Multivariable Control Systems ECSE 6460 Fall 2009 Lecture

Multivariable Control Systems ECSE 6460 Fall 2009 Lecture 26: 4 December 2009

Uncertain MIMO Plant Gp(s)

Frequency Response

Feedback Control Loop + - K(s) Gp(s) • practically zero steady state error for step reference • rise time less than 1 second • overshoot less than 5% • robustly for any value of c

Menu for the day l Representing the uncertainty a la unstructured uncertainty l a la structured uncertainty l l Assessment of robust stability a la unstructured uncertainty l a la structured uncertainty l

Representing Uncertainty

Unstructured Uncertainty + G (s) Wi (s) Output uncertainty D(s) Wo (s) D(s) + Input uncertainty + + G (s)

Output Uncertainty Wo (s) G (s) We need to select a weight such that: Output uncertainty D(s) + +

Input Uncertainty Wi (s) D(s) Input uncertainty + + We need to select a weight such that: G (s)

Weight Wo(s) Refer to uncert 1. m

Structured Uncertainty

Structured Uncertainty Ws (s) G (s) D relax to + +

Relaxation from real to complex uncertainty

Robust Stability

Assessment of Robustness l In this topic, we first design a controller for the nominal system, then we assess the robustness properties of the system. + - l K(s) Gp(s) Two control designs are explored, decentralized inverse-based, and H-infinity synthesis.

Decentralized Inverse Based on nominal plant model Refer to decent. m

Nominal Performance

H synthesis for nominal plant l Use S/T synthesis with: Refer to hinf_ex. m

Nominal Performance

Robust Stability Assessment Unstructured Uncertainty Wo (s) K(s) + - G (s) D(s) + +

Robust Stability Assessment Wo (s) K(s) - G (s) + +

Robust Stability Assessment Refer to decent. m

Robust Stability Assessment Refer to hinf_ex. m

Robust Stability Assessment Structured Uncertainty Ws (s) K(s) + - D G (s) + +

Robust Stability Assessment Structured Uncertainty Ws (s) K(s) + - G (s) + +

Structured Singular Value Peak at 0. 130 For decentralized controller

Perturbed system responses

Structured Singular Value Peak at 0. 150

Perturbed system responses

Robust Performance

Nominal Performance Refer to hinf_ex. m

Robust Performance Assessment Unstructured Uncertainty K(s) + - Wo (s) G (s) + + + WP (s)

Robust Performance Assessment Unstructured Uncertainty K(s) + - Wo (s) G (s) + + + WP (s)

Robust Performance Assessment DP D N

Robust Performance Assessment

Robust Performance Assessment Structured Uncertainty Ws (s) + K(s) + - G (s) + + WP (s)

Robust Performance Assessment Structured Uncertainty Ws (s) + K(s) + - G (s) + + WP (s)

Robust Performance Assessment DP D N

Robust Performance Assessment