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| '''Full state feedback''' (FSF), or '''pole placement''', is a method employed in [[feedback]] control system theory to place the [[closed-loop pole]]s of a plant in pre-determined locations in the [[s-plane]].<ref name="Sontag1998">*{{cite book | | The name of the author is Numbers but it's not the most masucline title out there. California is our birth place. Supervising is my profession. One of the extremely very best issues in the globe for me is to do aerobics and now I'm attempting to earn money with it.<br><br>Feel free to surf to my blog post - [http://www.blaze16.com/blog/255682 www.blaze16.com] |
| | last = Sontag
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| | first = Eduardo
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| | authorlink = Eduardo D. Sontag
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| | year = 1998
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| | title = Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition
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| | publisher = Springer
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| | isbn = 0-387-98489-5
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| }}</ref> Placing poles is desirable because the location of the poles corresponds directly to the [[eigenvalue]]s of the system, which control the characteristics of the response of the system. The system must be considered [[controllable]] in order to implement this method.
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| ==Principle<ref>[http://ctms.engin.umich.edu/CTMS/index.php?example=Introduction§ion=ControlStateSpace#24 Control Design Using Pole Placement]</ref>==
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| If the closed-loop input-output transfer function can be represented by a state space equation, see [[State space (controls)]],
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| :<math>\dot{\underline{x}}=\mathbf{A}\underline{x}+\mathbf{B}\underline{u}; </math>
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| :<math>\underline{y} = \mathbf{C}\underline{x}+\mathbf{D}\underline{u}</math>
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| then the poles of the system are the roots of the characteristic equation given by
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| :<math>\left|s\textbf{I}-\textbf{A}\right|=0.</math>
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| Full state feedback is utilized by commanding the input vector <math>\underline{u}</math>. Consider an input proportional (in the matrix sense) to the state vector,
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| :<math>\underline{u}=-\mathbf{K}\underline{x}</math>.
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| Substituting into the state space equations above,
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| :<math>\dot{\underline{x}}=(\mathbf{A}-\mathbf{B}\mathbf{K})\underline{x}; </math>
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| :<math>\underline{y} = (\mathbf{C}-\mathbf{D}\mathbf{K})\underline{x}.</math>
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| The roots of the FSF system are given by the characteristic equation, <math>\det\left[s\textbf{I}-\left(\textbf{A}-\textbf{B}\textbf{K}\right)\right]</math>. Comparing the terms of this equation with those of the desired characteristic equation yields the values of the feedback matrix <math>\textbf{K}</math> which force the closed-loop eigenvalues to the pole locations specified by the desired characteristic equation.
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| == Example of FSF ==
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| Consider a control system given by the following state space equations
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| :<math>\dot{\underline{x}}=\begin{bmatrix}0 & 1 \\ -2 & -3\end{bmatrix}\underline{x}+\begin{bmatrix} 0 \\ 1\end{bmatrix}\underline{u}</math>
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| The uncontrolled system has closed-loop poles at <math>s=-1</math> and <math>s=-2</math>. Suppose, for considerations of the response, we wish the controlled system eigenvalues to be located at <math>s=-1</math> and <math>s=-5</math>. The desired characteristic equation is then <math>s^2+6s+5=0</math>.
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| Following the procedure given above, <math>\mathbf{K}=\begin{bmatrix} k_1 & k_2\end{bmatrix}</math>, and the FSF controlled system characteristic equation is
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| :<math>\left|s\mathbf{I}-\left(\mathbf{A}-\mathbf{B}\mathbf{K}\right)\right|=\det\begin{bmatrix}s & -1 \\ 2+k_1 & s+3+k_2 \end{bmatrix}=s^2+(3+k_2)s+(2+k_1)</math>.
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| Upon setting this characteristic equation equal to the desired characteristic equation, we find
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| :<math>\mathbf{K}=\begin{bmatrix}3 & 3\end{bmatrix}</math>.
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| Therefore, setting <math>\underline{u}=-\mathbf{K}\underline{x}</math> forces the closed-loop poles to the desired locations, affecting the response as desired.
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| '''NOTE:''' This only works for Single-Input systems. Multiple input systems will have a '''K''' matrix that is not unique. Choosing, therefore, the best '''K''' values is not trivial. Recommend using a [[linear-quadratic regulator]] for such applications.
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| ==References==
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| {{reflist}}
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| ==See also==
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| *[[Pole splitting]]
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| *[[Step response]]
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| == External links ==
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| *[http://reference.wolfram.com/mathematica/ref/StateFeedbackGains.html Mathematica function to compute the state feedback gains]
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| [[Category:Control theory]]
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The name of the author is Numbers but it's not the most masucline title out there. California is our birth place. Supervising is my profession. One of the extremely very best issues in the globe for me is to do aerobics and now I'm attempting to earn money with it.
Feel free to surf to my blog post - www.blaze16.com