Concepts of State, State Variables, and State Model

In control system analysis, especially for dynamic systems, the concepts of state, state variables, and state models are fundamental for understanding and modeling system behavior.

1. State of a System

The state of a system at any given time is the minimum set of variables that fully describe the system’s condition and behavior independent of past inputs.

In simple terms, the state gives a snapshot of the system — it captures all necessary information to determine future behavior, given future inputs.

2. State Variables

State variables are the set of variables required to describe the state of a dynamic system. These variables typically correspond to energy-storing elements in physical systems.

• The number of state variables equals the order of the system.
• They are functions of time, denoted by:

X(t) = [x₁(t), x₂(t), ..., xₙ(t)]T

3. State Model (State-Space Representation)

The state model (or state-space model) is a mathematical representation of a physical system using first-order differential equations. It relates:

• State variables
• Inputs
• Outputs

Standard Form (Continuous-Time):

𝑋̇(t) = A·X(t) + B·U(t)
Y(t) = C·X(t) + D·U(t)

Where:

• X(t): State vector (n × 1)
• U(t): Input vector (m × 1)
• Y(t): Output vector (p × 1)
• A: System matrix (n × n)
• B: Input matrix (n × m)
• C: Output matrix (p × n)
• D: Feedthrough matrix (p × m)

Why Use State-Space Models?

• Applicable to MIMO (Multiple Input Multiple Output) systems
• Works for time-varying and nonlinear systems
• Useful for modern control design and simulation (e.g., MATLAB)
• Efficient for computer-based control implementation

Example:

Consider the second-order system:

𝑦̈ + 5𝑦̇ + 6y = u(t)

Let:

x₁ = y
x₂ = 𝑦̇

Then the state-space model becomes:

𝑥̇₁ = x₂
𝑥̇₂ = -6x₁ - 5x₂ + u(t)

Or in matrix form:

[𝑥̇₁]
[𝑥̇₂] = [ 0   1 ] [x₁] + [ 0 ] u(t)
         [-6 -5 ] [x₂]   [ 1 ]

y = [1  0] [x₁
           x₂]

This is a simple example of how a higher-order differential equation is converted into a first-order vector-matrix form for modern control analysis.

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