Mikhail V. Khlebnikov

Optimization of Linear Systems Subject to Bounded Exogenous Disturbances: The Invariant Ellipsoid Technique

This survey covers a variety of results associated with control of systems subjected to arbitrary bounded exogenous disturbances. The method of invariant ellipsoids reduces the design of optimal controllers to finding the smallest invariant ellipsoid of the closed-loop dynamical system. The main tool of this approach is the linear matrix inequality technique. This simple yet versatile approach has high potential in extensions and generalizations; it is equally applicable to both the continuous and discrete time versions of the problems.

Petersen's lemma on matrix uncertainty and its generalizations

Various generalizations and refinements are proposed for a well-known result on robust matrix sign-definiteness, which is extensively exploited in quadratic stability, design of robustly quadratically stabilizing controllers, robust LQR-problem, etc. The main emphasis is put on formulating the results in terms of linear matrix inequalities.

Suppression of bounded exogenous disturbances: A linear dynamic output controller

In the paper, we study a problem of constructing a linear dynamic output controller for suppressing bounded exogenous perturbations in a linear control system. We propose an approach based on a method of invariant ellipsoids and technique of linear matrix inequalities. A control of gyroplatform and two-mass system is given as an example.

Robust filtering under nonrandom disturbances: The invariant ellipsoid approach

We present a simple and universal observer-based approach to solving the problem of robust filtering of unknown-but-bounded exogenous disturbances. The heart of this approach is the method of invariant ellipsoids. Application of this technique allows for a reformulation of the original problem in terms of linear matrix inequalities with reduction to semidefinite programming and one-dimensional optimization, which are easy to solve numerically. Continuous-time and discrete-time cases are studied in equal detail. The efficacy of the approach is demonstrated via the double pendulum example.

Sparse feedback in linear control systems

We consider a classical problem of linear static state feedback design in the linear system ẋ = Ax + Bu subject to a nonstandard constraint that the control vector u = Kx has as many zero components as possible.A simple approach to approximate solutions of such kind of nonconvex problems is proposed, which is based on convexification. The problem reduces to the minimization of special matrix norms subject to the constraints in the form of linear matrix inequalities (LMIs).The approach can be generalized to numerous problems of robust and optimal control that admit a “sparse” reformulation. To the best of our knowledge, both the solution and the problem formulation are new.

A nonfragile controller for suppressing exogenous disturbances

We suggest an approach to the problem of building a nonfragile controller, i.e., one that would allow for variation in its parameters, in order to suppress bounded exogenous disturbances in a linear dynamic system. We consider both continuous and discrete version of the problem and also its robust version. As an example, we study the control problem for a double oscillator.

Large deviations in linear control systems with nonzero initial conditions

Research in the transient response in linear systems with nonzero initial conditions was initiated by A.A. Feldbaum in his pioneering work [1] as early as in 1948. However later, studies in this direction have faded down, and since then, the notion of transient process basically means the response of the system with zero initial conditions to the unit step input. A breakthrough in this direction is associated with the paper [2] by R.N. Izmailov, where large deviations of the trajectories from the origin were shown to be unavoidable if the poles of the closed-loop system are shifted far to the left in the complex plane.In this paper we continue the analysis of this phenomenon for systems with nonzero initial conditions. Namely, we propose a more accurate estimate of the magnitude of the peak and show that the effect of large deviations may be observed for different root locations. We also present an upper bound on deviations by using the linear matrix inequality (LMI) technique. This same approach is then applied to the design of a stabilizing linear feedback aimed at diminishing deviations in the closed-loop system. Related problems are also discussed, e.g., such as analysis of the transient response of systems with zero initial conditions and exogenous disturbances in the form of either unit step function or harmonic signal.

New generalizations of the Petersen lemma

Consideration was given to the generalizations of the Petersen lemma of matrix uncertainty which is one of the main technical facilities for handling the structured matrix uncertainties used to advantage in various robust formulations of the problems of stabilization and control.

Quadratic stabilization of bilinear control systems

In this paper, a stabilization problem of bilinear control systems is considered. Using the linear matrix inequality technique and quadratic Lyapunov functions, an approach is proposed to the construction of the so-called stabilizability ellipsoid such that the trajectories of the closed-loop system emanating from any point inside this ellipsoid asymptotically tend to the origin. The approach allows for an efficient construction of nonconvex approximations to stabilizability domains of bilinear systems.The results are extended to robust formulations of the problem, where the system matrix is subjected to structured uncertainty.

Nonfragile controllers for rejection of exogenous disturbances

We consider design of stabilizing controllers for linear systems subjected to persistent exogenous disturbances. The performance index for the closed-loop system is taken in the form of the “size” of the invariant (bounding) ellipsoid for its output. The optimal controller that attains the minimal size is shown to be fragile in the sense that small variations of its coefficients lead to a dramatic degradation of the quality or even to the loss of stability. We show how to design a nonfragile stabilizing controller that tolerates variations of its parameters and yields much smaller ellipsoids. The approach is exemplified through a well-known benchmark control problem for a mechanical two-mass-spring system.