控制系统的相对稳定性

　　用劳斯判据
　　如果系统闭环特征根均在 s 左半平面，且和虚轴有一段距离，则系统有一定的 稳定裕量 。虚轴左移σ，令 z = s + σ ，
　　将 s = z - σ 代入系统特征式 ，
　　得到 z 的方程式，采用劳斯判据， 可知距离虚轴 σ 以右是否有根。
　　对于现代控制理论涉及的更广泛类型的系统，通常采用李 雅普诺夫稳定性判据 。
　　李雅普诺夫第一方法 又称间接法，它是通过系统状态方程的解来判断系统的稳定性。
　　李雅普诺夫第二方法 又称直接法，它不通过系统状态方程的解来判断系统的稳定性，而是借助李雅普诺夫函数对稳定性作出判断，是从 广义能量的观点进行稳定性分析 的。例如有阻尼的振动系统能量连续减小（总能量对时间的导数是负定的），系统会逐渐停止在平衡状态，系统是稳定的。
　　内模控制
　　这是把外部作用信号的动力学模型植入控制器来构成高精度反馈控制系统的一种设计。 内模原理 指出，任何一个能良好地抵消外部扰动或跟踪参考输入信号的反馈控制系统，其反馈回路必须包含一个与外部输入信号相同的动力学模型。这个内部模型称为内模。
　　内模原理的建立，为完全消除外部扰动对控制系统运动的影响，并使系统实现对任意形式参考输入信号的无稳态误差的跟踪提供了理论依据。内模原理已在线性定常系统和随动系统的综合设计中得到有效的应用。
　　系统在稳定的前提下，在控制器中包含一个纯积分环节可实现对阶跃信号的完全跟踪或抑制；包含二个纯积分环节可实现对斜坡信号的完全跟踪或抑制；均可作为 内模原理 的特例。
　　内模控制 方法是Garcia和Morari于1982年首先正式提出。内模控制除了具有能消除不可测干扰的优点外，还有简单、跟踪调节性能好、鲁棒性强等优点。内模控制进一步推广到非线性系统。
　　内模控制还和诸如预测控制、最优控制、自适应控制、模糊控制相结合，使其不断得到改进并广泛应用于工程实践中。
　　内模控制 （Internal Model Control,简记为IMC）是一种基于被控过程的内部模型的控制方法。
　　翻译成英文：
　　Relative stability of the control system
　　Routh criterion
　　If the closed-loop characteristic roots of the system are on the left half plane of s, and there is a certain distance from the imaginary axis, the system has a certain stability margin. The imaginary axis is shifted to the left by σ, let z=s+σ.
　　Substitute s=z-σ into the characteristic formula of the system,
　　Obtain the equation of z and use the Routh criterion to know whether there is a root to the right of the distance imaginary axis σ.
　　For the wider types of systems involved in modern control theory, the Lyapunov stability criterion is usually used.
　　The Lyapunov first method is also called the indirect method, which judges the stability of the system through the solution of the system state equation.
　　The second method of Lyapunov is also called the direct method. It does not judge the stability of the system through the solution of the system equation of state, but uses the Lyapunov function to judge the stability.
　　It analyzes the stability from the perspective of generalized energy. . For example, the energy of a damped vibration system decreases continuously (the derivative of total energy with respect to time is negatively definite), the system will gradually stop in the equilibrium state, and the system is stable.
　　Internal Model Control
　　This is a design in which the dynamic model of externally acting signals is implanted into the controller to form a high-precision feedback control system.
　　The internal model principle points out that for any feedback control system that can well offset external disturbances or track the reference input signal, its feedback loop must include a dynamic model that is the same as the external input signal. This internal model is called the internal model.
　　The establishment of the internal model principle provides a theoretical basis for completely eliminating the influence of external disturbance on the motion of the control system, and enabling the system to track the instability error of any form of reference input signal.
　　The principle of internal model has been effectively applied in the comprehensive design of linear steady system and servo system.
　　Under the premise of stability, the controller includes a pure integral link to achieve complete tracking or suppression of the step signal;
　　includes two pure integral links to achieve complete tracking or suppression of the ramp signal; both can be used as internal models A special case of the principle.
　　The internal model control method was first formally proposed by Garcia and Morari in 1982.
　　In addition to the advantages of eliminating undetectable interference, internal model control also has the advantages of simplicity, good tracking adjustment performance, and strong robustness. Internal model control is further extended to nonlinear systems.
　　Internal model control is also combined with predictive control, optimal control, adaptive control, and fuzzy control to make it continuously improved and widely used in engineering practice.
　　Internal Model Control (IMC) is a control method based on the internal model of the controlled process.
　　参考资料：百度
　　英文翻译：Google翻译
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