Full state feedback
Encyclopedia
Full state feedback or pole placement, is a method employed in feedback
Feedback
Feedback describes the situation when output from an event or phenomenon in the past will influence an occurrence or occurrences of the same Feedback describes the situation when output from (or information about the result of) an event or phenomenon in the past will influence an occurrence or...

 control system theory to place the closed-loop pole
Closed-loop pole
Closed-loop poles are the positions of the poles of a closed-loop transfer function in the s-plane. The open-loop transfer function is equal to the product of all transfer function blocks in the forward path in the block diagram...

s of a plant in pre-determined locations in the s-plane. Placing poles is desirable because the location of the poles corresponds directly to the eigenvalues 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.

If the closed-loop input-output transfer function can be represented by a state space equation, see State space (controls)
State space (controls)
In control engineering, a state space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations...

,



then the poles of the system are the roots of the characteristic equation given by


Full state feedback is utilized by commanding the input vector . Consider an input proportional (in the matrix sense) to the state vector,
.

Substituting into the state space equations above,



The roots of the FSF system are given by the characteristic equation, . Comparing the terms of this equation with those of the desired characteristic equation yields the values of the feedback matrix which force the closed-loop eigenvalues to the pole locations specified by the desired characteristic equation.

Example of FSF

Consider a control system given by the following state space equations


The uncontrolled system has closed-loop poles at and . Suppose, for considerations of the response, we wish the controlled system eigenvalues to be located at and . The desired characteristic equation is then .

Following the procedure given above, , and the FSF controlled system characteristic equation is
.

Upon setting this characteristic equation equal to the desired characteristic equation, we find
.

Therefore, setting forces the closed-loop poles to the desired locations, affecting the response as desired.

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
Linear-quadratic regulator
The theory of optimal control is concerned with operating a dynamic system at minimum cost. The case where the system dynamics are described by a set of linear differential equations and the cost is described by a quadratic functional is called the LQ problem...

for such applications.
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