What Is State Feedback?

State feedback is a control strategy in which the input to a dynamical system is computed as a linear combination of all the system's state variables. Rather than reacting only to a measured output, a state feedback controller uses the full internal state to determine the control action at each instant, allowing the closed-loop dynamics to be shaped with a high degree of precision. The technique is a cornerstone of modern control theory and is applicable to any system whose state can be measured or reliably estimated.

The foundation of state feedback lies in the state-space representation of a dynamical system, expressed as the matrix equations relating state derivatives, inputs, and outputs. When a system is controllable, a feedback gain matrix can be chosen to place the closed-loop eigenvalues at any desired locations in the complex plane, directly determining the transient and steady-state behavior of the system.

Control Law and Gain Design

The state feedback control law takes the form u = −Kx, where u is the control input, x is the state vector, and K is the feedback gain matrix. Selecting K is the central design task. The pole placement design technique described by Rutgers University shows how, for a controllable system, the characteristic polynomial of the closed-loop matrix (A − BK) can be set to match any desired polynomial by solving for the appropriate K. For single-input systems, Ackermann's formula provides a direct closed-form solution. For multi-input systems, the problem is under-determined, and additional criteria such as robustness or minimum gain are imposed to select among the feasible solutions.

Linear Quadratic Regulation

The linear quadratic regulator (LQR) is an optimal state feedback design method that chooses K by minimizing a quadratic cost function penalizing both state deviation and control effort. The cost function weights are tuning parameters that let a designer express relative priorities between fast convergence and small actuator inputs. The optimal gain is obtained by solving the algebraic Riccati equation, a standard numerical procedure available in control toolboxes. LQR has the desirable property of guaranteed stability margins: the resulting closed-loop system has at least 60 degrees of phase margin and infinite gain margin. These robustness certificates make LQR a preferred design method when the system model is accurate and the cost weights are chosen thoughtfully. The IEEE paper on data-driven pole placement for linear systems extends classical gain design to settings where a model is not known in advance and must be inferred from measured trajectories.

Observer-Based State Feedback

Full state feedback requires access to all state variables, but in practice many states are not directly measured by sensors. In this case, a state estimator (observer) is paired with the feedback controller. The separation principle, which holds for linear time-invariant systems, guarantees that the observer and the controller can be designed independently without sacrificing closed-loop stability: the estimator is designed to reconstruct the state accurately, and the feedback gain is designed as if the true state were available. The combination is called an observer-based controller or a dynamic output-feedback controller. The Control Tutorials for MATLAB and Simulink introduction to state-space methods illustrates how pole placement and LQR designs extend naturally to the observer-based setting.

Applications

State feedback has applications in a wide range of disciplines, including:

• Aircraft and spacecraft attitude control, where multiple angular states must be regulated simultaneously
• Industrial robot joint control, using full state feedback across position and velocity states
• Automotive active suspension and stability control systems
• Power converter regulation, where DC-link voltage and inductor current are controlled jointly
• Chemical process control, where temperature, pressure, and concentration states are fed back to actuators