經典控制理論

經典控制理論（Classical control theory）是控制理論中的分支，以拉普拉斯变换為控制系統的分析工具，探討有輸入的动力系统之特性、以及反馈對系統特性的影響。

控制理論中常見的目標是要控制一系統（稱為受控體），使其輸出可以依照一希望的控制信號（稱為參考信號，可能是定值或是變動量）。為了實現此一目的，會設計控制器來監控輸出，並且和參考信號比較。實際輸出和參考信號的差（稱為誤差信號）會反馈到控制器中，再由控制器產生受控體的輸入信號，使受控體的實際輸出接近參考信號。

經典控制理論主要處理线性时不变的單一輸入單一輸出系統，可以計算系統輸入信號及輸出信號的拉普拉斯变换，而系統的传递函数和輸入信號及輸出信號的拉普拉斯变换有關。

反馈

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To overcome the limitations of the 開迴路控制器, classical control theory introduces 反馈. A closed-loop controller uses feedback to control states or 负反馈s of a 动力系统. Its name comes from the information path in the system: process inputs (e.g., 電壓 applied to an 电动机) have an effect on the process outputs (e.g., speed or torque of the motor), which is measured with 传感器s and processed by the controller; the result (the control signal) is "fed back" as input to the process, closing the loop.

Closed-loop controllers have the following advantages over 開迴路控制器s:

• disturbance rejection (such as hills in the cruise control example above)
• guaranteed performance even with 数学模型 uncertainties, when the model structure does not match perfectly the real process and the model parameters are not exact
• 不稳定性 processes can be stabilized
• reduced sensitivity to parameter variations
• improved reference tracking performance

In some systems, closed-loop and open-loop control are used simultaneously. In such systems, the open-loop control is termed 前馈控制 and serves to further improve reference tracking performance.

A common closed-loop controller architecture is the PID控制器.

經典控制及現代控制

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A Physical system can be modeled in the "time domain", where the response of a given system is a function of the various inputs, the previous system values, and time. As time progresses, the state of the system and its response change. However, time-domain models for systems are frequently modeled using high-order differential equations which can become impossibly difficult for humans to solve and some of which can even become impossible for modern computer systems to solve efficiently.

To counteract this problem, classical control theory uses the 拉普拉斯变换 to change an Ordinary Differential Equation (ODE) in the time domain into a regular algebraic polynomial in the transform domain. Once a given system has been converted into the transform domain it can be manipulated with greater ease.

現代控制理论, instead of changing domains to avoid the complexities of time-domain ODE mathematics, converts the differential equations into a system of lower-order time domain equations called 状态空间, which can then be manipulated using techniques from linear algebra.

拉普拉斯变换

主条目：拉普拉斯变换

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Classical control theory uses the Laplace transform to model the systems and signals. The Laplace transform is a frequency-domain approach for continuous time signals irrespective of whether the system is stable or unstable. The Laplace transform of a 函数 f(t), defined for all 实数s t ≥ 0, is the function F(s), which is a unilateral transform defined by

$F(s)=\int _{0}^{\infty }e^{-st}f(t)\,dt$

where s is a 复数 (数学) frequency parameter

$s=\sigma +i\omega$, with real numbers σ and ω.

閉迴路傳遞函數

更多信息：閉迴路傳遞函數

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The output of the system y(t) is fed back through a sensor measurement F to the reference value r(t). The controller C then takes the error e (difference) between the reference and the output to change the inputs u to the system under control P. This is shown in the figure. This kind of controller is a closed-loop controller or feedback controller.

This is called a single-input-single-output (SISO) control system; MIMO (i.e., Multi-Input-Multi-Output) systems, with more than one input/output, are common. In such cases variables are represented through 向量 instead of simple 标量 (数学) values. For some 分佈式參數系統 the vectors may be infinite-向量空间的维数 (typically functions).

A simple feedback control loop

If we assume the controller C, the plant P, and the sensor F are 線性關係 and 时不变系统 (i.e., elements of their 传递函数 C(s), P(s), and F(s) do not depend on time), the systems above can be analysed using the 拉普拉斯变换 on the variables. This gives the following relations:

$Y(s)=P(s)U(s)\,\!$

$U(s)=C(s)E(s)\,\!$

$E(s)=R(s)-F(s)Y(s).\,\!$

Solving for Y(s) in terms of R(s) gives

$Y(s)=\left({\frac {P(s)C(s)}{1+F(s)P(s)C(s)}}\right)R(s)=H(s)R(s).$

The expression $H(s)={\frac {P(s)C(s)}{1+F(s)P(s)C(s)}}$ is referred to as the closed-loop transfer function of the system. The numerator is the forward (open-loop) gain from r to y, and the denominator is one plus the gain in going around the feedback loop, the so-called loop gain. If $|P(s)C(s)|\gg 1$, i.e., it has a large 范数 with each value of s, and if $|F(s)|\approx 1$, then Y(s) is approximately equal to R(s) and the output closely tracks the reference input.

PID控制器

更多信息：PID控制器

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The PID控制器 is probably the most-used feedback control design. PID is an initialism for Proportional-Integral-Derivative, referring to the three terms operating on the error signal to produce a control signal. If u(t) is the control signal sent to the system, y(t) is the measured output and r(t) is the desired output, and tracking error $e(t)=r(t)-y(t)$, a PID controller has the general form

$u(t)=K_{P}e(t)+K_{I}\int e(t){\text{d}}t+K_{D}{\frac {\text{d}}{{\text{d}}t}}e(t).$

The desired closed loop dynamics is obtained by adjusting the three parameters $K_{P}$, $K_{I}$ and $K_{D}$, often iteratively by "tuning" and without specific knowledge of a plant model. Stability can often be ensured using only the proportional term. The integral term permits the rejection of a step disturbance (often a striking specification in 过程控制). The derivative term is used to provide damping or shaping of the response. PID controllers are the most well established class of control systems: however, they cannot be used in several more complicated cases, especially if multiple-input multiple-output systems (MIMO) systems are considered.

Applying Laplace transformation results in the transformed PID controller equation

$u(s)=K_{P}e(s)+K_{I}{\frac {1}{s}}e(s)+K_{D}se(s)$

$u(s)=\left(K_{P}+K_{I}{\frac {1}{s}}+K_{D}s\right)e(s)$

with the PID controller transfer function

$C(s)=\left(K_{P}+K_{I}{\frac {1}{s}}+K_{D}s\right).$

There exists a nice example of the closed-loop system discussed above. If we take

PID controller transfer function in series form

$C(s)=K\left(1+{\frac {1}{sT_{i}}}\right)(1+sT_{d})$

1st order filter in feedback loop

$F(s)={\frac {1}{1+sT_{f}}}$

linear actuator with filtered input

$P(s)={\frac {A}{1+sT_{p}}}$, A = const

and insert all this into expression for closed-loop transfer function H(s), then tuning is very easy: simply put

$K={\frac {1}{A}},T_{i}=T_{f},T_{d}=T_{p}$

and get H(s) = 1 identically.

For practical PID controllers, a pure differentiator is neither physically realisable nor desirable due to amplification of noise and resonant modes in the system. Therefore, a phase-lead compensator type approach is used instead, or a differentiator with low-pass roll-off.

工具

經典控制理論中會使用許多工具來分析系統，並且設計控制器。工具包括有根軌跡圖、奈奎斯特稳定判据、波德圖、增益裕度及相位裕度等。

相關條目

• 状态空间