#LyX 1.6.7 created this file. For more info see http://www.lyx.org/ \lyxformat 345 \begin_document \begin_header \textclass scrartcl \begin_preamble \newcommand\blabl{} \end_preamble \use_default_options false \begin_modules theorems-ams theorems-ams-extended \end_modules \language english \inputencoding auto \font_roman lmodern \font_sans default \font_typewriter default \font_default_family default \font_sc false \font_osf false \font_sf_scale 100 \font_tt_scale 100 \graphics default \paperfontsize default \spacing single \use_hyperref true \pdf_bookmarks true \pdf_bookmarksnumbered false \pdf_bookmarksopen false \pdf_bookmarksopenlevel 1 \pdf_breaklinks false \pdf_pdfborder false \pdf_colorlinks false \pdf_backref false \pdf_pdfusetitle true \papersize default \use_geometry false \use_amsmath 1 \use_esint 0 \cite_engine basic \use_bibtopic false \paperorientation portrait \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \defskip medskip \quotes_language english \papercolumns 1 \papersides 1 \paperpagestyle default \tracking_changes false \output_changes false \author "" \author "" \end_header \begin_body \begin_layout Standard \begin_inset FormulaMacro \newcommand{\isa}[1]{i_{\alpha#1}} {i_{\alpha#1}} \end_inset \begin_inset FormulaMacro \newcommand{\isb}[1]{i_{\beta#1}} {i_{\beta#1}} \end_inset \begin_inset FormulaMacro \newcommand{\Dt}{\Delta t} {\Delta t} \end_inset \begin_inset FormulaMacro \newcommand{\om}{\omega} {\omega} \end_inset \begin_inset FormulaMacro \newcommand{\th}{\vartheta} {\vartheta} \end_inset \begin_inset FormulaMacro \newcommand{\usa}[1]{u_{\alpha#1}} {u_{\alpha#1}} \end_inset \begin_inset FormulaMacro \newcommand{\usb}[1]{u_{\beta#1}} {u_{\beta#1}} \end_inset \end_layout \begin_layout Title PMSM system description \end_layout \begin_layout Section Model of PMSM Drive \end_layout \begin_layout Standard Permanent magnet synchronous machine (PMSM) drive with surface magnets on the rotor is described by conventional equations of PMSM in the stationary reference frame: \begin_inset Formula \begin{align} \frac{d\isa{}}{dt} & =-\frac{R_{s}}{L_{s}}\isa{}+\frac{\Psi_{PM}}{L_{s}}\omega_{me}\sin\th+\frac{\usa{}}{L_{s}},\nonumber \\ \frac{d\isb{}}{dt} & =-\frac{R_{s}}{L_{s}}\isb{}-\frac{\Psi_{PM}}{L_{s}}\omega_{me}\cos\th+\frac{\usb{}}{L_{s}},\label{eq:simulator}\\ \frac{d\om}{dt} & =\frac{k_{p}p_{p}^{2}\Psi_{pm}}{J}\left(\isb{}\cos(\th)-\isa{}\sin(\th)\right)-\frac{B}{J}\om-\frac{p_{p}}{J}T_{L},\nonumber \\ \frac{d\th}{dt} & =\omega_{me}.\nonumber \end{align} \end_inset Here, \begin_inset Formula $\isa{}$ \end_inset , \begin_inset Formula $\isb{}$ \end_inset , \begin_inset Formula $\usa{}$ \end_inset and \begin_inset Formula $\usb{}$ \end_inset represent stator current and voltage in the stationary reference frame, respectively; \begin_inset Formula $\om$ \end_inset is electrical rotor speed and \begin_inset Formula $\th$ \end_inset is electrical rotor position. \begin_inset Formula $R_{s}$ \end_inset and \begin_inset Formula $L_{s}$ \end_inset is stator resistance and inductance respectively, \begin_inset Formula $\Psi_{pm}$ \end_inset is the flux of permanent magnets on the rotor, \begin_inset Formula $B$ \end_inset is friction and \begin_inset Formula $T_{L}$ \end_inset is load torque, \begin_inset Formula $J$ \end_inset is moment of inertia, \begin_inset Formula $p_{p}$ \end_inset is the number of pole pairs, \begin_inset Formula $k_{p}$ \end_inset is the Park constant. \end_layout \begin_layout Standard The sensor-less control scenario arise when sensors of the speed and position ( \begin_inset Formula $\om$ \end_inset and \begin_inset Formula $\th$ \end_inset ) are missing (from various reasons). Then, the only observed variables are: \begin_inset Formula \begin{equation} y_{t}=\left[\begin{array}{c} \isa{}(t),\isb{}(t),\usa{}(t),\usb{}(t)\end{array}\right].\label{eq:obs}\end{equation} \end_inset Which are, however, observed only up to some precision. \end_layout \begin_layout Standard Discretization of the model ( \begin_inset CommandInset ref LatexCommand ref reference "eq:simulator" \end_inset ) was performed using Euler method with the following result: \begin_inset Formula \begin{align*} \isa{,t+1} & =(1-\frac{R_{s}}{L_{s}}\Dt)\isa{,t}+\frac{\Psi_{pm}}{L_{s}}\Dt\omega_{t}\sin\vartheta_{e,t}+\usa{,t}\frac{\Dt}{L_{s}},\\ \isb{,t+1} & =(1-\frac{R_{s}}{L_{s}}\Dt)\isb{,t}-\frac{\Psi_{pm}}{L_{s}}\Dt\omega_{t}\cos\vartheta_{t}+\usb{,t}\frac{\Dt}{L_{s}},\\ \om_{t+1} & =(1-\frac{B}{J}\Dt)\om_{t}+\Dt\frac{k_{p}p_{p}^{2}\Psi_{pm}}{J}\left(\isb{,t}\cos(\th_{t})-\isa{,t}\sin(\th_{t})\right)-\frac{p_{p}}{J}T_{L}\Dt,\\ \vartheta_{t+1} & =\vartheta_{t}+\omega_{t}\Dt.\end{align*} \end_inset In this work, we consider parameters of the model known, we can make the following substitutions to simplify notation, \begin_inset Formula $a=1-\frac{R_{s}}{L_{s}}\Dt$ \end_inset , \begin_inset Formula $b=\frac{\Psi_{pm}}{L_{s}}\Dt$ \end_inset , \begin_inset Formula $c=\frac{\Dt}{L_{s}}$ \end_inset , \begin_inset Formula $d=1-\frac{B}{J}\Dt$ \end_inset , \family roman \series medium \shape up \size normal \emph off \bar no \noun off \color none \begin_inset Formula $e=\Dt\frac{k_{p}p_{p}^{2}\Psi_{pm}}{J}$ \end_inset , which results in a simplified model: \family default \series default \shape default \size default \emph default \bar default \noun default \color inherit \begin_inset Formula \begin{align} \isa{,t+1} & =a\,\isa{,t}+b\omega_{t}\sin\vartheta_{t}+c\usa{,t},\nonumber \\ \isb{,t+1} & =a\,\isb{,t}-b\omega_{t}\cos\vartheta_{t}+c\usb{,t},\label{eq:model-simple}\\ \om_{t+1} & =d\om_{t}+e\left(\isb{,t}\cos(\th_{t})-\isa{,t}\sin(\th_{t})\right),\nonumber \\ \vartheta_{t+1} & =\vartheta_{t}+\omega_{t}\Dt.\nonumber \end{align} \end_inset \end_layout \begin_layout Standard The above equations can be aggregated into state \begin_inset Formula $x_{t}=[\isa{,t},\isb{,t},\om_{t},\th_{t}]$ \end_inset will be denoted as \begin_inset Formula $x_{t+1}=g(x_{t},u_{t})$ \end_inset . \end_layout \begin_layout Subsection Transformation to d-q coordinates \end_layout \begin_layout Standard For many applications, it is advantageous to consider altervative coordinate system denoted d-q as follows \begin_inset Formula \begin{eqnarray*} \left[\begin{array}{c} d\\ q\end{array}\right] & = & \left[\begin{array}{cc} \cos\vartheta & \sin\vartheta\\ -\sin\vartheta & \cos\vartheta\end{array}\right]\left[\begin{array}{c} \alpha\\ \beta\end{array}\right]\\ \left[\begin{array}{c} \alpha\\ \beta\end{array}\right] & = & \left[\begin{array}{cc} \cos\vartheta & -\sin\vartheta\\ \sin\vartheta & \cos\vartheta\end{array}\right]\left[\begin{array}{c} d\\ q\end{array}\right]\end{eqnarray*} \end_inset Under this transformation, the whole model ( \begin_inset CommandInset ref LatexCommand ref reference "eq:model" \end_inset ) can be transformed into d-q coordinates. \end_layout \begin_layout Standard In this text, we will transform only one single quantity, \begin_inset Formula $L_{d}$ \end_inset and \begin_inset Formula $L_{q}$ \end_inset for which it holds \begin_inset Formula $L_{d}=kL_{q}$ \end_inset . Then, \begin_inset Formula \begin{eqnarray*} \left[\begin{array}{c} L_{\alpha}\\ L_{\beta}\end{array}\right] & = & \left[\begin{array}{cc} \cos\vartheta & -\sin\vartheta\\ \sin\vartheta & \cos\vartheta\end{array}\right]\left[\begin{array}{c} L_{d}\\ L_{q}\end{array}\right].\\ & = & L_{d}\left[\begin{array}{cc} \cos\vartheta & -\sin\vartheta\\ \sin\vartheta & \cos\vartheta\end{array}\right]\left[\begin{array}{c} k\\ 1\end{array}\right]\\ & = & L\left[\begin{array}{cc} k\cos\vartheta & -\sin\vartheta\\ k\sin\vartheta & \cos\vartheta\end{array}\right]=L\left[\begin{array}{c} k_{c\vartheta}\\ k_{s\vartheta}\end{array}\right].\end{eqnarray*} \end_inset Then, model of the drive is changed to \begin_inset Formula \begin{align*} \isa{,t+1} & =(1-\frac{R_{s}}{L_{s}k_{c\vartheta}}\Dt)\isa{,t}+\frac{\Psi_{pm}}{L_{s}k_{c\vartheta}}\Dt\omega_{t}\sin\vartheta_{e,t}+\usa{,t}\frac{\Dt}{L_{s}k_{c\vartheta}},\\ \isb{,t+1} & =(1-\frac{R_{s}}{L_{s}k_{s\vartheta}}\Dt)\isb{,t}-\frac{\Psi_{pm}}{L_{s}k_{s\vartheta}}\Dt\omega_{t}\cos\vartheta_{t}+\usb{,t}\frac{\Dt}{L_{s}k_{s\vartheta}},\\ \om_{t+1} & =(1-\frac{B}{J}\Dt)\om_{t}+\Dt\frac{k_{p}p_{p}^{2}\Psi_{pm}}{J}\left(\isb{,t}\cos(\th_{t})-\isa{,t}\sin(\th_{t})\right)-\frac{p_{p}}{J}T_{L}\Dt,\\ \vartheta_{t+1} & =\vartheta_{t}+\omega_{t}\Dt.\end{align*} \end_inset Transformation to full d-q \begin_inset Formula \begin{eqnarray*} i_{d,t+1} & = & (1-\frac{R_{s}}{L_{d}}\Dt)i_{d,t}+\frac{L_{q}}{L_{d}}i_{q,t}\Dt\omega_{t}+u_{d,t}\frac{\Dt}{L_{d}},\\ i_{q,t+1} & = & -\frac{L_{d}}{L_{q}}\Dt\omega_{t}i_{d,t}+(1-\frac{R_{s}}{L_{q}}\Dt)i_{q,t}-\frac{\Psi_{pm}}{L_{q}}\Dt\omega_{t}+u_{q,t}\frac{\Dt}{L_{q}},\\ \omega_{t+1} & = & \underbrace{(1-\frac{B}{J}\Dt)}_{\approx1}\om_{t}+\Dt\frac{k_{p}p_{p}^{2}}{J}((L_{d}-L_{q})i_{d}+\Psi_{pm})i_{q}\\ \vartheta_{t+1} & = & \vartheta_{t}+\Delta t\omega_{t}\end{eqnarray*} \end_inset \end_layout \begin_layout Standard Observation: \begin_inset Formula \[ \left[\begin{array}{c} i_{\alpha}\\ i_{\beta}\end{array}\right]=\left[\begin{array}{cc} \cos\vartheta & -\sin\vartheta\\ \sin\vartheta & \cos\vartheta\end{array}\right]\left[\begin{array}{c} i_{d}\\ i_{q}\end{array}\right]+e_{t}\] \end_inset \end_layout \begin_layout Subsection Gaussian model of disturbances \end_layout \begin_layout Standard This model is motivated by the well known Kalman filter, which is optimal for linear system with Gaussian noise. Hence, we model all disturbances to have covariance matrices \begin_inset Formula $Q_{t}$ \end_inset and \begin_inset Formula $R_{t}$ \end_inset for the state \begin_inset Formula $x_{t}$ \end_inset and observations \begin_inset Formula $y_{t}$ \end_inset respectively. \begin_inset Formula \begin{align} x_{t+1} & \sim\mathcal{N}(g(x_{t}),Q_{t})\label{eq:model}\\ y_{t} & \sim\mathcal{N}([\isa{,t},\isb{,t}]',R_{t})\nonumber \end{align} \end_inset \end_layout \begin_layout Standard Under this assumptions, Bayesian estimation of the state, \begin_inset Formula $x_{t}$ \end_inset , can be approximated by so called Extended Kalman filter which approximates posterior density of the state by a Gaussian \begin_inset Formula \[ f(x_{t}|y_{1}\ldots y_{t})=\mathcal{N}(\hat{x}_{t},S_{t}).\] \end_inset Its sufficient statistics \begin_inset Formula $S_{t}=\left[\hat{x}_{t},P_{t}\right]$ \end_inset is evaluated recursively as follows: \begin_inset Formula \begin{eqnarray} \hat{x}_{t} & = & g(\hat{x}_{t-1})-K\left(y_{t}-h(\hat{x}_{t-1})\right).\label{eq:ekf_mean}\\ R_{y} & = & CP_{t-1}C'+R_{t},\nonumber \\ K & = & P_{t-1}C'R_{y}^{-1},\nonumber \\ S_{t} & = & P_{t-1}-P_{t-1}C'R_{y}^{-1}CP_{t-1},\\ P_{t} & = & AS_{t}A'+Q_{t}.\label{eq:ekf_cov}\end{eqnarray} \end_inset where \begin_inset Formula $A=\frac{d}{dx_{t}}g(x_{t})$ \end_inset , \begin_inset Formula $C=\frac{d}{dx_{t}}h(x_{t})$ \end_inset , \begin_inset Formula $g(x_{t})$ \end_inset is model ( \begin_inset CommandInset ref LatexCommand ref reference "eq:model-simple" \end_inset ) and \begin_inset Formula $h(x_{t})$ \end_inset direct observation of \begin_inset Formula $y_{t}=[\isa{,t},\isb{,t}]$ \end_inset , i.e. \begin_inset Formula \[ A=\left[\begin{array}{cccc} a & 0 & b\sin\th & b\om\cos\th\\ 0 & a & -b\cos\th & b\om\sin\th\\ -e\sin\th & e\cos\th & d & -e(\isb{}\sin\th+\isa{}\cos\th)\\ 0 & 0 & \Dt & 1\end{array}\right],\quad C=\left[\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\end{array}\right]\] \end_inset \end_layout \begin_layout Standard \begin_inset Formula \[ B=\left[\begin{array}{cc} c & 0\\ 0 & c\\ 0 & 0\\ 0 & 0\end{array}\right]\] \end_inset \end_layout \begin_layout Standard Covariance matrices of the system \begin_inset Formula $Q$ \end_inset and \begin_inset Formula $R$ \end_inset are supposed to be known. \end_layout \begin_layout Subsubsection Reduced order version \end_layout \begin_layout Standard Equations ( \begin_inset CommandInset ref LatexCommand ref reference "eq:model" \end_inset ) ca be restructured by considering \begin_inset Formula $i_{s\alpha}$ \end_inset and \begin_inset Formula $i_{s\beta}$ \end_inset as external observations. Then the state variables are \begin_inset Formula $x_{t}=[\omega_{t},\vartheta_{t}]$ \end_inset and as follows: \begin_inset Formula \begin{align} \om_{t+1} & =d\om_{t}+e\left(\isb{,t}\cos(\th_{t})-\isa{,t}\sin(\th_{t})\right),\label{eq:rord-state}\\ \vartheta_{t+1} & =\vartheta_{t}+\omega_{t}\Dt.\nonumber \end{align} \end_inset and the onbservation equations are \begin_inset Formula \begin{align} \isa{,t+1} & =a\,\isa{,t}+b\omega_{t}\sin\vartheta_{t}+c\usa{,t},\nonumber \\ \isb{,t+1} & =a\,\isb{,t}-b\omega_{t}\cos\vartheta_{t}+c\usb{,t},\label{eq:rord-obs}\end{align} \end_inset These equations are used by the EKF to update estimates of mean values. The new matrices \begin_inset Formula $A$ \end_inset and \begin_inset Formula $C$ \end_inset are \begin_inset Formula \[ A=\left[\begin{array}{cc} d & -e(\isb{}\sin\th+\isa{}\cos\th)\\ \Dt & 1\end{array}\right],\quad C=\left[\begin{array}{cc} b\sin\th & b\om\cos\th\\ -b\cos\th & b\om\sin\th\end{array}\right].\] \end_inset \end_layout \begin_layout Subsection Test system \end_layout \begin_layout Standard A real PMSM system on which the algorithms will be tested has parameters: \end_layout \begin_layout Standard \begin_inset Formula \begin{eqnarray*} R_{s} & = & 0.28;\\ L_{s} & = & 0.003465;\\ \Psi_{pm} & = & 0.1989;\\ k_{p} & = & 1.5\\ p & = & 4.0;\\ J & = & 0.04;\\ \Delta t & = & 0.000125\end{eqnarray*} \end_inset which yields \begin_inset Formula \begin{eqnarray*} a & = & 0.9898\\ b & = & 0.0072\\ c & = & 0.0361\\ d & = & 1\\ e & = & 0.0149\end{eqnarray*} \end_inset The covaraince matrices \begin_inset Formula $Q$ \end_inset and \begin_inset Formula $R$ \end_inset are assumed to be known. For the initial tests, we can use the following values: \end_layout \begin_layout Standard \begin_inset Formula \begin{eqnarray*} Q & = & \mathrm{diag}(0.0013,0.0013,5e-6,1e-10),\\ R & = & \mathrm{diag}(0.0006,0.0006).\end{eqnarray*} \end_inset \end_layout \begin_layout Standard Limits: \begin_inset Formula \begin{align*} u_{\alpha,max} & =50V, & u_{\alpha,min} & =-50V,\\ u_{\beta,max} & =50V. & u_{\beta,min} & =-50V,\end{align*} \end_inset \end_layout \begin_layout Standard Perhaps better: \begin_inset Formula \[ u_{\alpha}^{2}+u_{\beta}^{2}<100^{2}.\] \end_inset \end_layout \begin_layout Standard \end_layout \begin_layout Section Control \end_layout \begin_layout Standard The task is to reach predefined speed \begin_inset Formula $\overline{\omega}_{t}$ \end_inset . \end_layout \begin_layout Standard For simplicity, we will assume additive loss function: \begin_inset Formula \begin{eqnarray*} l(x_{t},u_{t}) & = & (\omega_{t}-\overline{\omega}_{t})^{2}+q(\usa{,t}^{2}+\usb{,t}^{2}).\\ & = & (\omega_{t}-\overline{\omega}_{t})\Xi(\omega_{t}-\overline{\omega}_{t})+[\usa t,\usb t]\underbrace{\left[\begin{array}{cc} \upsilon & 0\\ 0 & \upsilon\end{array}\right]}_{\Upsilon}\left[\begin{array}{c} \usa t\\ \usb t\end{array}\right]\end{eqnarray*} \end_inset Here, \begin_inset Formula $\Upsilon$ \end_inset is the chosen penalization of the inputs, which remains to be tuned. \end_layout \begin_layout Standard \series bold Note \series default : classical notation of penalization matrices is \begin_inset Formula $Q$ \end_inset and \begin_inset Formula $R$ \end_inset , but it conflicts wit \begin_inset Formula $Q$ \end_inset and \begin_inset Formula $R$ \end_inset in ( \begin_inset CommandInset ref LatexCommand ref reference "eq:model" \end_inset ). \end_layout \begin_layout Standard Following the standard dynamic programming approach, optimization of the loss function can be done recursively, as follows: \begin_inset Formula \[ V(x_{t-1},u_{t-1})=\arg\min_{u_{t}}\mathsf{E}_{f(x_{t},y_{t}|x_{t-1})}\left\{ l(x_{t},u_{t})+V(x_{t},u_{t})\right\} ,\] \end_inset where \begin_inset Formula $V(x_{t},u_{t})$ \end_inset is the Bellman function. Since the model evolution is stochastic, we can reformulate it in terms of sufficient statistics, \begin_inset Formula $S$ \end_inset as follows: \begin_inset Formula \[ V(S_{t-1})=\min_{u_{t}}\mathsf{E}_{f(x_{t},y_{t}|x_{t-1})}\left\{ l(x_{t},u_{t})+V(S_{t})\right\} .\] \end_inset \end_layout \begin_layout Standard Representation of the Bellman function depends on chosen approximation. \end_layout \begin_layout Subsection LQG control \end_layout \begin_layout Standard Control of linear state-space model with Gaussian noise \begin_inset Formula \begin{eqnarray*} x_{t} & = & Ax_{t-1}+Bu_{t}+Q^{\frac{1}{2}}v_{t},\\ y_{t} & = & Cx_{t}+Du_{t}+R^{\frac{1}{2}}w_{t}.\end{eqnarray*} \end_inset to minimize loss function \begin_inset Formula \[ L_{t}=(x_{t}-\overline{x}_{t})'\Xi(x_{t}-\overline{x}_{t})+u_{t}'\Upsilon u_{t}.\] \end_inset \end_layout \begin_layout Standard Optimal solution in the sense of dynamic programming on horizon \begin_inset Formula $t+h$ \end_inset is: \begin_inset Newline newline \end_inset \begin_inset Formula \begin{eqnarray*} u_{t} & = & L_{t}\left(\hat{x}_{t}-\overline{x}_{t}\right),\\ L_{t} & = & -(B'S_{t+1}B+\Upsilon)^{-1}B'S_{t+1}A,\\ S_{t} & = & A'(S_{t+1}-S_{t+1}B(B'S_{t+1}B+\Upsilon)^{-1}B'S_{t+1})A+\Xi,\end{eqnarray*} \end_inset This solution is certainty equivalent, i.e. only the first moment, \begin_inset Formula $\hat{x}$ \end_inset , of the Kalman filter is used. \end_layout \begin_layout Subsection PI control \end_layout \begin_layout Standard The classical control is based on transformation to \begin_inset Formula $dq$ \end_inset reference frame: \begin_inset Formula \begin{eqnarray*} i_{d} & = & i_{\alpha}\cos(\vartheta)+i_{\beta}\sin(\vartheta),\\ i_{q} & = & i_{\beta}\cos(\vartheta)-i_{\alpha}\sin(\vartheta).\end{eqnarray*} \end_inset Desired \begin_inset Formula $i_{q}$ \end_inset current, \begin_inset Formula $\overline{i}_{q}$ \end_inset , is derived using PI controller \begin_inset Formula \[ \overline{i}_{q}=PI(\overline{\omega}-\omega,P_{i},I_{i}).\] \end_inset This current needs to be achieved through voltages \begin_inset Formula $u_{d},u_{q}$ \end_inset which are again obtained from a PI controller \begin_inset Formula \begin{eqnarray*} u_{d} & = & PI(-i_{d},P_{u},I_{u}),\\ u_{q} & = & PI(\overline{i}_{q}-i_{q},P_{u},I_{u}).\end{eqnarray*} \end_inset These are compensated (for some reason) as follows: \begin_inset Formula \begin{eqnarray*} u_{d} & = & u_{d}-L_{S}\omega\overline{i}_{q},\\ u_{q} & = & u_{q}+\Psi_{pm}\omega.\end{eqnarray*} \end_inset Conversion to \begin_inset Formula $u_{\alpha},u_{\beta}$ \end_inset is \begin_inset Formula \begin{align*} u_{\alpha} & =|U|\cos(\phi), & u_{\beta} & =|U|\sin(\phi)\\ |U| & =\sqrt{u_{d}^{2}+u_{q}^{2}}, & \phi & =\begin{cases} \arctan(\frac{u_{q}}{u_{d}})+\vartheta & u_{d}\ge0\\ \arctan(\frac{u_{q}}{u_{d}})+\pi+\vartheta & u_{d}<0\end{cases}\end{align*} \end_inset \end_layout \begin_layout Standard PI controller is defined as follows: \begin_inset Formula \begin{eqnarray*} x & = & PI(\epsilon,P,I)\\ & = & P\epsilon+I(S_{t-1}+\epsilon)\\ S_{t} & = & S_{t-1}+\epsilon\end{eqnarray*} \end_inset Constants for the system: \begin_inset Formula \[ P_{i}=3,\,\, I_{i}=0.00375,\,\, P_{u}=20,\,\, I_{u}=0.5.\] \end_inset The requested values for \begin_inset Formula $\omega$ \end_inset should be kept in interval \begin_inset Formula $<-30,30>$ \end_inset . \end_layout \begin_layout Subsection Cautious LQG control \end_layout \begin_layout Standard Uncertainty in \begin_inset Formula $A$ \end_inset . \end_layout \begin_layout Standard Sigma points: \begin_inset Formula $x^{(i)}=\hat{x}+hv_{i}$ \end_inset , \begin_inset Formula $v_{i}$ \end_inset are eigenvectors of \begin_inset Formula $P$ \end_inset . \end_layout \begin_layout Standard \begin_inset Formula \begin{eqnarray*} E\{x'A(x)QA(x)x\} & = & \frac{1}{n}\sum x'A(x^{(i)})QA^{(i)}(x)x\\ & = & x'Zx'\end{eqnarray*} \end_inset \end_layout \begin_layout Standard Uncented transform... \end_layout \begin_layout Subsection Poor-man's dual LQG control \end_layout \begin_layout Standard Various heuristic solutions to dual extension of LQG has been proposed. Most of them is based on approximation of the loss function \begin_inset Formula \[ L_{t}=(x_{t}-\overline{x}_{t})'\Xi(x_{t}-\overline{x}_{t})+(u_{t}-\overline{u}_{t})'\Upsilon(u_{t}-\overline{u}_{t})+DUAL\_TERM.\] \end_inset where DUAL_TERM is typically a function of \begin_inset Formula $P_{t+2}$ \end_inset . \end_layout \begin_layout Standard To be continued... \end_layout \begin_layout Subsection Test Scenarios \end_layout \begin_layout Standard With almost full information, design of the control strategy should be almost trivial: \begin_inset Formula \begin{eqnarray*} \hat{\isa{}} & = & 0,\,\hat{\isb{}}=0,\hat{\omega}=1,\th=\frac{\pi}{2},\\ P_{t} & = & \mathrm{diag}([0.01,0.01,0.01,0.01]).\end{eqnarray*} \end_inset \end_layout \begin_layout Standard The difficulty arise with growing initial covariance matrix: \begin_inset Formula \begin{eqnarray*} \hat{\isa{}} & = & 0,\,\hat{\isb{}}=0,\hat{\omega}=1,\th=\frac{\pi}{2},\\ P_{t} & = & \mathrm{diag}([0.01,0.01,0.01,1]).\end{eqnarray*} \end_inset \end_layout \begin_layout Standard Or even worse: \begin_inset Formula \begin{eqnarray*} \hat{\isa{}} & = & 0,\,\hat{\isb{}}=0,\hat{\omega}=1,\th=\frac{\pi}{2},\\ P_{t} & = & \mathrm{diag}([0.01,0.01,0.01,10]).\end{eqnarray*} \end_inset ==\SpecialChar \- = \end_layout \begin_layout Standard The requested value \begin_inset Formula $\overline{\omega}_{t}=1.0015.$ \end_inset \end_layout \begin_layout Conjecture It is sufficient to consider hyper-state \begin_inset Formula $H=[\hat{i}_{\alpha},\hat{i}_{\beta},\hat{\omega},\hat{\vartheta},P(3,3),P(4,4)]$ \end_inset . \end_layout \begin_layout Conjecture \begin_inset CommandInset bibtex LatexCommand bibtex bibfiles "bibtex/vs,bibtex/vs-world,bibtex/world_classics,bibtex/world,new_bib_PS" options "plain" \end_inset \end_layout \end_body \end_document