For the synthesis problem for an invariant tracking system for a nonlinear automated control object under incomplete measurements, we develop a decompositional procedure of observer state synthesis with sigmoidal correcting influences in order to get current estimates of the unmeasured state variables and existing uncertainties. This observer in the pre-limit situation possesses the advantages of an observer with discontinuous correcting influences operating in sliding mode; in particular, it lets us estimate external influences without introducing their dynamical model. Unlike a sliding mode observer whose order has been extended due to filters over discontinuous correcting influences, the dimension of this observer equals the dimension of the control object, and in a microprocessor implementation this observer ensures better quality (smoothness) of the signals being estimated.

For the problem of providing invariance in outputs of the system with external perturbations not belonging to the control space, a solution was proposed based on the effect of vibrolinearization of the relay elements. The developed relay vortex algorithms of control were used in the closed-loop system for generation of damping oscillating modes with unlimited growth in oscillation frequency which enabled relay vibrolinearization. At that, realized was the theoretically infinite coefficient of linearization enabling asymptotic invariance of the output to a wide class of external perturbations.

We study the static state feedback design problem for linear systems under given constraints on the magnitude of the control input. A quadratic performance index is constructed such that it is optimized by the synthesized control. The issues of fragility of the controller are analyzed; this phenomenon relates to the loss of the stabilizability property of a controller under small variations of its parameters. Linear matrix inequalities are used as a main technical tool.

We consider a classical problem of linear static state feedback design in the linear system subject to a nonstandard constraint that the control vector 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.

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.