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Aizerman's Conjecture
In nonlinear control, Aizerman's conjecture or Aizerman problem states that a linear system in feedback with a sector nonlinearity would be stable if the linear system is stable for any linear gain of the sector. This conjecture was proven false but led to the (valid) sufficient criteria on absolute stability. Mathematical statement of Aizerman's conjecture (Aizerman problem) ''Consider a system with one scalar nonlinearity'' : \frac=Px+qf(e),\quad e=r^*x \quad x\in\mathbb R^n, ''where P is a constant n×n-matrix, q, r are constant n-dimensional vectors, ∗ is an operation of transposition, f(e) is scalar function, and f(0)=0. Suppose that the nonlinearity f is sector bounded, meaning that for some real'' k_1 and k_2 with k_1 |
Nonlinear Control
Nonlinear control theory is the area of control theory which deals with systems that are nonlinear, time-variant, or both. Control theory is an interdisciplinary branch of engineering and mathematics that is concerned with the behavior of dynamical systems with inputs, and how to modify the output by changes in the input using feedback, feedforward, or signal filtering. The system to be controlled is called the "plant". One way to make the output of a system follow a desired reference signal is to compare the output of the plant to the desired output, and provide feedback to the plant to modify the output to bring it closer to the desired output. Control theory is divided into two branches. Linear control theory applies to systems made of devices which obey the superposition principle. They are governed by linear differential equations. A major subclass is systems which in addition have parameters which do not change with time, called ''linear time invariant'' (LTI ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear Control
Nonlinear control theory is the area of control theory which deals with systems that are nonlinear, time-variant, or both. Control theory is an interdisciplinary branch of engineering and mathematics that is concerned with the behavior of dynamical systems with inputs, and how to modify the output by changes in the input using feedback, feedforward, or signal filtering. The system to be controlled is called the "plant". One way to make the output of a system follow a desired reference signal is to compare the output of the plant to the desired output, and provide feedback to the plant to modify the output to bring it closer to the desired output. Control theory is divided into two branches. Linear control theory applies to systems made of devices which obey the superposition principle. They are governed by linear differential equations. A major subclass is systems which in addition have parameters which do not change with time, called ''linear time invariant'' (LTI ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Attractor
In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain close even if slightly disturbed. In finite-dimensional systems, the evolving variable may be represented algebraically as an ''n''-dimensional vector. The attractor is a region in ''n''-dimensional space. In physical systems, the ''n'' dimensions may be, for example, two or three positional coordinates for each of one or more physical entities; in economic systems, they may be separate variables such as the inflation rate and the unemployment rate. If the evolving variable is two- or three-dimensional, the attractor of the dynamic process can be represented geometrically in two or three dimensions, (as for example in the three-dimensional case depicted to the right). An attractor can be a point, a finite set of points, a curve, a m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hidden Oscillation
In the bifurcation theory, a bounded oscillation that is born without loss of stability of stationary set is called a hidden oscillation. In nonlinear control theory, the birth of a hidden oscillation in a time-invariant control system with bounded states means crossing a boundary, in the domain of the parameters, where local stability of the stationary states implies global stability (see, e.g. Kalman's conjecture). If a hidden oscillation (or a set of such hidden oscillations filling a compact subset of the phase space of the dynamical system) attracts all nearby oscillations, then it is called a hidden attractor. For a dynamical system with a unique equilibrium point that is globally attractive, the birth of a hidden attractor corresponds to a qualitative change in behaviour from monostability to bi-stability. In the general case, a dynamical system may turn out to be multistable and have coexisting local attractors in the phase space. While trivial attractors, i.e. stable equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kalman's Conjecture
Kalman's conjecture or Kalman problem is a disproved conjecture on absolute stability of nonlinear control system with one scalar nonlinearity, which belongs to the sector of linear stability. Kalman's conjecture is a strengthening of Aizerman's conjecture and is a special case of Markus–Yamabe conjecture. This conjecture was proven false but led to the (valid) sufficient criteria on absolute stability. Mathematical statement of Kalman's conjecture (Kalman problem) In 1957 R. E. Kalman in his paper stated the following: If ''f''(''e'') in Fig. 1 is replaced by constants ''K'' corresponding to all possible values of ''f'''(''e''), and it is found that the closed-loop system is stable for all such ''K'', then it intuitively clear that the system must be monostable; i.e., all transient solutions will converge to a unique, stable critical point. Kalman's statement can be reformulated in the following conjecture: Consider a system with one scalar nonlinearity'' : \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hidden Oscillation
In the bifurcation theory, a bounded oscillation that is born without loss of stability of stationary set is called a hidden oscillation. In nonlinear control theory, the birth of a hidden oscillation in a time-invariant control system with bounded states means crossing a boundary, in the domain of the parameters, where local stability of the stationary states implies global stability (see, e.g. Kalman's conjecture). If a hidden oscillation (or a set of such hidden oscillations filling a compact subset of the phase space of the dynamical system) attracts all nearby oscillations, then it is called a hidden attractor. For a dynamical system with a unique equilibrium point that is globally attractive, the birth of a hidden attractor corresponds to a qualitative change in behaviour from monostability to bi-stability. In the general case, a dynamical system may turn out to be multistable and have coexisting local attractors in the phase space. While trivial attractors, i.e. stable equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
