State Space Analysis

Why it matters

State Space Analysis is essential for modern control systems, particularly when dealing with multiple-input and multiple-output (MIMO) systems. It provides a comprehensive framework for modeling, analyzing, and designing control systems in a way that is more versatile than traditional methods like transfer functions.

Key ideas

• State Variables: These are variables that represent the state of the system. The number of independent states is the realization order; a minimal realization has the input–output system order.
• 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 a matrix equation of the form x' = Ax + Bu, where x is the state vector, u is the input vector, A is the system matrix, and B is the input matrix.
• Output Equation: Relates the state vector to the output vector using y = Cx + Du, where y is the output vector, C is the output matrix, and D is the feedthrough (or direct transmission) matrix.
• Controllability and Observability: Key properties that determine whether a system's state can be fully controlled or observed from the inputs and outputs.

Formulas

• State Equation: x' = Ax + Bu
  • x': Derivative of the state vector (state variables per time unit)
  • A: System matrix (entries carry units determined by the chosen state/input/output scaling)
  • x: State vector (state variables)
  • B: Input matrix (entries carry units determined by the chosen state/input/output scaling)
  • u: Input vector (input variables)
• Output Equation: y = Cx + Du
  • y: Output vector (output variables)
  • C: Output matrix (entries carry units determined by the chosen state/input/output scaling)
  • D: Feedthrough matrix (entries carry units determined by the chosen state/input/output scaling)
  • u: Input vector (input variables)

Worked example

Given a system with matrices:

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

Find the state space representation and determine if the system is controllable.

1. State Space Representation:

   • State Equation: x' = Ax + Bu
     • x' = [[0, 1], [-2, -3]]x + [[0], [1]]u
   • Output Equation: y = Cx + Du
     • y = [1, 0]x + [0]u

2. Controllability:

   • Form the controllability matrix Q = [B, AB]
   • Calculate AB = A * B = [[0, 1], [-2, -3]] * [[0], [1]] = [[1], [-3]]
   • Q = [[0, 1], [1, -3]]
   • Check the rank of Q. If rank(Q) = n (number of state variables), the system is controllable.
   • Rank of Q is 2, which equals the number of state variables.

Answer: The system is controllable since det(Q) = −1. Its observability matrix [C; CA] = [[1,0],[0,1]] also has rank 2. The transfer function is C(sI−A)^−1B+D = 1/(s²+3s+2).

Common mistakes

• Confusing the order of matrices in multiplication.
• Forgetting to check the rank of the controllability matrix.
• Misidentifying state variables and input/output variables.

For GATE EC

Questions often involve deriving the state space representation from a given system, analyzing controllability and observability, and converting between state space and transfer function representations. Practice forming and analyzing the controllability and observability matrices.

Quick check

1. What is the state equation in state space analysis?
2. How do you determine if a system is controllable?
3. What is the role of the matrix D in the output equation?

Answers: 1. x' = Ax + Bu; 2. By checking the rank of the controllability matrix; 3. It represents the direct transmission from input to output.