15.2 Control system design

Control system design is a crucial part of engineering applications. It involves creating controllers that regulate system behavior, from simple PID controllers to advanced adaptive and nonlinear strategies. These techniques aim to optimize performance, stability, and robustness in various engineering systems.

Optimal control methods take design further by minimizing cost functions while meeting constraints. This includes techniques like the Linear Quadratic Regulator and Model Predictive Control. These approaches use mathematical models to predict and optimize system behavior, balancing performance and efficiency in real-world applications.

Controller Design Techniques

PID Controller Tuning and Gain Scheduling

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• PID controllers combine proportional, integral, and derivative actions to minimize error in a control system
• Proportional action provides immediate response to error, integral action eliminates steady-state error, and derivative action improves transient response
• Ziegler-Nichols method determines initial PID gains based on system's critical gain and period
• Cohen-Coon method uses process reaction curve to calculate controller parameters
• Gain scheduling adjusts controller parameters based on operating conditions
• Implements multiple linear controllers for different operating points
• Interpolates between controllers to handle transitions between operating regions

Adaptive and Nonlinear Control Strategies

• Adaptive control modifies controller parameters in real-time to maintain performance as system characteristics change
• Model Reference Adaptive Control (MRAC) adjusts controller to match desired reference model behavior
• Self-Tuning Regulators (STR) estimate system parameters and update controller accordingly
• Nonlinear control addresses systems with inherent nonlinearities
• Feedback linearization transforms nonlinear systems into linear equivalents for easier control
• Sliding mode control provides robust performance for uncertain nonlinear systems
• Backstepping design constructs stabilizing controllers for nonlinear systems in a recursive manner

Optimal Control Methods

Optimal Control Theory and Applications

• Optimal control minimizes a cost function while satisfying system constraints
• Pontryagin's Maximum Principle provides necessary conditions for optimal control
• Dynamic programming solves optimal control problems by breaking them into smaller subproblems
• Applications include trajectory optimization (spacecraft maneuvers) and resource allocation (energy management)
• Model predictive control (MPC) uses system model to predict future behavior and optimize control actions
• MPC solves finite-horizon optimal control problem at each time step
• Receding horizon strategy implements only first control action and repeats optimization

Linear Quadratic Regulator and State Feedback

• Linear Quadratic Regulator (LQR) minimizes quadratic cost function for linear systems
• Cost function balances state deviation and control effort
• Algebraic Riccati Equation solves for optimal feedback gain matrix
• LQR provides guaranteed stability margins and robustness properties
• State feedback control uses full state information to place closed-loop poles
• Ackermann's formula calculates state feedback gains for single-input systems
• Bass-Gura method determines feedback gains for multi-input systems

System Analysis and Representation

State-Space Representation and Analysis

• State-space models describe system dynamics using first-order differential equations
• State variables represent internal system conditions
• State equation: $\dot{x} = Ax + Bu$ relates state derivatives to current state and input
• Output equation: $y = Cx + Du$ relates system outputs to state and input
• Controllability determines ability to drive system to desired state
• Observability assesses ability to reconstruct state from output measurements
• Kalman decomposition separates system into controllable and observable subsystems

Stability Analysis and Robustness Considerations

• Stability ensures bounded system response to bounded inputs
• Lyapunov stability theory analyzes nonlinear system stability without solving equations of motion
• Direct method constructs Lyapunov function to prove stability
• Indirect method linearizes system around equilibrium point for local stability analysis
• Robust control maintains stability and performance despite system uncertainties
• H-infinity control minimizes worst-case disturbance amplification
• Mu-synthesis addresses structured uncertainties in system model
• Robustness measures include gain margin, phase margin, and sensitivity functions