1 / 19

Linear Matrix Inequalities in System and Control Theory

Linear Matrix Inequalities in System and Control Theory. Solmaz Sajjadi Kia Adviser: Prof. Jabbari System, Dynamics and Control Seminar UCI, MAE Dept. April 14, 2008. Linear Matrix Inequality (LMI). Set of n polynomial inequalities in x , e.g., Convex constraint on x.

merlin
Download Presentation

Linear Matrix Inequalities in System and Control Theory

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Linear Matrix Inequalities in System and Control Theory Solmaz Sajjadi Kia Adviser: Prof. Jabbari System, Dynamics and Control Seminar UCI, MAE Dept. April 14, 2008

  2. Linear Matrix Inequality (LMI) • Set of n polynomial inequalities in x, e.g., • Convex constraint on x

  3. Matrices as Variable Multiple LMIs

  4. LMI Problems Feasibility Minimization Problem

  5. How do we cast our control problems in LMI form? We rely on quadratic function V(x)=x’Px Three Useful Properties to Cast Problems in Convex LMI From • Congruent Transformation • S-Procedure • Schur Complement

  6. Congruent transformation

  7. Stable State Feedback Synthesis Problem

  8. S Procedure • Three Useful Properties to Cast Problems in Convex LMI From • Congruent Transformation • S-Procedure • Schur Complement

  9. Reachable Set/Invariant Set for Peak Bound Disturbance • The reachable set (from zero): is the set of points the state vector can reach with zero initial condition, given some limitations on the disturbance. • The invariant set: is the set that the state vector does not leave once it is inside of it, again given some limits on the disturbance.

  10. Reachable Set/Invariant Set for Peak Bound Disturbance Ellipsoidal Estimate Peak Bound Disturbance

  11. Linear (thus convex) Verses Nonlinear Convex inequality Three Useful Properties to Cast Problems in Convex LMI From Nonlinear (convex) inequalities are converted to LMI form using Schur Complement • Congruent Transformation • S-Procedure • Schur Complement

  12. H∞ or L2 Gain

  13. Norm of a vector in an ellipsoid Find Max of ||u||=||Kx|| for x in {x| xTPx≤c2 }

  14. A Saturation Problem Problem: Synthesis/Analysis of a Bounded State Feedback Controller (||u||<umax) exposed wT(t)w(t)≤w2max • Analysis: What is the largest disturbance this system can tolerate with K • Synthesis: Find a K such that controller never saturates

  15. Analysis: What is the largest disturbance (e.g. wmax) the system can tolerate ? xTPx<wTmax -umax=Kx umax=Kx

  16. Synthesis: Find a K such that controller never saturates xTPx<wTmax Kx=-umax Kx=umax

  17. Good Reference • Prof. Jabbari’s Note on LMIs • S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, “Linear Matrix Inequalities in Systems and Control Theory”

More Related