear autonomous singularly perturbed differential system \[ dx/dt = A_{11} x_1 + A_{12} x_2, \qquad \varepsilon dx_2 /dt = A_{21} x_1 + A_{22} x_2,\tag{*} \] where \(x_1 \in \mathbb{R}^{n_1} , x_2 \in \mathbb{R}^{n_2}, \varepsilon\) is a small positive parameter. The authors derive an algorithm to find a positive number \(\varepsilon^*\) such that the equilibrium point \(x_1 =0, x_2 =0\) of \((*)\) is asymptotically stable for \(0 < \varepsilon \leq \varepsilon^*\). The approach is based on the representation of \((*)\) in the form \(dx/dt =A(\varepsilon) x\), \(x \in \mathbb{R}^{n_1 + n_2}= \mathbb{R}^n\) and uses results by A. T. Fuller [J. Math. Anal. Appl. 23, 71-98 (1968; Zbl 0157.15705)] to estimate the spectrum of \(A(\varepsilon)\) by means of an \(n^2 \times n^2\)-matrix \(D(\varepsilon)\). Finally, the authors consider the problem to improve \(\varepsilon^*\) by applying a linear state feedback controller. They give some “learning” algorithm to attack this problem.


34D15 Singular perturbations of ordinary differential equations
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
34H05 Control problems involving ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations


Zbl 0157.15705
Full Text: DOI


[1] DOI: 10.1109/9.59817 · Zbl 0721.93059
[2] DOI: 10.1016/0167-6911(88)90059-X · Zbl 0655.93060
[3] DOI: 10.1016/0022-247X(68)90116-9 · Zbl 0157.15705
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[8] DOI: 10.1016/0005-1098(79)90021-9 · Zbl 0407.93033
[9] DOI: 10.1109/9.250478 · Zbl 0774.93053
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