Application of Randomized Algorithm in Robust Eigenstructure Assignment Titas Bera, M. S. Bhat and Debasish Ghose Abstract— Several robust eigenstructure assignment meth- ods for MIMO systems try to exploit non-uniqueness of controller matrix to optimize certain performance index. In general these methods lead to computational intractability or become too conservative design. Therefore the scope of using randomized algorithms as an alternative exists. Based on well established statistical learning theory algorithms this paper provides methodologies to design robust controller under structured uncertainty by eigenstructure assignment method. Also a Monte Carlo based computational procedure for a low sensitivity regional pole assignment problem is illustrated. With randomized methods, a controller can be easily found working satisfactorily for most of the time, while the price to pay for is its probabilistic completeness. I. INTRODUCTION Eigenstructure assignment is a well established method that directly incorporates classical time domain design spec- ifications. Initiated by Wonham [1] and later, Moore [2] described the non-uniqueness of the controller which can assign a self conjugate set of eigenvalues for MIMO system. The seminal paper of Srinathkumar [3] clearly describes the freedom available to the designer in selecting maxi- mum number of eigenvalues and number of entries in each corresponding eigenvector. There exist several iterative and non-iterative methods which exploits this freedom to design appropriate controller for meeting the design specifications. Like, since the early design periods, attempts have been made to assign eigenvalues having low sensitivity with respect to perturbation in system matrices for better robustness. These can be thought of some what indirect approaches to design a robust controller. In these methods, the sensitivity measure considered in general, is the overall eigenvalue condition number which is a function of quadratic norm of modal matrix. There exists several methods for the problem of exact eigenvalue placement with minimum overall eigenvalue condition number. One can find in [4] details of low sensitiv- ity eigenstructure assignment methods for exact eigenvalue location. In practice, it is hardly required to place the eigenvalue exactly. Instead, often the design requirement is to place the poles in pre-specified regions in the complex plane. A low sensitivity eigenvalue assignment within a specific This work is partially supported by UKIERI Project Titas Bera is a research scholar in the Department of Aerospace Engineering, Indian Institute of Science, Bangalore, India titasbera@aero.iisc.ernet.in M.S. Bhat is with Department of Aerospace Engineering, Indian Institute of Science, Bangalore, India msbdcl@aero.iisc.ernet.in Debasish Ghose is with Department of Aerospace En- gineering, Indian Institute of Science, Bangalore, India dghose@aero.iisc.ernet.in region is considered by Lam and Tam [5]. Their approach is based on gradient based minimization technique to find the optimum pole location, which corresponds to a minimum Frobenius condition number of the closed loop state matrix. In general such a design is over conservative when com- pared to designs based on minimization of quadratic overall eigenvalue condition number. Note that quadratic condition number minimization with respect to a relaxed pole location is in general a non convex problem. In contrast to minimizing overall eigenvalue sensitivity, several direct attempts have been made to ensure both time domain and frequency domain robust stability and performance measures. This is done by combining frequency domain stability and performance measures along with time domain design specification to realize joint optimum robust- ness. In [4] and [6], minimization of an infinity norm of mixed sensitivity measure over all possible controller is done using genetic algorithm and gradient based methods. But as shown by researchers, H ∞ norm minimization methods results in very conservative controller design. From early 1990’s control researchers began to consider the computational complexity of several control problems. One can find a survey of existing results regarding computa- tional complexity of many robust control problems in [7]. In particular, simultaneous stabilization of linear systems with static state/output feedback when the controller parameters are constrained is proved to be NP-Hard. (Clearly one can relate the problem of simultaneous stabilization of linear systems by constrained controller to the problem of robust constrained eigenstructure assignment methods, since in the later case, the designer generally put constraints on some entries of desired eigenvector matrix to shape the time response/ mode decoupling). Details can be found in [8] and [9]. With such pessimistic result, an alternative approach that is recently gaining popularity is to use randomized algo- rithm to break the curse of dimensionality. In general these algorithm are not required to work in all cases, but rather most of the cases. These algorithms have polynomial time complexity. The price one pays is that, these algorithms are only probabilistically complete. Early attempt to bring randomization into control is initiated by Ray and Stengel [10]. More recently Khargonekar and Tikku [11], Tempo and Dabbane [12] and Vidyasagar [13] contributed to this field. With this motivation, in this paper a randomized approach to low sensitivity eigenstructure assignment for relaxed pole location is developed. Also, application of random- ized algorithm originated from statistical learning theory