LMIs in Control/Stability Analysis/D-Stability

Definition Consider . The matrix is -stable if and only if there exists , where , such that , or equivalent , where is the Kroenecker product, The eigenvalues of a -stable matrix lie within the LMI region , which is defined as , where , , , and is the complex conjugate of . Conic Sector Region Stability via the Dilation Lemma Consider and . The matrix satisfies , where , if and only if there exist and , where , such that . Equivalently, the matrix satisfies if and only if there exist and , and , where , such that . Moreover, for every that satisfies , and are solutions to α-Region Stability via the Dilation Lemma Consider and . The matrix satisfies , where if and only if there exist and , where , such that . Equivalently, the matrix satisfies if and only if there exist , , and , where , such that . Moreover, for every that satisfies and are solutions to . Circular Region Stability via the Dilation Lemma Consider , , and , where . The matrix satisfies , where , if and only if there exist and , where , such that . Equivalently, the matrix satisfies if and only if there exist , , and , where , such that Moreover, for every that satisfies and are solutions to External Links LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet. LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd. LMI Properties and Applications in Systems, Stability, and Control Theory - A downloadable book on LMIs by Ryan James Caverly and James Richard Forbes.