Reduced spatial order model reference adaptive control of spatially varying distributed parameter systems of parabolic and hyperbolic types

Abstract

This paper presents control laws for distributed parameter systems of parabolic and hyperbolic types with unknown spatially varying parameters. These laws, based on the model reference adaptive control approach, guarantee asymptotic tracking of the output of the reference model by the output of the plant for arbitrary time invariant, but spatially varying reference input. The novel capabilities of the algorithms proposed are providing reduced sensitivity to measurement noise due to the reduced order of the spatial di!erentiation of the output data and permitting on-line estimation of the spatially varying plant parameters, constructively enforceable through the reference input and/or boundary conditions. The parameter estimation is carried out by means of an auxiliary system with the time-varying parameters that simultaneously converge in¸ to plant parameters when appropriate input signals in the reference model are used. The orthogonal expansions of these time-varying parameters, which can be computed by passing the auxiliary system parameters through the integrator block, converge to the plant parameters pointwise if the latter are su$ciently smooth. The parameter convergence is obtained by combining the adaptation laws with su$ciently rich input signals, referred to as generators of persistent excitation, which guarantee the existence of a unique steady state for the parameter errors.

Spatiotemporal adaptive state feedback control of a linear parabolic partial differential equation

“…So approximate controllability results (e.g., Theorem 4.2.1 on Page 162 43 ) help determine the integer n and also ensure the bound of the decomposition intervals Δx i , i ∈  . On the other hand, the LMIs (13) are always feasible with small enough value of 𝜗 i , i ∈  , appropriate control gains k i (x, t), i ∈  and design parameters p > 0, 0 < 𝜚 i < 1, i ∈  (see their equivalent form given in the inequalities (31)). The value of 𝜗 i , i ∈  will become smaller and smaller as the number of actuators (the integer n) increases.…”

“…The control gains k i (x, t), i ∈  in (5) are chosen as the form (32), where 𝜒 1 (x) = 0.2, 𝜒 2 (x) = 0.3 and λ(x, t) is governed by (9) subject to the boundary conditions (10) and the initial condition (11). It has been checked that the above chosen parameter values satisfy the inequalities (31). That is, the space-time varying LMIs (13) are fulfilled for the above chosen parameter values.…”

Systematization of Approaches to Adaptive Boundary Stabilization of PDEs

“…The early efforts on adaptive control of distributed parameter systems were using tuning of a scalar gain to a high level to stabilize some classes of (relatively degree one) infinite dimensional plants (see the survey by Logemann and Townley [2] for an exhaustive list of references). Model reference (MRAC) type schemes were designed by Hong and Bentsman [3], Bohm et al [4], and Bentsman and Orlov [5]. While the focus in these papers is on functional, spatially dependent parametric uncertainty and the proofs of identifiability, the control is distributed in the PDE domain, allowing access to all the uncertain terms, akin to an infinite set of parallel first-order systems.…”

On Identifiability of Linear Infinite-Dimensional Systems

“…Then the system has the same Markov parameters CAnB, n = 1,2, Thus, in analogy to the finite-dimensional case, the identifiability of the infinite-dimensional system is guaranteed if it is in a canonical form such that if confined to this form the operators A, B, C are uniquely determined by the Markov parameters. The major challenge will be an explicit definition of the canonical form of a linear Hilbert space-valued system (I), (2).…”