LMI Pole Regions for a Robust Discrete-Time Pole Placement Controller Design

Abstract

Herein, robust pole placement controller design for linear uncertain discrete time dynamic systems is addressed. The adopted approach uses the so called "D regions" where the closed loop system poles are determined to lie. The discrete time pole regions corresponding to the prescribed damping of the resulting closed loop system are studied. The key issue is to determine the appropriate convex approximation to the originally non-convex discrete-time system pole region, so that numerically efficient robust controller design algorithms based on Linear Matrix Inequalities (LMI) can be used. Several alternatives for relatively simple inner approximations and their corresponding LMI descriptions are presented. The developed LMI region for the prescribed damping can be arbitrarily combined with other LMI pole limitations (e.g., stability degree). Simple algorithms to calculate the matrices for LMI representation of the proposed convex pole regions are provided in a concise way. The results and their use in a robust controller design are illustrated on a case study of a laboratory magnetic levitation system.

Linear Matrix Inequalities for an Iterative Solution of Robust Output Feedback Control of Systems with Bounded and Stochastic Uncertainty

“…If the matrix C has full row rank, the algebraic constraint in (12) ensures that M has full rank and that it is therefore invertible. Then, the resulting controller gain is given by [33]:…”

“…Set up the LMI constraint (13) for each vertex v of the polytopic uncertainty model (stability of the control law) together with the equality constraint (12) for the output feedback gain Set up the LMI constraint (27) for each system output y m,i (stability and eigenvalue constraints of the output filters) Solve all LMIs above in an initialization stage and revert the linearizing changes of coordinates according to ( 14) and ( 26) to obtain the controller gains K y from Corollary 1 and the filter gains k f,i from Corollary 2 Store the matrices and gainsQ = Q,Q…”

“…Set up the LMI constraint (13) for each vertex v of the polytopic uncertainty model (stability of the control law) together with the equality constraint (12) for the output feedback gain Set up the LMI constraint (27) for each system output y m,i (stability and eigenvalue constraints of the output filters) Set up all LMI constraints (41) as functions of the previous filter gainsǩ f,i Specify the cost function J in (38), where all variables marked by( •) symbols correspond to the previously computed resultsQ,Q f,i ,Ǩ y , andǩ f,i , and add the penalty term α • J −J 2 , α > 0, to the cost function to prevent excessive gain variations between two subsequent iteration steps Solve all LMIs above, and revert the linearizing changes of the coordinates according to (14) and ( 26) to obtain the controller gains K y from Corollary 1 and the filter gains k f,i from Corollary 2…”

“…For the discrete-time counterpart, cf. [ 11 , 12 ]. In addition to continuous-time systems with an integer-order time derivative, LMI techniques have recently also become an active topic of research if robust controllers and observers are to be designed for fractional-order systems with uncertainty [ 13 , 14 , 15 , 16 ].…”

Discrete-Time Pole-Region Robust Controller for Magnetic Levitation Plant

“…For the discrete-time systems, however, the pole region for the prescribed relative damping is no longer convex, which causes a specific problem and can be sensed as a kind of asymmetry with the continuous-time counterpart. To simplify the controller design, some variants of a convex approximation of the non-convex cardioid pole-region have been presented [12][13][14][15][16]. In [12], an inner circle is proposed, which in [15] is enlarged to an inner ellipse.…”