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The secrets of nonlinear control theory: Why are real-world systems so complex?
In the field of control theory, nonlinear control theory is undoubtedly a challenging branch. This theory deals primarily with systems that are nonlinear, time-varying, or both. Control theory is an interdisciplinary subject that spans engineering and mathematics and is concerned with the behavior of dynamic systems and how the output can be modified to achieve a desired effect through feedback, feedforward, or signal filtering.
The "plant" in the control system is the object that needs to be controlled. This is done by comparing the output to a desired reference signal and passing a feedback signal back to the plant, which adjusts its output until it approaches the desired result.
Control theory can be divided into two main branches: linear control theory and nonlinear control theory. Linear control theory focuses on systems that obey the superposition principle, which are usually described by linear differential equations. Nonlinear control theory covers a wider range of system types because almost all real control systems are nonlinear. These complex nonlinear systems are often described by nonlinear differential equations and require more rigorous mathematical techniques to handle.
Some characteristics of nonlinear systems include not obeying the superposition principle, having multiple isolated equilibrium points, and exhibiting limit cycles, bifurcations, or chaotic behaviors.
The techniques for dealing with these nonlinear systems include: describing function method, phase plane method, Lyapunov stability analysis, etc. Control design techniques have also evolved and can be divided into several categories. Some techniques attempt to treat the system as linear within a limited operating range and apply existing linear design techniques, while others attempt to use auxiliary nonlinear feedback to make the system linear for control design purposes.
For example, a heating system controlled by a thermostat is a typical nonlinear control system. In this heating setting, the system shuts down once the set "off" temperature is reached. This on-off response makes the entire system unable to control temperature as accurately as a linear device. When the temperature drops below the on setting, the heater starts, energy input causes the temperature to rise, and when it reaches the off setting again it turns off again, forming a continuous cycle. This phenomenon of fluctuation around the ideal temperature is called a limit cycle and shows the characteristics of a nonlinear control system.
Even if the plant itself is linear, nonlinear controllers can exhibit easier implementation, faster speed, higher accuracy, or lower control energy, making their design process more valuable.
The analysis and control of nonlinear systems present many challenges, but these challenges also promote the development of related technologies. Since the complexity of nonlinear systems increases the difficulty of designing control systems, researchers often use digital simulation languages to simulate the operational modes of these systems on computers to explore their behavior.
In the analysis of nonlinear feedback systems, Lur'e's problem is one of the early important analytical tools. This problem explores the stability of systems composed of linear and nonlinear feedback. Understanding how to bridge the gap between linearity and nonlinearity can help engineers better design effective control systems.
In addition to the Lur'e problem, there are also important theoretical results in the study of absolute stability, such as the circular criterion and the Popov criterion. These theories not only show the complexity of nonlinear control, but also reveal the wonderful behavior of the system. Place. These studies not only enrich our understanding of nonlinear systems, but also promote the development of corresponding technologies.
In summary, the challenges faced by nonlinear control theory are as diverse and complex as the complexity of the real world. So can we find a more intuitive and simple way to understand and design the control methods of these nonlinear systems? ?
