## Stability of a Time Varying System

Consider the linear time varying equation e(t) = /1 u*(t )e(t), (H.1) with 0 < umin < u*(t) < umax V t ◦ 0 (H.2) and such that A = – A/L is a stability matrix with П a negative number greater than the largest real part of its eigenvalues. Use the Gronwall-Bellman inequality Rugh […]

## Characterization of the Eigenvalues of the Linearized Tracking Dynamics

Proof of Proposition 6.1 The tracking dynamics is described by the following system of differential equations Xn = 0 Xn-1 = аR(1 – –Xn-2) xn xn—1 Xn-2 = aR(1 — X7-—Xn—3) (G.1) xn xn 1

## Warped Time Optimization

In this appendix the underlying control laws (i. e., the control law assuming that the plant parameters are known) used in the WARTIC-i/o and WARTIC-state control algorithms, described in Chap. 5, are deduced. F. 1 Proof of the WARTIC-i/o Control Law Hereafter we prove Eq. (5.16) that yields a closed-form expression for the WARTIC – […]

## Derivation of the MUSMAR Dual Algorithm

For the sake of simplifying the notation, the mean conditioned on the observations E {-j Ok} is denoted by E {■}. E. 1 Deduction of Equation (3.137) Since, and E {u2(k + i _ 1)} = E I ^i_1u(k) + f[_1s(k)j = E |^2_1 u2(k) + 2u(k)/M-1p[_1s(k) + sT(k)p-1pT’_1s(k)^ = E {m2-1} u2(k) + 2u(k)E […]

## MUSMAR as a Newton Minimization Algorithm

In Mosca et al. (1989) Propositions 1 and 2 of Chap. 3 that characterize the possible convergence points of MUSMAR and the direction it yields for the progression of the controller gains are proved using the ODE method for studying the convergence of stochastic algorithms. In this appendix we use a different method that has […]

## MUSMAR Models

In this appendix we explain how the predictive models used by MUSMAR, (3.62) are obtained from the ARX model (3.9). As explained in Sect. 3.2.4 the MUSMAR adaptive control algorithm restricts the future control samples (with respect to the present discrete time denoted k), from time k + 1 up to t + T — […]

## Recursive Least Squares Deduction

In order to deduce Eqs. (3.73)-(3.75) start by considering the following lemma: Matrix inversion lemma For A, B, C and D matrices of convenient dimensions such that the indicated inver­sions exist, it holds that Proof Right multiply the right hand side of (B.1) by A + BCD to get ^A-1 – A-1 [DA-1 B + […]

## Solution of the DCSF PDE

This appendix addresses the issue of solving the PDE (2.5). A.1 Homogeneous Equation Start by considering the homogeneous equation to which the following ODE relating the independent variables t and x is associated (A.2) The first integral of (A.2) is a relation of the form p(x, t) = C for C an arbitrary constant, satisfied […]