Linear Matrix Inequality (LMI)
Contributed by Ljubo Vlacic.
- A. M. Lyapunov published his seminal work known as Lyapunov
theory (). 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.
- 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 (, ).
- 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 (, , ).
- The KYP lemma resulted in Popov criterion, Circle criterion,
Tsypkin criterion (, ) and
- V. A. Yakubovich published many papers (,
, ) 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
- 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 (, ) and Sandberg (, , ) and quadratic optimal
- The idea of having a computer search for a Lyapunov function
appeared in the literature.
- By then, it was known that LMI appearing in the KYP lemma
could also be solved by solving a certain algebraic Riccati equation ().
- Early 1970s
- B. D. O. Anderson and S. Vongpanitlerd noted the difficulty
in solving the LMI directly ().
- 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 ().
- 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 ().
- 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 (,
- 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 ().
- 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