The state space representation of a first-order system is given as ẋ = -x + u y = x Where, x is the state variable, u is the control input and y is the controlled output. Let u = -K.x be the control law, where K is the controller gain. To place a closed-loop pole at -2, the value of K is ______

The correct answer is 1.

Let's derive the solution step-by-step.

We are given the state space representation of a first-order system as:
$\dot{x}=-x+u$
$y=x$
where $x$ is the state variable, $u$ is the control input, and $y$ is the controlled output.

The system is controlled using a state feedback control law:
$u=-Kx$
where $K$ is the controller gain.

Substituting the control law $u=-Kx$ into the state equation:
$\dot{x}=-x-Kx$
Factoring out $x$ on the right-hand side gives:
$\dot{x}=-(K+1)x$

The system is of the form $\dot{x}=Ax$, where the closed-loop system matrix $A$ is:
$A=-(K+1)$

The characteristic equation of a state model is determined by:
$|sI-A|=0$
Since this is a first-order system, $I=1$, and the characteristic equation simplifies directly to:
$s-A=0$
Substituting $A=-(K+1)$:
$s-[-(K+1\left)\right]=0$
$s+K+1=0$

Therefore, the closed-loop pole is located at:
$s=-(K+1)$

We are given that the desired closed-loop pole is located at $-2$. Setting $s=-2$:
$-2=-(K+1)$
$2=K+1$
$K=1$

Thus, the required value of the controller gain $K$ is 1.