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author Daniele Nicolodi <nicolodi@science.unitn.it>
date Wed, 23 Nov 2011 19:22:13 +0100
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1 <!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
2 <HTML>
3 <HEAD>
4 <META HTTP-EQUIV="CONTENT-TYPE" CONTENT="text/html; charset=windows-1252">
5 <TITLE></TITLE>
6 <META NAME="GENERATOR" CONTENT="OpenOffice.org 3.1 (Win32)">
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8 <META NAME="CHANGEDBY" CONTENT="Adrien G">
9 <META NAME="CHANGED" CONTENT="20090827;17521500">
10 </HEAD>
11 <BODY LANG="en-US" DIR="LTR">
12 <P>
13 <H2>Why use state space modeling?</H2>
14 <P>State space modeling is efficient to simulate systems with large
15 dimensionality, be it in terms of inputs, outputs, or pole/zeros.
16 Adding nonlinearities to a model is also easier as in the frequency
17 domain, however there is no such capability in the toolbox yet.
18 Another reason to use them is to build complex parametric models,
19 where intricated parameters make impossible the symbolic calculation
20 of the transfer function coefficients &ndash; as a matter of fact
21 the transfer function can be computed out of a determinant involving
22 the A matrix, explaining the complexity of the calculation.</P>
23 <P>For tasks such as identification, state space modeling is a
24 computationally rather heavy, especially if colored noise is
25 involved in the process.</P>
26 <P>State space models can be converted into a matrix of transfer
27 function in the s- or the z-domain. The functions in the toolbox
28 that enable this are ss2pzmodel, ss2miir, ss2rational.</P>
29 <H2>Generalities on State Space Modeling</H2>
30 <P><FONT COLOR="#000000">In order to familiarize with state space
31 modeling, this help page is a mere copy of the wiki page.</FONT></P>
32 <P><FONT COLOR="#000000">In control engineering, a </FONT><FONT COLOR="#000000"><B>state
33 space representation</B></FONT><FONT COLOR="#000000"> is a
34 mathematical model of a physical system as a set of input, output
35 and state variables related by first-order <A HREF="http://en.wikipedia.org/wiki/Differential_equation">differential
36 equations</A>. To abstract from the number of inputs, outputs </FONT>and<FONT COLOR="#000000">
37 states, the variables are expressed as vectors and the differential
38 and algebraic equations are written in matrix form (the last one can
39 be done when the <A HREF="http://en.wikipedia.org/wiki/Dynamical_system">dynamical
40 system</A> is linear and time invariant). The state space
41 representation (also known as the &quot;time-domain approach&quot;)
42 provides a convenient and compact way to model and analyze systems
43 with multiple inputs and outputs. With </FONT><FONT COLOR="#000000"><I>p</I></FONT><FONT COLOR="#000000">
44 inputs and </FONT><FONT COLOR="#000000"><I>q</I></FONT><FONT COLOR="#000000">
45 outputs, we would otherwise have to write down <A HREF="http://en.wikipedia.org/wiki/Laplace_transform">Laplace
46 transforms</A> to encode all the information about a system. Unlike
47 the frequency domain approach, the use of the state space
48 representation is not limited to systems with linear components and
49 zero initial conditions. &quot;State space&quot; refers to the space
50 whose axes are the state variables. The state of the system can be
51 represented as a vector within that space.</FONT></P>
52 <H2><FONT COLOR="#000000">State variables</FONT></H2>
53 <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">
54 </FONT>
55 </H2>
56 <P>Typical state space model</P>
57 <P><FONT COLOR="#000000">The internal <A HREF="http://en.wikipedia.org/wiki/State_variable">state
58 variables</A> are the smallest possible subset of system variables
59 that can represent the entire state of the system at any given time.
60 State variables must be linearly independent; a state variable
61 cannot be a linear combination of other state variables. The minimum
62 number of state variables required to represent a given system, </FONT><FONT COLOR="#000000"><I>n</I></FONT><FONT COLOR="#000000">,
63 is usually equal to the order of the system's defining differential
64 equation. If the system is represented in transfer function form,
65 the minimum number of state variables is equal to the order of the
66 transfer function's denominator after it has been reduced to a
67 proper fraction. It is important to understand that converting a
68 state space realization to a transfer function form may lose some
69 internal information about the system, and may provide a description
70 of a system which is stable, when the state-space realization is
71 unstable at certain points. In electric circuits, the number of
72 state variables is often, though not always, the same as the number
73 of energy storage elements in the circuit such as <A HREF="http://en.wikipedia.org/wiki/Capacitor">capacitors</A>
74 and <A HREF="http://en.wikipedia.org/wiki/Inductor">inductors</A>.</FONT></P>
75 <H2><FONT COLOR="#000000">Linear systems</FONT></H2>
76 <P>The most general state-space representation of a linear system
77 with <I>p</I> inputs, <I>q</I> outputs and <I>n</I> state variables
78 is written in the following form:</P>
79 <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>
80 </P>
81 <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>
82 </P>
83 <P>where:</P>
84 <UL>
85 <LI><P><IMG SRC="images/State_space_fichiers/0b46dcfa3789d90891d7d22aa4bf1b83.png" NAME="images6" ALT="x(\cdot)" ALIGN=BOTTOM WIDTH=31 HEIGHT=21 BORDER=0>
86 is called the &quot;state vector&quot;,&nbsp;
87 <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>;
88 </P>
89 <LI><P><IMG SRC="images/State_space_fichiers/89df6576cd198dc2449bf59e97a61df3.png" NAME="images8" ALT="y(\cdot)" ALIGN=BOTTOM WIDTH=30 HEIGHT=21 BORDER=0>
90 is called the &quot;output vector&quot;,&nbsp;
91 <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>;
92 </P>
93 <LI><P><IMG SRC="images/State_space_fichiers/918aa6368974b6014859c56e1ef6762c.png" NAME="images10" ALT="u(\cdot)" ALIGN=BOTTOM WIDTH=31 HEIGHT=21 BORDER=0>
94 is called the &quot;input (or control) vector&quot;,&nbsp;
95 <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>;
96 </P>
97 <LI><P><IMG SRC="images/State_space_fichiers/b53a04dbe423d4e39eaedadf54e3ee31.png" NAME="images12" ALT="A(\cdot)" ALIGN=BOTTOM WIDTH=35 HEIGHT=21 BORDER=0>
98 is the &quot;state matrix&quot;,&nbsp;
99 <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>,
100 </P>
101 <LI><P><IMG SRC="images/State_space_fichiers/8f528e3252e95ac5edef108971a1d43f.png" NAME="images14" ALT="B(\cdot)" ALIGN=BOTTOM WIDTH=36 HEIGHT=21 BORDER=0>
102 is the &quot;input matrix&quot;,&nbsp;
103 <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>,
104 </P>
105 <LI><P><IMG SRC="images/State_space_fichiers/d4d99b70fae67ddcd7923dbe5dbb6424.png" NAME="images16" ALT="C(\cdot)" ALIGN=BOTTOM WIDTH=34 HEIGHT=21 BORDER=0>
106 is the &quot;output matrix&quot;,&nbsp;
107 <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>,
108 </P>
109 <LI><P><IMG SRC="images/State_space_fichiers/9828f5f36ec8af68cc289bbf41a2c751.png" NAME="images18" ALT="D(\cdot)" ALIGN=BOTTOM WIDTH=37 HEIGHT=21 BORDER=0>
110 is the &quot;feedthrough (or feedforward) matrix&quot; (in cases
111 where the system model does not have a direct feedthrough,
112 <IMG SRC="images/State_space_fichiers/9828f5f36ec8af68cc289bbf41a2c751.png" NAME="images19" ALT="D(\cdot)" ALIGN=BOTTOM WIDTH=37 HEIGHT=21 BORDER=0>
113 is the zero matrix),&nbsp;
114 <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>,
115 </P>
116 <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>.
117 </P>
118 </UL>
119 <P><FONT COLOR="#000000">In this general formulation, all matrices
120 are allowed to be time-variant (i.e., their elements can depend on
121 time); however, in the common <A HREF="http://en.wikipedia.org/wiki/LTI_system">LTI</A>
122 case, matrices will be time invariant. The time variable </FONT><FONT COLOR="#000000"><I>t</I></FONT><FONT COLOR="#000000">
123 can be a &quot;continuous&quot; (e.g.,
124 <IMG SRC="images/State_space_fichiers/6080cfaa75ff6363d282e29a14b46632.png" NAME="images22" ALT="t \in \mathbb{R}" ALIGN=BOTTOM WIDTH=45 HEIGHT=15 BORDER=0>)
125 or discrete (e.g.,
126 <IMG SRC="images/State_space_fichiers/d3bd3b5e80beacc6f4a28dacabf2c0c5.png" NAME="images23" ALT="t \in \mathbb{Z}" ALIGN=BOTTOM WIDTH=45 HEIGHT=15 BORDER=0>).
127 In the latter case, the time variable is usually indicated as </FONT><FONT COLOR="#000000"><I>k</I></FONT><FONT COLOR="#000000">.
128 <A HREF="http://en.wikipedia.org/wiki/Hybrid_system">Hybrid systems</A>
129 allow for time domains that have both continuous and discrete parts.
130 Depending on the assumptions taken, the state-space model
131 representation can assume the following forms:</FONT></P>
132 <TABLE DIR="LTR" CELLPADDING=4 CELLSPACING=2>
133 <TR VALIGN=TOP>
134 <TD>
135 <P><B>System type</B></P>
136 </TD>
137 <TD>
138 <P><B>State-space model</B></P>
139 </TD>
140 </TR>
141 <TR VALIGN=TOP>
142 <TD>
143 <P>Continuous time-invariant</P>
144 </TD>
145 <TD>
146 <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>
147 </TD>
148 </TR>
149 <TR VALIGN=TOP>
150 <TD>
151 <P>Continuous time-variant</P>
152 </TD>
153 <TD>
154 <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>
155 </TD>
156 </TR>
157 <TR VALIGN=TOP>
158 <TD>
159 <P>Discrete time-invariant</P>
160 </TD>
161 <TD>
162 <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>
163 </TD>
164 </TR>
165 <TR VALIGN=TOP>
166 <TD>
167 <P>Discrete time-variant</P>
168 </TD>
169 <TD>
170 <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>
171 </TD>
172 </TR>
173 <TR VALIGN=TOP>
174 <TD>
175 <P>Laplace domain of<BR>continuous time-invariant</P>
176 </TD>
177 <TD>
178 <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>
179 </TD>
180 </TR>
181 <TR VALIGN=TOP>
182 <TD>
183 <P>Z-domain of<BR>discrete time-invariant</P>
184 </TD>
185 <TD>
186 <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>
187 </TD>
188 </TR>
189 </TABLE>
190 <P><FONT COLOR="#000000">Example: Continuous-time LTI case</FONT>
191 </P>
192 <P><FONT COLOR="#000000">Stability and natural response
193 characteristics of a continuous-time <A HREF="http://en.wikipedia.org/wiki/LTI_system">LTI
194 system</A> (i.e., linear with matrices that are constant with
195 respect to time) can be studied from the <A HREF="http://en.wikipedia.org/wiki/Eigenvalue">eigenvalues</A>
196 of the matrix </FONT><FONT COLOR="#000000"><B>A</B></FONT><FONT COLOR="#000000">.
197 The stability of a time-invariant state-space model can be
198 determined by looking at the system's <A HREF="http://en.wikipedia.org/wiki/Transfer_function">transfer
199 function</A> in factored form. It will then look something like
200 this:</FONT></P>
201 <DL>
202 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
203 <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>
204 </DD></DL>
205 <P>
206 <FONT COLOR="#000000">The denominator of the transfer function is
207 equal to the <A HREF="http://en.wikipedia.org/wiki/Characteristic_polynomial">characteristic
208 polynomial</A> found by taking the <A HREF="http://en.wikipedia.org/wiki/Determinant">determinant</A>
209 of </FONT><FONT COLOR="#000000"><I>sI</I></FONT><FONT COLOR="#000000">
210 &minus; </FONT><FONT COLOR="#000000"><I>A</I></FONT><FONT COLOR="#000000">,</FONT></P>
211 <DL>
212 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
213 <IMG SRC="images/State_space_fichiers/05e74baab950e092f7ef8acbf4654c84.png" NAME="images37" ALT="\mathbf{\lambda}(s) = |sI - A|" ALIGN=BOTTOM WIDTH=129 HEIGHT=21 BORDER=0>.
214 </DD></DL>
215 <P>
216 <FONT COLOR="#000000">The roots of this polynomial (the <A HREF="http://en.wikipedia.org/wiki/Eigenvalue">eigenvalues</A>)
217 are the system transfer function's <A HREF="http://en.wikipedia.org/wiki/Complex_pole">poles</A>
218 (i.e., the <A HREF="http://en.wikipedia.org/wiki/Singularity">singularities</A>
219 where the transfer function's magnitude is unbounded). These poles
220 can be used to analyze whether the system is <A HREF="http://en.wikipedia.org/wiki/Exponential_stability">asymptotically
221 stable</A> or <A HREF="http://en.wikipedia.org/wiki/Marginal_stability">marginally
222 stable</A>. An alternative approach to determining stability, which
223 does not involve calculating eigenvalues, is to analyze the system's
224 <A HREF="http://en.wikipedia.org/wiki/Lyapunov_stability">Lyapunov
225 stability</A>.</FONT></P>
226 <P><FONT COLOR="#000000">The zeros found in the numerator of
227 <IMG SRC="images/State_space_fichiers/e75eebaaa95472ea1407f43448f37ab8.png" NAME="images38" ALT="\textbf{G}(s)" ALIGN=BOTTOM WIDTH=39 HEIGHT=21 BORDER=0>
228 can similarly be used to determine whether the system is <A HREF="http://en.wikipedia.org/wiki/Minimum_phase">minimum
229 phase</A>.</FONT></P>
230 <P><FONT COLOR="#000000">The system may still be </FONT><FONT COLOR="#000000"><B>input&ndash;output
231 stable</B></FONT><FONT COLOR="#000000"> (see <A HREF="http://en.wikipedia.org/wiki/BIBO_stability">BIBO
232 stable</A>) even though it is not internally stable. This may be the
233 case if unstable poles are canceled out by zeros (i.e., if those
234 singularities in the transfer function are <A HREF="http://en.wikipedia.org/wiki/Removable_singularity">removable</A>).</FONT></P>
235 <H3 STYLE="border: none; padding: 0cm"><FONT COLOR="#000000">Controllability</FONT></H3>
236 <P><FONT COLOR="#000000">Main article: <A HREF="http://en.wikipedia.org/wiki/Controllability">Controllability</A></FONT></P>
237 <P><FONT COLOR="#000000">Thus, state controllability condition
238 implies that it is possible&nbsp;&ndash; by admissible inputs&nbsp;&ndash;
239 to steer the states from any initial value to any final value within
240 some finite time window. A continuous time-invariant linear
241 state-space model is </FONT><FONT COLOR="#000000"><B>controllable</B></FONT><FONT COLOR="#000000">
242 <A HREF="http://en.wikipedia.org/wiki/Iff">if and only if</A></FONT></P>
243 <DL>
244 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
245 <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>
246 </DD></DL>
247 <H3 STYLE="border: none; padding: 0cm">
248 <FONT COLOR="#000000">Observability</FONT></H3>
249 <P><FONT COLOR="#000000">Main article: <A HREF="http://en.wikipedia.org/wiki/Observability">Observability</A></FONT></P>
250 <P>Observability is a measure for how well internal states of a
251 system can be inferred by knowledge of its external outputs. The
252 observability and controllability of a system are mathematical duals
253 (i.e., as controllablity provides that an input is available that
254 brings any initial state to any desired final state, observability
255 provides that knowing an output trajectory provides enough
256 information to predict the initial state of the system).</P>
257 <P>A continuous time-invariant linear state-space model is
258 <B>observable</B> if and only if</P>
259 <DL>
260 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
261 <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>
262 </DD></DL>
263 <P>
264 <FONT COLOR="#000000">(<A HREF="http://en.wikipedia.org/wiki/Rank_%28linear_algebra%29">Rank</A>
265 is the number of linearly independent rows in a matrix.)</FONT></P>
266 <H3 STYLE="border: none; padding: 0cm"><FONT COLOR="#000000">Transfer
267 function</FONT></H3>
268 <P><FONT COLOR="#000000">The &quot;<A HREF="http://en.wikipedia.org/wiki/Transfer_function">transfer
269 function</A>&quot; of a continuous time-invariant linear state-space
270 model can be derived in the following way:</FONT></P>
271 <P><FONT COLOR="#000000">First, taking the <A HREF="http://en.wikipedia.org/wiki/Laplace_transform">Laplace
272 transform</A> of</FONT></P>
273 <DL>
274 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
275 <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>
276 </DD></DL>
277 <P>
278 yields</P>
279 <DL>
280 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
281 <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>
282 </DD></DL>
283 <P>
284 Next, we simplify for
285 <IMG SRC="images/State_space_fichiers/0762e07773e4f1f30137d7915c154851.png" NAME="images43" ALT="\mathbf{X}(s)" ALIGN=BOTTOM WIDTH=40 HEIGHT=21 BORDER=0>,
286 giving</P>
287 <DL>
288 <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>
289 </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
290 <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>
291 </DD></DL>
292 <P>
293 this is substituted for
294 <IMG SRC="images/State_space_fichiers/0762e07773e4f1f30137d7915c154851.png" NAME="images46" ALT="\mathbf{X}(s)" ALIGN=BOTTOM WIDTH=40 HEIGHT=21 BORDER=0>
295 in the output equation</P>
296 <DL>
297 <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>,
298 giving
299 </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
300 <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>
301 </DD></DL>
302 <P>
303 <FONT COLOR="#000000">Because the <A HREF="http://en.wikipedia.org/wiki/Transfer_function">transfer
304 function</A>
305 <IMG SRC="images/State_space_fichiers/046cdad9e9789e63b5196b2be94110e7.png" NAME="images49" ALT="\mathbf{G}(s)" ALIGN=BOTTOM WIDTH=40 HEIGHT=21 BORDER=0>
306 is defined as the ratio of the output to the input of a system, we
307 take</FONT></P>
308 <DL>
309 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
310 <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>
311 </DD></DL>
312 <P>
313 and substitute the previous expression for
314 <IMG SRC="images/State_space_fichiers/336fec49a4b9e27e1bc99139f4023267.png" NAME="images51" ALT="\mathbf{Y}(s)" ALIGN=BOTTOM WIDTH=40 HEIGHT=21 BORDER=0>
315 with respect to
316 <IMG SRC="images/State_space_fichiers/3f0af3cafb3dcd9039883b8047986fc2.png" NAME="images52" ALT="\mathbf{U}(s)" ALIGN=BOTTOM WIDTH=40 HEIGHT=21 BORDER=0>,
317 giving</P>
318 <DL>
319 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
320 <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>
321 </DD></DL>
322 <P>
323 Clearly
324 <IMG SRC="images/State_space_fichiers/046cdad9e9789e63b5196b2be94110e7.png" NAME="images54" ALT="\mathbf{G}(s)" ALIGN=BOTTOM WIDTH=40 HEIGHT=21 BORDER=0>
325 must have <I>q</I> by <I>p</I> dimensionality, and thus has a total
326 of <I>qp</I> elements. So for every input there are <I>q</I>
327 transfer functions with one for each output. This is why the
328 state-space representation can easily be the preferred choice for
329 multiple-input, multiple-output (MIMO) systems.</P>
330 <H3 STYLE="border: none; padding: 0cm"><FONT COLOR="#000000">Canonical
331 realizations</FONT></H3>
332 <P><FONT COLOR="#000000">Any given transfer function which is
333 <A HREF="http://en.wikipedia.org/wiki/Strictly_proper">strictly
334 proper</A> can easily be transferred into state-space by the
335 following approach (this example is for a 4-dimensional,
336 single-input, single-output system)):</FONT></P>
337 <P>Given a transfer function, expand it to reveal all coefficients
338 in both the numerator and denominator. This should result in the
339 following form:</P>
340 <DL>
341 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
342 <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>.
343 </DD></DL>
344 <P>
345 The coefficients can now be inserted directly into the state-space
346 model by the following approach:</P>
347 <DL>
348 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
349 <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>
350 </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
351 <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>.
352 </DD></DL>
353 <P>
354 This state-space realization is called <B>controllable canonical
355 form</B> because the resulting model is guaranteed to be
356 controllable (i.e., because the control enters a chain of
357 integrators, it has the ability to move every state).</P>
358 <P>The transfer function coefficients can also be used to construct
359 another type of canonical form</P>
360 <DL>
361 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
362 <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>
363 </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
364 <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>.
365 </DD></DL>
366 <P>
367 This state-space realization is called <B>observable canonical form</B>
368 because the resulting model is guaranteed to be observable (i.e.,
369 because the output exits from a chain of integrators, every state
370 has an effect on the output).</P>
371 <H3 STYLE="border: none; padding: 0cm"><FONT COLOR="#000000">Proper
372 transfer functions</FONT></H3>
373 <P><FONT COLOR="#000000">Transfer functions which are only <A HREF="http://en.wikipedia.org/wiki/Proper_transfer_function">proper</A>
374 (and not <A HREF="http://en.wikipedia.org/wiki/Strictly_proper">strictly
375 proper</A>) can also be realised quite easily. The trick here is to
376 separate the transfer function into two parts: a strictly proper
377 part and a constant.</FONT></P>
378 <DL>
379 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
380 <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>
381 </DD></DL>
382 <P>
383 The strictly proper transfer function can then be transformed into a
384 canonical state space realization using techniques shown above. The
385 state space realization of the constant is trivially
386 <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>.
387 Together we then get a state space realization with matrices <I>A</I>,<I>B</I>
388 and <I>C</I> determined by the strictly proper part, and matrix <I>D</I>
389 determined by the constant.</P>
390 <P><BR>Here is an example to clear things up a bit:</P>
391 <DL>
392 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
393 <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>
394 </DD></DL>
395 <P>
396 which yields the following controllable realization</P>
397 <DL>
398 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
399 <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>
400 </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
401 <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>
402 </DD></DL>
403 <P>
404 Notice how the output also depends directly on the input. This is
405 due to the
406 <IMG SRC="images/State_space_fichiers/b00b33e9ecc56576da0c0710f7fd548e.png" NAME="images65" ALT="\textbf{G}(\infty)" ALIGN=BOTTOM WIDTH=51 HEIGHT=20 BORDER=0>
407 constant in the transfer function.</P>
408 <H3 STYLE="border: none; padding: 0cm"><FONT COLOR="#000000">Feedback</FONT></H3>
409 <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>
410 </P>
411 <P>Typical state space model with feedback</P>
412 <P><FONT COLOR="#000000">A common method for feedback is to multiply
413 the output by a matrix </FONT><FONT COLOR="#000000"><I>K</I></FONT><FONT COLOR="#000000">
414 and setting this as the input to the system:
415 <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>.
416 Since the values of </FONT><FONT COLOR="#000000"><I>K</I></FONT><FONT COLOR="#000000">
417 are unrestricted the values can easily be negated for <A HREF="http://en.wikipedia.org/wiki/Negative_feedback">negative
418 feedback</A>. The presence of a negative sign (the common notation)
419 is merely a notational one and its absence has no impact on the end
420 results.</FONT></P>
421 <DL>
422 <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>
423 </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
424 <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>
425 </DD></DL>
426 <P>
427 becomes</P>
428 <DL>
429 <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>
430 </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
431 <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>
432 </DD></DL>
433 <P>
434 solving the output equation for
435 <IMG SRC="images/State_space_fichiers/8bd1b0065e60113616b6750730e820f4.png" NAME="images72" ALT="\mathbf{y}(t)" ALIGN=BOTTOM WIDTH=34 HEIGHT=21 BORDER=0>
436 and substituting in the state equation results in</P>
437 <DL>
438 <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>
439 </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
440 <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>
441 </DD></DL>
442 <P>
443 <FONT COLOR="#000000">The advantage of this is that the <A HREF="http://en.wikipedia.org/wiki/Eigenvalues">eigenvalues</A>
444 of </FONT><FONT COLOR="#000000"><I>A</I></FONT><FONT COLOR="#000000">
445 can be controlled by setting </FONT><FONT COLOR="#000000"><I>K</I></FONT><FONT COLOR="#000000">
446 appropriately through eigendecomposition of
447 <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>.
448 This assumes that the open-loop system is <A HREF="http://en.wikipedia.org/wiki/Controllability">controllable</A>
449 or that the unstable eigenvalues of </FONT><FONT COLOR="#000000"><I>A</I></FONT><FONT COLOR="#000000">
450 can be made stable through appropriate choice of </FONT><FONT COLOR="#000000"><I>K</I></FONT><FONT COLOR="#000000">.</FONT></P>
451 <P>One fairly common simplification to this system is removing <I>D</I>
452 and setting <I>C</I> to identity, which reduces the equations to</P>
453 <DL>
454 <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>
455 </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
456 <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>
457 </DD></DL>
458 <P>
459 This reduces the necessary eigendecomposition to just <I>A</I> + <I>BK</I>.</P>
460 <H3 STYLE="border: none; padding: 0cm"><FONT COLOR="#000000">Feedback
461 with setpoint (reference) input</FONT></H3>
462 <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>
463 </P>
464 <P>Output feedback with set point</P>
465 <P>In addition to feedback, an input, <I>r</I>(<I>t</I>), can be
466 added such that
467 <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>
468 <DL>
469 <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>
470 </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
471 <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>
472 </DD></DL>
473 <P>
474 becomes</P>
475 <DL>
476 <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>
477 </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
478 <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>
479 </DD></DL>
480 <P>
481 solving the output equation for
482 <IMG SRC="images/State_space_fichiers/8bd1b0065e60113616b6750730e820f4.png" NAME="images84" ALT="\mathbf{y}(t)" ALIGN=BOTTOM WIDTH=34 HEIGHT=21 BORDER=0>
483 and substituting in the state equation results in</P>
484 <DL>
485 <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>
486 </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
487 <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>
488 </DD></DL>
489 <P>
490 One fairly common simplification to this system is removing <I>D</I>,
491 which reduces the equations to</P>
492 <DL>
493 <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>
494 </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
495 <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>
496 </DD></DL>
497 <H3 STYLE="border: none; padding: 0cm">
498 <FONT COLOR="#000000">Moving object example</FONT></H3>
499 <P><FONT COLOR="#000000">A classical linear system is that of
500 one-dimensional movement of an object. The <A HREF="http://en.wikipedia.org/wiki/Newton%27s_laws_of_motion">Newton's
501 laws of motion</A> for an object moving horizontally on a plane and
502 attached to a wall with a spring</FONT></P>
503 <DL>
504 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
505 <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>
506 </DD></DL>
507 <P>
508 where</P>
509 <UL>
510 <LI><P STYLE="margin-bottom: 0cm; border: none; padding: 0cm"><I>y</I>(<I>t</I>)
511 is position;
512 <IMG SRC="images/State_space_fichiers/4721de091b5d15bb9809fa41a5e40db7.png" NAME="images90" ALT="\dot y(t)" ALIGN=BOTTOM WIDTH=32 HEIGHT=21 BORDER=0>
513 is velocity;
514 <IMG SRC="images/State_space_fichiers/d822334addb31f4d4fe1683e609e2742.png" NAME="images91" ALT="\ddot{y}(t)" ALIGN=BOTTOM WIDTH=32 HEIGHT=21 BORDER=0>
515 is acceleration
516 </P>
517 <LI><P STYLE="margin-bottom: 0cm; border: none; padding: 0cm"><I>u</I>(<I>t</I>)
518 is an applied force
519 </P>
520 <LI><P STYLE="margin-bottom: 0cm; border: none; padding: 0cm"><I>k</I><SUB>1</SUB>
521 is the viscous friction coefficient
522 </P>
523 <LI><P STYLE="margin-bottom: 0cm; border: none; padding: 0cm"><I>k</I><SUB>2</SUB>
524 is the spring constant
525 </P>
526 <LI><P><I>m</I> is the mass of the object
527 </P>
528 </UL>
529 <P>The state equation would then become</P>
530 <DL>
531 <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>
532 </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
533 <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>
534 </DD></DL>
535 <P>
536 where</P>
537 <UL>
538 <LI><P STYLE="margin-bottom: 0cm; border: none; padding: 0cm"><I>x</I><SUB>1</SUB>(<I>t</I>)
539 represents the position of the object
540 </P>
541 <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>
542 is the velocity of the object
543 </P>
544 <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>
545 is the acceleration of the object
546 </P>
547 <LI><P>the output
548 <IMG SRC="images/State_space_fichiers/8bd1b0065e60113616b6750730e820f4.png" NAME="images96" ALT="\mathbf{y}(t)" ALIGN=BOTTOM WIDTH=34 HEIGHT=21 BORDER=0>
549 is the position of the object
550 </P>
551 </UL>
552 <P><FONT COLOR="#000000">The <A HREF="http://en.wikipedia.org/wiki/Controllability">controllability</A>
553 test is then</FONT></P>
554 <DL>
555 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
556 <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>
557 </DD></DL>
558 <P>
559 which has full rank for all <I>k</I><SUB>1</SUB> and <I>m</I>.</P>
560 <P><FONT COLOR="#000000">The <A HREF="http://en.wikipedia.org/wiki/Observability">observability</A>
561 test is then</FONT></P>
562 <DL>
563 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
564 <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>
565 </DD></DL>
566 <P>
567 which also has full rank. Therefore, this system is both
568 controllable and observable.</P>
569 <H2 STYLE="border: none; padding: 0cm"><FONT COLOR="#000000">Nonlinear
570 systems</FONT></H2>
571 <P>The more general form of a state space model can be written as
572 two functions.</P>
573 <DL>
574 <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>
575 </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
576 <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>
577 </DD></DL>
578 <P>
579 The first is the state equation and the latter is the output
580 equation. If the function
581 <IMG SRC="images/State_space_fichiers/f6980ec56e3e7fcce7a143b507a53126.png" NAME="images101" ALT="f(\cdot,\cdot,\cdot)" ALIGN=BOTTOM WIDTH=59 HEIGHT=21 BORDER=0>
582 is a linear combination of states and inputs then the equations can
583 be written in matrix notation like above. The <I>u</I>(<I>t</I>)
584 argument to the functions can be dropped if the system is unforced
585 (i.e., it has no inputs).</P>
586 <H3 STYLE="border: none; padding: 0cm"><FONT COLOR="#000000">Pendulum
587 example</FONT></H3>
588 <P><FONT COLOR="#000000">A classic nonlinear system is a simple
589 unforced <A HREF="http://en.wikipedia.org/wiki/Pendulum">pendulum</A></FONT></P>
590 <DL>
591 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
592 <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>
593 </DD></DL>
594 <P>
595 where</P>
596 <UL>
597 <LI><P STYLE="margin-bottom: 0cm; border: none; padding: 0cm">&theta;(<I>t</I>)
598 is the angle of the pendulum with respect to the direction of
599 gravity
600 </P>
601 <LI><P STYLE="margin-bottom: 0cm; border: none; padding: 0cm"><I>m</I>
602 is the mass of the pendulum (pendulum rod's mass is assumed to be
603 zero)
604 </P>
605 <LI><P STYLE="margin-bottom: 0cm; border: none; padding: 0cm"><I>g</I>
606 is the gravitational acceleration
607 </P>
608 <LI><P STYLE="margin-bottom: 0cm; border: none; padding: 0cm"><I>k</I>
609 is coefficient of friction at the pivot point
610 </P>
611 <LI><P><I>l</I> is the radius of the pendulum (to the center of
612 gravity of the mass <I>m</I>)
613 </P>
614 </UL>
615 <P>The state equations are then</P>
616 <DL>
617 <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>
618 </DD><DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
619 <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>
620 </DD></DL>
621 <P>
622 where</P>
623 <UL>
624 <LI><P STYLE="margin-bottom: 0cm; border: none; padding: 0cm"><I>x</I><SUB>1</SUB>(<I>t</I>):
625 = &theta;(<I>t</I>) is the angle of the pendulum
626 </P>
627 <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>
628 is the rotational velocity of the pendulum
629 </P>
630 <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>
631 is the rotational acceleration of the pendulum
632 </P>
633 </UL>
634 <P>Instead, the state equation can be written in the general form</P>
635 <DL>
636 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
637 <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>
638 </DD></DL>
639 <P>
640 <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
641 points</A> of a system are when
642 <IMG SRC="images/State_space_fichiers/4009ed416fcf36f8b7ec3dbabe9fa791.png" NAME="images108" ALT="\dot{x} = 0" ALIGN=BOTTOM WIDTH=46 HEIGHT=15 BORDER=0>
643 and so the equilibrium points of a pendulum are those that satisfy</FONT></P>
644 <DL>
645 <DD STYLE="margin-left: 0cm; margin-bottom: 0.5cm; border: none; padding: 0cm">
646 <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>
647 </DD></DL>
648 <P>
649 for integers <I>n</I>.</P>
650 </SPAN><BR><BR>
651 </P>
652 <P><BR><BR>
653 </P>
654 <P><BR><BR>
655 </P>
656 </BODY>
657 </HTML>