2014-10-23 Miroslav Fikar * chap63.tex: --- chap63.tex.~1~ 2014-10-16 10:52:16.000000000 +0200 +++ chap63.tex 2014-10-23 21:54:19.366947199 +0200 @@ -418,7 +418,7 @@ the steady state is a complex function of parameters, but can be approximated as \begin{equation} - T_{\epsilon} \approx \frac{ \log \left( p \sqrt{1-\zeta^2} \right)}{ + T_{\epsilon} \approx \frac{ \ln \left( p \sqrt{1-\zeta^2} \right)}{ \zeta\omega_0} \end{equation} Maximum overshoot is given by the relation @@ -445,7 +445,7 @@ performance indices and their relations to controller parameters and vice versa. Integral cost function value is inversely proportional to - $\omega_0^2$. Maximum overshoot and all time indices are inversely + $\omega_0^2$. All time indices are inversely proportional to $\omega_0$. Transient response of the closed-loop system to setpoints or disturbances is improved by increasing $\omega_0$. Further, maximum overshoot or damping coefficient 2014-10-16 Miroslav Fikar * chap63.tex: wrong formula for maximum overshoot --- chap63.tex.~1~ 2007-01-03 00:43:39.000000000 +0100 +++ chap63.tex 2014-10-16 10:52:16.000000000 +0200 @@ -423,7 +423,7 @@ \end{equation} Maximum overshoot is given by the relation \begin{equation} - e_{\max} = e^{-\pi \zeta \sqrt{1-\zeta^2}} = \sqrt{\zeta_d} + e_{\max} = e^{-\pi \zeta / \sqrt{1-\zeta^2}} = \sqrt{\zeta_d} \end{equation} and occurs at the time \begin{equation} 2013-11-27 Miroslav Fikar * chap64.tex: typos, untranslated strings --- chap64.tex.~1~ 2004-09-19 13:53:10.000000000 +0200 +++ chap64.tex 2013-11-27 14:01:54.000000000 +0100 @@ -1230,7 +1230,7 @@ sets smaller overshoot, PID$_2$ is underdamped} \label{tab:6znkmit} \begin{tabular}{llll} - Regul�tor & $Z_R$ & $T_I$ & $T_D$ \\ \hline + Controller& $Z_R$ & $T_I$ & $T_D$ \\ \hline P & $0.5 Z_{Rk}$ & & \\ PI & $0.4 Z_{Rk}$ & $0.8 T_k$ & \\ PID & $0.6 Z_{Rk}$ & $0.5 T_k$ & $0.125 T_k$ \\ \hline @@ -1276,7 +1276,7 @@ \caption{The Ziegler-Nichols controller tuning based on the process step response} \label{tab:6znpch} \begin{tabular}{llll} - Regul�tor & $Z_R$ & $T_I$ & $T_D$ \\ \hline + Controller& $Z_R$ & $T_I$ & $T_D$ \\ \hline P & $\frac{1}{Z} \frac{T_n}{T_u}$ & & \\ PI & $\frac{0.9}{Z} \frac{T_n}{T_u}$ & $3.33T_u$ & \\ PID & $\frac{1.2}{Z} \frac{T_n}{T_u}$ & $2 T_u$ & $0.5 T_u$ \\ \hline @@ -1315,7 +1315,7 @@ speaking, Ziegler-Nichols methods have the following in common: \begin{itemize} \item transient responses are fairly oscillatory, -\item not suitable for tume-delay systems, +\item not suitable for time-delay systems, \item difficult tuning. \end{itemize} 2013-01-12 Miroslav Fikar * chap55.tex: typo B -> \Gamma --- chap55.tex.~1~ 2004-09-10 20:47:52.000000000 +0200 +++ chap55.tex 2013-01-12 15:38:03.978108303 +0100 @@ -274,7 +274,7 @@ \ve{x}(2) &=& \ve{\Phi}\ve{x}(1) +\ve{\Gamma}\ve{u}(1)\\ %\label{eq:5540} &=& \ve{\Phi}^{2}\ve{x}(0) +\ve{\Phi}\ve{\Gamma}\ve{u}(0)+ - \ve{B}\ve{u}(1) + \ve{\Gamma}\ve{u}(1) \end{eqnarray} Further continuation gives 2013-01-07 Miroslav Fikar * chap52.tex: p. 197, (5.36), initial value of the Z-transform --- chap52.tex.~1~ 2004-11-14 14:46:12.000000000 +0100 +++ chap52.tex 2013-01-07 20:37:39.032307712 +0100 @@ -118,7 +118,7 @@ For an initial function value holds \begin{equation} %\label{eq:5217} -\lim_{k\rightarrow 0}f(kT_{s})=\lim_{z\rightarrow \infty}\frac{z-1}{z}F(z) +\lim_{k\rightarrow 0}f(kT_{s})=\lim_{z\rightarrow \infty} F(z) \end{equation} 2012-04-03 Miroslav Fikar * chappred3.tex: replaced all 't+' by 'k+'. E.g. y(t+1) -> y(k+1) 2012-03-05 Miroslav Fikar * chapopt6.tex (u): Page 1580, (4.6.75): delete minus sign in equation @@ -612,7 +612,7 @@ or \begin{equation} \label{eq:opt0674} - u = - \frac{q(s)}{p(s)}\left( {w - y} \right) + u = \frac{q(s)}{p(s)}\left( {w - y} \right) \end{equation} The choice of $ \tilde {w} $ from~(\ref{eq:opt0672}) changes the closed-loop system in Fig.~\ref{fig:opt065} from a 2012-02-25 Miroslav Fikar * opt065.eps: Fig. 4.6.5, page 156, change: p(s)/o(s) -> q(s)/o(s) 2012-02-24 Miroslav Fikar * chapopt6.tex: page 150, (4.6.5) typo CL -> LC \end{pmatrix} &=& \begin{pmatrix} {\ve{A} - \ve{BK}} & {\ve{BK}} \\ - \ve{0} & {\ve{A} - \ve{CL}} + \ve{0} & {\ve{A} - \ve{LC}} \end{pmatrix} \begin{pmatrix} \ve{x}(t) \\ \ve{e}(t) 2012-02-22 Miroslav Fikar * chapopt5.tex, page 148, equations (4.5.34), (4.5.35), (4.5.38), missing transpose @@ -276,7 +276,7 @@ \label{eq:opt0533} \ve{\dot{z}}(t) - \ve{\dot{N}}(t)\ve{\lambda} (t) \\ \mbox{} - \ve{N}(t)\left[ \ve{C}^T\ve{S}^{-1}\ve{y}(t) - - \ve{CS}^{-1}\ve{C}\left( \ve{z}(t) - \ve{N}(t)\ve{\lambda }(t) \right) + - \ve{C}^T\ve{S}^{-1}\ve{C}\left( \ve{z}(t) - \ve{N}(t)\ve{\lambda }(t) \right) - \ve{A}^T\ve{\lambda}(t) \right]\\ = \ve{A}\left[ {\ve{z}(t) - \ve{N}(t)\ve{\lambda}(t)} \right] - \ve{V\lambda}(t) @@ -286,7 +286,7 @@ \label{eq:opt0534} \ve{\dot{z}}(t) - \ve{N}(t)\ve{C}^T\ve{S}^{ - 1}\left( {\ve{y}(t) - \ve{Cz}(t)} \right) - \ve{Az}(t) \\ = \left[ {\ve{\dot{N}}(t) - \ve{N}(t)\ve{A}^T - \ve{AN}(t) - + \ve{N}(t)\ve{CS}^{ - 1}\ve{CN}(t) - \ve{V}} \right]\ve{\lambda }(t) + + \ve{N}(t)\ve{C}^T\ve{S}^{-1}\ve{CN}(t) - \ve{V}} \right]\ve{\lambda }(t) \end{multline} We can choose $\ve{z}(t)$ and $\ve{N}(t)$ such that \begin{eqnarray} @@ -297,7 +297,7 @@ \ve{z}(0) &=& \ve{\bar{x}}_0 \\ \label{eq:opt0538a} \ve{V} &=& \ve{\dot{N}}(t) - \ve{N}(t)\ve{A}^T - \ve{AN}(t) + -\ve{N}(t)\ve{CS}^{ - 1}\ve{CN}(t) \\ +\ve{N}(t)\ve{C}^T\ve{S}^{-1}\ve{CN}(t) \\ \label{eq:opt0537} \ve{N}(0) &=& \ve{N}_0 \end{eqnarray} * chapopt5.tex, page 147, equation (4.5.27), wrong sign @@ -222,7 +222,7 @@ $\ve{x}(0)$ is free as well and thus \begin{equation} \label{eq:opt0526} -\ve{x}(0) = \ve{\bar {x}}_0 + \ve{N}_0 \ve{\lambda}(0) +\ve{x}(0) = \ve{\bar {x}}_0 - \ve{N}_0 \ve{\lambda}(0) \end{equation} Optimal control follows from the optimality condition 2011-05-23 Miroslav Fikar * chappred3.tex: page 216, equation (5.3.61) untranslated word from Slovak: inak $\to$ otherwise @@ -648,7 +648,7 @@ \bar{u}(k-i+j)= \left\{ \begin{array}{ll} u_f(k-1) & j \geq i \\ - u_f(k-i+j) & \mbox{inak} + u_f(k-i+j) & \mbox{otherwise} \end{array} \right. \end{equation} 2010-11-03 Miroslav Fikar * chapopt4.tex (I): page 137, typo: sufficent -> sufficient 2009-08-14 Miroslav Fikar * chapad3.tex: page 237, equation (6.3.1), typo in numerator of transfer function 48c48 < G(s) = \frac{b_{s1} s + a_{s0} }{a_{s2} s^2 + a_{s1} s + 1} --- > G(s) = \frac{b_{s1} s + b_{s0} }{a_{s2} s^2 + a_{s1} s + 1} 2007-05-16 * chapopt9.tex (section{$H_2$ Optimal Control}): changed \ve{C}(s) for \ve{R}(s) several times in order not to interfere with \ve{C}. Also in fig:opt091. (section{$H_2$ Optimal Control}): forgotten right parenthesis: equations~(\ref{eq:opt093}),~(\ref{eq:opt0912}),~(\ref{eq:opt0913}), and~(\ref{eq:opt0914}, as well as to equations~(\ref{eq:opt093}),~(\ref{eq:opt0912}),~(\ref{eq:opt0913}), and~(\ref{eq:opt0914}), as well as * chapopt7.tex (subsection{Polynomial LQ Control Design with Observer for): typo: delete exclamation mark (!): 113,114c113,114 < \begin{thm}[LQ control with observer]\index{theorem!LQ control with observer}! < \label{thm:opt07:lqobserver} --- > \begin{thm}[LQ control with observer]\index{theorem!LQ control with observer}% > \label{thm:opt07:lqobserver}% * chapopt5.tex (subsection{Kalman Filter}): change \ve{\xi}_0(t) to \ve{\xi}_x(t): 112c112 < \ve{\dot{x}}(t) = \ve{Ax}(t) + \ve{\xi}_0(t) --- > \ve{\dot{x}}(t) = \ve{Ax}(t) + \ve{\xi}_x(t) 114c114 < where $\ve{\xi}_0(t)$ is $n$-dimensional stochastic process vector. --- > where $\ve{\xi}_x(t)$ is $n$-dimensional stochastic process vector. 118c118 < \ex{\ve{\xi}_0(t) } &=& \ve{0} --- > \ex{\ve{\xi}_x(t) } &=& \ve{0} 120,121c120,121 < \cov{ \ve{\xi}_0(t),\ve{\xi}_0(\tau) } &=& \ex{ \ve{\xi}_0(t)\ve{\xi}^T_0(\tau ) } = \ve{V}\delta(t - \tau) < \end{eqnarray} --- > \cov{ \ve{\xi}_x(t),\ve{\xi}_x(\tau) } &=& \ex{ \ve{\xi}_x(t)\ve{\xi}^T_x(\tau ) } = \ve{V}\delta(t - \tau) > \end{eqnarray} * chapopt1.tex (section{Problem of Optimal Control and Principle of Minimum}): Hamiltionian function -> Hamiltion function 2007-03-16 Jan Mikles * chapopt9.tex (section{$H_2$ Optimal Control}): Missing transposition in eq:opt0913 (4.10.13): \ve{D}_{12} \ve{C}_1 -> \ve{D}_{12}^T \ve{C}_1 * chapopt7.tex (subsection{Polynomial LQ Design with State Estimation for): typo below eq:opt0740 (4.8.41): matrix -> equation To find such a matrix, we will transform the Riccati matrix as follows -> To find such a matrix, we will transform the Riccati equation as follows. 2007-01-30 Miroslav Fikar * chapad3.tex (subsection{Continuous-Time Adaptive LQ Control of a Second Order System}): typo inicialisation -> initialisation 2x 2007-01-24 Jan Mikles * chapopt10.tex (subsection{Parametrisation of Stabilising Controllers for): sign change in numerator of (4.7.51), page 181: \begin{equation} \label{eq:opt1051} \ve{R}(s) = (\ve{\tilde {X}}_L (s) - \ve{\tilde {T}}(s)\ve{\tilde {B}}_L (s))^{ - 1}(\ve{\tilde {Y}}_L (s) - \ve{\tilde {T}}(s)\ve{\tilde {A}}_L(s)) \end{equation} change to \begin{equation} \label{eq:opt1051} \ve{R}(s) = (\ve{\tilde {X}}_L (s) - \ve{\tilde {T}}(s)\ve{\tilde {B}}_L (s))^{ - 1}(\ve{\tilde {Y}}_L (s) + \ve{\tilde {T}}(s)\ve{\tilde {A}}_L(s)) \end{equation} 2007-01-19 * chapopt2: Fig. 4.2.3: superfluous ... after second exchanger 2007-01-02 * minor changes in case of chapter/session titles 2006-12-27 * chappred5.tex (subsection{Finite Terminal Penalty}): p. 224^13, typo: matrix W is not bold * chapopt10.tex (subsubsection{Dead-Beat Control}): p. 185_16, typo: same situation is for he -> same situation is for the 2006-03-20 Miroslav Fikar * obr6uro.pic(3.1 Closed-loop system, Fig. 3.1.1): Actuator should be a part of process 2006-02-03 Miroslav Fikar * chapopt3.tex (subsection{Tracking Problem}): p. 134, (4.3.20) superfluous (t_f): \begin{eqnarray} \label{eq:opt0318} \ve{P}(t_f) &=& \ve{C}^T\ve{Q}_{yt_f} \ve{C} \\ \label{eq:opt0319} \ve{\gamma }(t_f) &=& \ve{C}^T(t_f) \ve{Q}_{yt_f } \ve{w}(t_f) \end{eqnarray} change for \begin{eqnarray} \label{eq:opt0318} \ve{P}(t_f) &=& \ve{C}^T\ve{Q}_{yt_f} \ve{C} \\ \label{eq:opt0319} \ve{\gamma }(t_f) &=& \ve{C}^T \ve{Q}_{yt_f } \ve{w}(t_f) \end{eqnarray} 2005-03-15 Miroslav Fikar * chappred3.tex (Calculation of the Optimal Control): equation (5.3.20) has no formula with it - deleted 2005-03-12 Miroslav Fikar * chappred5.tex (Derivation of CRHPC, (5.5.8), p. 221): added reference to the matrix inversion lemma: The block matrix inversion formula states -> The block matrix inversion formula states (see its proof of Lemma~\ref{thm:id:invers} on page~\pageref{thm:id:invers}) * chappred5.tex (Derivation of CRHPC, (5.5.8), p. 221): Incorrect sign in the element (2,2) of the inverse matrix: -\Delta -> \Delta \begin{equation} \left( \begin{array}{cc} A^{-1} & D \\ C & B \end{array} \right)^{-1} = \left( \begin{array}{cc} A+ AD\Delta CA & -AD \Delta\\ -\Delta CA & -\Delta \end{array} \right),\quad \Delta^{-1} = B - CAD \end{equation} -> \begin{equation} \left( \begin{array}{cc} A^{-1} & D \\ C & B \end{array} \right)^{-1} = \left( \begin{array}{cc} A+ AD\Delta CA & -AD \Delta\\ -\Delta CA & \Delta \end{array} \right),\quad \Delta^{-1} = B - CAD \end{equation} 2005-03-09 Lubos Cirka * chappred3.tex (Derivation of the Predictor from State-Space Models): (5.3.53) and (5.3.54): matrix C must be inside: \begin{equation} \ve{G} = \ve{\bar{C}} \begin{pmatrix} \ve{\bar{B}} & \ve{0} & \ldots & \ldots & \ve{0} \\ \ve{\bar{A}\bar{B}} & \ve{\bar{B}} & \ve{0} & \ldots & \ve{0} \\ \vdots & & \ddots & \ddots & \vdots \\ \vdots & & & \ve{\bar{B}} & \ve{0} \\ \ve{\bar{A}}^{N_2-1}\ve{\bar{B}} &\ldots & & \ldots & \ve{\bar{B}} \\ \end{pmatrix} \end{equation} and \begin{equation} \ve{y}_0 = \ve{\bar{C}} \begin{pmatrix} \ve{\bar{A}} \\ \ve{\bar{A}}^2 \\ \vdots \\ \ve{\bar{A}}^{N_2} \end{pmatrix} \ve{\bar{x}}(k) \end{equation} -> \begin{equation} \ve{G} = \begin{pmatrix} \ve{\bar{C}\bar{B}} & \ve{0} & \ldots & \ldots & \ve{0} \\ \ve{\bar{C}\bar{A}\bar{B}} & \ve{\bar{C}\bar{B}} & \ve{0} & \ldots & \ve{0} \\ \vdots & & \ddots & \ddots & \vdots \\ \vdots & & & \ve{\bar{C}\bar{B}} & \ve{0} \\ \ve{\bar{C}\bar{A}}^{N_2-1}\ve{\bar{B}} &\ldots & & \ldots & \ve{\bar{C}\bar{B}} \\ \end{pmatrix} \end{equation} and \begin{equation} \ve{y}_0 = \begin{pmatrix} \ve{\bar{C}\bar{A}} \\ \ve{\bar{C}\bar{A}}^2 \\ \vdots \\ \ve{\bar{C}\bar{A}}^{N_2} \end{pmatrix} \ve{\bar{x}}(k) \end{equation} 2005-02-03 Miroslav Fikar * chap53.tex (p. 29_3, Example 1.3.3, section{Discrete-Time Transfer Functions}): T_s + 2 -> T_s + 1 \begin{equation*} G(s)=\frac{Z_2}{(T_1s+1)(T_2 s+2)},\qquad T_1\ne T_2 \end{equation*} -> \begin{equation*} G(s)=\frac{Z_2}{(T_1s+1)(T_2 s+1)},\qquad T_1\ne T_2 \end{equation*} 2005-02-03 Martin Herceg * chap53.tex (p. 30^6-30^9, Example 1.3.3, section{Discrete-Time Transfer Functions}): b1, b2, a1, a2 \begin{eqnarray*} b_1 & = & Z_2 T_1 T_2 \left[-\left(e^{-\frac{T_s}{T_1}} + e^{-\frac {T_s}{T_2}}\right) -\frac{T_1(1+e^{-\frac{T_{s}}{T_{2}}})}{T_2-T_1} +\frac{T_2(1+e^{-\frac {T_{s}}{T_{1}}})}{T_2-T_1} \right]\\ b_2 & = & Z_2 T_1 T_2 \left[ e^{-\frac{T_s}{T_1}} e^{-\frac{T_s}{T_2}} +\frac{T_1e^{-\frac{T_s}{T_2}}}{T_2-T_1} - \frac{T_2e^{-\frac{T_s}{T_1}}}{T_2-T_1} \right] \\ a_1 & = &-\left(e^{-\frac{T_s}{T_1}} + e^{-\frac{T_s}{T_1}}\right)\\ a_2 & = & e^{-\frac{T_s}{T_1}} e^{-\frac{T_s}{T_1}} \end{eqnarray*} -> \begin{eqnarray*} b_1 & = & Z_2 \left[-\left(e^{-\frac{T_s}{T_1}} + e^{-\frac {T_s}{T_2}}\right) -\frac{T_1(1+e^{-\frac{T_{s}}{T_{2}}})}{T_2-T_1} +\frac{T_2(1+e^{-\frac {T_{s}}{T_{1}}})}{T_2-T_1} \right]\\ b_2 & = & Z_2 \left[ e^{-\frac{T_s}{T_1}} e^{-\frac{T_s}{T_2}} +\frac{T_1e^{-\frac{T_s}{T_2}}}{T_2-T_1} - \frac{T_2e^{-\frac{T_s}{T_1}}}{T_2-T_1} \right] \\ a_1 & = &-\left(e^{-\frac{T_s}{T_1}} + e^{-\frac{T_s}{T_2}}\right)\\ a_2 & = & e^{-\frac{T_s}{T_1}} e^{-\frac{T_s}{T_2}} \end{eqnarray*} 2005-01-11 Martin Herceg * chappred7.tex, p.229, Fig. 5.7.1: N1 -> N2 2005-01-05 Martin Herceg * chappred3.tex (subsection{Closed-loop Relations}): p. 218, (5.3.40)] missing term: &=& C \frac{ A\Delta + \sum_{j=N_1}^{N_2} k_j z^{j-1}(B-G_j)}{\sum_{j=N_1}^{N_2} k_j} change for &=& C \frac{ A\Delta + \sum_{j=N_1}^{N_2} k_j z^{j-1}(B - A\Delta G_j)}{\sum_{j=N_1}^{N_2} k_j} 2004-11-17 Miroslav Fikar * chap54.tex (section{Input-Output Discrete-Time Models -- Difference Equations}), p30, (1.4.6): delete $q^{-d}()$: B(q^{-1})=q^{-d}(b_1q^{-1}+b_2q^{-2} + \cdots + b_m q^{-m}) change for B(q^{-1})=b_1q^{-1}+b_2q^{-2} + \cdots + b_m q^{-m} 2004-11-14 Miroslav Fikar * chap52.tex (subsubsection*{Partial Fraction Expansion}), example 1.2.1., p26_3: added material before: \begin{equation*} f(kT_{s)}=\frac{5}{3}\left(1-e^{-0,916k}\right), \qquad k=0,\, 1,\, 2,\, \ldots \end{equation*} new: \begin{equation*} f(kT_{s)}=\frac{5}{3}\left(1-e^{-0,916k}\right) = \frac{5}{3}\left(1-\left(\frac{2}{5}\right)^k \right), \qquad k=0,\, 1,\, 2,\, \ldots \end{equation*} * errata.tex: created file for changes in the manuscript 2004-11-09 Miroslav Fikar * Book printed. New log starts.