Abstract
Handling delays and uncertain parameters in control systems is an interesting and challenging class of problems. In this paper, we consider the problem of “bounded-input bounded-output stabilizing” a class of single-input single-output, linear time-varying plant models with a time-delay margin as large as desired and a considerable amount of uncertainty in the input matrix of the state-space model. The proposed controller, while periodic and mildly nonlinear, is of low complexity; it tolerates slow variations in the delay and the elements of the input matrix as well as occasional jumps in these parameters, and guarantees that the effect of the initial condition decays exponentially to zero, even in the presence of noise.
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Notes
If the closed-loop system were LTI, then this kind of stability is simply the classical bounded-input bounded-output (BIBO) notion of stability.
Roughly speaking, by this we mean that the estimate corresponds to a system for which we have maintained controllability.
Normally, \(\mathrm{sgn}(0)\) is defined to be zero.
For the plant model to be causal, we need y(t) to depend solely on x(t) and \(\{ u ( \theta ) : \theta \in [0, t] \}\) for \(t \ge 0\). This will be guaranteed if \(\tau (t)\) is always non-negative.
We have imposed a technical requirement that these variables are continuous from the right to simplify the proofs.
We delay d by \(\overline{\tau }\) seconds, so that d from time \(-\overline{\tau }\) to \(\infty \) is included in the norm of \(\overline{w}\).
Note that if there is no noise and we have zero initial conditions, then for every \(\tau \in [0, \overline{\tau }]\), we have that \(u(t-\tau ) = 0\) for \(t \in [kT, kT+T_1)\); this property is critical for the design of the state estimator.
If \(\tau (t)\) is constant for all \(t \ge 0\), then \(a[k] = T_1+\tau \) and \(b[k] = T_1+T_2+\tau \) for all \(k \ge 0\).
Checking to see if \(\check{B}_d[k] \) lies in \(\hat{\mathcal{B}}_d\) is easy: simply compute the norm of \(\check{B}_d[k] \) and the determinant of the related controllability matrix.
For every \(B_d \in \hat{\mathcal {B}_d}\), we have that \((A_d, B_d)\) is controllable, which implies that \(F(B_d)\) is unique.
We choose \(\rho > 0\) so that both terms on the RHS of (21) have the same sign; this means, in particular, that \(\nu [k] = 0\) iff \(\chi [k] = 0\).
Since the Eq. (23) contains both \(B_d[k]\) and \(B_d[k-1]\), clearly this solution depends on both quantities.
Note that \(\check{B}_d\) may not belong to \(\ell _\infty \); however, this signal is intermediary in nature and is used in the description of K to enhance clarity.
This holds even if there are switches in B(t) and \(\tau (t)\).
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This work was supported by the Natural Sciences and Engineering Research Council of Canada via a research grant.
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Gaudette, D.L., Miller, D.E. Adaptive stabilization of a class of time-varying systems with an uncertain delay. Math. Control Signals Syst. 28, 16 (2016). https://doi.org/10.1007/s00498-016-0166-5
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DOI: https://doi.org/10.1007/s00498-016-0166-5