Conference Paper

Controllability and observability of uncertain systems: A robust measure

Dept. of Control & Dynamical Syst., California Inst. of Technol., Pasadena, CA, USA
DOI: 10.1109/CDC.2009.5399774 Conference: Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009. Proceedings of the 48th IEEE Conference on
Source: IEEE Xplore

ABSTRACT This paper deals with the class of polynomially uncertain continuous-time linear time-invariant (LTI) systems whose uncertainties belong to a semi-algebraic set. The objective is to determine the minimum of the smallest singular value of the controllability or observability Gramian over the uncertainty region. This provides a quantitative measure for the robust controllability or observability degree of the system. To this end, it is shown that the problem can be recast as a sum-of-squares (SOS) problem. In the special case when the uncertainty region is polytopic, the corresponding SOS formulation can be simplified significantly. One can apply the proposed method to any large-scale interconnected system to identify those inputs and outputs that are more effective in controlling the system. This enables the designer to simplify the control structure by ignoring those inputs and outputs whose contribution to the overall control operation is relatively weak. A numerical example is presented to demonstrate the efficacy of the results.

  • [Show abstract] [Hide abstract]
    ABSTRACT: This paper deals with the robust stability of discrete-time linear time-invariant systems with parametric uncertainties belonging to a semialgebraic set. It is asserted that the robust stability of any system over any semialgebraic set (satisfying a mild condition) is equivalent to solvability of a semidefinite programming (SDP) problem, which can be handled using the available software tools. The particular case of a semialgebraic set associated with a polytope is then investigated, and a computationally appealing method is proposed to attain the SDP problem by means of a sampling technique, introduced recently in the literature. Furthermore, it is shown that the current result encompasses the ones presented in some of the recent works. The efficacy of the proposed method is demonstrated through some illustrative examples, and the results are compared to some of the existing methods.
    IEEE Transactions on Automatic Control 03/2008; · 2.72 Impact Factor
  • [Show abstract] [Hide abstract]
    ABSTRACT: The robust stability of uncertain linear systems in polytopic domains is investigated in this paper. The main contribution is to provide a systematic procedure for generating sufficient robust stability linear matrix inequality conditions based on homogeneous polynomially parameter-dependent Lyapunov matrix functions of arbitrary degree on the uncertain parameters. The conditions exploit the positivity of the uncertain parameters, being constructed in such a way that: as the degree of the polynomial increases, the number of linear matrix inequalities and free variables increases and the test becomes less conservative; if a feasible solution exists for a certain degree, the conditions will also be verified for larger degrees. For any given degree, the feasibility of a set of linear matrix inequalities defined at the vertices of the polytope assures the robust stability. Both continuous and discrete-time uncertain systems are addressed, as illustrated by numerical examples.
    Systems & Control Letters. 01/2006; 55:52-61.
  • [Show abstract] [Hide abstract]
    ABSTRACT: In previous works we have proposed a robust stability condition for linear time-invariant discrete-time systems which makes use of a Lyapunov function with linear dependence on the uncertain parameters. This condition is expressed as a set of linear matrix inequalities (LMI) where an additional variable is kept common to all LMI. These features have enabled the development of successful robust filtering and control algorithms. In this short note we investigate possible extensions of this stability condition to handle Lyapunov functions with arbitrary parameter dependence while keeping a variable common to all LMI. By showing that feasibility of the original condition is indeed necessary for the existence of a family of robust stability conditions where the Lyapunov function can have arbitrary dependence on the uncertain parameters, we conclude that no such extensions are possible.
    Systems & Control Letters 11/2005; 54(11):1131–1134. · 1.67 Impact Factor


Available from