State Variables

In classical control, a system is described through its transfer function, which relates input and output in the frequency domain. But transfer functions hide the internal dynamics of a system. State variables are the minimum set of variables that, along with the future inputs, completely determine the future behavior of a system. This concept forms the foundation of modern control theory.

Core Concept Explanation

Every physical system stores energy in some form. A capacitor stores electrical energy, an inductor stores magnetic energy, a mass stores kinetic energy, and a spring stores potential energy. The variables that describe the energy stored at any instant are the natural candidates for state variables. The number of independent energy storage elements in a system equals the order of the system, which also equals the number of state variables required.

The key property of state variables is their completeness. If the state vector x(t0) at some initial time t0 is known, and the input u(t) for all t greater than t0 is known, then the output y(t) for all future time can be uniquely determined. No additional past history of the system is needed. This is called the state property or Markov property of the state.

State variables are not unique. Any n independent linear combinations of a valid set of state variables form another equally valid set. This freedom is exploited when transforming state equations into canonical forms such as the controllable canonical form or observable canonical form, which simplify analysis and controller design.

Mathematical Expression

For an nth-order linear time-invariant system, the state is described by n first-order differential equations grouped as the state equation: dx/dt = A x(t) + B u(t), where x is the n times 1 state vector, u is the r times 1 input vector, A is the n times n system matrix, and B is the n times r input matrix. The output equation is y(t) = C x(t) + D u(t), where C is the p times n output matrix and D is the p times r feedthrough matrix.

For a single-input single-output system described by the nth-order ODE d^n y/dt^n + a_(n-1) d^(n-1)y/dt^(n-1) + ... + a0 y = b0 u, the simplest choice of state variables is x1 = y, x2 = dy/dt, x3 = d2y/dt2, ..., xn = d^(n-1)y/dt^(n-1). These are called phase variables.

Practical Understanding

State variable representation is powerful because it handles multi-input multi-output (MIMO) systems naturally, whereas transfer functions are limited to SISO systems or require matrix transfer functions. Modern control design methods such as pole placement, linear quadratic regulator (LQR), and Kalman filtering are all formulated in state space form. These methods cannot be conveniently expressed using classical transfer function techniques.

In a series RLC circuit driven by a voltage source, the natural state variables are the capacitor voltage Vc(t) and the inductor current iL(t). These two variables together capture all energy in the system at any instant. Knowing them along with future input completely determines future circuit behavior, including all voltages and currents throughout the circuit.

Solved Numerical Example

Given:
A mechanical system: mass M = 1 kg, damper B = 3 N.s/m, spring K = 2 N/m
Force input f(t), output is position x(t)

Why this formula applies:
Second-order mechanical system has 2 energy storage elements (mass = KE, spring = PE)
So 2 state variables are needed.

Differential equation:
M * x''(t) + B * x'(t) + K * x(t) = f(t)
1 * x'' + 3 * x' + 2 * x = f(t)

State variable selection:
x1 = x(t) (position)
x2 = x'(t) (velocity)

State equations:
x1' = x2
x2' = -2*x1 - 3*x2 + f(t)

Matrix form:
[x1']   [0   1] [x1]   [0]
[x2'] = [-2 -3] [x2] + [1] * f(t)

Output equation:
y = [1  0] [x1; x2]

Final Answer:
A = [[0,1],[-2,-3]], B = [0,1]^T, C = [1,0], D = 0
System order = 2, number of state variables = 2

Exam Tip: The number of state variables always equals the system order, which equals the number of independent energy storage elements. For an nth-order transfer function with denominator degree n, exactly n state variables are needed. Non-minimum phase systems or repeated poles do not change this count.

Mechanism Diagram

Summary of State Variable Properties

• State variables must be linearly independent of each other. Redundant variables are not allowed.
• The state at t0 plus input for t greater than t0 completely determines all future outputs.
• Natural state variables come from energy storage: capacitor voltage, inductor current, position, velocity.
• Phase variables (y, y', y'', ...) are always a valid state variable choice for any ODE.
• State variables are not observable directly as output unless the output equation maps them appropriately.

Quick Revision

• State variables are the minimum set of variables that, with future inputs, completely determine future system behavior.
• Number of state variables = system order = number of independent energy storage elements.
• Phase variables: x1=y, x2=y', x3=y'', ..., xn = y^(n-1) for an nth-order system.
• State variables are not unique. Any non-singular linear transformation of a valid set is also valid.
• State equation: dx/dt = Ax + Bu. Output equation: y = Cx + Du.
• Common trap: confusing the number of state variables with the number of inputs or outputs in MIMO systems.
• GATE relevance: questions often ask to identify state variables, form state equations from ODEs, or identify the system order from the state matrix size.