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| author | Daniele Nicolodi <nicolodi@science.unitn.it> |
|---|---|
| date | Mon, 05 Dec 2011 16:20:06 +0100 |
| parents | f0afece42f48 |
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN"> <HTML> <HEAD> <META HTTP-EQUIV="CONTENT-TYPE" CONTENT="text/html; charset=windows-1252"> <TITLE></TITLE> <META NAME="GENERATOR" CONTENT="OpenOffice.org 3.1 (Win32)"> <META NAME="CREATED" CONTENT="0;0"> <META NAME="CHANGEDBY" CONTENT="Adrien G"> <META NAME="CHANGED" CONTENT="20090827;17521500"> </HEAD> <BODY LANG="en-US" DIR="LTR"> <P> <H2>Why use state space modeling?</H2> <P>State space modeling is efficient to simulate systems with large dimensionality, be it in terms of inputs, outputs, or pole/zeros. Adding nonlinearities to a model is also easier as in the frequency domain, however there is no such capability in the toolbox yet. Another reason to use them is to build complex parametric models, where intricated parameters make impossible the symbolic calculation of the transfer function coefficients – as a matter of fact the transfer function can be computed out of a determinant involving the A matrix, explaining the complexity of the calculation.</P> <P>For tasks such as identification, state space modeling is a computationally rather heavy, especially if colored noise is involved in the process.</P> <P>State space models can be converted into a matrix of transfer function in the s- or the z-domain. The functions in the toolbox that enable this are ss2pzmodel, ss2miir, ss2rational.</P> <H2>Generalities on State Space Modeling</H2> <P><FONT COLOR="#000000">In order to familiarize with state space modeling, this help page is a mere copy of the wiki page.</FONT></P> <P><FONT COLOR="#000000">In control engineering, a </FONT><FONT COLOR="#000000"><B>state space representation</B></FONT><FONT COLOR="#000000"> is a mathematical model of a physical system as a set of input, output and state variables related by first-order <A HREF="http://en.wikipedia.org/wiki/Differential_equation">differential equations</A>. To abstract from the number of inputs, outputs </FONT>and<FONT COLOR="#000000"> states, the variables are expressed as vectors and the differential and algebraic equations are written in matrix form (the last one can be done when the <A HREF="http://en.wikipedia.org/wiki/Dynamical_system">dynamical system</A> is linear and time invariant). The state space representation (also known as the "time-domain approach") provides a convenient and compact way to model and analyze systems with multiple inputs and outputs. With </FONT><FONT COLOR="#000000"><I>p</I></FONT><FONT COLOR="#000000"> inputs and </FONT><FONT COLOR="#000000"><I>q</I></FONT><FONT COLOR="#000000"> outputs, we would otherwise have to write down <A HREF="http://en.wikipedia.org/wiki/Laplace_transform">Laplace transforms</A> to encode all the information about a system. Unlike the frequency domain approach, the use of the state space representation is not limited to systems with linear components and zero initial conditions. "State space" refers to the space whose axes are the state variables. The state of the system can be represented as a vector within that space.</FONT></P> <H2><FONT COLOR="#000000">State variables</FONT></H2> <H2><FONT COLOR="#000080"><A HREF="http://en.wikipedia.org/wiki/File:Typical_State_Space_model.png"><FONT COLOR="#000080"><IMG SRC="images/State_space_fichiers/Typical_State_Space_model.png" NAME="images3" ALIGN=BOTTOM WIDTH=382 HEIGHT=154 BORDER=1></FONT></A></FONT><FONT COLOR="#000000"> </FONT> </H2> <P>Typical state space model</P> <P><FONT COLOR="#000000">The internal <A HREF="http://en.wikipedia.org/wiki/State_variable">state variables</A> are the smallest possible subset of system variables that can represent the entire state of the system at any given time. State variables must be linearly independent; a state variable cannot be a linear combination of other state variables. The minimum number of state variables required to represent a given system, </FONT><FONT COLOR="#000000"><I>n</I></FONT><FONT COLOR="#000000">, is usually equal to the order of the system's defining differential equation. If the system is represented in transfer function form, the minimum number of state variables is equal to the order of the transfer function's denominator after it has been reduced to a proper fraction. It is important to understand that converting a state space realization to a transfer function form may lose some internal information about the system, and may provide a description of a system which is stable, when the state-space realization is unstable at certain points. In electric circuits, the number of state variables is often, though not always, the same as the number of energy storage elements in the circuit such as <A HREF="http://en.wikipedia.org/wiki/Capacitor">capacitors</A> and <A HREF="http://en.wikipedia.org/wiki/Inductor">inductors</A>.</FONT></P> <H2><FONT COLOR="#000000">Linear systems</FONT></H2> <P>The most general state-space representation of a linear system with <I>p</I> inputs, <I>q</I> outputs and <I>n</I> state variables is written in the following form:</P> <P><IMG SRC="images/State_space_fichiers/036212a5fe0e6fded881a8f2d536cbb1.png" NAME="images4" ALT="\dot{\mathbf{x}}(t) = A(t) \mathbf{x}(t) + B(t) \mathbf{u}(t)" ALIGN=BOTTOM WIDTH=227 HEIGHT=21 BORDER=0> </P> <P><IMG SRC="images/State_space_fichiers/1d1151bf91e0a3f42167568730cead9f.png" NAME="images5" ALT="\mathbf{y}(t) = C(t) \mathbf{x}(t) + D(t) \mathbf{u}(t)" ALIGN=BOTTOM WIDTH=229 HEIGHT=21 BORDER=0> </P> <P>where:</P> <UL> <LI><P><IMG SRC="images/State_space_fichiers/0b46dcfa3789d90891d7d22aa4bf1b83.png" NAME="images6" ALT="x(\cdot)" ALIGN=BOTTOM WIDTH=31 HEIGHT=21 BORDER=0> is called the "state vector", <IMG SRC="images/State_space_fichiers/76bfa39f75341b43cc349cbf72261ac6.png" NAME="images7" ALT="x(t) \in \mathbb{R}^n" ALIGN=BOTTOM WIDTH=80 HEIGHT=21 BORDER=0>; </P> <LI><P><IMG SRC="images/State_space_fichiers/89df6576cd198dc2449bf59e97a61df3.png" NAME="images8" ALT="y(\cdot)" ALIGN=BOTTOM WIDTH=30 HEIGHT=21 BORDER=0> is called the "output vector", <IMG SRC="images/State_space_fichiers/592f30a36abb23d30f67aeac77f10a63.png" NAME="images9" ALT="y(t) \in \mathbb{R}^q" ALIGN=BOTTOM WIDTH=78 HEIGHT=21 BORDER=0>; </P> <LI><P><IMG SRC="images/State_space_fichiers/918aa6368974b6014859c56e1ef6762c.png" NAME="images10" ALT="u(\cdot)" ALIGN=BOTTOM WIDTH=31 HEIGHT=21 BORDER=0> is called the "input (or control) vector", <IMG SRC="images/State_space_fichiers/09a537199dfe82dc35d0050c72850b4c.png" NAME="images11" ALT="u(t) \in \mathbb{R}^p" ALIGN=BOTTOM WIDTH=78 HEIGHT=21 BORDER=0>; </P> <LI><P><IMG SRC="images/State_space_fichiers/b53a04dbe423d4e39eaedadf54e3ee31.png" NAME="images12" ALT="A(\cdot)" ALIGN=BOTTOM WIDTH=35 HEIGHT=21 BORDER=0> is the "state matrix", <IMG SRC="images/State_space_fichiers/d67f69962a6411d08f1ba642d8c75617.png" NAME="images13" ALT="\operatorname{dim}[A(\cdot)] = n \times n" ALIGN=BOTTOM WIDTH=153 HEIGHT=21 BORDER=0>, </P> <LI><P><IMG SRC="images/State_space_fichiers/8f528e3252e95ac5edef108971a1d43f.png" NAME="images14" ALT="B(\cdot)" ALIGN=BOTTOM WIDTH=36 HEIGHT=21 BORDER=0> is the "input matrix", <IMG SRC="images/State_space_fichiers/a4d9e024860226d6ad283d1d217f1c31.png" NAME="images15" ALT="\operatorname{dim}[B(\cdot)] = n \times p" ALIGN=BOTTOM WIDTH=152 HEIGHT=21 BORDER=0>, </P> <LI><P><IMG SRC="images/State_space_fichiers/d4d99b70fae67ddcd7923dbe5dbb6424.png" NAME="images16" ALT="C(\cdot)" ALIGN=BOTTOM WIDTH=34 HEIGHT=21 BORDER=0> is the "output matrix", <IMG SRC="images/State_space_fichiers/452963fb0f31641c763dab57d77f6d15.png" NAME="images17" ALT="\operatorname{dim}[C(\cdot)] = q \times n" ALIGN=BOTTOM WIDTH=151 HEIGHT=21 BORDER=0>, </P> <LI><P><IMG SRC="images/State_space_fichiers/9828f5f36ec8af68cc289bbf41a2c751.png" NAME="images18" ALT="D(\cdot)" ALIGN=BOTTOM WIDTH=37 HEIGHT=21 BORDER=0> is the "feedthrough (or feedforward) matrix" (in cases where the system model does not have a direct feedthrough, <IMG SRC="images/State_space_fichiers/9828f5f36ec8af68cc289bbf41a2c751.png" NAME="images19" ALT="D(\cdot)" ALIGN=BOTTOM WIDTH=37 HEIGHT=21 BORDER=0> is the zero matrix), <IMG SRC="images/State_space_fichiers/5b6e715515525947881d6fb6ea56fb15.png" NAME="images20" ALT="\operatorname{dim}[D(\cdot)] = q \times p" ALIGN=BOTTOM WIDTH=151 HEIGHT=21 BORDER=0>, </P> <LI><P><IMG SRC="images/State_space_fichiers/b6b4d433fef1cae095423823ab60d1e5.png" NAME="images21" ALT="\dot{\mathbf{x}}(t) := \frac{\operatorname{d}}{\operatorname{d}t} \mathbf{x}(t)" ALIGN=BOTTOM WIDTH=120 HEIGHT=42 BORDER=0>. </P> </UL> <P><FONT COLOR="#000000">In this general formulation, all matrices are allowed to be time-variant (i.e., their elements can depend on time); however, in the common <A HREF="http://en.wikipedia.org/wiki/LTI_system">LTI</A> case, matrices will be time invariant. The time variable </FONT><FONT COLOR="#000000"><I>t</I></FONT><FONT COLOR="#000000"> can be a "continuous" (e.g., <IMG SRC="images/State_space_fichiers/6080cfaa75ff6363d282e29a14b46632.png" NAME="images22" ALT="t \in \mathbb{R}" ALIGN=BOTTOM WIDTH=45 HEIGHT=15 BORDER=0>) or discrete (e.g., <IMG SRC="images/State_space_fichiers/d3bd3b5e80beacc6f4a28dacabf2c0c5.png" NAME="images23" ALT="t \in \mathbb{Z}" ALIGN=BOTTOM WIDTH=45 HEIGHT=15 BORDER=0>). In the latter case, the time variable is usually indicated as </FONT><FONT COLOR="#000000"><I>k</I></FONT><FONT COLOR="#000000">. <A HREF="http://en.wikipedia.org/wiki/Hybrid_system">Hybrid systems</A> allow for time domains that have both continuous and discrete parts. Depending on the assumptions taken, the state-space model representation can assume the following forms:</FONT></P> <TABLE DIR="LTR" CELLPADDING=4 CELLSPACING=2> <TR VALIGN=TOP> <TD> <P><B>System type</B></P> </TD> <TD> <P><B>State-space model</B></P> </TD> </TR> <TR VALIGN=TOP> <TD> <P>Continuous time-invariant</P> </TD> <TD> <P><IMG SRC="images/State_space_fichiers/ddfd74546a0e35f9ec054af2ecd3f2fa.png" NAME="images24" ALT="\dot{\mathbf{x}}(t) = A \mathbf{x}(t) + B \mathbf{u}(t)" ALIGN=BOTTOM WIDTH=182 HEIGHT=21 BORDER=0><BR><IMG SRC="images/State_space_fichiers/d0ac09f5cde2ce822ecc3e369692d04b.png" NAME="images25" ALT="\mathbf{y}(t) = C \mathbf{x}(t) + D \mathbf{u}(t)" ALIGN=BOTTOM WIDTH=184 HEIGHT=21 BORDER=0></P> </TD> </TR> <TR VALIGN=TOP> <TD> <P>Continuous time-variant</P> </TD> <TD> <P><IMG SRC="images/State_space_fichiers/95f05c22aefb06c919d8fb5ce8b26689.png" NAME="images26" ALT="\dot{\mathbf{x}}(t) = \mathbf{A}(t) \mathbf{x}(t) + \mathbf{B}(t) \mathbf{u}(t)" ALIGN=BOTTOM WIDTH=230 HEIGHT=21 BORDER=0><BR><IMG SRC="images/State_space_fichiers/b2f523062e3020870da5e52c293a6a49.png" NAME="images27" ALT="\mathbf{y}(t) = \mathbf{C}(t) \mathbf{x}(t) + \mathbf{D}(t) \mathbf{u}(t)" ALIGN=BOTTOM WIDTH=230 HEIGHT=21 BORDER=0></P> </TD> </TR> <TR VALIGN=TOP> <TD> <P>Discrete time-invariant</P> </TD> <TD> <P><IMG SRC="images/State_space_fichiers/ab56c0fbbdf9445c187021a5df2553fa.png" NAME="images28" ALT="\mathbf{x}(k+1) = A \mathbf{x}(k) + B \mathbf{u}(k)" ALIGN=BOTTOM WIDTH=227 HEIGHT=21 BORDER=0><BR><IMG SRC="images/State_space_fichiers/7da081c23726f8324878080112ec1af3.png" NAME="images29" ALT="\mathbf{y}(k) = C \mathbf{x}(k) + D \mathbf{u}(k)" ALIGN=BOTTOM WIDTH=195 HEIGHT=21 BORDER=0></P> </TD> </TR> <TR VALIGN=TOP> <TD> <P>Discrete time-variant</P> </TD> <TD> <P><IMG SRC="images/State_space_fichiers/9fae6c7e6be7c79bde9cff343e96e853.png" NAME="images30" ALT="\mathbf{x}(k+1) = \mathbf{A}(k) \mathbf{x}(k) + \mathbf{B}(k) \mathbf{u}(k)" ALIGN=BOTTOM WIDTH=281 HEIGHT=21 BORDER=0><BR><IMG SRC="images/State_space_fichiers/693a070bff55c952ca4b781867656bcf.png" NAME="images31" ALT="\mathbf{y}(k) = \mathbf{C}(k) \mathbf{x}(k) + \mathbf{D}(k) \mathbf{u}(k)" ALIGN=BOTTOM WIDTH=248 HEIGHT=21 BORDER=0></P> </TD> </TR> <TR VALIGN=TOP> <TD> <P>Laplace domain of<BR>continuous time-invariant</P> </TD> <TD> <P><IMG SRC="images/State_space_fichiers/73704ff87694600b875bcdbe9dd47f82.png" NAME="images32" ALT="s \mathbf{X}(s) = A \mathbf{X}(s) + B \mathbf{U}(s)" ALIGN=BOTTOM WIDTH=211 HEIGHT=21 BORDER=0><BR><IMG SRC="images/State_space_fichiers/b4edd4ea37a52ac0cfb97ef2cf8b6166.png" NAME="images33" ALT="\mathbf{Y}(s) = C \mathbf{X}(s) + D \mathbf{U}(s)" ALIGN=BOTTOM WIDTH=206 HEIGHT=21 BORDER=0></P> </TD> </TR> <TR VALIGN=TOP> <TD> <P>Z-domain of<BR>discrete time-invariant</P> </TD> <TD> <P><IMG SRC="images/State_space_fichiers/f40fb3cdcef75a4ffb787848e34f1a5d.png" NAME="images34" ALT="z \mathbf{X}(z) = A \mathbf{X}(z) + B \mathbf{U}(z)" ALIGN=BOTTOM WIDTH=216 HEIGHT=21 BORDER=0><BR><IMG SRC="images/State_space_fichiers/db506ab5a492b5f2ce0d1645db661c8e.png" NAME="images35" ALT="\mathbf{Y}(z) = C \mathbf{X}(z) + D \mathbf{U}(z)" ALIGN=BOTTOM WIDTH=208 HEIGHT=21 BORDER=0></P> </TD> </TR> </TABLE> <P><FONT COLOR="#000000">Example: Continuous-time LTI case</FONT> </P> <P><FONT COLOR="#000000">Stability and natural response characteristics of a continuous-time <A HREF="http://en.wikipedia.org/wiki/LTI_system">LTI system</A> (i.e., linear with matrices that are constant with respect to time) can be studied from the <A HREF="http://en.wikipedia.org/wiki/Eigenvalue">eigenvalues</A> of the matrix </FONT><FONT COLOR="#000000"><B>A</B></FONT><FONT COLOR="#000000">. The stability of a time-invariant state-space model can be determined by looking at the system's <A HREF="http://en.wikipedia.org/wiki/Transfer_function">transfer function</A> in factored form. It will then look something like this:</FONT></P> <DL> <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/3fa715cc96511bc510aabcb71c1f2781.png" NAME="images36" ALT=" \textbf{G}(s) = k \frac{ (s - z_{1})(s - z_{2})(s - z_{3}) }{ (s - p_{1})(s - p_{2})(s - p_{3})(s - p_{4}) }" ALIGN=BOTTOM WIDTH=346 HEIGHT=48 BORDER=0> </DD></DL> <P> <FONT COLOR="#000000">The denominator of the transfer function is equal to the <A HREF="http://en.wikipedia.org/wiki/Characteristic_polynomial">characteristic polynomial</A> found by taking the <A HREF="http://en.wikipedia.org/wiki/Determinant">determinant</A> of </FONT><FONT COLOR="#000000"><I>sI</I></FONT><FONT COLOR="#000000"> − </FONT><FONT COLOR="#000000"><I>A</I></FONT><FONT COLOR="#000000">,</FONT></P> <DL> <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/05e74baab950e092f7ef8acbf4654c84.png" NAME="images37" ALT="\mathbf{\lambda}(s) = |sI - A|" ALIGN=BOTTOM WIDTH=129 HEIGHT=21 BORDER=0>. </DD></DL> <P> <FONT COLOR="#000000">The roots of this polynomial (the <A HREF="http://en.wikipedia.org/wiki/Eigenvalue">eigenvalues</A>) are the system transfer function's <A HREF="http://en.wikipedia.org/wiki/Complex_pole">poles</A> (i.e., the <A HREF="http://en.wikipedia.org/wiki/Singularity">singularities</A> where the transfer function's magnitude is unbounded). These poles can be used to analyze whether the system is <A HREF="http://en.wikipedia.org/wiki/Exponential_stability">asymptotically stable</A> or <A HREF="http://en.wikipedia.org/wiki/Marginal_stability">marginally stable</A>. An alternative approach to determining stability, which does not involve calculating eigenvalues, is to analyze the system's <A HREF="http://en.wikipedia.org/wiki/Lyapunov_stability">Lyapunov stability</A>.</FONT></P> <P><FONT COLOR="#000000">The zeros found in the numerator of <IMG SRC="images/State_space_fichiers/e75eebaaa95472ea1407f43448f37ab8.png" NAME="images38" ALT="\textbf{G}(s)" ALIGN=BOTTOM WIDTH=39 HEIGHT=21 BORDER=0> can similarly be used to determine whether the system is <A HREF="http://en.wikipedia.org/wiki/Minimum_phase">minimum phase</A>.</FONT></P> <P><FONT COLOR="#000000">The system may still be </FONT><FONT COLOR="#000000"><B>input–output stable</B></FONT><FONT COLOR="#000000"> (see <A HREF="http://en.wikipedia.org/wiki/BIBO_stability">BIBO stable</A>) even though it is not internally stable. This may be the case if unstable poles are canceled out by zeros (i.e., if those singularities in the transfer function are <A HREF="http://en.wikipedia.org/wiki/Removable_singularity">removable</A>).</FONT></P> <H3 STYLE="border: none; padding: 0cm"><FONT COLOR="#000000">Controllability</FONT></H3> <P><FONT COLOR="#000000">Main article: <A HREF="http://en.wikipedia.org/wiki/Controllability">Controllability</A></FONT></P> <P><FONT COLOR="#000000">Thus, state controllability condition implies that it is possible – by admissible inputs – to steer the states from any initial value to any final value within some finite time window. A continuous time-invariant linear state-space model is </FONT><FONT COLOR="#000000"><B>controllable</B></FONT><FONT COLOR="#000000"> <A HREF="http://en.wikipedia.org/wiki/Iff">if and only if</A></FONT></P> <DL> <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/43a31067bb4883d16e9670dee45cafa6.png" NAME="images39" ALT="\operatorname{rank}\begin{bmatrix}B& AB& A^{2}B& ...& A^{n-1}B\end{bmatrix} = n" ALIGN=BOTTOM WIDTH=319 HEIGHT=24 BORDER=0> </DD></DL> <H3 STYLE="border: none; padding: 0cm"> <FONT COLOR="#000000">Observability</FONT></H3> <P><FONT COLOR="#000000">Main article: <A HREF="http://en.wikipedia.org/wiki/Observability">Observability</A></FONT></P> <P>Observability is a measure for how well internal states of a system can be inferred by knowledge of its external outputs. The observability and controllability of a system are mathematical duals (i.e., as controllablity provides that an input is available that brings any initial state to any desired final state, observability provides that knowing an output trajectory provides enough information to predict the initial state of the system).</P> <P>A continuous time-invariant linear state-space model is <B>observable</B> if and only if</P> <DL> <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/0fda525c5450269a5ee9e6a429c318d8.png" NAME="images40" ALT="\operatorname{rank}\begin{bmatrix}C\\ CA\\ ...\\ CA^{n-1}\end{bmatrix} = n" ALIGN=BOTTOM WIDTH=161 HEIGHT=97 BORDER=0> </DD></DL> <P> <FONT COLOR="#000000">(<A HREF="http://en.wikipedia.org/wiki/Rank_%28linear_algebra%29">Rank</A> is the number of linearly independent rows in a matrix.)</FONT></P> <H3 STYLE="border: none; padding: 0cm"><FONT COLOR="#000000">Transfer function</FONT></H3> <P><FONT COLOR="#000000">The "<A HREF="http://en.wikipedia.org/wiki/Transfer_function">transfer function</A>" of a continuous time-invariant linear state-space model can be derived in the following way:</FONT></P> <P><FONT COLOR="#000000">First, taking the <A HREF="http://en.wikipedia.org/wiki/Laplace_transform">Laplace transform</A> of</FONT></P> <DL> <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/ddfd74546a0e35f9ec054af2ecd3f2fa.png" NAME="images41" ALT="\dot{\mathbf{x}}(t) = A \mathbf{x}(t) + B \mathbf{u}(t)" ALIGN=BOTTOM WIDTH=182 HEIGHT=21 BORDER=0> </DD></DL> <P> yields</P> <DL> <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/73704ff87694600b875bcdbe9dd47f82.png" NAME="images42" ALT="s\mathbf{X}(s) = A \mathbf{X}(s) + B \mathbf{U}(s)" ALIGN=BOTTOM WIDTH=211 HEIGHT=21 BORDER=0> </DD></DL> <P> Next, we simplify for <IMG SRC="images/State_space_fichiers/0762e07773e4f1f30137d7915c154851.png" NAME="images43" ALT="\mathbf{X}(s)" ALIGN=BOTTOM WIDTH=40 HEIGHT=21 BORDER=0>, giving</P> <DL> <DD STYLE="margin-left: 0cm; border: none; padding: 0cm"><IMG SRC="images/State_space_fichiers/f28999fa2e82ade6d97d98696ca7ff3a.png" NAME="images44" ALT="(s\mathbf{I} - A)\mathbf{X}(s) = B\mathbf{U}(s)" ALIGN=BOTTOM WIDTH=194 HEIGHT=21 BORDER=0> </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/5b52cc12764f99ab8e3fcfab7bb54706.png" NAME="images45" ALT="\mathbf{X}(s) = (s\mathbf{I} - A)^{-1}B\mathbf{U}(s)" ALIGN=BOTTOM WIDTH=214 HEIGHT=23 BORDER=0> </DD></DL> <P> this is substituted for <IMG SRC="images/State_space_fichiers/0762e07773e4f1f30137d7915c154851.png" NAME="images46" ALT="\mathbf{X}(s)" ALIGN=BOTTOM WIDTH=40 HEIGHT=21 BORDER=0> in the output equation</P> <DL> <DD STYLE="margin-left: 0cm; border: none; padding: 0cm"><IMG SRC="images/State_space_fichiers/b4edd4ea37a52ac0cfb97ef2cf8b6166.png" NAME="images47" ALT="\mathbf{Y}(s) = C\mathbf{X}(s) + D\mathbf{U}(s)" ALIGN=BOTTOM WIDTH=206 HEIGHT=21 BORDER=0>, giving </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/97826155d4972cbddde77a96957af4b6.png" NAME="images48" ALT="\mathbf{Y}(s) = C((s\mathbf{I} - A)^{-1}B\mathbf{U}(s)) + D\mathbf{U}(s)" ALIGN=BOTTOM WIDTH=327 HEIGHT=23 BORDER=0> </DD></DL> <P> <FONT COLOR="#000000">Because the <A HREF="http://en.wikipedia.org/wiki/Transfer_function">transfer function</A> <IMG SRC="images/State_space_fichiers/046cdad9e9789e63b5196b2be94110e7.png" NAME="images49" ALT="\mathbf{G}(s)" ALIGN=BOTTOM WIDTH=40 HEIGHT=21 BORDER=0> is defined as the ratio of the output to the input of a system, we take</FONT></P> <DL> <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/64c51e4be94f51d2064e88ccd44c5927.png" NAME="images50" ALT="\mathbf{G}(s) = \mathbf{Y}(s) / \mathbf{U}(s)" ALIGN=BOTTOM WIDTH=159 HEIGHT=21 BORDER=0> </DD></DL> <P> and substitute the previous expression for <IMG SRC="images/State_space_fichiers/336fec49a4b9e27e1bc99139f4023267.png" NAME="images51" ALT="\mathbf{Y}(s)" ALIGN=BOTTOM WIDTH=40 HEIGHT=21 BORDER=0> with respect to <IMG SRC="images/State_space_fichiers/3f0af3cafb3dcd9039883b8047986fc2.png" NAME="images52" ALT="\mathbf{U}(s)" ALIGN=BOTTOM WIDTH=40 HEIGHT=21 BORDER=0>, giving</P> <DL> <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/510b6fe3ab8ef640745bfc65e481fdb6.png" NAME="images53" ALT="\mathbf{G}(s) = C(s\mathbf{I} - A)^{-1}B + D" ALIGN=BOTTOM WIDTH=229 HEIGHT=23 BORDER=0> </DD></DL> <P> Clearly <IMG SRC="images/State_space_fichiers/046cdad9e9789e63b5196b2be94110e7.png" NAME="images54" ALT="\mathbf{G}(s)" ALIGN=BOTTOM WIDTH=40 HEIGHT=21 BORDER=0> must have <I>q</I> by <I>p</I> dimensionality, and thus has a total of <I>qp</I> elements. So for every input there are <I>q</I> transfer functions with one for each output. This is why the state-space representation can easily be the preferred choice for multiple-input, multiple-output (MIMO) systems.</P> <H3 STYLE="border: none; padding: 0cm"><FONT COLOR="#000000">Canonical realizations</FONT></H3> <P><FONT COLOR="#000000">Any given transfer function which is <A HREF="http://en.wikipedia.org/wiki/Strictly_proper">strictly proper</A> can easily be transferred into state-space by the following approach (this example is for a 4-dimensional, single-input, single-output system)):</FONT></P> <P>Given a transfer function, expand it to reveal all coefficients in both the numerator and denominator. This should result in the following form:</P> <DL> <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/cc96db353659ef3ff0408ce45112743e.png" NAME="images55" ALT=" \textbf{G}(s) = \frac{n_{1}s^{3} + n_{2}s^{2} + n_{3}s + n_{4}}{s^{4} + d_{1}s^{3} + d_{2}s^{2} + d_{3}s + d_{4}}" ALIGN=BOTTOM WIDTH=298 HEIGHT=47 BORDER=0>. </DD></DL> <P> The coefficients can now be inserted directly into the state-space model by the following approach:</P> <DL> <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/612dae4aec269355721c7ad6ec438adf.png" NAME="images56" ALT="\dot{\textbf{x}}(t) = \begin{bmatrix} -d_{1}& -d_{2}& -d_{3}& -d_{4}\\ 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0 \end{bmatrix}\textbf{x}(t) + \begin{bmatrix} 1\\ 0\\ 0\\ 0\\ \end{bmatrix}\textbf{u}(t)" ALIGN=BOTTOM WIDTH=398 HEIGHT=97 BORDER=0> </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/b578227cbffe36298e451dddad03316a.png" NAME="images57" ALT=" \textbf{y}(t) = \begin{bmatrix} n_{1}& n_{2}& n_{3}& n_{4} \end{bmatrix}\textbf{x}(t)" ALIGN=BOTTOM WIDTH=238 HEIGHT=25 BORDER=0>. </DD></DL> <P> This state-space realization is called <B>controllable canonical form</B> because the resulting model is guaranteed to be controllable (i.e., because the control enters a chain of integrators, it has the ability to move every state).</P> <P>The transfer function coefficients can also be used to construct another type of canonical form</P> <DL> <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/48a1d333782665d7903fe3f142cbeed6.png" NAME="images58" ALT="\dot{\textbf{x}}(t) = \begin{bmatrix} -d_{1}& 1& 0& 0\\ -d_{2}& 0& 1& 0\\ -d_{3}& 0& 0& 1\\ -d_{4}& 0& 0& 0 \end{bmatrix}\textbf{x}(t) + \begin{bmatrix} n_{1}\\ n_{2}\\ n_{3}\\ n_{4} \end{bmatrix}\textbf{u}(t)" ALIGN=BOTTOM WIDTH=337 HEIGHT=97 BORDER=0> </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/79354cdcf381d3986dafa970009f10b4.png" NAME="images59" ALT=" \textbf{y}(t) = \begin{bmatrix} 1& 0& 0& 0 \end{bmatrix}\textbf{x}(t)" ALIGN=BOTTOM WIDTH=198 HEIGHT=25 BORDER=0>. </DD></DL> <P> This state-space realization is called <B>observable canonical form</B> because the resulting model is guaranteed to be observable (i.e., because the output exits from a chain of integrators, every state has an effect on the output).</P> <H3 STYLE="border: none; padding: 0cm"><FONT COLOR="#000000">Proper transfer functions</FONT></H3> <P><FONT COLOR="#000000">Transfer functions which are only <A HREF="http://en.wikipedia.org/wiki/Proper_transfer_function">proper</A> (and not <A HREF="http://en.wikipedia.org/wiki/Strictly_proper">strictly proper</A>) can also be realised quite easily. The trick here is to separate the transfer function into two parts: a strictly proper part and a constant.</FONT></P> <DL> <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/4634759df4f7389f67ac2d972c937ee6.png" NAME="images60" ALT=" \textbf{G}(s) = \textbf{G}_{SP}(s) + \textbf{G}(\infty)" ALIGN=BOTTOM WIDTH=205 HEIGHT=21 BORDER=0> </DD></DL> <P> The strictly proper transfer function can then be transformed into a canonical state space realization using techniques shown above. The state space realization of the constant is trivially <IMG SRC="images/State_space_fichiers/f66dfcf330a8db1668e1fefff7ef58bb.png" NAME="images61" ALT="\textbf{y}(t) = \textbf{G}(\infty)\textbf{u}(t)" ALIGN=BOTTOM WIDTH=146 HEIGHT=20 BORDER=0>. Together we then get a state space realization with matrices <I>A</I>,<I>B</I> and <I>C</I> determined by the strictly proper part, and matrix <I>D</I> determined by the constant.</P> <P><BR>Here is an example to clear things up a bit:</P> <DL> <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/53d2d208c9f9c252174c85b1e0c7bce3.png" NAME="images62" ALT=" \textbf{G}(s) = \frac{s^{2} + 3s + 3}{s^{2} + 2s + 1} = \frac{s + 2}{s^{2} + 2s + 1} + 1" ALIGN=BOTTOM WIDTH=322 HEIGHT=45 BORDER=0> </DD></DL> <P> which yields the following controllable realization</P> <DL> <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/d9679b5a5a38271e615b871f76c79c38.png" NAME="images63" ALT="\dot{\textbf{x}}(t) = \begin{bmatrix} -2& -1\\ 1& 0\\ \end{bmatrix}\textbf{x}(t) + \begin{bmatrix} 1\\ 0\end{bmatrix}\textbf{u}(t)" ALIGN=BOTTOM WIDTH=270 HEIGHT=48 BORDER=0> </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/278467bd0ca464c472b4e47477bdaaa5.png" NAME="images64" ALT=" \textbf{y}(t) = \begin{bmatrix} 1& 2\end{bmatrix}\textbf{x}(t) + \begin{bmatrix} 1\end{bmatrix}\textbf{u}(t)" ALIGN=BOTTOM WIDTH=231 HEIGHT=25 BORDER=0> </DD></DL> <P> Notice how the output also depends directly on the input. This is due to the <IMG SRC="images/State_space_fichiers/b00b33e9ecc56576da0c0710f7fd548e.png" NAME="images65" ALT="\textbf{G}(\infty)" ALIGN=BOTTOM WIDTH=51 HEIGHT=20 BORDER=0> constant in the transfer function.</P> <H3 STYLE="border: none; padding: 0cm"><FONT COLOR="#000000">Feedback</FONT></H3> <P STYLE="margin-bottom: 0cm; border: none; padding: 0cm"><FONT COLOR="#000080"><A HREF="http://en.wikipedia.org/wiki/File:Typical_State_Space_model_with_feedback.png"><FONT COLOR="#000080"><IMG SRC="images/State_space_fichiers/Typical_State_Space_model_with_feedback.png" NAME="images66" ALIGN=BOTTOM WIDTH=359 HEIGHT=224 BORDER=1></FONT></A></FONT> </P> <P>Typical state space model with feedback</P> <P><FONT COLOR="#000000">A common method for feedback is to multiply the output by a matrix </FONT><FONT COLOR="#000000"><I>K</I></FONT><FONT COLOR="#000000"> and setting this as the input to the system: <IMG SRC="images/State_space_fichiers/3b8be748204028e004ff5140c7ed2e78.png" NAME="images67" ALT="\mathbf{u}(t) = K \mathbf{y}(t)" ALIGN=BOTTOM WIDTH=112 HEIGHT=21 BORDER=0>. Since the values of </FONT><FONT COLOR="#000000"><I>K</I></FONT><FONT COLOR="#000000"> are unrestricted the values can easily be negated for <A HREF="http://en.wikipedia.org/wiki/Negative_feedback">negative feedback</A>. The presence of a negative sign (the common notation) is merely a notational one and its absence has no impact on the end results.</FONT></P> <DL> <DD STYLE="margin-left: 0cm; border: none; padding: 0cm"><IMG SRC="images/State_space_fichiers/ddfd74546a0e35f9ec054af2ecd3f2fa.png" NAME="images68" ALT="\dot{\mathbf{x}}(t) = A \mathbf{x}(t) + B \mathbf{u}(t)" ALIGN=BOTTOM WIDTH=182 HEIGHT=21 BORDER=0> </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/d0ac09f5cde2ce822ecc3e369692d04b.png" NAME="images69" ALT="\mathbf{y}(t) = C \mathbf{x}(t) + D \mathbf{u}(t)" ALIGN=BOTTOM WIDTH=184 HEIGHT=21 BORDER=0> </DD></DL> <P> becomes</P> <DL> <DD STYLE="margin-left: 0cm; border: none; padding: 0cm"><IMG SRC="images/State_space_fichiers/75e73a990aed8d68c4ea0dfe1dc04823.png" NAME="images70" ALT="\dot{\mathbf{x}}(t) = A \mathbf{x}(t) + B K \mathbf{y}(t)" ALIGN=BOTTOM WIDTH=200 HEIGHT=21 BORDER=0> </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/417f9e2993d831bf2ec0777be3f684d9.png" NAME="images71" ALT="\mathbf{y}(t) = C \mathbf{x}(t) + D K \mathbf{y}(t)" ALIGN=BOTTOM WIDTH=202 HEIGHT=21 BORDER=0> </DD></DL> <P> solving the output equation for <IMG SRC="images/State_space_fichiers/8bd1b0065e60113616b6750730e820f4.png" NAME="images72" ALT="\mathbf{y}(t)" ALIGN=BOTTOM WIDTH=34 HEIGHT=21 BORDER=0> and substituting in the state equation results in</P> <DL> <DD STYLE="margin-left: 0cm; border: none; padding: 0cm"><IMG SRC="images/State_space_fichiers/57650b1210f16a5bcd9a545c351f0be8.png" NAME="images73" ALT="\dot{\mathbf{x}}(t) = \left(A + B K \left(I - D K\right)^{-1} C \right) \mathbf{x}(t)" ALIGN=BOTTOM WIDTH=312 HEIGHT=26 BORDER=0> </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/78413a2591cd295ceda503bc6b2ea91b.png" NAME="images74" ALT="\mathbf{y}(t) = \left(I - D K\right)^{-1} C \mathbf{x}(t)" ALIGN=BOTTOM WIDTH=216 HEIGHT=24 BORDER=0> </DD></DL> <P> <FONT COLOR="#000000">The advantage of this is that the <A HREF="http://en.wikipedia.org/wiki/Eigenvalues">eigenvalues</A> of </FONT><FONT COLOR="#000000"><I>A</I></FONT><FONT COLOR="#000000"> can be controlled by setting </FONT><FONT COLOR="#000000"><I>K</I></FONT><FONT COLOR="#000000"> appropriately through eigendecomposition of <IMG SRC="images/State_space_fichiers/ff1b77ecd88e145dd3c106bdc6e98811.png" NAME="images75" ALT="\left(A + B K \left(I - D K\right)^{-1} C \right)" ALIGN=BOTTOM WIDTH=212 HEIGHT=30 BORDER=0>. This assumes that the open-loop system is <A HREF="http://en.wikipedia.org/wiki/Controllability">controllable</A> or that the unstable eigenvalues of </FONT><FONT COLOR="#000000"><I>A</I></FONT><FONT COLOR="#000000"> can be made stable through appropriate choice of </FONT><FONT COLOR="#000000"><I>K</I></FONT><FONT COLOR="#000000">.</FONT></P> <P>One fairly common simplification to this system is removing <I>D</I> and setting <I>C</I> to identity, which reduces the equations to</P> <DL> <DD STYLE="margin-left: 0cm; border: none; padding: 0cm"><IMG SRC="images/State_space_fichiers/3468897c1e237cff1ab38682f9b78693.png" NAME="images76" ALT="\dot{\mathbf{x}}(t) = \left(A + B K \right) \mathbf{x}(t)" ALIGN=BOTTOM WIDTH=184 HEIGHT=21 BORDER=0> </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/67a23f55dcbb3f93c3c5c8fc3681952b.png" NAME="images77" ALT="\mathbf{y}(t) = \mathbf{x}(t)" ALIGN=BOTTOM WIDTH=94 HEIGHT=21 BORDER=0> </DD></DL> <P> This reduces the necessary eigendecomposition to just <I>A</I> + <I>BK</I>.</P> <H3 STYLE="border: none; padding: 0cm"><FONT COLOR="#000000">Feedback with setpoint (reference) input</FONT></H3> <P STYLE="margin-bottom: 0cm; border: none; padding: 0cm"><FONT COLOR="#000080"><A HREF="http://en.wikipedia.org/wiki/File:Typical_State_Space_model_with_feedback_and_input.png"><FONT COLOR="#000080"><IMG SRC="images/State_space_fichiers/Typical_State_Space_model_with_feedback_and_input.png" NAME="images78" ALIGN=BOTTOM WIDTH=446 HEIGHT=240 BORDER=1></FONT></A></FONT> </P> <P>Output feedback with set point</P> <P>In addition to feedback, an input, <I>r</I>(<I>t</I>), can be added such that <IMG SRC="images/State_space_fichiers/c821af7d0e7d7d7b2a163898969ff496.png" NAME="images79" ALT="\mathbf{u}(t) = -K \mathbf{y}(t) + \mathbf{r}(t)" ALIGN=BOTTOM WIDTH=183 HEIGHT=21 BORDER=0>.</P> <DL> <DD STYLE="margin-left: 0cm; border: none; padding: 0cm"><IMG SRC="images/State_space_fichiers/ddfd74546a0e35f9ec054af2ecd3f2fa.png" NAME="images80" ALT="\dot{\mathbf{x}}(t) = A \mathbf{x}(t) + B \mathbf{u}(t)" ALIGN=BOTTOM WIDTH=182 HEIGHT=21 BORDER=0> </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/d0ac09f5cde2ce822ecc3e369692d04b.png" NAME="images81" ALT="\mathbf{y}(t) = C \mathbf{x}(t) + D \mathbf{u}(t)" ALIGN=BOTTOM WIDTH=184 HEIGHT=21 BORDER=0> </DD></DL> <P> becomes</P> <DL> <DD STYLE="margin-left: 0cm; border: none; padding: 0cm"><IMG SRC="images/State_space_fichiers/3d79549a0fc64810b943e3cb57e13af8.png" NAME="images82" ALT="\dot{\mathbf{x}}(t) = A \mathbf{x}(t) - B K \mathbf{y}(t) + B \mathbf{r}(t)" ALIGN=BOTTOM WIDTH=272 HEIGHT=21 BORDER=0> </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/798f03a10bb4be76f6abf2ac6910a396.png" NAME="images83" ALT="\mathbf{y}(t) = C \mathbf{x}(t) - D K \mathbf{y}(t) + D \mathbf{r}(t)" ALIGN=BOTTOM WIDTH=274 HEIGHT=21 BORDER=0> </DD></DL> <P> solving the output equation for <IMG SRC="images/State_space_fichiers/8bd1b0065e60113616b6750730e820f4.png" NAME="images84" ALT="\mathbf{y}(t)" ALIGN=BOTTOM WIDTH=34 HEIGHT=21 BORDER=0> and substituting in the state equation results in</P> <DL> <DD STYLE="margin-left: 0cm; border: none; padding: 0cm"><IMG SRC="images/State_space_fichiers/38d70e22f115d9278eb21de4e4e484e6.png" NAME="images85" ALT="\dot{\mathbf{x}}(t) = \left(A - B K \left(I + D K\right)^{-1} C \right) \mathbf{x}(t) + B \left(I - K \left(I + D K\right)^{-1}D \right) \mathbf{r}(t)" ALIGN=BOTTOM WIDTH=588 HEIGHT=25 BORDER=0> </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/37aad76449d0f436741f30b04413c4cd.png" NAME="images86" ALT="\mathbf{y}(t) = \left(I + D K\right)^{-1} C \mathbf{x}(t) + \left(I + D K\right)^{-1} D \mathbf{r}(t)" ALIGN=BOTTOM WIDTH=394 HEIGHT=24 BORDER=0> </DD></DL> <P> One fairly common simplification to this system is removing <I>D</I>, which reduces the equations to</P> <DL> <DD STYLE="margin-left: 0cm; border: none; padding: 0cm"><IMG SRC="images/State_space_fichiers/680eb3b0a3e7da89e78ed453cd1667ad.png" NAME="images87" ALT="\dot{\mathbf{x}}(t) = \left(A - B K C \right) \mathbf{x}(t) + B \mathbf{r}(t)" ALIGN=BOTTOM WIDTH=271 HEIGHT=21 BORDER=0> </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/2f8e4ed7d6ee668acb3731a715a41e64.png" NAME="images88" ALT="\mathbf{y}(t) = C \mathbf{x}(t)" ALIGN=BOTTOM WIDTH=109 HEIGHT=21 BORDER=0> </DD></DL> <H3 STYLE="border: none; padding: 0cm"> <FONT COLOR="#000000">Moving object example</FONT></H3> <P><FONT COLOR="#000000">A classical linear system is that of one-dimensional movement of an object. The <A HREF="http://en.wikipedia.org/wiki/Newton%27s_laws_of_motion">Newton's laws of motion</A> for an object moving horizontally on a plane and attached to a wall with a spring</FONT></P> <DL> <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/4ac69961a4f494cd12470e4005c7b6c9.png" NAME="images89" ALT="m \ddot{y}(t) = u(t) - k_1 \dot{y}(t) - k_2 y(t)" ALIGN=BOTTOM WIDTH=258 HEIGHT=21 BORDER=0> </DD></DL> <P> where</P> <UL> <LI><P STYLE="margin-bottom: 0cm; border: none; padding: 0cm"><I>y</I>(<I>t</I>) is position; <IMG SRC="images/State_space_fichiers/4721de091b5d15bb9809fa41a5e40db7.png" NAME="images90" ALT="\dot y(t)" ALIGN=BOTTOM WIDTH=32 HEIGHT=21 BORDER=0> is velocity; <IMG SRC="images/State_space_fichiers/d822334addb31f4d4fe1683e609e2742.png" NAME="images91" ALT="\ddot{y}(t)" ALIGN=BOTTOM WIDTH=32 HEIGHT=21 BORDER=0> is acceleration </P> <LI><P STYLE="margin-bottom: 0cm; border: none; padding: 0cm"><I>u</I>(<I>t</I>) is an applied force </P> <LI><P STYLE="margin-bottom: 0cm; border: none; padding: 0cm"><I>k</I><SUB>1</SUB> is the viscous friction coefficient </P> <LI><P STYLE="margin-bottom: 0cm; border: none; padding: 0cm"><I>k</I><SUB>2</SUB> is the spring constant </P> <LI><P><I>m</I> is the mass of the object </P> </UL> <P>The state equation would then become</P> <DL> <DD STYLE="margin-left: 0cm; border: none; padding: 0cm"><IMG SRC="images/State_space_fichiers/c3b6d7d7cd783c7f6a6dd2ff7478f8a0.png" NAME="images92" ALT="\left[ \begin{matrix} \mathbf{\dot{x_1}}(t) \\ \mathbf{\dot{x_2}}(t) \end{matrix} \right] = \left[ \begin{matrix} 0 & 1 \\ -\frac{k_2}{m} & -\frac{k_1}{m} \end{matrix} \right] \left[ \begin{matrix} \mathbf{x_1}(t) \\ \mathbf{x_2}(t) \end{matrix} \right] + \left[ \begin{matrix} 0 \\ \frac{1}{m} \end{matrix} \right] \mathbf{u}(t)" ALIGN=BOTTOM WIDTH=357 HEIGHT=49 BORDER=0> </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/7771c47e6e2f82d6138ee2e2ec920c7e.png" NAME="images93" ALT="\mathbf{y}(t) = \left[ \begin{matrix} 1 & 0 \end{matrix} \right] \left[ \begin{matrix} \mathbf{x_1}(t) \\ \mathbf{x_2}(t) \end{matrix} \right]" ALIGN=BOTTOM WIDTH=176 HEIGHT=49 BORDER=0> </DD></DL> <P> where</P> <UL> <LI><P STYLE="margin-bottom: 0cm; border: none; padding: 0cm"><I>x</I><SUB>1</SUB>(<I>t</I>) represents the position of the object </P> <LI><P STYLE="margin-bottom: 0cm; border: none; padding: 0cm"><IMG SRC="images/State_space_fichiers/93381b17950c251e433b768a2e2ba6b3.png" NAME="images94" ALT="x_2(t) = \dot{x_1}(t)" ALIGN=BOTTOM WIDTH=108 HEIGHT=21 BORDER=0> is the velocity of the object </P> <LI><P STYLE="margin-bottom: 0cm; border: none; padding: 0cm"><IMG SRC="images/State_space_fichiers/08b851e43484faf6400aa16c3632cda5.png" NAME="images95" ALT="\dot{x_2}(t) = \ddot{x_1}(t)" ALIGN=BOTTOM WIDTH=108 HEIGHT=21 BORDER=0> is the acceleration of the object </P> <LI><P>the output <IMG SRC="images/State_space_fichiers/8bd1b0065e60113616b6750730e820f4.png" NAME="images96" ALT="\mathbf{y}(t)" ALIGN=BOTTOM WIDTH=34 HEIGHT=21 BORDER=0> is the position of the object </P> </UL> <P><FONT COLOR="#000000">The <A HREF="http://en.wikipedia.org/wiki/Controllability">controllability</A> test is then</FONT></P> <DL> <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/1b66d8f8c0227384d70806878463f8ce.png" NAME="images97" ALT="\left[ \begin{matrix} B & AB \end{matrix} \right] = \left[ \begin{matrix} \left[ \begin{matrix} 0 \\ \frac{1}{m} \end{matrix} \right] & \left[ \begin{matrix} 0 & 1 \\ -\frac{k_2}{m} & -\frac{k_1}{m} \end{matrix} \right] \left[ \begin{matrix} 0 \\ \frac{1}{m} \end{matrix} \right] \end{matrix} \right] = \left[ \begin{matrix} 0 & \frac{1}{m} \\ \frac{1}{m} & \frac{k_1}{m^2} \end{matrix} \right]" ALIGN=BOTTOM WIDTH=422 HEIGHT=49 BORDER=0> </DD></DL> <P> which has full rank for all <I>k</I><SUB>1</SUB> and <I>m</I>.</P> <P><FONT COLOR="#000000">The <A HREF="http://en.wikipedia.org/wiki/Observability">observability</A> test is then</FONT></P> <DL> <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/05e20d83a2969f3b9e48fef4bacc8d3c.png" NAME="images98" ALT="\left[ \begin{matrix} C \\ CA \end{matrix} \right] = \left[ \begin{matrix} \left[ \begin{matrix} 1 & 0 \end{matrix} \right] \\ \left[ \begin{matrix} 1 & 0 \end{matrix} \right] \left[ \begin{matrix} 0 & 1 \\ -\frac{k_2}{m} & -\frac{k_1}{m} \end{matrix} \right] \end{matrix} \right] = \left[ \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right]" ALIGN=BOTTOM WIDTH=340 HEIGHT=74 BORDER=0> </DD></DL> <P> which also has full rank. Therefore, this system is both controllable and observable.</P> <H2 STYLE="border: none; padding: 0cm"><FONT COLOR="#000000">Nonlinear systems</FONT></H2> <P>The more general form of a state space model can be written as two functions.</P> <DL> <DD STYLE="margin-left: 0cm; border: none; padding: 0cm"><IMG SRC="images/State_space_fichiers/0902bfc977a4f7f7c7cc8bc54ac93634.png" NAME="images99" ALT="\mathbf{\dot{x}}(t) = \mathbf{f}(t, x(t), u(t))" ALIGN=BOTTOM WIDTH=174 HEIGHT=21 BORDER=0> </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/f5d907725ffb26ae9fd08b9615298756.png" NAME="images100" ALT="\mathbf{y}(t) = \mathbf{h}(t, x(t), u(t))" ALIGN=BOTTOM WIDTH=178 HEIGHT=21 BORDER=0> </DD></DL> <P> The first is the state equation and the latter is the output equation. If the function <IMG SRC="images/State_space_fichiers/f6980ec56e3e7fcce7a143b507a53126.png" NAME="images101" ALT="f(\cdot,\cdot,\cdot)" ALIGN=BOTTOM WIDTH=59 HEIGHT=21 BORDER=0> is a linear combination of states and inputs then the equations can be written in matrix notation like above. The <I>u</I>(<I>t</I>) argument to the functions can be dropped if the system is unforced (i.e., it has no inputs).</P> <H3 STYLE="border: none; padding: 0cm"><FONT COLOR="#000000">Pendulum example</FONT></H3> <P><FONT COLOR="#000000">A classic nonlinear system is a simple unforced <A HREF="http://en.wikipedia.org/wiki/Pendulum">pendulum</A></FONT></P> <DL> <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/7880de515445681f6e12f7c482ddc9ef.png" NAME="images102" ALT="ml\ddot\theta(t)= -mg\sin\theta(t) - kl\dot\theta(t)" ALIGN=BOTTOM WIDTH=258 HEIGHT=25 BORDER=0> </DD></DL> <P> where</P> <UL> <LI><P STYLE="margin-bottom: 0cm; border: none; padding: 0cm">θ(<I>t</I>) is the angle of the pendulum with respect to the direction of gravity </P> <LI><P STYLE="margin-bottom: 0cm; border: none; padding: 0cm"><I>m</I> is the mass of the pendulum (pendulum rod's mass is assumed to be zero) </P> <LI><P STYLE="margin-bottom: 0cm; border: none; padding: 0cm"><I>g</I> is the gravitational acceleration </P> <LI><P STYLE="margin-bottom: 0cm; border: none; padding: 0cm"><I>k</I> is coefficient of friction at the pivot point </P> <LI><P><I>l</I> is the radius of the pendulum (to the center of gravity of the mass <I>m</I>) </P> </UL> <P>The state equations are then</P> <DL> <DD STYLE="margin-left: 0cm; border: none; padding: 0cm"><IMG SRC="images/State_space_fichiers/0cec7d908b5ca88fb84845636a5aa5b2.png" NAME="images103" ALT="\dot{x_1}(t) = x_2(t)" ALIGN=BOTTOM WIDTH=108 HEIGHT=21 BORDER=0> </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/66a8c07009561c21dcf5481d01b376f3.png" NAME="images104" ALT="\dot{x_2}(t) = - \frac{g}{l}\sin{x_1}(t) - \frac{k}{m}{x_2}(t)" ALIGN=BOTTOM WIDTH=254 HEIGHT=41 BORDER=0> </DD></DL> <P> where</P> <UL> <LI><P STYLE="margin-bottom: 0cm; border: none; padding: 0cm"><I>x</I><SUB>1</SUB>(<I>t</I>): = θ(<I>t</I>) is the angle of the pendulum </P> <LI><P STYLE="margin-bottom: 0cm; border: none; padding: 0cm"><IMG SRC="images/State_space_fichiers/5cd1da7ba81f4b866e8174ab86e3c5c8.png" NAME="images105" ALT="x_2(t) :=\dot{x_1}(t)" ALIGN=BOTTOM WIDTH=112 HEIGHT=21 BORDER=0> is the rotational velocity of the pendulum </P> <LI><P><IMG SRC="images/State_space_fichiers/7bbc064941778c6e2e4d88bf4152c728.png" NAME="images106" ALT="\dot{x_2} =\ddot{x_1}" ALIGN=BOTTOM WIDTH=63 HEIGHT=17 BORDER=0> is the rotational acceleration of the pendulum </P> </UL> <P>Instead, the state equation can be written in the general form</P> <DL> <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/e0da7bf257cb650885319ce4b5833bd8.png" NAME="images107" ALT="\dot{x}(t) = \left( \begin{matrix} \dot{x_1}(t) \\ \dot{x_2}(t) \end{matrix} \right) = \mathbf{f}(t, x(t)) = \left( \begin{matrix} x_2(t) \\ - \frac{g}{l}\sin{x_1}(t) - \frac{k}{m}{x_2}(t) \end{matrix} \right)." ALIGN=BOTTOM WIDTH=473 HEIGHT=49 BORDER=0> </DD></DL> <P> <FONT COLOR="#000000">The <A HREF="http://en.wikipedia.org/wiki/Mechanical_equilibrium">equilibrium</A>/<A HREF="http://en.wikipedia.org/wiki/Stationary_point">stationary points</A> of a system are when <IMG SRC="images/State_space_fichiers/4009ed416fcf36f8b7ec3dbabe9fa791.png" NAME="images108" ALT="\dot{x} = 0" ALIGN=BOTTOM WIDTH=46 HEIGHT=15 BORDER=0> and so the equilibrium points of a pendulum are those that satisfy</FONT></P> <DL> <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm"> <IMG SRC="images/State_space_fichiers/f8f06079d8991d2831f60ec5a190f11b.png" NAME="images109" ALT="\left( \begin{matrix} x_1 \\ x_2 \end{matrix} \right) = \left( \begin{matrix} n\pi \\ 0 \end{matrix} \right)" ALIGN=BOTTOM WIDTH=120 HEIGHT=48 BORDER=0> </DD></DL> <P> for integers <I>n</I>.</P> </SPAN><BR><BR> </P> <P><BR><BR> </P> <P><BR><BR> </P> </BODY> </HTML>