State Feedback Control

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

State feedback is a control system design technique that requires measuring all state variables of a system. It involves feeding back all system states to the system input through a constant feedback gain matrix ($K$).

A mathematical model known as the 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:

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

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

The matrices A, B, C, and D define the system 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 primary objective is to place the eigenvalues of the closed-loop system matrix (A − BK) at specific locations in the complex plane to achieve desired system properties, such as stability and a faster response. 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 altering the system poles through feedback, engineers can enhance 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, its transient response fails to meet design requirements, the pole placement technique allows us to reassign the eigenvalues. The stability and transient response of the closed-loop system are determined by the eigenvalues of the matrix $A-BK$, known as the closed-loop eigenvalues. These differ from the open-loop eigenvalues of the matrix $A$.

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

Using the pole placement technique with state feedback, the poles of the closed-loop system are moved to desired locations to achieve not only system stability but also required performance criteria. The primary goal of this technique is to determine the optimal feedback gain matrix, $K$, that places the poles at the desired positions. The poles of the closed-loop system are its eigenvalues. Thus, by using the matrix $K$ to alter pole locations, the eigenvalues of the expression $A-BK$ are modified. Therefore, designing a state feedback controller involves two steps: selecting the desired poles and computing the corresponding K matrix.

Several methods exist for computing the optimal K value. The most commonly used are the Coefficient Matching method and Ackermann’s formula.

The Coefficient Matching method is a fundamental approach for state feedback design.

The logic involves comparing the system’s characteristic equation with the desired characteristic equation and determining the optimal K value by equating coefficients. The system’s characteristic equation is denoted as $\alpha (s)$ and is given by:

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

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 desired pole locations and is computed as:

${\alpha }_d(s)=(s-p_1)(s-p_2)...(s-p_n)$

The values $p_i$ represent the desired poles. After computing the equations $\alpha (s)$ and ${\alpha }_d(s)$, they are compared ($\alpha (s)={\alpha }_d(s)$), and the elements of the K vector are determined, yielding the optimal K value.

Another widely used method in the literature is Ackermann’s formula. The K vector is computed primarily using the following formula:

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

In this formula, the vector $[0\dots 1]$ is a (1×n) vector with the last element equal to 1 and all other elements equal to 0. The matrix $C$ is the controllability matrix, computed as $[B,AB,...,A^{n-1}B]$, and ${\alpha }_d(s)$ is the desired characteristic equation.