Script capital H2 reset controller design for linear systems using piecewise quadratic Lyapunov functions

This study presents reset controller design conditions for multiple-input and multiple-output (MIMO) linear systems in terms of linear matrix inequalities (LMIs). The design conditions were developed using a piecewise quadratic Lyapunov function (PWQLF), leading to less conservative conditions than using a common Lyapunov function. Achieving LMI relaxation of the PWQLF-based reset controller design conditions for MIMO linear systems is challenging because (1) dividing the flow set of the reset condition is highly complex, (2) equality constraints, which make the PWQLF continuous on the boundaries between adjacent flow sets, prevent the use of congruence transformation in matrix inequalities, and (3) indefiniteness of the matrices for regional constraints also makes it difficult to apply standard LMI relaxation techniques. To overcome these difficulties, this study introduces a two-dimensional projection of the plant's output measurements and reset controller's states to construct a reset condition, allowing the PWQLF to be tractable in MIMO linear systems. Then, a new stability analysis condition is derived using the elimination lemma, overcoming the challenges in LMI relaxation of the controller design attributed to the equality constraints and indefinite matrices. An LMI-based reset controller design condition is then proposed based on the proposed stability analysis. To account for transient behavior, Script capital H2$$ {\mathscr{H}}_2 $$ tracking reset controller design problem is addressed. A numerical example is provided to demonstrate the effectiveness of the proposed method.