Control System Design in State Space

Why it matters

State space design is crucial for modern control systems as it provides a comprehensive framework for modeling, analyzing, and designing complex systems. It is particularly useful for systems with multiple inputs and outputs, making it essential for advanced engineering applications.

Key ideas

• State Space Representation: A method of representing dynamic systems using a set of first-order differential equations. It provides a time-domain approach to system analysis.
• State Variables: Variables that represent the system's state at any given time. They are used to describe the system's behavior.
• State Equations: Equations that describe the relationship between the state variables, inputs, and outputs of the system.
• Controllability and Observability: Key properties that determine whether a system can be controlled or observed from its inputs and outputs.
• State Feedback Control: A control strategy where the state variables are fed back into the system to achieve desired performance.
• Pole Placement: A method used in state feedback control to place the poles of the closed-loop system in desired locations for stability and performance.

Formulas

• State Space Representation: ẋ = Ax + Bu
  • ẋ: Derivative of the state vector (state change rate), unit: depends on the system
  • A: System matrix, unit: depends on the system
  • x: State vector, unit: depends on the system
  • B: Input matrix, unit: depends on the system
  • u: Input vector, unit: depends on the system
• Output Equation: y = Cx + Du
  • y: Output vector, unit: depends on the system
  • C: Output matrix, unit: depends on the system
  • D: Feedthrough (or direct transmission) matrix, unit: depends on the system

Worked example

Given a system with matrices:

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

Use u = −Kx (no reference input). Design a state feedback controller to place the poles of the closed-loop system at -1 and -2.

1. Determine the characteristic equation of the desired closed-loop system:

   • Desired poles: -1, -2
   • Characteristic equation: (s + 1)(s + 2) = s² + 3s + 2

2. Calculate the controllability matrix:

   • Controllability Matrix = [B, AB] = [[0, 1], [1, -3]]
   • Verify controllability: The matrix is full rank (rank = 2), so the system is controllable.

3. Determine the state feedback gain K:

   • Use pole placement method: K = [k1, k2]
   • Solve A - BK = [[0, 1], [-2, -3]] - [[0], [1]] * [k1, k2] = [[0, 1], [-2-k1, -3-k2]]
   • Match with desired characteristic equation: s² + (3+k2)s + (2+k1) = s² + 3s + 2
   • Solve for k1 and k2: k1 = 0, k2 = 0

4. State feedback gain:

   • K = [0, 0]

Final Answer: The state feedback gain K is [0, 0], because the open-loop poles already equal the requested −1 and −2. For a nontrivial target −4 and −5 instead, matching s²+9s+20 gives K = [18,6]. State feedback requires available state measurements or an observer.

Common mistakes

• Not verifying the controllability of the system before attempting pole placement.
• Incorrectly calculating the controllability matrix, leading to wrong conclusions about system controllability.
• Misplacing poles due to algebraic errors in solving the characteristic equation.

For GATE EC

Questions often involve determining the controllability and observability of a system, designing state feedback controllers, and performing pole placement. Practice solving problems related to state space representation and state feedback control design.

Quick check

1. What is the purpose of state feedback control?
2. How do you verify if a system is controllable?
3. What is the role of the D matrix in state space representation?

Answers: 1. To achieve desired system performance by feeding back state variables. 2. By checking the rank of the controllability matrix. 3. It represents the direct transmission path from input to output.