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Multiple Model Reference Adaptive Control with Blending for Non-Square Multivariable Systems

Alex Lovi, Baris Fidan, Christopher Nielsen

Abstract

In this paper we develop a multiple model reference adaptive controller (MMRAC) with blending. The systems under consideration are non-square, i.e., the number of inputs is not equal to the number of states; multi-input, linear, time-invariant with uncertain parameters that lie inside of a known, compact, and convex set. Moreover, the full state of the plant is available for feedback. A multiple model online identification scheme for the plant's state and input matrices is developed that guarantees the estimated parameters converge to the underlying plant model under the assumption of persistence of excitation. Using an exact matching condition, the parameter estimates are used in a control law such that the plant's states asymptotically track the reference signal generated by a state-space model reference. The control architecture is proven to provide boundedness of all closed-loop signals and to asymptotically drive the state tracking error to zero. Numerical simulations illustrate the stability and efficacy of the proposed MMRAC scheme.

Multiple Model Reference Adaptive Control with Blending for Non-Square Multivariable Systems

Abstract

In this paper we develop a multiple model reference adaptive controller (MMRAC) with blending. The systems under consideration are non-square, i.e., the number of inputs is not equal to the number of states; multi-input, linear, time-invariant with uncertain parameters that lie inside of a known, compact, and convex set. Moreover, the full state of the plant is available for feedback. A multiple model online identification scheme for the plant's state and input matrices is developed that guarantees the estimated parameters converge to the underlying plant model under the assumption of persistence of excitation. Using an exact matching condition, the parameter estimates are used in a control law such that the plant's states asymptotically track the reference signal generated by a state-space model reference. The control architecture is proven to provide boundedness of all closed-loop signals and to asymptotically drive the state tracking error to zero. Numerical simulations illustrate the stability and efficacy of the proposed MMRAC scheme.
Paper Structure (14 sections, 7 theorems, 73 equations, 7 figures)

This paper contains 14 sections, 7 theorems, 73 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Corner Model Selection
  4. Satisfying the Matching Conditions for the Corner Models
  5. Obtaining the sets S and S'
  6. Example
  7. Online Parameter Identification
  8. Multiple-Model Parameter Identifier Design
  9. Stability and Convergence of the Identification Process
  10. Multiple Model Reference Adaptive Control
  11. Multiple Model Reference Adaptive Control Design
  12. Stability Analysis of the Entire System
  13. Simulations
  14. Conclusions

Key Result

Proposition 1

Suppose that the plant pr:plant and reference model pr:model_reference are such that pr:assumption_geometric is satisfied. If there exists a set $\mathcal{S}$ that satisfies pr:assumption_convex_hull, then there exists a set $\mathcal{S}^\prime$ that also satisfies pr:assumption_convex_hull, and fur

Figures (7)

  • Figure 1: Graphic illustration of the projection process to select corner models.
  • Figure 2: State of the system $x_p(t)$, and state of the reference model $x_r(t)$.
  • Figure 3: Control efforts for MMRAC and a single model MRAC.
  • Figure 4: Euclidean norm of the tracking error for MMRAC and a single model MRAC.
  • Figure 5: Semilog plot for the norm 2 of the tracking errors, and the linear regressions for MMRAC and a single model.
  • ...and 2 more figures

Theorems & Definitions (15)

  • Remark 1
  • Proposition 1
  • proof
  • Remark 2
  • Lemma 1: See ioannou_adaptive_2006, Barbalat's Lemma
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • ...and 5 more