JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech III Year LINEAR SYSTEMS ANALYSIS Previous Question Paper (February/March - 2022 - R16)

A static system, also known as a memoryless system, is one where the output at any instant depends only on the input at that same instant. It does not store information about past inputs or predict future inputs. Mathematically, for a system with input x(t) and output y(t), the relationship is y(t) = f(x(t)). A simple example is a resistive circuit where the voltage-current relationship is v(t) = i(t)R. Since the voltage at any time t depends only on the current at that exact time, it is a static system.

A dynamic system is one where the output at any instant depends on the current input as well as past inputs or future inputs. These systems possess memory and are typically described by differential or difference equations. For example, a capacitor is a dynamic system because the voltage across it depends on the integral of the current over time: v(t) = (1/C) ∫ i(τ) dτ from -∞ to t. Because the output depends on the history of the input, the capacitor is a dynamic system.

To derive the state-variable equations for a circuit with state variables v₁, v₂, and v₅ (representing voltages across capacitors C₁, C₂, and C₅), we apply Kirchhoff's Current Law (KCL) at each node associated with the capacitors. The normal-form equations are expressed as a system of first-order differential equations in the form dx/dt = Ax + Bu.

1. Identify the current through each capacitor using i = C(dv/dt).
2. Apply KCL at the nodes to express the derivative of each state variable (dv₁/dt, dv₂/dt, dv₅/dt) as a linear combination of the state variables and the independent input sources.
3. Arrange the resulting equations into the matrix form:
   [dv₁/dt; dv₂/dt; dv₅/dt] = [A₁₁ A₁₂ A₁₃; A₂₁ A₂₂ A₂₃; A₃₁ A₃₂ A₃₃] [v₁; v₂; v₅] + [B₁; B₂; B₃] u(t)

The specific coefficients A_ij and B_i are determined by the values of the resistors and capacitors present in the circuit topology provided in the figure.