Robust stability and performance analysis for multiple model adaptive controllers

Abstract

Abstract-For an Estimation Based Multiple Model SwitchedAdaptive Control (EMMSAC) algorithm controlling a MIMO minimal LTI plant, lp, 1 ≤ p ≤ ∞ bounds on the gain from the input and output disturbances to the internal signals are obtained which are invariant to the number of models in the plant model set. For a compact uncertainty set it is shown that a realisable EMMSAC algorithm achieves robust stability for any plant within the uncertainty set.

Gain bounds for multiple model switched adaptive control of general MIMO LTI systems

“…The (lengthy) proof is omitted, see [1]. 2 This bound from the external disturbances to the internal signals is the direct generalisation of Theorem 4.1 in [2] from the class of dead beat stabilisable SISO systems to the class of MIMO minimal LTI systems.…”

Gain bounds for multiple model switched adaptive control of general MIMO LTI systems

“…The (lengthy) proof is omitted, see [1]. 2 This bound from the external disturbances to the internal signals is the direct generalisation of Theorem 4.1 in [2] from the class of dead beat stabilisable SISO systems to the class of MIMO minimal LTI systems.…”

Switching control for parameter identifiability of uncertain systems

“…Proof of Theorem 3: Since Θ L is ǫ-dense in Θ, there exists at least oneθ * ∈ Θ L such that |θ −θ * | ≤ ǫ. For such aθ * , one has Combining the two latter inequalities, we obtain α(|θ −θ L |) |χ 0 | ≤ β(ǫ)|χ 0 | + 2 z (f) 2,I which can be written as (20), Proposition 2 and the fact that any class K function is invertible.…”

Robust Stability for Multiple Model Adaptive Control: Part I—The Framework

“…Proof: The proof can be found in [6] and is related to previous work on the deterministic interpretation of the Kalman Filter, see e.g. [27], [22].…”

“…Tighter expressions for α and β can be found in [6]. If additionally K : P → C is an (asymptotic) stabilising design, it can be shown that Assumption 2(2) holds since asymptotic stability implies l r stability 1 ≤ r ≤ ∞.…”

Gain bounds for multiple model switched adaptive control of general MIMO LTI systems

“…The (lengthy) proof is omitted, see [1]. 2 This bound from the external disturbances to the internal signals is the direct generalisation of Theorem 4.1 in [2] from the class of dead beat stabilisable SISO systems to the class of MIMO minimal LTI systems.…”

Switching control for parameter identifiability of uncertain systems

“…Proof of Theorem 3: Since Θ L is ǫ-dense in Θ, there exists at least oneθ * ∈ Θ L such that |θ −θ * | ≤ ǫ. For such aθ * , one has Combining the two latter inequalities, we obtain α(|θ −θ L |) |χ 0 | ≤ β(ǫ)|χ 0 | + 2 z (f) 2,I which can be written as (20), Proposition 2 and the fact that any class K function is invertible.…”

Robust Stability for Multiple Model Adaptive Control: Part I—The Framework

“…Proof: The proof can be found in [6] and is related to previous work on the deterministic interpretation of the Kalman Filter, see e.g. [27], [22].…”

“…Tighter expressions for α and β can be found in [6]. If additionally K : P → C is an (asymptotic) stabilising design, it can be shown that Assumption 2(2) holds since asymptotic stability implies l r stability 1 ≤ r ≤ ∞.…”

Gain bounds for multiple model switched adaptive control of general MIMO LTI systems

“…The (lengthy) proof is omitted, see [1]. 2 This bound from the external disturbances to the internal signals is the direct generalisation of Theorem 4.1 in [2] from the class of dead beat stabilisable SISO systems to the class of MIMO minimal LTI systems.…”

Switching control for parameter identifiability of uncertain systems

“…Proof of Theorem 3: Since Θ L is ǫ-dense in Θ, there exists at least oneθ * ∈ Θ L such that |θ −θ * | ≤ ǫ. For such aθ * , one has Combining the two latter inequalities, we obtain α(|θ −θ L |) |χ 0 | ≤ β(ǫ)|χ 0 | + 2 z (f) 2,I which can be written as (20), Proposition 2 and the fact that any class K function is invertible.…”

Robust Stability for Multiple Model Adaptive Control: Part I—The Framework

“…Proof: The proof can be found in [6] and is related to previous work on the deterministic interpretation of the Kalman Filter, see e.g. [27], [22].…”

“…Tighter expressions for α and β can be found in [6]. If additionally K : P → C is an (asymptotic) stabilising design, it can be shown that Assumption 2(2) holds since asymptotic stability implies l r stability 1 ≤ r ≤ ∞.…”