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Small-gain Theorem
In nonlinear systems, the formalism of input-output stability is an important tool in studying the stability of interconnected systems since the gain of a system directly relates to how the norm of a signal increases or decreases as it passes through the system. The small-gain theorem gives a sufficient condition for finite-gain \mathcal stability of the feedback connection. The small gain theorem was proved by George Zames in 1966. It can be seen as a generalization of the Nyquist stability criterion, Nyquist criterion to non-linear time-varying MIMO systems (systems with multiple inputs and multiple outputs). ''Theorem''. Assume two stable systems S_1 and S_2 are connected in a feedback loop, then the closed loop system is input-output stable if \, S_1\, \cdot \, S_2\, < 1 and both and are stable by themselves. (This norm is typically the H-infinity_methods_in_control_theory, -norm, the size of the largest s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Feedback Connection Between Two Systems
Feedback occurs when outputs of a system are routed back as inputs as part of a chain of cause-and-effect that forms a circuit or loop. The system can then be said to ''feed back'' into itself. The notion of cause-and-effect has to be handled carefully when applied to feedback systems: History Self-regulating mechanisms have existed since antiquity, and the idea of feedback had started to enter economic theory in Britain by the 18th century, but it was not at that time recognized as a universal abstraction and so did not have a name. The first ever known artificial feedback device was a float valve, for maintaining water at a constant level, invented in 270 BC in Alexandria, Egypt. This device illustrated the principle of feedback: a low water level opens the valve, the rising water then provides feedback into the system, closing the valve when the required level is reached. This then reoccurs in a circular fashion as the water level fluctuates. Centrifugal governors were u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear Systems
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems. Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one. In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear combination of the un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George Zames
George Zames (January 7, 1934 – August 10, 1997) was a Polish-Canadian control theorist and professor at McGill University, Montreal, Quebec, Canada. Zames is known for his fundamental contributions to the theory of robust control, and was credited for the development of various well-known results such as small-gain theorem, passivity theorem, circle criterion in input–output form, and most famously, H-infinity methods. Biography Childhood George Zames was born on January 7, 1934 in Łódź, Poland to a Jewish family. Growing up in Warsaw, Zames and his family escaped the city at the onset of World War II, and moved to Kobe (Japan), through Lithuania and Siberia, and finally to the Anglo-French International Settlement in Shanghai. Zames indicated later that he and his family owe their lives to the transit visa provided by the Japanese Consul to Lithuania, Chiune Sugihara. In Shanghai, Zames continued his schooling, and in 1948, the family emigrated to Canada. Educati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nyquist Stability Criterion
In control theory and stability theory, the Nyquist stability criterion or Strecker–Nyquist stability criterion, independently discovered by the German electrical engineer at Siemens in 1930 and the Swedish-American electrical engineer Harry Nyquist at Bell Telephone Laboratories in 1932, is a graphical technique for determining the stability of a dynamical system. Because it only looks at the Nyquist plot of the open loop systems, it can be applied without explicitly computing the poles and zeros of either the closed-loop or open-loop system (although the number of each type of right-half-plane singularities must be known). As a result, it can be applied to systems defined by non- rational functions, such as systems with delays. In contrast to Bode plots, it can handle transfer functions with right half-plane singularities. In addition, there is a natural generalization to more complex systems with multiple inputs and multiple outputs, such as control systems for airplan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MIMO Systems
In radio, multiple-input and multiple-output, or MIMO (), is a method for multiplying the capacity of a radio link using multiple transmission and receiving antennas to exploit multipath propagation. MIMO has become an essential element of wireless communication standards including IEEE 802.11n (Wi-Fi 4), IEEE 802.11ac (Wi-Fi 5), HSPA+ (3G), WiMAX, and Long Term Evolution (LTE). More recently, MIMO has been applied to power-line communication for three-wire installations as part of the ITU G.hn standard and of the HomePlug AV2 specification. At one time, in wireless the term "MIMO" referred to the use of multiple antennas at the transmitter and the receiver. In modern usage, "MIMO" specifically refers to a class of techniques for sending and receiving more than one data signal simultaneously over the same radio channel by exploiting multipath propagation. Additionally, modern MIMO usage often refers to multiple data signals sent to different receivers (with one or more receive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
H-infinity Methods In Control Theory
''H''∞ (i.e. "''H''-infinity") methods are used in control theory to synthesize controllers to achieve stabilization with guaranteed performance. To use ''H''∞ methods, a control designer expresses the control problem as a mathematical optimization problem and then finds the controller that solves this optimization. ''H''∞ techniques have the advantage over classical control techniques in that ''H''∞ techniques are readily applicable to problems involving multivariate systems with cross-coupling between channels; disadvantages of ''H''∞ techniques include the level of mathematical understanding needed to apply them successfully and the need for a reasonably good model of the system to be controlled. It is important to keep in mind that the resulting controller is only optimal with respect to the prescribed cost function and does not necessarily represent the best controller in terms of the usual performance measures used to evaluate controllers such as settling time, ener ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Input-to-state Stability
Input-to-state stability (ISS)Eduardo D. Sontag. Mathematical Control Theory: Finite-Dimensional Systems. Springer-Verlag, London, 1998Hassan K. Khalil. Nonlinear Systems. Prentice Hall, 2002.Andrii Mironchenko. Input-to-state stability Springer, 2023. is a stability notion widely used to study stability of nonlinear with external inputs. Roughly speaking, a control system is ISS if it is globally asymptotically stable in the absence of external inputs and if its trajectories are bounded by a function of the size of the input for all sufficiently large times. The importance of ISS is due to the fact that the concept has bridged the gap between [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
