Control Theory Lecture

Overview of Control Theory Progression

• State Space Form: Converting differential equations to state space form using matrices.
• System Stability: Checking stability in state space form through eigenvalues.
• Control Design for Stability: Choosing control laws to stabilize the system, ensuring closed-loop system stability (negative eigenvalues).

Topics Covered Previously

1. State Space Transformation: Differential equations to state space.
2. Stability Check: Eigenvalues.
3. Control Design: For closed-loop stability.
4. Transfer Functions & Discrete Systems: Key concepts in control systems.

New Topic: LQR (Linear Quadratic Regulator)

• Purpose: Practical tool for designing control laws to stabilize systems in a state space form.
• Comparison to Other Methods: Alternative to pole placement, widely used in control theory.
• Introduction of Optimality: Beyond stability, aiming for the best performance.

Optimality Concept

• Definition: System performance measured based on how control error decays to zero.
• Cost Function: Integral over time to measure system performance, penalizing non-zero states and control inputs.
• Optimal Cost: The minimum possible cost for given initial conditions and control policy.
• Optimal Control Policy (π*): Control policy that minimizes the cost.

Dynamics and Control Policy

• Dynamic Equation: ( \dot{x} = f(x, u) )
• Control Policy: ( u = \pi(x, t) )
• Cost Function: ( J(x_0, \pi) = \int_0^{\infty} g(x(t), u(t)) dt )

Hamilton-Jacobi-Bellman (HJB) Equation

• Formulation: Optimal control theory using HJB equation.
• Application: Applies to both linear and nonlinear systems.

Practical Implementation – LQR

• System: Linear Time-Invariant (LTI) systems.
• State-Space Representation: ( x(t) = Ax(t) + Bu(t) )
• Quadratic Cost: ( J = \int_0^{\infty} (x^TQx + u^TRu) dt )
• Positive Definite Matrices: Q and R must be positive definite.

Derivation and Solution of LQR

• Optimal Cost Function: Quadratic in form ( J = x^T S x )
• Partial Derivative: ( \frac{\partial J}{\partial x} )
• LQR Control Law: ( u(t) = -Kx(t) ) where ( K = R^{-1} B^T S )
• Algebraic Riccati Equation (ARE): Used to solve for S matrix.

Computational Tools

• MATLAB LQR Function: lqr(A, B, Q, R).
• Python SciPy Solve: scipy.linalg.solve_continuous_are(A, B, Q, R).

Comparison: LQR vs. Pole Placement

• Pole Placement: Direct control of system eigenvalues, potentially high control effort.
• LQR: Indirect control through cost function, likely to result in reasonable control gains.

Summary

• LQR: An essential tool in control theory, balancing good system performance and reasonable control effort.
• Optimality: A stricter measure than stability alone.
• Mathematical Foundation: HJB equation and Algebraic Riccati Equation are central to understanding LQR.

Further Study

• Discrete LQR Systems: Seminar focus for next session.
• Application: Recommended to practice derivations for deeper understanding.