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author Daniele Nicolodi <nicolodi@science.unitn.it>
date Wed, 23 Nov 2011 19:22:13 +0100
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11 <title>Statespace models (LTPDA Toolbox)</title>
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21 <p style="font-size:1px;">&nbsp;</p>
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23 <table class="nav" summary="Navigation aid" border="0" width=
24 "100%" cellpadding="0" cellspacing="0">
25 <tr>
26 <td valign="baseline"><b>LTPDA Toolbox</b></td><td><a href="../helptoc.html">contents</a></td>
27
28 <td valign="baseline" align="right"><a href=
29 "smodel_introduction.html"><img src="b_prev.gif" border="0" align=
30 "bottom" alt="Introduction to parametric models in LTPDA"></a>&nbsp;&nbsp;&nbsp;<a href=
31 "ssm_introduction.html"><img src="b_next.gif" border="0" align=
32 "bottom" alt="Introduction to Statespace Models with LTPDA"></a></td>
33 </tr>
34 </table>
35
36 <h1 class="title"><a name="f3-12899" id="f3-12899"></a>Statespace models</h1>
37 <hr>
38
39 <p>
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51 <P>
52 <H2>Why use state space modeling?</H2>
53 <P>State space modeling is efficient to simulate systems with large
54 dimensionality, be it in terms of inputs, outputs, or pole/zeros.
55 Adding nonlinearities to a model is also easier as in the frequency
56 domain, however there is no such capability in the toolbox yet.
57 Another reason to use them is to build complex parametric models,
58 where intricated parameters make impossible the symbolic calculation
59 of the transfer function coefficients &ndash; as a matter of fact
60 the transfer function can be computed out of a determinant involving
61 the A matrix, explaining the complexity of the calculation.</P>
62 <P>For tasks such as identification, state space modeling is a
63 computationally rather heavy, especially if colored noise is
64 involved in the process.</P>
65 <P>State space models can be converted into a matrix of transfer
66 function in the s- or the z-domain. The functions in the toolbox
67 that enable this are ss2pzmodel, ss2miir, ss2rational.</P>
68 <H2>Generalities on State Space Modeling</H2>
69 <P><FONT COLOR="#000000">In order to familiarize with state space
70 modeling, this help page is a mere copy of the wiki page.</FONT></P>
71 <P><FONT COLOR="#000000">In control engineering, a </FONT><FONT COLOR="#000000"><B>state
72 space representation</B></FONT><FONT COLOR="#000000"> is a
73 mathematical model of a physical system as a set of input, output
74 and state variables related by first-order <A HREF="http://en.wikipedia.org/wiki/Differential_equation">differential
75 equations</A>. To abstract from the number of inputs, outputs </FONT>and<FONT COLOR="#000000">
76 states, the variables are expressed as vectors and the differential
77 and algebraic equations are written in matrix form (the last one can
78 be done when the <A HREF="http://en.wikipedia.org/wiki/Dynamical_system">dynamical
79 system</A> is linear and time invariant). The state space
80 representation (also known as the &quot;time-domain approach&quot;)
81 provides a convenient and compact way to model and analyze systems
82 with multiple inputs and outputs. With </FONT><FONT COLOR="#000000"><I>p</I></FONT><FONT COLOR="#000000">
83 inputs and </FONT><FONT COLOR="#000000"><I>q</I></FONT><FONT COLOR="#000000">
84 outputs, we would otherwise have to write down <A HREF="http://en.wikipedia.org/wiki/Laplace_transform">Laplace
85 transforms</A> to encode all the information about a system. Unlike
86 the frequency domain approach, the use of the state space
87 representation is not limited to systems with linear components and
88 zero initial conditions. &quot;State space&quot; refers to the space
89 whose axes are the state variables. The state of the system can be
90 represented as a vector within that space.</FONT></P>
91 <H2><FONT COLOR="#000000">State variables</FONT></H2>
92 <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">
93 </FONT>
94 </H2>
95 <P>Typical state space model</P>
96 <P><FONT COLOR="#000000">The internal <A HREF="http://en.wikipedia.org/wiki/State_variable">state
97 variables</A> are the smallest possible subset of system variables
98 that can represent the entire state of the system at any given time.
99 State variables must be linearly independent; a state variable
100 cannot be a linear combination of other state variables. The minimum
101 number of state variables required to represent a given system, </FONT><FONT COLOR="#000000"><I>n</I></FONT><FONT COLOR="#000000">,
102 is usually equal to the order of the system's defining differential
103 equation. If the system is represented in transfer function form,
104 the minimum number of state variables is equal to the order of the
105 transfer function's denominator after it has been reduced to a
106 proper fraction. It is important to understand that converting a
107 state space realization to a transfer function form may lose some
108 internal information about the system, and may provide a description
109 of a system which is stable, when the state-space realization is
110 unstable at certain points. In electric circuits, the number of
111 state variables is often, though not always, the same as the number
112 of energy storage elements in the circuit such as <A HREF="http://en.wikipedia.org/wiki/Capacitor">capacitors</A>
113 and <A HREF="http://en.wikipedia.org/wiki/Inductor">inductors</A>.</FONT></P>
114 <H2><FONT COLOR="#000000">Linear systems</FONT></H2>
115 <P>The most general state-space representation of a linear system
116 with <I>p</I> inputs, <I>q</I> outputs and <I>n</I> state variables
117 is written in the following form:</P>
118 <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>
119 </P>
120 <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>
121 </P>
122 <P>where:</P>
123 <UL>
124 <LI><P><IMG SRC="images/State_space_fichiers/0b46dcfa3789d90891d7d22aa4bf1b83.png" NAME="images6" ALT="x(\cdot)" ALIGN=BOTTOM WIDTH=31 HEIGHT=21 BORDER=0>
125 is called the &quot;state vector&quot;,&nbsp;
126 <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>;
127 </P>
128 <LI><P><IMG SRC="images/State_space_fichiers/89df6576cd198dc2449bf59e97a61df3.png" NAME="images8" ALT="y(\cdot)" ALIGN=BOTTOM WIDTH=30 HEIGHT=21 BORDER=0>
129 is called the &quot;output vector&quot;,&nbsp;
130 <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>;
131 </P>
132 <LI><P><IMG SRC="images/State_space_fichiers/918aa6368974b6014859c56e1ef6762c.png" NAME="images10" ALT="u(\cdot)" ALIGN=BOTTOM WIDTH=31 HEIGHT=21 BORDER=0>
133 is called the &quot;input (or control) vector&quot;,&nbsp;
134 <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>;
135 </P>
136 <LI><P><IMG SRC="images/State_space_fichiers/b53a04dbe423d4e39eaedadf54e3ee31.png" NAME="images12" ALT="A(\cdot)" ALIGN=BOTTOM WIDTH=35 HEIGHT=21 BORDER=0>
137 is the &quot;state matrix&quot;,&nbsp;
138 <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>,
139 </P>
140 <LI><P><IMG SRC="images/State_space_fichiers/8f528e3252e95ac5edef108971a1d43f.png" NAME="images14" ALT="B(\cdot)" ALIGN=BOTTOM WIDTH=36 HEIGHT=21 BORDER=0>
141 is the &quot;input matrix&quot;,&nbsp;
142 <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>,
143 </P>
144 <LI><P><IMG SRC="images/State_space_fichiers/d4d99b70fae67ddcd7923dbe5dbb6424.png" NAME="images16" ALT="C(\cdot)" ALIGN=BOTTOM WIDTH=34 HEIGHT=21 BORDER=0>
145 is the &quot;output matrix&quot;,&nbsp;
146 <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>,
147 </P>
148 <LI><P><IMG SRC="images/State_space_fichiers/9828f5f36ec8af68cc289bbf41a2c751.png" NAME="images18" ALT="D(\cdot)" ALIGN=BOTTOM WIDTH=37 HEIGHT=21 BORDER=0>
149 is the &quot;feedthrough (or feedforward) matrix&quot; (in cases
150 where the system model does not have a direct feedthrough,
151 <IMG SRC="images/State_space_fichiers/9828f5f36ec8af68cc289bbf41a2c751.png" NAME="images19" ALT="D(\cdot)" ALIGN=BOTTOM WIDTH=37 HEIGHT=21 BORDER=0>
152 is the zero matrix),&nbsp;
153 <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>,
154 </P>
155 <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>.
156 </P>
157 </UL>
158 <P><FONT COLOR="#000000">In this general formulation, all matrices
159 are allowed to be time-variant (i.e., their elements can depend on
160 time); however, in the common <A HREF="http://en.wikipedia.org/wiki/LTI_system">LTI</A>
161 case, matrices will be time invariant. The time variable </FONT><FONT COLOR="#000000"><I>t</I></FONT><FONT COLOR="#000000">
162 can be a &quot;continuous&quot; (e.g.,
163 <IMG SRC="images/State_space_fichiers/6080cfaa75ff6363d282e29a14b46632.png" NAME="images22" ALT="t \in \mathbb{R}" ALIGN=BOTTOM WIDTH=45 HEIGHT=15 BORDER=0>)
164 or discrete (e.g.,
165 <IMG SRC="images/State_space_fichiers/d3bd3b5e80beacc6f4a28dacabf2c0c5.png" NAME="images23" ALT="t \in \mathbb{Z}" ALIGN=BOTTOM WIDTH=45 HEIGHT=15 BORDER=0>).
166 In the latter case, the time variable is usually indicated as </FONT><FONT COLOR="#000000"><I>k</I></FONT><FONT COLOR="#000000">.
167 <A HREF="http://en.wikipedia.org/wiki/Hybrid_system">Hybrid systems</A>
168 allow for time domains that have both continuous and discrete parts.
169 Depending on the assumptions taken, the state-space model
170 representation can assume the following forms:</FONT></P>
171 <TABLE DIR="LTR" CELLPADDING=4 CELLSPACING=2>
172 <TR VALIGN=TOP>
173 <TD>
174 <P><B>System type</B></P>
175 </TD>
176 <TD>
177 <P><B>State-space model</B></P>
178 </TD>
179 </TR>
180 <TR VALIGN=TOP>
181 <TD>
182 <P>Continuous time-invariant</P>
183 </TD>
184 <TD>
185 <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>
186 </TD>
187 </TR>
188 <TR VALIGN=TOP>
189 <TD>
190 <P>Continuous time-variant</P>
191 </TD>
192 <TD>
193 <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>
194 </TD>
195 </TR>
196 <TR VALIGN=TOP>
197 <TD>
198 <P>Discrete time-invariant</P>
199 </TD>
200 <TD>
201 <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>
202 </TD>
203 </TR>
204 <TR VALIGN=TOP>
205 <TD>
206 <P>Discrete time-variant</P>
207 </TD>
208 <TD>
209 <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>
210 </TD>
211 </TR>
212 <TR VALIGN=TOP>
213 <TD>
214 <P>Laplace domain of<BR>continuous time-invariant</P>
215 </TD>
216 <TD>
217 <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>
218 </TD>
219 </TR>
220 <TR VALIGN=TOP>
221 <TD>
222 <P>Z-domain of<BR>discrete time-invariant</P>
223 </TD>
224 <TD>
225 <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>
226 </TD>
227 </TR>
228 </TABLE>
229 <P><FONT COLOR="#000000">Example: Continuous-time LTI case</FONT>
230 </P>
231 <P><FONT COLOR="#000000">Stability and natural response
232 characteristics of a continuous-time <A HREF="http://en.wikipedia.org/wiki/LTI_system">LTI
233 system</A> (i.e., linear with matrices that are constant with
234 respect to time) can be studied from the <A HREF="http://en.wikipedia.org/wiki/Eigenvalue">eigenvalues</A>
235 of the matrix </FONT><FONT COLOR="#000000"><B>A</B></FONT><FONT COLOR="#000000">.
236 The stability of a time-invariant state-space model can be
237 determined by looking at the system's <A HREF="http://en.wikipedia.org/wiki/Transfer_function">transfer
238 function</A> in factored form. It will then look something like
239 this:</FONT></P>
240 <DL>
241 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
242 <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>
243 </DD></DL>
244 <P>
245 <FONT COLOR="#000000">The denominator of the transfer function is
246 equal to the <A HREF="http://en.wikipedia.org/wiki/Characteristic_polynomial">characteristic
247 polynomial</A> found by taking the <A HREF="http://en.wikipedia.org/wiki/Determinant">determinant</A>
248 of </FONT><FONT COLOR="#000000"><I>sI</I></FONT><FONT COLOR="#000000">
249 &minus; </FONT><FONT COLOR="#000000"><I>A</I></FONT><FONT COLOR="#000000">,</FONT></P>
250 <DL>
251 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
252 <IMG SRC="images/State_space_fichiers/05e74baab950e092f7ef8acbf4654c84.png" NAME="images37" ALT="\mathbf{\lambda}(s) = |sI - A|" ALIGN=BOTTOM WIDTH=129 HEIGHT=21 BORDER=0>.
253 </DD></DL>
254 <P>
255 <FONT COLOR="#000000">The roots of this polynomial (the <A HREF="http://en.wikipedia.org/wiki/Eigenvalue">eigenvalues</A>)
256 are the system transfer function's <A HREF="http://en.wikipedia.org/wiki/Complex_pole">poles</A>
257 (i.e., the <A HREF="http://en.wikipedia.org/wiki/Singularity">singularities</A>
258 where the transfer function's magnitude is unbounded). These poles
259 can be used to analyze whether the system is <A HREF="http://en.wikipedia.org/wiki/Exponential_stability">asymptotically
260 stable</A> or <A HREF="http://en.wikipedia.org/wiki/Marginal_stability">marginally
261 stable</A>. An alternative approach to determining stability, which
262 does not involve calculating eigenvalues, is to analyze the system's
263 <A HREF="http://en.wikipedia.org/wiki/Lyapunov_stability">Lyapunov
264 stability</A>.</FONT></P>
265 <P><FONT COLOR="#000000">The zeros found in the numerator of
266 <IMG SRC="images/State_space_fichiers/e75eebaaa95472ea1407f43448f37ab8.png" NAME="images38" ALT="\textbf{G}(s)" ALIGN=BOTTOM WIDTH=39 HEIGHT=21 BORDER=0>
267 can similarly be used to determine whether the system is <A HREF="http://en.wikipedia.org/wiki/Minimum_phase">minimum
268 phase</A>.</FONT></P>
269 <P><FONT COLOR="#000000">The system may still be </FONT><FONT COLOR="#000000"><B>input&ndash;output
270 stable</B></FONT><FONT COLOR="#000000"> (see <A HREF="http://en.wikipedia.org/wiki/BIBO_stability">BIBO
271 stable</A>) even though it is not internally stable. This may be the
272 case if unstable poles are canceled out by zeros (i.e., if those
273 singularities in the transfer function are <A HREF="http://en.wikipedia.org/wiki/Removable_singularity">removable</A>).</FONT></P>
274 <H3 STYLE="border: none; padding: 0cm"><FONT COLOR="#000000">Controllability</FONT></H3>
275 <P><FONT COLOR="#000000">Main article: <A HREF="http://en.wikipedia.org/wiki/Controllability">Controllability</A></FONT></P>
276 <P><FONT COLOR="#000000">Thus, state controllability condition
277 implies that it is possible&nbsp;&ndash; by admissible inputs&nbsp;&ndash;
278 to steer the states from any initial value to any final value within
279 some finite time window. A continuous time-invariant linear
280 state-space model is </FONT><FONT COLOR="#000000"><B>controllable</B></FONT><FONT COLOR="#000000">
281 <A HREF="http://en.wikipedia.org/wiki/Iff">if and only if</A></FONT></P>
282 <DL>
283 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
284 <IMG SRC="images/State_space_fichiers/43a31067bb4883d16e9670dee45cafa6.png" NAME="images39" ALT="\operatorname{rank}\begin{bmatrix}B&amp; AB&amp; A^{2}B&amp; ...&amp; A^{n-1}B\end{bmatrix} = n" ALIGN=BOTTOM WIDTH=319 HEIGHT=24 BORDER=0>
285 </DD></DL>
286 <H3 STYLE="border: none; padding: 0cm">
287 <FONT COLOR="#000000">Observability</FONT></H3>
288 <P><FONT COLOR="#000000">Main article: <A HREF="http://en.wikipedia.org/wiki/Observability">Observability</A></FONT></P>
289 <P>Observability is a measure for how well internal states of a
290 system can be inferred by knowledge of its external outputs. The
291 observability and controllability of a system are mathematical duals
292 (i.e., as controllablity provides that an input is available that
293 brings any initial state to any desired final state, observability
294 provides that knowing an output trajectory provides enough
295 information to predict the initial state of the system).</P>
296 <P>A continuous time-invariant linear state-space model is
297 <B>observable</B> if and only if</P>
298 <DL>
299 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
300 <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>
301 </DD></DL>
302 <P>
303 <FONT COLOR="#000000">(<A HREF="http://en.wikipedia.org/wiki/Rank_%28linear_algebra%29">Rank</A>
304 is the number of linearly independent rows in a matrix.)</FONT></P>
305 <H3 STYLE="border: none; padding: 0cm"><FONT COLOR="#000000">Transfer
306 function</FONT></H3>
307 <P><FONT COLOR="#000000">The &quot;<A HREF="http://en.wikipedia.org/wiki/Transfer_function">transfer
308 function</A>&quot; of a continuous time-invariant linear state-space
309 model can be derived in the following way:</FONT></P>
310 <P><FONT COLOR="#000000">First, taking the <A HREF="http://en.wikipedia.org/wiki/Laplace_transform">Laplace
311 transform</A> of</FONT></P>
312 <DL>
313 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
314 <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>
315 </DD></DL>
316 <P>
317 yields</P>
318 <DL>
319 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
320 <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>
321 </DD></DL>
322 <P>
323 Next, we simplify for
324 <IMG SRC="images/State_space_fichiers/0762e07773e4f1f30137d7915c154851.png" NAME="images43" ALT="\mathbf{X}(s)" ALIGN=BOTTOM WIDTH=40 HEIGHT=21 BORDER=0>,
325 giving</P>
326 <DL>
327 <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>
328 </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
329 <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>
330 </DD></DL>
331 <P>
332 this is substituted for
333 <IMG SRC="images/State_space_fichiers/0762e07773e4f1f30137d7915c154851.png" NAME="images46" ALT="\mathbf{X}(s)" ALIGN=BOTTOM WIDTH=40 HEIGHT=21 BORDER=0>
334 in the output equation</P>
335 <DL>
336 <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>,
337 giving
338 </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
339 <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>
340 </DD></DL>
341 <P>
342 <FONT COLOR="#000000">Because the <A HREF="http://en.wikipedia.org/wiki/Transfer_function">transfer
343 function</A>
344 <IMG SRC="images/State_space_fichiers/046cdad9e9789e63b5196b2be94110e7.png" NAME="images49" ALT="\mathbf{G}(s)" ALIGN=BOTTOM WIDTH=40 HEIGHT=21 BORDER=0>
345 is defined as the ratio of the output to the input of a system, we
346 take</FONT></P>
347 <DL>
348 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
349 <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>
350 </DD></DL>
351 <P>
352 and substitute the previous expression for
353 <IMG SRC="images/State_space_fichiers/336fec49a4b9e27e1bc99139f4023267.png" NAME="images51" ALT="\mathbf{Y}(s)" ALIGN=BOTTOM WIDTH=40 HEIGHT=21 BORDER=0>
354 with respect to
355 <IMG SRC="images/State_space_fichiers/3f0af3cafb3dcd9039883b8047986fc2.png" NAME="images52" ALT="\mathbf{U}(s)" ALIGN=BOTTOM WIDTH=40 HEIGHT=21 BORDER=0>,
356 giving</P>
357 <DL>
358 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
359 <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>
360 </DD></DL>
361 <P>
362 Clearly
363 <IMG SRC="images/State_space_fichiers/046cdad9e9789e63b5196b2be94110e7.png" NAME="images54" ALT="\mathbf{G}(s)" ALIGN=BOTTOM WIDTH=40 HEIGHT=21 BORDER=0>
364 must have <I>q</I> by <I>p</I> dimensionality, and thus has a total
365 of <I>qp</I> elements. So for every input there are <I>q</I>
366 transfer functions with one for each output. This is why the
367 state-space representation can easily be the preferred choice for
368 multiple-input, multiple-output (MIMO) systems.</P>
369 <H3 STYLE="border: none; padding: 0cm"><FONT COLOR="#000000">Canonical
370 realizations</FONT></H3>
371 <P><FONT COLOR="#000000">Any given transfer function which is
372 <A HREF="http://en.wikipedia.org/wiki/Strictly_proper">strictly
373 proper</A> can easily be transferred into state-space by the
374 following approach (this example is for a 4-dimensional,
375 single-input, single-output system)):</FONT></P>
376 <P>Given a transfer function, expand it to reveal all coefficients
377 in both the numerator and denominator. This should result in the
378 following form:</P>
379 <DL>
380 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
381 <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>.
382 </DD></DL>
383 <P>
384 The coefficients can now be inserted directly into the state-space
385 model by the following approach:</P>
386 <DL>
387 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
388 <IMG SRC="images/State_space_fichiers/612dae4aec269355721c7ad6ec438adf.png" NAME="images56" ALT="\dot{\textbf{x}}(t) = \begin{bmatrix} -d_{1}&amp; -d_{2}&amp; -d_{3}&amp; -d_{4}\\ 1&amp; 0&amp; 0&amp; 0\\ 0&amp; 1&amp; 0&amp; 0\\ 0&amp; 0&amp; 1&amp; 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>
389 </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
390 <IMG SRC="images/State_space_fichiers/b578227cbffe36298e451dddad03316a.png" NAME="images57" ALT=" \textbf{y}(t) = \begin{bmatrix} n_{1}&amp; n_{2}&amp; n_{3}&amp; n_{4} \end{bmatrix}\textbf{x}(t)" ALIGN=BOTTOM WIDTH=238 HEIGHT=25 BORDER=0>.
391 </DD></DL>
392 <P>
393 This state-space realization is called <B>controllable canonical
394 form</B> because the resulting model is guaranteed to be
395 controllable (i.e., because the control enters a chain of
396 integrators, it has the ability to move every state).</P>
397 <P>The transfer function coefficients can also be used to construct
398 another type of canonical form</P>
399 <DL>
400 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
401 <IMG SRC="images/State_space_fichiers/48a1d333782665d7903fe3f142cbeed6.png" NAME="images58" ALT="\dot{\textbf{x}}(t) = \begin{bmatrix} -d_{1}&amp; 1&amp; 0&amp; 0\\ -d_{2}&amp; 0&amp; 1&amp; 0\\ -d_{3}&amp; 0&amp; 0&amp; 1\\ -d_{4}&amp; 0&amp; 0&amp; 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>
402 </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
403 <IMG SRC="images/State_space_fichiers/79354cdcf381d3986dafa970009f10b4.png" NAME="images59" ALT=" \textbf{y}(t) = \begin{bmatrix} 1&amp; 0&amp; 0&amp; 0 \end{bmatrix}\textbf{x}(t)" ALIGN=BOTTOM WIDTH=198 HEIGHT=25 BORDER=0>.
404 </DD></DL>
405 <P>
406 This state-space realization is called <B>observable canonical form</B>
407 because the resulting model is guaranteed to be observable (i.e.,
408 because the output exits from a chain of integrators, every state
409 has an effect on the output).</P>
410 <H3 STYLE="border: none; padding: 0cm"><FONT COLOR="#000000">Proper
411 transfer functions</FONT></H3>
412 <P><FONT COLOR="#000000">Transfer functions which are only <A HREF="http://en.wikipedia.org/wiki/Proper_transfer_function">proper</A>
413 (and not <A HREF="http://en.wikipedia.org/wiki/Strictly_proper">strictly
414 proper</A>) can also be realised quite easily. The trick here is to
415 separate the transfer function into two parts: a strictly proper
416 part and a constant.</FONT></P>
417 <DL>
418 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
419 <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>
420 </DD></DL>
421 <P>
422 The strictly proper transfer function can then be transformed into a
423 canonical state space realization using techniques shown above. The
424 state space realization of the constant is trivially
425 <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>.
426 Together we then get a state space realization with matrices <I>A</I>,<I>B</I>
427 and <I>C</I> determined by the strictly proper part, and matrix <I>D</I>
428 determined by the constant.</P>
429 <P><BR>Here is an example to clear things up a bit:</P>
430 <DL>
431 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
432 <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>
433 </DD></DL>
434 <P>
435 which yields the following controllable realization</P>
436 <DL>
437 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
438 <IMG SRC="images/State_space_fichiers/d9679b5a5a38271e615b871f76c79c38.png" NAME="images63" ALT="\dot{\textbf{x}}(t) = \begin{bmatrix} -2&amp; -1\\ 1&amp; 0\\ \end{bmatrix}\textbf{x}(t) + \begin{bmatrix} 1\\ 0\end{bmatrix}\textbf{u}(t)" ALIGN=BOTTOM WIDTH=270 HEIGHT=48 BORDER=0>
439 </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
440 <IMG SRC="images/State_space_fichiers/278467bd0ca464c472b4e47477bdaaa5.png" NAME="images64" ALT=" \textbf{y}(t) = \begin{bmatrix} 1&amp; 2\end{bmatrix}\textbf{x}(t) + \begin{bmatrix} 1\end{bmatrix}\textbf{u}(t)" ALIGN=BOTTOM WIDTH=231 HEIGHT=25 BORDER=0>
441 </DD></DL>
442 <P>
443 Notice how the output also depends directly on the input. This is
444 due to the
445 <IMG SRC="images/State_space_fichiers/b00b33e9ecc56576da0c0710f7fd548e.png" NAME="images65" ALT="\textbf{G}(\infty)" ALIGN=BOTTOM WIDTH=51 HEIGHT=20 BORDER=0>
446 constant in the transfer function.</P>
447 <H3 STYLE="border: none; padding: 0cm"><FONT COLOR="#000000">Feedback</FONT></H3>
448 <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>
449 </P>
450 <P>Typical state space model with feedback</P>
451 <P><FONT COLOR="#000000">A common method for feedback is to multiply
452 the output by a matrix </FONT><FONT COLOR="#000000"><I>K</I></FONT><FONT COLOR="#000000">
453 and setting this as the input to the system:
454 <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>.
455 Since the values of </FONT><FONT COLOR="#000000"><I>K</I></FONT><FONT COLOR="#000000">
456 are unrestricted the values can easily be negated for <A HREF="http://en.wikipedia.org/wiki/Negative_feedback">negative
457 feedback</A>. The presence of a negative sign (the common notation)
458 is merely a notational one and its absence has no impact on the end
459 results.</FONT></P>
460 <DL>
461 <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>
462 </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
463 <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>
464 </DD></DL>
465 <P>
466 becomes</P>
467 <DL>
468 <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>
469 </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
470 <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>
471 </DD></DL>
472 <P>
473 solving the output equation for
474 <IMG SRC="images/State_space_fichiers/8bd1b0065e60113616b6750730e820f4.png" NAME="images72" ALT="\mathbf{y}(t)" ALIGN=BOTTOM WIDTH=34 HEIGHT=21 BORDER=0>
475 and substituting in the state equation results in</P>
476 <DL>
477 <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>
478 </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
479 <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>
480 </DD></DL>
481 <P>
482 <FONT COLOR="#000000">The advantage of this is that the <A HREF="http://en.wikipedia.org/wiki/Eigenvalues">eigenvalues</A>
483 of </FONT><FONT COLOR="#000000"><I>A</I></FONT><FONT COLOR="#000000">
484 can be controlled by setting </FONT><FONT COLOR="#000000"><I>K</I></FONT><FONT COLOR="#000000">
485 appropriately through eigendecomposition of
486 <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>.
487 This assumes that the open-loop system is <A HREF="http://en.wikipedia.org/wiki/Controllability">controllable</A>
488 or that the unstable eigenvalues of </FONT><FONT COLOR="#000000"><I>A</I></FONT><FONT COLOR="#000000">
489 can be made stable through appropriate choice of </FONT><FONT COLOR="#000000"><I>K</I></FONT><FONT COLOR="#000000">.</FONT></P>
490 <P>One fairly common simplification to this system is removing <I>D</I>
491 and setting <I>C</I> to identity, which reduces the equations to</P>
492 <DL>
493 <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>
494 </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
495 <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>
496 </DD></DL>
497 <P>
498 This reduces the necessary eigendecomposition to just <I>A</I> + <I>BK</I>.</P>
499 <H3 STYLE="border: none; padding: 0cm"><FONT COLOR="#000000">Feedback
500 with setpoint (reference) input</FONT></H3>
501 <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>
502 </P>
503 <P>Output feedback with set point</P>
504 <P>In addition to feedback, an input, <I>r</I>(<I>t</I>), can be
505 added such that
506 <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>
507 <DL>
508 <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>
509 </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
510 <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>
511 </DD></DL>
512 <P>
513 becomes</P>
514 <DL>
515 <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>
516 </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
517 <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>
518 </DD></DL>
519 <P>
520 solving the output equation for
521 <IMG SRC="images/State_space_fichiers/8bd1b0065e60113616b6750730e820f4.png" NAME="images84" ALT="\mathbf{y}(t)" ALIGN=BOTTOM WIDTH=34 HEIGHT=21 BORDER=0>
522 and substituting in the state equation results in</P>
523 <DL>
524 <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>
525 </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
526 <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>
527 </DD></DL>
528 <P>
529 One fairly common simplification to this system is removing <I>D</I>,
530 which reduces the equations to</P>
531 <DL>
532 <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>
533 </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
534 <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>
535 </DD></DL>
536 <H3 STYLE="border: none; padding: 0cm">
537 <FONT COLOR="#000000">Moving object example</FONT></H3>
538 <P><FONT COLOR="#000000">A classical linear system is that of
539 one-dimensional movement of an object. The <A HREF="http://en.wikipedia.org/wiki/Newton%27s_laws_of_motion">Newton's
540 laws of motion</A> for an object moving horizontally on a plane and
541 attached to a wall with a spring</FONT></P>
542 <DL>
543 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
544 <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>
545 </DD></DL>
546 <P>
547 where</P>
548 <UL>
549 <LI><P STYLE="margin-bottom: 0cm; border: none; padding: 0cm"><I>y</I>(<I>t</I>)
550 is position;
551 <IMG SRC="images/State_space_fichiers/4721de091b5d15bb9809fa41a5e40db7.png" NAME="images90" ALT="\dot y(t)" ALIGN=BOTTOM WIDTH=32 HEIGHT=21 BORDER=0>
552 is velocity;
553 <IMG SRC="images/State_space_fichiers/d822334addb31f4d4fe1683e609e2742.png" NAME="images91" ALT="\ddot{y}(t)" ALIGN=BOTTOM WIDTH=32 HEIGHT=21 BORDER=0>
554 is acceleration
555 </P>
556 <LI><P STYLE="margin-bottom: 0cm; border: none; padding: 0cm"><I>u</I>(<I>t</I>)
557 is an applied force
558 </P>
559 <LI><P STYLE="margin-bottom: 0cm; border: none; padding: 0cm"><I>k</I><SUB>1</SUB>
560 is the viscous friction coefficient
561 </P>
562 <LI><P STYLE="margin-bottom: 0cm; border: none; padding: 0cm"><I>k</I><SUB>2</SUB>
563 is the spring constant
564 </P>
565 <LI><P><I>m</I> is the mass of the object
566 </P>
567 </UL>
568 <P>The state equation would then become</P>
569 <DL>
570 <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 &amp; 1 \\ -\frac{k_2}{m} &amp; -\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>
571 </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
572 <IMG SRC="images/State_space_fichiers/7771c47e6e2f82d6138ee2e2ec920c7e.png" NAME="images93" ALT="\mathbf{y}(t) = \left[ \begin{matrix} 1 &amp; 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>
573 </DD></DL>
574 <P>
575 where</P>
576 <UL>
577 <LI><P STYLE="margin-bottom: 0cm; border: none; padding: 0cm"><I>x</I><SUB>1</SUB>(<I>t</I>)
578 represents the position of the object
579 </P>
580 <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>
581 is the velocity of the object
582 </P>
583 <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>
584 is the acceleration of the object
585 </P>
586 <LI><P>the output
587 <IMG SRC="images/State_space_fichiers/8bd1b0065e60113616b6750730e820f4.png" NAME="images96" ALT="\mathbf{y}(t)" ALIGN=BOTTOM WIDTH=34 HEIGHT=21 BORDER=0>
588 is the position of the object
589 </P>
590 </UL>
591 <P><FONT COLOR="#000000">The <A HREF="http://en.wikipedia.org/wiki/Controllability">controllability</A>
592 test is then</FONT></P>
593 <DL>
594 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
595 <IMG SRC="images/State_space_fichiers/1b66d8f8c0227384d70806878463f8ce.png" NAME="images97" ALT="\left[ \begin{matrix} B &amp; AB \end{matrix} \right] = \left[ \begin{matrix} \left[ \begin{matrix} 0 \\ \frac{1}{m} \end{matrix} \right] &amp; \left[ \begin{matrix} 0 &amp; 1 \\ -\frac{k_2}{m} &amp; -\frac{k_1}{m} \end{matrix} \right] \left[ \begin{matrix} 0 \\ \frac{1}{m} \end{matrix} \right] \end{matrix} \right] = \left[ \begin{matrix} 0 &amp; \frac{1}{m} \\ \frac{1}{m} &amp; \frac{k_1}{m^2} \end{matrix} \right]" ALIGN=BOTTOM WIDTH=422 HEIGHT=49 BORDER=0>
596 </DD></DL>
597 <P>
598 which has full rank for all <I>k</I><SUB>1</SUB> and <I>m</I>.</P>
599 <P><FONT COLOR="#000000">The <A HREF="http://en.wikipedia.org/wiki/Observability">observability</A>
600 test is then</FONT></P>
601 <DL>
602 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
603 <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 &amp; 0 \end{matrix} \right] \\ \left[ \begin{matrix} 1 &amp; 0 \end{matrix} \right] \left[ \begin{matrix} 0 &amp; 1 \\ -\frac{k_2}{m} &amp; -\frac{k_1}{m} \end{matrix} \right] \end{matrix} \right] = \left[ \begin{matrix} 1 &amp; 0 \\ 0 &amp; 1 \end{matrix} \right]" ALIGN=BOTTOM WIDTH=340 HEIGHT=74 BORDER=0>
604 </DD></DL>
605 <P>
606 which also has full rank. Therefore, this system is both
607 controllable and observable.</P>
608 <H2 STYLE="border: none; padding: 0cm"><FONT COLOR="#000000">Nonlinear
609 systems</FONT></H2>
610 <P>The more general form of a state space model can be written as
611 two functions.</P>
612 <DL>
613 <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>
614 </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
615 <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>
616 </DD></DL>
617 <P>
618 The first is the state equation and the latter is the output
619 equation. If the function
620 <IMG SRC="images/State_space_fichiers/f6980ec56e3e7fcce7a143b507a53126.png" NAME="images101" ALT="f(\cdot,\cdot,\cdot)" ALIGN=BOTTOM WIDTH=59 HEIGHT=21 BORDER=0>
621 is a linear combination of states and inputs then the equations can
622 be written in matrix notation like above. The <I>u</I>(<I>t</I>)
623 argument to the functions can be dropped if the system is unforced
624 (i.e., it has no inputs).</P>
625 <H3 STYLE="border: none; padding: 0cm"><FONT COLOR="#000000">Pendulum
626 example</FONT></H3>
627 <P><FONT COLOR="#000000">A classic nonlinear system is a simple
628 unforced <A HREF="http://en.wikipedia.org/wiki/Pendulum">pendulum</A></FONT></P>
629 <DL>
630 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
631 <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>
632 </DD></DL>
633 <P>
634 where</P>
635 <UL>
636 <LI><P STYLE="margin-bottom: 0cm; border: none; padding: 0cm">&theta;(<I>t</I>)
637 is the angle of the pendulum with respect to the direction of
638 gravity
639 </P>
640 <LI><P STYLE="margin-bottom: 0cm; border: none; padding: 0cm"><I>m</I>
641 is the mass of the pendulum (pendulum rod's mass is assumed to be
642 zero)
643 </P>
644 <LI><P STYLE="margin-bottom: 0cm; border: none; padding: 0cm"><I>g</I>
645 is the gravitational acceleration
646 </P>
647 <LI><P STYLE="margin-bottom: 0cm; border: none; padding: 0cm"><I>k</I>
648 is coefficient of friction at the pivot point
649 </P>
650 <LI><P><I>l</I> is the radius of the pendulum (to the center of
651 gravity of the mass <I>m</I>)
652 </P>
653 </UL>
654 <P>The state equations are then</P>
655 <DL>
656 <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>
657 </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
658 <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>
659 </DD></DL>
660 <P>
661 where</P>
662 <UL>
663 <LI><P STYLE="margin-bottom: 0cm; border: none; padding: 0cm"><I>x</I><SUB>1</SUB>(<I>t</I>):
664 = &theta;(<I>t</I>) is the angle of the pendulum
665 </P>
666 <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>
667 is the rotational velocity of the pendulum
668 </P>
669 <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>
670 is the rotational acceleration of the pendulum
671 </P>
672 </UL>
673 <P>Instead, the state equation can be written in the general form</P>
674 <DL>
675 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
676 <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>
677 </DD></DL>
678 <P>
679 <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
680 points</A> of a system are when
681 <IMG SRC="images/State_space_fichiers/4009ed416fcf36f8b7ec3dbabe9fa791.png" NAME="images108" ALT="\dot{x} = 0" ALIGN=BOTTOM WIDTH=46 HEIGHT=15 BORDER=0>
682 and so the equilibrium points of a pendulum are those that satisfy</FONT></P>
683 <DL>
684 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
685 <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>
686 </DD></DL>
687 <P>
688 for integers <I>n</I>.</P>
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