Summary#
This training provided an introduction to several key topics in classical control theory:
Branches of Control Theory: Control theory encompasses a broad range of analytical tools and design techniques, including classical, optimal, robust, adaptive, and nonlinear control. Each branch makes different assumptions about system models and controller architecture to address various control challenges.
Connections to Machine Learning: Modern machine learning and classical control share deep theoretical links. Framing problems as dynamical systems opens up new ways to analyze neural network training and design adaptive controllers. Reinforcement learning in particular connects closely to optimal control theory.
Reinforcement Learning Parallels: Both fields develop strategies to optimize sequential decision making problems modeled as Markov decision processes. Dynamic programming, policy iteration, and value function techniques are fundamental in each field. Key differences include control’s focus on stability guarantees and explicitly modeling system dynamics.
Classical Control Limitations: While powerful for analyzing linear time-invariant (LTI) systems, classical control has inherent limitations. Tradeoffs between performance metrics cannot be independently tuned. Fundamental constraints exist for unstable and non-minimum phase systems. Applicability is limited for complex nonlinear dynamics and plants with unstructured uncertainty.
Limitation of Linear Control Systems#
While linear time-invariant (LTI) control systems provide a powerful set of analytical tools, they have inherent limitations that restrict their performance in real-world applications.
Tuning for optimal performance involves balancing sometimes competing objectives (response speed vs robustness), which LTI control cannot achieve independently.
Inability to Independently Tune Closed-Loop Performance#
The closed-loop dynamics of an LTI system are characterized by a set of response metrics that cannot be independently adjusted by the controller design. These include:
Response Speed: The rate at which disturbances are rejected and the system recovers to the setpoint. Faster response risks increased overshoot.
Noise Sensitivity: Small fluctuations in sensor measurements can excite unmodeled high-frequency dynamics. Reducing noise sensitivity typically slows down the response.
Steady-State Error: Offsets between the measured output and desired setpoint under persistent disturbances. Minimizing offset requires increased control effort.
Control Effort: The size and rate of change of control signals required. Lower effort limits responsiveness and disturbance rejection.
Stability Robustness: Sensitivity to uncertainties in model parameters or unmodeled dynamics. Higher robustness usually requires slower control designs.
Fundamental Performance Limitations#
Certain performance limitations are fundamental consequences of the LTI system dynamics:
Overshoot in Unstable Systems: Eliminating overshoot requires preview control, not possible with feedback of current states only.
Undershoot in Non-minimum Phase Systems: These systems exhibit unavoidable undershoot for any stabilizing LTI controller.
Modeling Errors: Parametric uncertainty directly degrades closed-loop stability and frequency response. Structural errors introduce unmodeled dynamics that cannot be addressed.
Real-World Complexity#
Practical systems have complex dynamics that break the LTI assumptions required for most control techniques:
Nonlinear Behavior: Most physical systems exhibit nonlinear responses that cannot be globally modeled by an LTI approximation.
Time-Varying Parameters: Model parameters change over time due to wear, temperature variations, aging, etc.
Unstructured Uncertainty: Unexpected disturbances and operating conditions outside the model’s scope.
Constraints on States and Inputs: Actuators saturate and systems can only operate safely within state constraints.
For these reasons, more advanced control techniques are often needed to achieve robust high-performance control of real-world systems.
Conclusion#
Overall, classical control provides an essential starting point for understanding feedback control systems. When combined with modern estimation, optimization, and machine learning techniques, classical methods such as PID control continue to enable high-performance control for real-world applications.
If you want to learn about more advanced control topics and the use of machine learning for control systems, consider attending the Machine Learning Control Transferlab training.