Adaptive Control for Linear TimeInvariant Systems
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Abstract
Adaptive control of linear timeinvariant (LTI) systems deals with the control of LTI systems whose parameters are constant but otherwise completely unknown. In some cases, large norm bounds as to where the unknown parameters are located in the parameter space are also assumed to be known. In general, adaptive control deals with LTI plants which cannot be controlled with fixed gain controllers, i.e., nonadaptive control methods, and their parameters even though assumed constant for design and analysis purposes may change over time in an unpredictable manner. Most of the adaptive control approaches for LTI systems use the socalled certainty equivalence principle where a control law motivated from the known parameter case is combined with an adaptive law for estimating on line the unknown parameters. The control law could be associated with different control objectives and the adaptive law with different parameter estimation techniques. These combinations give rise to a wide class of adaptive control schemes. The two popular control objectives that led to a wide range of adaptive control schemes include model reference adaptive control (MRAC) and adaptive pole placement control (APPC). In MRAC, the control objective is for the plant output to track the output of a reference model, designed to represent the desired properties of the plant, for any reference input signal. APPC is more general and is based on control laws whose objective is to set the poles of the closed loop at desired locations chosen based on performance requirements. Another class of adaptive controllers for LTI systems that involves ideas from MRAC and APPC is based on multiple models, search methods, and switching logic. In this class of schemes, the unknown parameter space is partitioned to smaller subsets. For each subset, a parameter estimator or a stabilizing controller is designed or a combination of the two. The problem then is to identify which subset in the parameter space the unknown plant model belongs to and/or which controller is a stabilizing one and meets the control objective. A switching logic is designed based on different considerations to identify the most appropriate plant model or controller from the list of candidate plant models and/or controllers. In this entry, we briefly describe the above approaches to adaptive control for LTI systems.
Keywords
Adaptive Pole Placement Control (APPC) Model Reference Adaptive Control (MRAC) Unknown Parameter Space Indirect MRAC MRAC SchemeModel Reference Adaptive Control

Plant Assumptions: G _{ p }(s) is minimum phase, i.e., has stable zeros, its relative degree, n ^{∗} = number of poles − number of zeros, is known and an upper bound n on its order is also known. In addition, the sign of its highfrequency gain is known even though it can be relaxed with additional complexity.

Reference Model Assumptions: W _{ m }(s) has stable poles and zeros, its relative degree is equal to n ^{∗} that of the plant, and its order is equal or less to the one assumed for the plant, i.e., of n.
The above assumptions are also used to meet the control objective in the case of known parameters, and therefore the minimum phase and relative degree assumptions are characteristics of the control objective and do not arise because of adaptive control considerations. The relative degree matching is used to avoid the need to differentiate signals in the control law. The minimum phase assumption comes from the fact that the only way for the control law to force the closedloop plant transfer function to be equal to that of the reference model is to cancel the zeros of the plant using feedback and replace them with those of the reference model using a feedforward term. Such zero pole cancelations are possible if the zeros are stable, i.e., the plant is minimum phase; otherwise stability cannot be guaranteed for nonzero initial conditions and/or inexact cancelations.
 Control law:where \(\alpha \triangleq \alpha _{n2}(s) = {[{s}^{n2},{s}^{n3},\ldots ,s,1]}^{T}\) for n ≥ 2, and \(\alpha (s) \triangleq0\) for n = 1, and Λ(s) is a monic polynomial with stable roots and degree n − 1 having numerator of W _{ m }(s) as a factor.$$u_{p} =\theta _{ 1}^{T}(t) \frac{\alpha (s)} {\Lambda (s)}u_{p} +\theta _{ 2}^{T} \frac{\alpha (s)} {\Lambda (s)}y_{p} +\theta _{3}(t)y_{p} + c_{0}(t)r {=\theta }^{T}(t)\omega ,$$(4)
 Adaptive law:where Γ is a positive definite matrix referred to as the adaptive gain and \(\dot{\rho }=\gamma \epsilon \xi\), \(\epsilon= \frac{e_{1}\rho \xi } {m_{s}^{2}}\), \(m_{s}^{2} = 1 {+\phi }^{T}\phi + u_{f}^{2}\), \(\xi {=\theta }^{T}\phi + u_{f}\), \(\phi = W_{m}(s)\omega\), and u _{ f } = W _{ m }(s)u _{ p }.$$\dot{\theta }= \Gamma \epsilon \phi ,$$(5)
The stability properties of the above direct MRAC scheme which are typical for all classes of MRAC are the following (Ioannou and Fidan 2006; Ioannou and Sun 1996):(i) All signals in the closedloop plant are bounded, and the tracking error e _{1} converges to zero asymptotically and (ii) if the plant transfer function contains no zero pole cancelations and r is sufficiently rich of order 2n, i.e., it contains at least n distinct frequencies, then the parameter error \(\vert \tilde{\theta }\vert= \vert \theta {\theta }^{{\ast}}\vert \) and the tracking error e _{1} converge to zero exponentially fast.
Additional details on MRAC are presented in chapter 116 solely devoted to MRAC.
Adaptive Pole Placement Control
P1. G _{ p }(s) is strictly proper with known degree, and R _{ p }(s) is a monic polynomial whose degree n is known and Q _{ m }(s)Z _{ p }(s) and R _{ p }(s) are coprime.
Assumption P1 allows Z _{ p } and R _{ p } to be nonHurwitz in contrast to the MRAC case where Z _{ p } is required to be Hurwitz.
Search Methods, Multiple Models, and Switching Schemes
One of the drawbacks of APPC is the stabilizability condition which requires the estimated plant at each time t to satisfy the detectability and stabilizability condition that is necessary for the controller parameters to exist. Since the adaptive law cannot guarantee such a property, an approach emerged that involves the precalculation of a set of controllers based on the partitioning of the plant parameter space. The problem then becomes one of identifying which one of the controllers is the most appropriate one. The switching to the “best” possible controller could be based on some logic that is driven by some cost index, multiple estimation models, and other techniques (Fekri et al. 2007; Hespanha et al. 2003; Kuipers and Ioannou 2010; Morse 1996; Narendra and Balakrishnan 1997; Stefanovic and Safonov 2011). One of the drawbacks of this approach is that it is difficult if at all possible to find a finite set of stabilizing controllers that cover the whole unknown parameter space especially for highorder plants. If found its dimension may be so large that makes it impractical. Another drawback that is present in all adaptive schemes is that in the absence of persistently exciting signals which guarantee that the input/output data have sufficient information about the unknown plant parameters, there is no guarantee that the controller the scheme converged to is indeed a stabilizing one. In other words, if switching is disengaged or the adaptive law is switched off, there is no guarantee that a small disturbance will not drive the corresponding LTI scheme unstable. Nevertheless these techniques allow the incorporation of wellestablished robust control techniques in designing a priori the set of controller candidates. The problem is that if the plant parameters change in a way not accounted for a priori, no controller from the set may be stabilizing leading to an unstable system. More details can be found in chapter 119 on switching adaptive control.
Robust Adaptive Control
The MRAC and APPC schemes presented above are designed for LTI systems. Due to the adaptive law, the closedloop system is no longer LTI but nonlinear and time varying. It has been shown using simple examples that the pure integral action of the adaptive law could cause parameter drift in the presence of small disturbances and/or unmodeled dynamics (Ioannou and Fidan 2006; Ioannou and Kokotovic 1983; Ioannou and Sun 1996) which could then excite the unmodeled dynamics and lead to instability. Modifications to counteract these possible instabilities led to the field of robust adaptive control whose focus was to modify the adaptive law in order to guarantee robustness with respect to disturbances, unmodeled dynamics, timevarying parameters, classes of nonlinearities, etc., by using techniques such as normalizing signals, projection, fixed and switching sigma modification, etc. More details on this topic can be found in chapter 118 on robust adaptive control.
CrossReferences
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