# Category Adaptive Control of Solar Energy Collector Systems

## 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 (1996) to conclude that t \e(t)|| < \e(0)Wexp{J ||Au%(x)\dx}. 0 (H.3) Considering (H.2), this implies \e(t)|| < ||e(0)||enumint. (H.4) Since П < 0, this establishes asymptotic stability of (H.1). □

References

Goodwin GC, Sin KS (1984) Adaptive filtering prediction and control. Prentice-Hall, New York Ibragimov NH (1999) Elementary Lie group analysis and ordinary differential equations. Wiley, New York

MoscaE, Zappa G, Lemos JM (1989) Robustness of multipredictor adap...

## 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 X1 )

Xn—Xn — 1 h

The first equation is already linear and provides an eigenvalue at the origin. Lineariz­ing the other n — 1 equations around the equilibrium point defined by

r

Xi = — i r = 1

n

yields the (n — 1) x (n — 1) Jacobian matrix: —2n n 0 0 … 0 n nn 0 …0 —n 0 — n n … 0

—n 0 ………….. 0 n

In order to compute the eigenvalues of this Jacobian matrix, start by observing that it can be written as

J = n(—I + A) (G.3)

where I is the identity of order n — 1 and A[(n — 1) x (n — 1)] is the matrix

The characteristic polynomial of J is obtained from

1

det(sl — J ) = det(sl + I...

## 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 – i/o control law. For that sake, consider the predictive model (5.15) and assume that u is constant and equal to u (k) over the prediction horizon, yielding

ni

70(k + i) = au(k) 22 R(k – 1 + j) + a22 R(k – p)u(k – p) + вTin(k + i – n). j=i p=i

(F.1)

Assume now that the future values of radiation at time k + 1 up to time k + T (that are unknown at time k) are equal to R(k). Equation (F.1) becomes

n-i

T0(k + і) = au(k)R(k)i + y R(k – p)u(k – p) + вTin(k + і – n)...

## 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 _1pJ_^ s(k) + sT(k)E |Фі_1 pf_^ s(k)

= [A2_1 + u2(k) + 2u(k) lM-1ФІ-1 + s(k)

+ sT(k)E [фі_1ф[_1} s(k) (E.2)

the minimization of the cost function according to Eq. (3.133) leads to T

2 9u(k) 2 9u(k)

T "УA2 + aei + pfa 2-1 + pa,  i = 1

Since the data vector, z(k) = [u (k) sT (k)] T, used to estimate the predictive models parameters is common to all models (actually, the vector used is z(k – T) so the T-steps ahead predictor can use the last output available to perform the estimation an...

## 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 the advantage of not relying on the ODE method, being thus more comprehensive.

D. 1 Proof of Proposition 1

According to the control strategy used

y (t + j) « вj (Fk-1) + Hj (Fk-1, Fk)s (t) (D.1)

and

u(t + j _ 1) « дj-x(Fk-)n(t) + G’j_j(Ft-х, Fk)s(t), (D.2)

where

Hj (Fk_1, Ft) = p j (Fk_1) + в j (Fk_1 Ft) (D.3)

and

Gj _1( Fk_1, Ft) = фі _1( Fk_1) Ft (D.4)

Let Fk be computed according to (3.104, 3.105)...

## 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 — 1, to be given by a constant feedback of the pseudo-state, leaving u(k) free. In order to see how the predictive model (3.47) is modified by this assumption, start by observing that the pseudostate s(k) defined in (3.95) satisfies the dynamic state equation

s(k + 1) = 0ss (k) + Vsu (k) + eTe(k), (C.1)

in which

eT = [10 … 0], (C.2)

PT "

!n — 0(n— 1)xn (C 3)

01x(n+m)

0(m—1)xn Im — 1 — (m—1)x1

rs = [bo 0 … 010 … 0]T, (C.4)

and

PT = [—ax… — anbx… bm]. (C.5)

The matrix entries ...

## 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 + C-1] 1 DA-1^ (A + BCD)

= I – A-1 B^DA-1 B + C-1] 1 D

+ A-1 BCD – A-1 b[dA-1 B + C-1] 1 DA-1 BCD

= I + A-1 B [DA-1 B + C-1] 1 j[DA-1 B + C-1] CD – D – DA-1 BCdJ = I. Now, left multiply the right hand side of (B.1) by A + BCD to get    ■ DA-1 B + C-1

These quantities are related by the matrix regression model written for all the available data

z = ФLsft + V. (B.5)

With this notation the least squares functional (3.72) can be written Jls(\$) = 2 (z – &LS\$) MLS (z – <PL...

## Solution of the DCSF PDE

This appendix addresses the issue of solving the PDE (2.5).  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 by any solution x = x (t) of (A.2), where the function p is not identically constant for all the values of x and t. In other words, the function p is constant along each solution of (A.2), with the constant C depending on the solution. Since in the case of equation (a1e1) there are 2 independent variables, there is only one functionally independent integrals Ibragimov (1999). By integrating (A...