State-Space Analysis

Why it matters

State-Space Analysis is a powerful method used in control systems to model and analyze systems with multiple inputs and outputs. It is essential for designing modern control systems in electronics and telecommunications, enabling engineers to predict system behavior and ensure stability.

Key ideas

• State Variables: These are variables that represent the system's state at any given time. They provide a way to model dynamic systems.
• State-Space Representation: A mathematical model of a physical system as a set of input, output, and state variables related by first-order differential equations.
• State Equation: Describes the system dynamics using state variables.
• Output Equation: Relates the state variables to the system outputs.
• Controllability and Observability: Key properties that determine if a system's state can be controlled or observed through its inputs and outputs.

Formulas

• State Equation: x'(t) = A·x(t) + B·u(t)
  • x'(t): Derivative of the state vector (state change rate)
  • A: State matrix
  • x(t): State vector
  • B: Input matrix
  • u(t): Input vector
• Output Equation: y(t) = C·x(t) + D·u(t)
  • y(t): Output vector
  • C: Output matrix
  • D: Feedthrough (or direct transmission) matrix

Worked example

Given a system with matrices:

• A = [[0, 1], [-2, -3]]
• B = [[0], [1]]
• C = [1, 0]
• D = [0]

Find the output for u(t) = exp(t), t ≥ 0, with x(0) = [0,0]ᵀ.

1. State Equation: x'(t) = A·x(t) + B·u(t)

   • Substitute A, B, and u(t):
   • x'(t) = [[0, 1], [-2, -3]]·x(t) + [[0], [1]]·e^t

2. Output Equation: y(t) = C·x(t) + D·u(t)

   • Substitute C, D, and u(t):
   • y(t) = [1, 0]·x(t) + [0]·e^t

3. Since y = x1 and x2 = y', the scalar equation is y'' + 3y' + 2y = exp(t). With zero initial conditions, Y(s) = 1/[(s − 1)(s + 1)(s + 2)].

4. Partial fractions give Y(s) = (1/6)/(s − 1) − (1/2)/(s + 1) + (1/3)/(s + 2).

Final answer: y(t) = exp(t)/6 − exp(−t)/2 + exp(−2t)/3 for t ≥ 0. Also x1 = y and x2 = exp(t)/6 + exp(−t)/2 − 2exp(−2t)/3. Both initial states are zero. The growing response comes from the unbounded input; it does not contradict the plant’s stable poles −1 and −2.

Common mistakes

• Confusing the roles of matrices A, B, C, and D.
• Incorrectly setting up the state or output equations.
• Neglecting the importance of initial conditions in solving state equations.

For GATE EC

Questions often involve deriving state-space representations from given system descriptions or converting between state-space and transfer function representations. Practice solving differential equations and understanding controllability and observability.

Quick check

1. What is the purpose of state variables in state-space analysis?
2. How does the state matrix A affect the system dynamics?
3. What is the significance of the output matrix C?

Answers: 1. To represent the system's state at any time. 2. It defines the relationship between state variables and their derivatives. 3. It relates the state variables to the system outputs.