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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

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

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Matrices as Variable Multiple LMIs

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LMI Problems Feasibility Minimization Problem

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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

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Congruent transformation

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Stable State Feedback Synthesis Problem

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

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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.

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

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

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H ∞ or L 2 Gain

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Norm of a vector in an ellipsoid Find Max of ||u||=||Kx|| for x in {x| x T Px≤c 2 }

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A Saturation Problem Problem: Synthesis/Analysis of a Bounded State Feedback Controller (||u||__
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x T Px

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x T Px

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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”

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