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This work is proposed to continue the discussion of the problems, theoretical foundations and practical features of the construction and synthesis of robust control systems with high gain, allowing us to control multidimensional nonlinear dynamic objects of high dimensional with functional uncertainties. If problems could not be solved at the level where they appeared, it is necessary to rise the level of understanding of the laws of nature, or in the words of master Lui-Shi Chun Qiu (China, 3rd century BC): "The boy of five chi growth leads the bull by the bridle and the bull obeys him in everything. This is because the person in this case follows naturalness" (the laws of nature). The judo philosophy ("soft way") is based on the principles of using the power and energy of the opponent to achieve victory. The purpose of this work is to demonstrate the theoretical aspects and practical features of the methods of synthesis of optimal control systems by the criterion of maximum reproduction accuracy using the example of robust systems, which allow to control dynamic objects with functional uncertainties, including unstable objects, no minimal-phase objects, neutral objects and objects with differentiation properties. The simplicity (at the level of the engineer) and universality, mathematical rigor and physical validity of this approach is based on the judo philosophy: suppressing the dynamics of a functionally uncertain object and external disturbances by the infinitely large gain with the finite control signal and at the same time maintaining sustainability. Theoretically exhaustive solution of the problem of robust control is given by the idea of constructing systems that are stable with an unlimited increase of the gain coefficient. The sustainability properties are valid for optimal systems that were synthesized using quadratic quality functionals that do not explicitly depend on the control signal, and using a restriction on the control signal. It is significant that in contrast to continuous systems with un-measurable disturbances and not well known control object (in which the conditions of invariance imply the use of infinitely large gain), in relay (discontinuous) systems the equivalent effect is achieved with the help of finite control signal. A nice bonus is the highest accuracy which leads to mathematically zero error of regulation, thus all error coefficients (of position, speed, acceleration, acceleration derivative, etc.) is also equal to zero in the presence of external and internal interferences. In fact, the optimal accuracy control system is equivalent to a system with astatism of the n-th order: the regulator contains n serial connected integrators.

The problem of the synthesis of robust control systems with a high gain and, in particular, optimal by the criterion of quick action, which allow optimal control by the accuracy of regulation of multidimensional non-linear dynamic objects with functional uncertainties, is discussed. A method is proposed for the analytical construction of optimal control systems by the criterion of quick action for a wide class of multidimensional nonlinear dynamic objects with functional uncertainties, unstable objects; no minimal-phase objects, neutral object and objects with differentiation properties. Simplicity and universality, mathematical rigor and physical validity of this method consists in usingR.R Be llman’s method and decomposing the optimal by the criterion of quick action problem into a series of simple first-order simple problems of the same type. A theoretically comprehensive solution to the robust control problem is given by the idea of constructing systems that are stable with an unlimited increase in gain. In this case, optimal systems have stability properties. Such systems are synthesized using quadratic quality functionals that are not explicitly dependent on the control signal and the restriction on the control signal. It is significant that, in contrast to continuous systems with unmeasurable perturbations and a little-known object, in which the conditions of invariance require the use of infinitely large gains, in relay (discontinuous) systems, the equivalent effect is achieved using finite control actions. Since the performance problem is a particular problem of the accuracy of reproducing the input action on the control object, the established control error (including all error coefficients: by position, speed, acceleration, jerk, etc.) is theoretically strictly equal to zero if external and internal interference, acting only on the control object, but not on the control system, including sensors of state variables of the control object or the input signal of the task. However, due to the inertia of the object, there can be no talk of accuracy in the transient process of working out the input signal of the task, even if it is optimal in terms of the criterion of fast action.

The problem of synthesis of an optimal controlled system with a quadratic quality criterion having an infinite number of switching points at a finite time inter val is discussed. In the theor y of optimal control, this phenomenon is called the "Fuller phenomenon". For more than 60 years, the Fuller problem has been very attractive, relevant, and still unsolved, especially for non-linear multidimensional dynamical systems of high order, and even more so, with obtaining a solution in an explicit analytical form for practical implementation in a control system.The purpose of this work is to demonstrate the theoretical aspects and practical features of the method of synthesis of optimal control systems by the fast acting criterion by the example of solving problems related to the Fuller phenomenon.When solving these problems, we use in the classical variations calculus and the Pontryagin maximum principle of the method of introducing a new additional phase variable into consideration, which is defined to the integral quality criterion and expands the original phase vector of the object. As a result, if the best optimal control in terms of fast acting for the control object is known then this technique makes it ver y easy to get a worse optimal control in terms of accuracy by including the Fuller accuracy criterion in the dynamics of the control object. It should be note that an important acquisition here is to increase the accuracy to the optimal value and reduce the established control error to zero, with all error coefficients (in position, speed, acceleration, jerk, etc.) equal to zeroin the presence of external and internal interference.Statements and solutions of the classical and modified Fuller problems are presented. As illustrative examples, we consider the traditional problems of the synthesis of optimal control in terms of speed, solved in well-known methods.