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List of contents of this article:

• 1,How to improve the speed of the second-order system under damping
• 2, second-order differential system, how to eliminate q(t)
• 3、Features of the second-order simulation system
• 4,Champions League 5, Why is the second-order system often designed in control engineering

How to improve the speed of the second-order system in the case of lack of damping

1, which is a equilibrium point. The pole of the damping second-order system is oneThe equilibrium point, that is, the specific gravity of mass and damping, just balances the kinetic energy, which is a special state of a dynamic system, which can provide important information and insights to help understand the functions and behaviors of the system.

2. Delay time td: refers to the time required for the output response to reach 50% of the steady-state value for the first time. ( 2) Rising time tr: refers to the time required for the output response to rise to the steady-state value for the first time.

3. The speed of the second-order system is related to time. A form of second-order system control system is classified according to mathematical models. It is a system that can be represented as a second-order linear ordinary differential equation by a mathematical model. The form of the solution of the second-order system can be distinguished and divided by the denominator polynomial P(s) corresponding to the transfer function W(s).

second-order differential system, how to eliminate q(t)

second-order linear differential equations can actually be reduced by the differential The method is solved by order, but the process is slightly complicated, but the corresponding process can fully reflect the separation variable method.

The general solution formula of the second-order differential equation: y+py+qy=f(x), where p and q are real constants.The free term f(x) is a continuous function defined on interval I, that is, when y+py+qy=0, it is called the homogeneous linear differential equation of the second-order constant coefficient.

The general solution formula of second-order differential equations is as follows: The first one: from y2-y1=cos2x-sin2x is the solution corresponding to the homogeneous equation, cos2x and sin2x are both solutions of homogeneous equations, so the general solution of the equation that can be obtained is: y=C1cos2x+C2sin2 X-xsin2x.

Summary of the solution of second-order differential equations: The second-order differential equation can be transformed into first-order differential equations by substituting appropriate variables. Differential equations with this property are called descending differential equations, and the corresponding solution method is called descending order method.

According to: h+ah+bh = δ(t) Find the pulse response function h(t) of the system and discrete it, and the sampling interval is consistent with the interval of Z".

Characteristics of the second-order analog system

1. There are the following characteristics: energy saving 5% to 8%; reducing capacity to reduce investment in transformers, circuit breakers and cables; improve productivity and maintain continuous power supply; can dynamically filter out various harmonics, The harmonics in the unit can be completely absorbed; no resonance will be generated.

2. Characteristics of the second-order system: when ωωn, H(ω)≈1: when ωωm, H(ω)→0. The parameters affecting the dynamic characteristics of the second-order system are: natural frequency and damping ratio. Near w=ωn, the amplitude and frequency characteristics of the system are most affected by the damping ratio. When ω~wn, the system resonates.

3. The damping ratio of the second-order system determines its oscillation characteristics: at ξ 0, the step response diverges and the system is unstable; at ξ≥1, there is no oscillation, no overtuning, and the transition process is long; at 0ξ1, there is oscillation, the smaller the ξ, the more serious the oscillation, but the faster the response; at ξ=0, isoamplitude oscillation occurs.

4. The system has different characteristics. For example, sometimes the system can automatically judge the order according to the feedback information of the system. Generally speaking, the order of the system can be judged according to the characteristics of the system, such as the characteristics of the first-order system, the characteristics of the second-order system, etc.

Under what circumstances is the second-order system unstable

Changing the gain in the second-order system will not affect dynamic variables such as overmodulation of the system, but only the response time of the system.

When a second-order system (including all linear systems) has a right half-plane pole in the s-plane, the system is unstable. If you want to form an agitation link, the damping coefficient of the second-order system must be 0, or its poles must be on the virtual axis.

For a second-order system, when the damping ratio is less than 1, the system is stable; when the damping ratio is greater than 1, the system is stable; but when the damping ratio is equal to 1, the system is in a critical stable state, and stability problems caused by boundary conditions may occur. Therefore, the selection and adjustment of the damping ratio can be used to improve the stability of the system.

The following statement about the second-order system is correct () A. Described by second-order differential equations B. The damping coefficient is less than 0 and the system is unstable C. The second-order system after damping can be regarded as a series of two first-order systems D.There must be supertone E in the step response. In the case of under-damping, the smaller the damping coefficient, the stronger the shock.

Judgement conditions: when the system is stable: amplitude margin 1: phase angle margin 0; the larger the amplitude margin and phase angle margin, the more stable the system. When the critical boundary of the system is stable: amplitude margin = 1 phase angle margin = 0; when the system is stable: amplitude margin 1 phase angle margin 0.

Why is the second-order system often designed in control engineering

In control engineering, type 1 system is often called a first-order no-static system, type 2 system is a second-order no-static system, and type 3 system is The third-order static-free system means that they are static-free systems for unit step jump, unit slope and unit acceleration respectively.If the deviation is eliminated, it is a static-aberration-free system, which can still run and has output.

The transition process of the second-order system has a great impact on the stability and control performance of the system. Steady-state properties The steady-state properties of second-order systems are usually characterized by the steady-state error coefficient and the steady-state response curve. The steady-state error coefficient is the difference between output and input, including static error coefficient, dynamic error coefficient and speed error coefficient, etc.

The situation of two real roots corresponds to two serial first-order systems. If both roots are negative, it is a stable situation of non-periodic convergence. When a1=0, a20, that is, a pair of conjugated virtual roots, will cause a frequency-fixed isoamplitude oscillation, which is a manifestation of system instability.

The optimal system of second-order engineering means that the damping ratio of the system is 0 under the consideration of equilibrium stability and speed.707. The design goal is to limit the overmodulation and shorten the adjustment time. According to the relationship between performance and parameters, perfect stability and fast response cannot be achieved at the same time.

In addition, according to the online information query, the initial meaning of the second-order control system is very common in the practice of control engineering, such as the DC motor controlled by the armature, the RLC network and the mechanical displacement system composed of the spring-mass-damper.