State feedback control is a closed-loop control method based on measuring the states of the system's output and feeding them back, then comparing this information with the targeted input value.

Details and Operation of the Method

State feedback is a control system design technique where all state variables of a system are required to be measured. It involves feeding back all states of the system to the system input through a fixed feedback gain matrix ($K$).

A mathematical model known as state-space representation uses matrices and vectors to describe the dynamic behavior of linear systems. In state-space form, a linear time-invariant (LTI) system is represented as follows:

$$$\dot{x}(t) = Ax(t) + Bu(t)$

$y(t) = Cx(t) + Du(t)$

A, B, C, and D matrices define the system's dynamics, while u(t) is the input, y(t) is the output, and x(t) is the state vector. This form provides a foundation for analyzing and designing sophisticated control systems, such as state feedback controllers.

State feedback control involves determining the control input using feedback from the state vector:

$u(t)=−Kx(t)$

Here, K is the state feedback gain matrix. The main goal is to achieve desired system characteristics, such as a stable and faster system response, by placing the pole locations of the closed-loop system matrix (A − BK) at specific positions in the complex plane. State feedback control is one of the fundamental techniques in modern control theory, providing precise control over system dynamics by using state variables as feedback. By changing the pole placement of the system through feedback, engineers can improve stability, reduce transient responses, and optimize control performance.

The control of an n-dimensional system and the shape of its transient response are determined by the location of the eigenvalues of the system matrix $A$. If the system is unstable or, even if stable, does not meet transient response design requirements, we can reassign the locations of the eigenvalues with feedback control. The stability and transient response of the closed-loop system are determined by the eigenvalues of the $A−BK$ matrix, which are called closed-loop eigenvalues. These are different from the eigenvalues of the $A$ matrix, which are the open-loop system eigenvalues.

Pole Placement Technique

The pole-placement technique forms the basis of state feedback control. Research highlights its applications in various fields, including power systems, robotics, and aerospace engineering, due to its adaptability to various linear system configurations and robustness against perturbations.

The state feedback method using the pole placement technique stabilizes the system and moves the poles of the closed-loop system to desired locations to achieve the required performance criteria. The primary objective of this technique is to find the optimal feedback gain, $K$, that places the poles at the desired locations. The poles of the closed-loop system are the eigenvalues of the closed-loop system. So, in fact, with the help of the $K$ matrix, the pole locations are changed, thereby changing the eigenvalues of the $A−BK$ expression. Therefore, there are two steps to designing a state feedback controller: selecting the desired poles and calculating the corresponding K matrix. There are several methods for calculating the optimal K value. The most commonly used are the 'Equalizing Coefficients' method and the Ackermann formula.

Equalizing Coefficients

The Equalizing Coefficients method is a fundamental method for state feedback design. The logic here is to compare the characteristic equation of the system with the desired characteristic equation and find the optimal K value by equalizing the coefficients. The characteristic equation of the system is denoted by $\alpha(s)$ and is equal to:

$\alpha(s) = det(sI-(A-BK))$

This equation is a function of s and contains variables dependent on K (such as k₁, k₂, ..., k_n). The desired characteristic equation is a function of the new poles at which the system is to be placed, and it is calculated as follows:

$\alpha_d(s) = (s-p_1)(s-p_2)...(s-p_n)$

The values $p_i$ represent the desired poles. After calculating the equations $\alpha(s)$ and $\alpha_d(s)$, the elements of the K vector are found by comparing them ($\alpha(s) = \alpha_d(s)$) and the optimal K value is obtained.

Ackermann's Formula

Another widely used method in the literature is Ackermann's formula. The K vector is fundamentally calculated by the following formula:

$K = [0 ...1]{C}^{-1}\alpha_d(A)$

In this formula, the $[0…1]$ vector is a $(1xn)$ dimensional vector with the last element being 1 and the rest being 0. The ${C}$ matrix is the controllability matrix calculated as $[B, AB, ... ,A^{n-1}B]$, and $\alpha_d(s)$ is the desired characteristic equation.