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Linear Matrix Inequality (LMI)

Contributed by Ljubo Vlacic.
1890
  • A. M. Lyapunov published his seminal work known as Lyapunov theory ([1]). This is usually referred to as the first appearance of the Linear Matrix Inequality (LMI) in control theory; analytical solution of the Lyapunov LMI via Lyapunov equation.

  • 1940s
  • A. I. Lur’e and V. N. Postnikov were first to apply Lyapunov’s methods to some specific practical problems in control engineering, specifically the problem of stability of a control system with nonlinearity in the actuator. Small LMIs were solved by hand ([2], [3]).

  • Early 1960s
  • R. E. Kalman, V. A. Yakubovich and V. M. Popov managed to reduce the solution of the LMIs that arose in the problem of Lur’e to simple graphical criteria known as Kalman-Yakubovich-Popov (KYP) lemma ([4], [5], [6]).

  • 1964
  • The KYP lemma resulted in Popov criterion, Circle criterion, Tsypkin criterion ([7], [8]) and many variations.

  • 1962-1965
  • V. A. Yakubovich published many papers ([6], [9], [10]) highlighting the important role of LMIs in control theory; e.g. The solution of certain inequalities in automatic control theory (1962) and The method of matrix equalities in the stability theory of nonlinear control systems (1965).

  • Late 1960s
  • The Kalman-Yakubovich-Popov (KYP) lemma and extensions were extensively studied and found to be related to the idea of passivity, the small gain criteria introduced by Zames ([11], [12]) and Sandberg ([13], [14], [15]) and quadratic optimal control .

  • 1965
  • The idea of having a computer search for a Lyapunov function appeared in the literature.

  • 1970
  • By then, it was known that LMI appearing in the KYP lemma could also be solved by solving a certain algebraic Riccati equation ([16]).

  • Early 1970s
  • B. D. O. Anderson and S. Vongpanitlerd noted the difficulty in solving the LMI directly ([17]).

  • 1971
  • J. C. Willems in a paper on Quadratic Optimal Control pointed out that an LMI problem could be solved by studying the symmetric solutions of a certain Riccati equation ([19]).

  • 1976
  • H. P. Horisberger and P. R. Belanger observed that the existence of a quadratic Lyapunov function that simultaneously proves stability of a collection of linear systems is a convex problem involving LMIs ([22]).

  • 1982-1983
  • E. S. Pyantnitskii and V. I. Skorodinskii were perhaps the first to assert that many LMIs that arise in control and systems theory can be formulated as convex optimization problems which can be reliably solved by computer solutions for which no analytical solution was likely to be found. They were the first to formulate the search for a Lyapunov function as a convex optimization problem, and then apply an algorithm guaranteed to solve the optimization problem ([20], [21]).

  • 1984
  • N. Karmarkar introduced a new linear programming algorithm that solved linear programs in polynomial-time, like the ellipsoid method, but in contrast to the ellipsoid method, was also very efficient in practice ([24]).

  • 1988
  • Yu. Nesterov and A. Nemirovski developed interior-point methods that apply directly to convex problems involving matrix inequalities, and in particular, to the problems encountered in control theory ([25]).

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