State Space Representation

The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.

Consider an RLC circuit, a common second-order system. To analyze this circuit using the state-space approach, two simultaneous first-order differential equations are necessary. The state variables in this context are derived from the quantities differentiated in the derivative equations associated with the energy-storage elements, specifically the inductor and capacitor.

Kirchhoff's voltage and current laws are employed to formulate the state equations. Kirchhoff's voltage law (KVL) states that all electrical potential differences around a loop is zero, while Kirchhoff's current law (KCL) asserts that the sum of currents entering a junction equals the sum of currents leaving. These laws enable the expression of non-state variables as linear combinations of state variables and inputs.

In an RLC circuit, the state variables are the voltage across the capacitor VC and the current through the inductor iL. Kirchhoff's laws express the resistor current and other non-state variables in terms of VC and iL. These expressions are then substituted back into the circuit's original differential equations.

After deriving the state equations, the final step is to represent these equations in vector-matrix form, achieving the state-space representation. For an RLC circuit, this might involve defining the state vector x, the input vector u, the output vector y, and matrices A, B, C, and D such that:

This representation is essential for analyzing the dynamic behavior of the system and designing appropriate control strategies.

In summary, the state-space approach provides a robust framework for handling complex systems, extending beyond the capabilities of frequency-domain techniques by accommodating nonlinearities, time variations, and multiple inputs and outputs.