Corrections to Linear Systems - January 2002

p. xvi: line 2-3
should read: "...should also prove valuable to researchers and practitioners for self-study. Many simple examples..."

p. 15: Figure 1.3
eliminate line segment crossing capacitor $C_1$.

p. 24: Figure 1.9(a)
move $(t_0, x_0)$ closer to the center point.

p. 39: line after monkowski's Inequality
it should read $p = q = 2$.

p. 46: line 8
reduce space between $\vert x_i - x_{i0}\vert \leq b_i$ and $i=1, \dots, n$.

p. 69: line 21 - p. 70: line 4
eliminate paragraphs beginning with "We will view $y \in Y$ and $u \in U$ as system..." on p. 69, and ending with "...called the unit pulse response (or the unit impulse response) of a linear discrete-time system." Replace with:

We will view $y \in Y$ and $u \in U$ as system outputs and system inputs, respectively, and we let $T : U \rightarrow Y$ denote a linear transformation that relates $u$ to $y$. We first consider the case when $u(k) = 0$ for $k < k_0, k, k_0 \in Z$. Also, we assume that for $k > n \geq k_0$, the inputs $u(k)$ do not contribute to the system output at time $n$ (i.e., the system is causal). Under these assumptions, and in view of the linearity of $T$, and by invoking the representation of signals by (16.8), we obtain for $y = \{ y(n) \}, n \in Z$, the expression $y(n) = T (\sum_{k = - \infty}^{\infty} u(k) \delta (n - k)) = T (\sum_{k = k_0... ...) = \sum_{k=k_0}^n u(k) T(\delta(n-k)) = \sum_{k=k_0}^n h(n,k) u(k), n \geq k_0$, and $y(n) = 0, n < k_0$, where $T (\delta(n-k)) \stackrel{\Delta}{=} (T \delta)(n -k) \stackrel{\Delta}{=} h(n,k)$ represents the response of $T$ to a unit pulse (resp.,discrete-time impulse or unit sample) occurring at $n = k$.

When the assumptions in the preceding discussion are no longer valid, then a different argument than the one given above needs to be used to arrive at the system representation. Indeed, for infinite sums, the interchanging of the order of the summation operation $\sum$ with the linear transformation $T$ is no longer valid. We refer the reader to a paper by I. W. Sandberg (``A Representation Theorem for Linear Discrete-Space Systems'',IEEE Transactions on Circuits and Systems - I, Vol. 46, No. 5, pp. 578-580, May 1998) for a derivation of the representation of general linear discrete-time systems. In that paper it is shown that an extra term needs to be added to the right-hand side of equation (16.9), even in the representation of general, linear, time-invariant, causal, discrete-time systems. (In the proof, the Hahn-Banach Theorem (which is concerned with the extension of bounded linear functionals) is employed and the extra required term is given by $\lim_{l \rightarrow \infty} T(\sum_{k = - \infty}^{-c_l-1} u(k) \delta (n-k) + \sum_{k = c_l +1}^{\infty} u(k) \delta (n-k))$ with $c_l \rightarrow \infty$ as $l \rightarrow \infty$. For a statement and proof of the Hahn-Banach theorem, refer, e.g., to reference [12, pp. 367-370] given at the end of this chapter.) In that paper it is also pointed out, however, that cases with such extra non-zero terms are not necessa rily of importance in applications. In particular, if inputs and outputs are defined (to be non-zero) on just the non-negative integers, then for causal systems no additional term is needed (or more specifically, the extra term is zero), as seen in our earlier argument. In any event, throughout the present book we will concern ourselves with linear discrete -time systems which can be represented by equation (16.9) for the single-input/single-output case (and appropriate generalizations for multi-input/multi-output cases).

p. 71: line 4 from bottom of page
should read: "A discrete-time system described by (16.15) is said to be..."

p. 74: line 17-20
delete sentence beginning with "To put it another way..." and ending with "...sequence of functions is not even defined."

p. 112: table
change $j_i$ to $j_1$.

p.125: line 7 from bottom of page
should read "...we note that condition (2.102) written as $(f-g)^{l}(\lambda_i)=0$ implies that..."

p. 129: line 3 from bottom of page
change 1 in column 3, row 3 of $\bar{A}$ matrix to 2.

p. 130: line 13
change Subsection G to Subsection O.

p. 134: line 14
should read: "...primary decomposition theorem presented in Subsection M..."

p. 139: line 6 from bottom of page
change coefficient $\alpha$ to $\alpha_i$ (insert subscript $i$).

p. 140: line 2
change Subsection 3.1B to Subsection 2.2B.

p.146: line 6
should read: "Substituting $\phi (t_0)= \Phi (t_0,t_0) \alpha + \phi_p (t_0)$, we obtain $\alpha = \phi (t_0)- \phi_p (t_0)=x_0 - \phi_p (t_0)$."

p.155: Table 4.2
the fourth line of the second column should read $f(t-a) p(t-a), a>0$.

p. 168: line 9-10
should read: "...we take the Laplace transform of both sides of (6.12), we obtain..."

line 5 from bottom of page
should read: "...obtained (6.20) directly by taking the Laplace transform of $H(t)$ given in (6.12)."

p. 172: line 2
transfer function equation numerator should be $-5s - 1$ instead of $-3s - 1$.

p. 179: Table 7.2
in column 2, table heading should be $\{f(k)\}, k \geq 0$.

p. 184: line 4
should read: " $x(t_{k+1}) = \Phi (t_{k+1},t_k)x(t_k) +$..."

p. 194: line 9-10
should read: "...representations of a vector $v^1$..."

p. 198: line 20-22 (in Exercise 2.43)
should read: Plot the components of the solution $\phi$. For different initial conditions $x(0)=(a, b)^T$..."

p. 199 : line 17
$= v_i \tilde{v}$ should be $v_i \tilde{v_i}$.

p. 200: line 19 (in Exercise 2.53)
replace parentheses with square brackets on column vectors in (a). It should read: "For $x(0)=[1, 1, 1, 1]^T$ and for $u(t)=[1, 1]^T \: t \geq 0,$..."

p. 201: Exercise 2.58
replace in system 1 $[\begin{array}{cc} 1,& 0\end{array}]u(k)$ by $u(k)$.
delete in system 2 $[\begin{array}{cc} 0,& 1\end{array}]u(k)$.

p. 203: line 6
in part (b), (ii) should read: " $H(s) = \omega^2_n / (s^2 + 2 \zeta \omega_n s + \omega^2_n)$,".

p. 205: line 22 (in Problem 2.66(b))
should read: "...where $X \in R^{p \times n}$ and $Y \in R^{n \times m}$..."

p. 208: line 1
should be $x_2$ instead of $x_3$.

p. 208: Exercise 2.75
in entry $(4,5)$ should be $k_3$ instead of $k_2$.

line 4
in column 1, row 4 of matrix, replace $\frac{k_1}{m_1}$ with $\frac{k_1}{m_2}$.

p. 209: line 4 (in (a))
last element of vector should be $x_5 - x_3$ instead of $x_5 - x_1$.

p. 214: line 6
should read: "...will now be studied at length."

p. 230: line 12-17 from bottom of page
delete sentences beginning with ``This implies that the null space of $\dots$'' at line 17, ending with `` $\dots$ so $x_2^Tx_1 \neq 0$'', on line 12. Replace with:

This implies that the null space of $W_r\left( t_0, t_1 \right)$ is nonempty. $W_r$ is symmetric, and so the range of $W_r$ is the orthogonal complement of its null space (prove this). Thus for any $u \in {\cal R}\left(W_r\right)$ and $v \in {\cal N}\left(W_r\right)$, $u^T v = 0$. Also, we may write $x_1 = x_1^\prime + x_1^{\prime\prime}$ with $x^\prime_1 \in {\cal R}\left(W_r\right)$ and $x_1^{\prime\prime} \in{\cal N}\left(W_r\right)$ ( $x_1^{\prime\prime} \neq 0$ since $x_1 \notin {\cal R}\left(W_r\right)$). Then there exists $x_2 \in {\cal N}\left(W_r\right)$ such that $x_2^Tx_1^{\prime\prime} \neq 0$, which implies $x_2^Tx_1 \neq 0$.

p.236: line 13-18 from bottom of page
delete sentences beginning with ``Indeed, this assumption implies that the null space of $W_r$ is nonempty, $\dots$'' at line 18, ending with ``Therefore $x_2^Tx_1 \neq 0$`` on line 13. Replace with:

This implies that the null space of $W_r$ is nonempty. $W_r$ is symmetric, and so the range of $W_r$ is the orthogonal complement of its null space (prove this). Thus for any $u \in {\cal R}\left(W_r\right)$ and $v \in {\cal N}\left(W_r\right)$, $u^T v = 0$. Also, we may write $x_1 = x_1^\prime + x_1^{\prime\prime}$ with $x^\prime_1 \in {\cal R}\left(W_r\right)$ and $x_1^{\prime\prime} \in{\cal N}\left(W_r\right)$ ( $x_1^{\prime\prime} \neq 0$ since $x_1 \notin {\cal R}\left(W_r\right)$). Then there exists $x_2 \in {\cal N}\left(W_r\right)$ such that $x_2^Tx_1^{\prime\prime} \neq 0$, which implies $x_2^Tx_1 \neq 0$.

p. 241: line 5 from the bottom
should be $s_i I - A$ instead of $sI - A$.

p. 253: line 10 from bottom
should read: "...for every finite $t$. Therefore, in view of (3.22)..."

p. 254: line 9
should read: "...(3.24) is satisfied. In view of Lemma 3.6, (3.25) follows..."

p. 259: line 4
replace $R_m$ by $R^{pn}$.

p. 261: line 17
should read: "...implies that $R_{\bar{cn}} = \{0\}$ or that the system is constructible."

p. 262: line 18
should read: " $...[\tilde{y}^T (-n), \dots, \tilde{y}^T (-1)]^T$. Equation (3.56) must be solved for..."

p. 270: line 11-12 from bottom of page
should read: "Proof. For details of the proof, refer to $[8]$ and to R.E. Kalman, "On the Computation of the Reachable/Observable Canonical Form,"
SIAM J. Control and Optimization, vol. 20, no. 2, pp. 258-260, 1982, where further clarifications to $[8]$ and an updated method of selecting $Q$ are given."

p. 280: line 6 from bottom of page
insert space or comma between $qA^{i-1}B = 0$ and $i=1, \dots, n-1$.

p. 302: line 4
replace both occurrences of $\lambda (s)$ with $\Lambda (s)$.

p. 313: Figure 3.6
add distance $y$ demarcation to right of figure, extending from top to bottom of drawing.

p. 335: line 7
add filled box at end of line after "...which is Ackermann's formula, (2.21)."

p. 336: line 14
change subscript of $A$ from 12 to 22.

p. 338: line 20
should read: "with $\left[ \begin{array}{c} M_j \\ -D_j \end{array} \right]$ a...".

p. 376: line 26-28 (in Exercise 4.15)
remove indentations before "Remark," as it pertains to all four parts of the Exercise.

p. 379: Exercise 4.26(b)
should be $C = [\begin{array}{ccc} 1, & 0, & 1\end{array}]$ instead of $C = [\begin{array}{cc} 1, & 1\end{array}]$

p. 396: line 3-4
column 3, row 3 of the first and second matrices should be modified to be $CA^{2n-2}B$ and $\bar{C} \bar{A}^{2n-2} \bar{B}$, respectively.

p.405: line 7
replace $+x_1^{(n)}$ by $- x_1^{(n)} + x_1^{(n)}$

p.406: line 3
should read: " $H(s) = \frac{b_2 s^2 + b_1 s + b_0}{s^2 + a_1 s + a_0}$".

p. 409: line 19
row 2, column 2 of the $D(s)$ matrix should be "$s$" instead of 2.

p. 409: line 16 from bottom
entry $(2,2)$ in $D(s)$ should be $s$ instead of $2$.

p. 419: last line on page
in right hand side of equation (4.47), replace $\delta_{m_H(s)}$ with $m_H(s),$

p. 430: line 8 from bottom of page (in Exercise 5.22)
eliminate parentheses from denominator of equation for transfer function $H(s)$.

line 5 from bottom of page
replace $\dot{\hat{x}}$ with $\dot{x}$.

p. 431: Figure 5.10
add $S$ to upper right hand corner of block diagram.

line 7
part (b) should read: "Determine a state-space representation for the entire system $S$, using your answer in (a)."

p.478: line 22
equation (8.11) should read: " $\vert\vert\phi (t)\vert\vert \leq (\frac{c_2}{c_1})^{1/2} \vert\vert x_0\vert\vert e^{1/2 (\gamma / c_2)(t-t_0)}, \; \; t \geq t_0 \geq 0,$"

p. 507: line 5 from bottom of page
should read: "If we now assume that (10.23) is controllable (from-the-origin) and observable..."

p.536: line 15
should read: "Alternatively, it can be shown that any crd $G^{*}{_R}(s)$ of $P_1(s)$ and $P_2 (s)$ [or a cld $G{^*}{_L}(s)$..."

p. 547: line 8 from bottom of page
should read: "The Eliminant Matrix..."

p. 554: line 7
should read: "at $t=0$. (This is true if and only if $P^{-1}(q)$..."

p. 563: line 3
should read: "...the equilibrium $x=0$ of the free system $\dot{x}=Ax$ is asymptotically stable."

p. 610: Figure 7.8
in caption, move $r=0$ to end of previous line.

p. 613: line 12 from bottom of page
remove minus sign so to read $(X^{\prime}{_1}, Y^{\prime}{_1}) = (X^{\prime}{_0}, Y^{\prime}{_0}).$

p. 621: line 5 from bottom
should be ${\tilde{D}^\prime_2}^{-1}$ instead of ${\tilde{D}_2}^{-1}$ in $H_2$

p. 625: line 22
should be Theorem 4.21 instead of Theorem 4.20.

p. 625: line 5, equation (4.160)
should be

$$C = (X'_1 - K' \tilde{N}')^{-1}[-(X'_2 + K' \tilde{D}'), X'],$$

p. 625: line 6
should be ... Also, $U' U'^{-1} =$

p. 633: line 9 from bottom of page
should be Exercises 7.23, 7.26 instead of Exercises 7.23, 7.2b.

p. 637: line 9
should read: "...in the Fractional Approach to Design," Int. J. Control..."

p.639: line 11 (in Exercise 7.3(d))
should read: "(d) If $U(s)$ and $V(s)$ (in (c)) are unimodular..."

line 29-30 (in Exercise 7.6)
indent second and third lines of part (a).

line 34 (in Exercise 7.7)
indent second line of part (a).