Lecture Notes on Modern Control Theory and State Space Modeling

Overview

• Continuation from previous class on state space modeling in modern control theory.
• Discussion on advantages of modern control theory.
• Introduction of examples: LRC circuit and full state feedback or pole placement.

Modern Control Theory

• Advantages:
  • Provides time-domain analysis.
  • Effective for highly non-linear systems.
  • Optimization capabilities (e.g., minimizing fuel).
• Key equations:
  • State equation: ẋ = Ax + Bu
  • Output equation: Y = Cx + Du

Block Diagram Representation

• ẋ represents the derivative of the state vector.
• Integration of ẋ gives the state x.
• Input u is summed with Ax to represent the system dynamics.
• Output Y is derived from the state Cx and feedforward Du.

LRC Circuit Example

• Description of LRC circuit components: Resistance (R), Inductance (L), Capacitance (C).
• States defined for LRC circuit:
  • State 1: I (current)
  • State 2: Vc (voltage across capacitor)
• Key equations derived from circuit relations:
  • V1 = VR + VL + VC
  • VR = IR, VL = L (di/dt), VC = (1/C) ∫i dt
• State space representation:
  • ẋ = Ax + Bu where:

    • A matrix:

      [ A = \begin{bmatrix} -\frac{R}{L} & -\frac{1}{L} \ \frac{1}{C} & 0 \end{bmatrix} ]

    • Input vector: u

Full State Feedback or Pole Placement

• Definition of full state feedback:
  • Feedback includes all states (regardless of the number).
• Purpose of pole placement:
  • To stabilize the system by placing poles in the left half of the complex plane.
• Control law example:
  • u = -Kx leads to modified state equation: ẋ = (A - BK)x.
• Importance of eigenvalues and system stability:
  • Closed-loop eigenvalues determine stability.

Examples and Analyses

• Example of transfer function: 1/(s - 1)
  • Initially unstable, poles in right half-plane.
  • By adjusting gain k, stability can be achieved.
• Discussion on aircraft model complexity and state counts:
  • Aircraft can have higher-order state matrices (e.g., 12 states for full dynamics).
  • Control strategies can often be simplified to fewer states.

Conclusion and Next Steps

• Modern vs. Classical Control Theory:
  • Each approach has its advantages for different situations.
• Upcoming topics:
  • Real-time implementation in MATLAB.
  • Linearization of systems using Taylor series and Jacobian methods.
• Importance of feedback and gains in controlling system response.
• Next class will focus on practical examples in control system design.