Model Reference Adaptive Control Survey of Control Systems (MEM 800) Presented by Keith Sevcik

Concept Controller Model Adjustment Mechanism Plant Controller Parameters ymodel u yplant uc Design controller to drive plant response to mimic ideal response (error = yplant-ymodel => 0) Designer chooses: reference model, controller structure, and tuning gains for adjustment mechanism

MIT Rule Tracking error: Form cost function: Update rule: Change in is proportional to negative gradient of sensitivity derivative

MIT Rule Can chose different cost functions EX: From cost function and MIT rule, control law can be formed

MIT Rule EX: Adaptation of feedforward gain Π Adjustment Mechanism ymodel u yplant uc Π θ Reference Model Plant - +

MIT Rule For system where is unknown Goal: Make it look like using plant (note, plant model is scalar multiplied by plant)

MIT Rule Choose cost function: Write equation for error: Calculate sensitivity derivative: Apply MIT rule:

MIT Rule Gives block diagram: considered tuning parameter Π Adjustment Mechanism ymodel u yplant uc Π θ Reference Model Plant - +

MIT Rule NOTE: MIT rule does not guarantee error convergence or stability usually kept small Tuning crucial to adaptation rate and stability.

MRAC of Pendulum System d2 d1 dc T

MRAC of Pendulum Controller will take form: Model Adjustment Mechanism Controller Parameters ymodel u yplant uc

MRAC of Pendulum Following process as before, write equation for error, cost function, and update rule: sensitivity derivative

MRAC of Pendulum Assuming controller takes the form:

MRAC of Pendulum

MRAC of Pendulum If reference model is close to plant, can approximate:

MRAC of Pendulum From MIT rule, update rules are then:

MRAC of Pendulum Block Diagram Π ymodel e yplant uc θ1 Reference Model + - θ2

MRAC of Pendulum Simulation block diagram (NOTE: Modeled to reflect control of DC motor)

MRAC of Pendulum Simulation with small gamma = UNSTABLE!

MRAC of Pendulum Solution: Add PD feedback

MRAC of Pendulum Simulation results with varying gammas

LabVIEW VI Front Panel

LabVIEW VI Back Panel

Experimental Results

Experimental Results PD feedback necessary to stabilize system Deadzone necessary to prevent updating when plant approached model Often went unstable (attributed to inherent instability in system i.e. little damping) Much tuning to get acceptable response

Conclusions Given controller does not perform well enough for practical use More advanced controllers could be formed from other methods Modified (normalized) MIT Lyapunov direct and indirect Discrete modeling using Euler operator Modified MRAC methods Fuzzy-MRAC Variable Structure MRAC (VS-MRAC)