Control theory is concerned with influencing the behavior of dynamical systems through external inputs. In electrical engineering, circuits naturally serve as controlled systems, with voltages or currents acting as inputs and measurable quantities such as voltages or currents acting as outputs.
From this perspective, control theory builds directly on the dynamical systems viewpoint. Once a circuit has been modeled using ordinary differential equations, control questions can be posed in a precise mathematical way.
Controlled Dynamical Systems
A controlled dynamical system may be written abstractly as
where \(x(t)\) represents the state of the system and \(u(t)\) represents an externally applied control input. In electrical circuits, the state typically consists of capacitor voltages and inductor currents, while the input may be a voltage or current source.
Linear Control Systems
When circuit elements obey linear constitutive relations, the resulting controlled system takes the form
where the matrix \(A\) encodes the internal dynamics of the circuit and the matrix \(B\) describes how inputs influence the state. This formulation is known as the state-space representation.
An output equation is often introduced,
where \(y(t)\) represents a measured output of interest, such as a capacitor voltage or branch current.
Feedback Control
In feedback control, the input \(u(t)\) is chosen as a function of the system’s current state or output. A common choice is linear state feedback,
which yields the closed-loop system
By selecting the gain matrix \(K\), the eigenvalues of the closed-loop system matrix \(A - BK\) may be shaped to achieve desired stability and performance properties.
Stability and Poles
Stability of an electrical control system is determined by the location of the system’s poles. In the state-space formulation, poles correspond to the eigenvalues of the system matrix. In the Laplace domain, they correspond to the roots of the characteristic equation.
Feedback control allows these poles to be shifted, enabling engineers to stabilize otherwise unstable circuits or modify transient response characteristics such as settling time and overshoot.
Control Interpretation of Circuits
Many familiar circuit behaviors can be reinterpreted through a control-theoretic lens. Resistance introduces damping, capacitance and inductance introduce energy storage, and feedback loops arise naturally in amplifiers, oscillators, and power electronics.
From this viewpoint, circuit design becomes the problem of shaping system dynamics through feedback and input design rather than simply solving equations.
Beyond Linear Control
When nonlinear elements are present, the system dynamics become nonlinear. Control theory still applies, but analysis typically relies on linearization, Lyapunov methods, or numerical simulation. The underlying state-space structure, however, remains unchanged.
Control theory therefore provides a unifying framework connecting circuit analysis, dynamical systems, and system-level design.