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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
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| | 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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