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python拉氏变换拉氏变换（Laplace transform）是一种在控制理论、信号处理和其他工程领域中经常使用的数学工具。
它将一个函数f(t)转换为另一个函数F(s)，其中s是复变量。
在Python中，我们可以使用不同的库来进行拉氏变换的计算和分析，其中最常用的是SymPy和SciPy库。
在SymPy中，我们可以使用place_transform函数来进行拉氏变换的计算。
这个函数的语法是place_transform(f, t, s, noconds=False)，其中f是输入函数，t是输入函数的自变量，s是拉氏变换的新变量。
这个函数将返回拉氏变换后的结果。
例如：python.import sympy as sp.t, s = sp.symbols('t s')。
f = sp.exp(-at)。
F, _ = place_transform(f, t, s)。
这将计算出函数f(t)=exp(-at)的拉氏变换F(s)。
另外，在SciPy库中，我们也可以使用place_transform函数来进行拉氏变换的计算。
这个函数的语法是place_transform(func, t, s, side='two_sided')，其中func是输入函数，t是输入函数的自变量，s是拉氏变换的新变量。
这个函数将返回拉氏变换后的结果。
例如：python.import scipy.signal as signal.t = np.linspace(0, 1, 1000)。
func = np.exp(-at)。
s, Y = place_transform(func, t, s)。
这将计算出函数func=exp(-at)的拉氏变换Y(s)。
除了使用现有的库函数进行计算，我们还可以编写自己的算法来进行拉氏变换的计算。
拉氏变换的计算涉及到复杂的积分和代数运算，因此需要对数学知识有一定的了解才能进行准确的计算。
Absolute stabilityA system description that reveals whether a system is stable or not stable without consideration of other system attributes such as degree of stability. 312ActuatorThe device that causes the process to provide the output; the devices that provides the motive power to the process.All-pass networkA nonminimum phase system that passes all frequences with equal gain.BandwidthThe frequency at which the frequency response has declined 3dB from its low-frequency value.Block diagramUnidirectional, operational block that represents the transfer functions of the elements of the system.AutomationThe control of an industrial process by automatic means.Bode plotThe logarithm of magnitude of the transfer function is plotted versus the logarithm of \omiga, the frequency. The phase, \phi , of the transfer function is separately plotted versus the logarithm of the frequency.Canonical formA fundamental or basic form of the state variable model representation, including diagonal canonical form, Jordan canonical form.Characteristic equationThe relation formed by equating to zero the denominator of a transfer function.Closed loop feedback controlA system that uses a measurement of the output and compares it with the desired output.Closed loop transfer functionA ratio of the output signal to input signal for an interconnection of systems when all the feedback or feedforward loops have been closed or otherwise accounted for .Generally obtained by block diagram or signal flow graph reduction.CompensatorAn additional component or circuit that is inserted into the system to equalize or compensate for the performance deficiency.ComponentsThe parts,subsystems,or subassemblies that comprise a total system.Control systemAn interconnection of components forming a system configuration that will provide a desired response.Controllability matrixA linear system is controllable if and only if the controllable matrix Wc has full rank. For single input single output linear systems the system is controllable if and only if the determinantof the controllability matrix is nonzero.Controllable systemA system with unconstrained control input u that transfers any initial state x(0) to any other state x(t).DesignThe process of conceiving or inventing the forms, parts, and details of a system to achieve a reasoned purpose.DetectableA system in which the states that are unobservable are naturally stable.Differential equationAn equation including differentials of a function.Digital control systemA control system using digital signals and digital computer to control a process.Discrete time approximationAn approximation used to obtain the time response of a system based on the division of the time into small increments \delta t.Disturbance signalAn unwanted input signal that affects the system's output signal.Error signalThe difference between the desired output and actual output.EstimationThe difference between the actual state and the estimated state.Feedback signalA measure of the output of the system used as feedback tom control the system.Final valueThe value that output achieves after all the response have faded. Also referred to as the steady state.Frequency responseThe steady state response of a system to a sinusoidal inputsignal.Laplace transformA transformation of a function f(t) from the time domain into the complex frequency.Linear approximationAn approximate model that results in linear relationship between the output and the input of the device.Linear quadratic regulatorAn optimal controller designed to minimize the quadratic performance index.LinearizedMade linear or placed in a linear form.LocusA path or trajectory that is traced out as a parameter is changed.Mathematical modelsDescriptions of the behavior of a system using mathematics.Matrix exponential functionAn important matrix function defined as e^At=I+At+(At)^2+... that plays a role in in solution of linear constant coefficient differential equations.Minimum phaseAll the zeros of a transfer function lie in the left side of the s-plane.Negative feedbackThe output signal is fed back so that is subtracts from the input signal.Observability matrixA linear system is observable if and only if the observability matrix W o has full rank.Observable systemA system with an output that possesses a component due to each state variable.ObserverA dynamic system used to estimate the state of another dynamic system given knowledge of the system input and measurements of system outputs.Open loop control systemA system that utilizes a device to control the process without using feedback.Optimal control systemA system whose parameters are adjusted so that the performance index reaches an extremum value.OptimizationThe adjustment of the parameters to achieve the most favorable or advantageous design.Output equationThe algebraic equation that relates state vector x, and the input u, to the outputs y, through the relationship y=Cx+Du.OvershotThe amount of the system output response proceeds beyond the desired response.Parameter designA method of selecting one or two parameters using the root locus method.Peak timeThe time for a system to respond to a step input and rise to a peak response.Performance indexA quantitative measure of the performance of a system.Physical variablesThe state variables representing the physical variables of the system.PID controllerA widely used controller used in industry of the form G(s)=Kp+Ki/s+Kds.Pole placementA design methodology wherein the objective is to place the eigenvalues of the closed loop system in desired regions of the complex plane.PolesThe roots of the denominator polynomial(i.e. The roots of the characteristic equation) of the transfer function.Positive feedbackThe output signal is fed back so that it adds to the input signal.Reference inputThe input to a control system often representing the desired output.RobotA programmable computer integrated with a reprogrammable, multifunctional manipulator used for a variety of tasks.Robust controlA system that exhibits the desired performance in the presenceof significant plant uncertainty.Robust stability criterionA test for robustness with respect to multiplicative perturbations in which stability is guaranteed if |M|<|1+1/G|.Root locusThe locus or path of the roots traced out on the s plane as a parameter is changed.Routh Hurwitz criterionA criterion for determining the stability of a system by examing the characteristic equation of the transfer function.Sampled dataData obtained for the system variables only at discrete intervals, data obtained once every sampling period.SimulationA model of system used to investigate the behavior of a system by utilizing actual input signal.StabilityA performance measure of a system; a system is stable if all the poles of the transfer function have negative real parts.StabilizableA system in which the states that are not controllable are naturally stable.State differential equationThe differential equation for the state vector: x=Ax+bu.State transition matrix\phi (t), The matrix exponential function that describes the unforced response of the system.State variablesThe set of variables that describe the system.State vectorThe vector matrix containing all n state variable; x1, x2, ..., xn.State space representationA time domain model comprised of the state differential equation, x=Ax+Bu, y=Cx+Du.State variable feedbackWhen the control signal for the process is a direct function of all the state variables.Steady stateThe value that after all the transient constituents of the response have faded. Also referred to as final state.SynthesisThe process by which new physical configurations of are created. The combining of separate elements or devices to form a coherent whole.SystemAn interconnection of elements and devices for a desired purpose.Time delayA pure time delay, T so that events occurring at time t at onepoint in the system occurs at another point in the system at a later time (t+T).Time demainThe mathematical domain that incorporates the time response and the description of a system in terms of time t.Time varying control systemA system for which one or more parameters may varying with time.TradeoffThe need to make a judgment about how much compromise is made between conflicting criteria.Transfer functionUnder zero initial conditions, the ratio the Laplace transform of the output variables to the Laplace transform of the input variable.Transient responseThe constituent of the system response that disappears withtime.Unit impulseA test input consisting of an impulse of infinite amplitude and zero width, and having an area of unity, used to determine the impulse response.Zero order holdA mathematical model of a sample and data hold operation whose input output transfer function is represented byG=(1-e^-ts)/s.ZerosThe roots of numerator polynomial of the transfer function..。
