Abstract
This chapter deals with the time response of continuous-time linear time-invariant dynamic system. Seven interactive tools are included, to understand the time response of linear systems of different orders and complexities. Moreover, an interactive tool has also been incorporated to adjust linear models from experimental data.
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Notes
- 1.
The input of the systems and their representative models will be represented by u(t) and the corresponding outputs by y(t) in all cases.
- 2.
This is a logical condition in linear causal systems and in linearized systems where the variables represent deviations from equilibrium or operating point. It is assumed that the reader has prior knowledge of the Laplace transform and its properties, which are usually included in classical automatic control texts. An excellent summary of the most common techniques for analysis in the complex variable domain (Laplace transform, Fourier series expansion, and Fourier transform) can be found in Appendix B of Ref. [5].
- 3.
The convolution integral can also be used in time-varying systems, while Laplace transforms cannot be used for such systems.
- 4.
Always that there is no pole–zero cancellation (proof omitted).
- 5.
- 6.
Notice that \(\beta _\ell \) is used here to represent the time constant associated with each of the m zeros of the transfer function to simplify the writing. The usual notation uses \(\tau _z\) to represent the time constant associated to a zero and \(\tau _p\) in the case of a pole.
- 7.
Notice that the interactive tools using this format introduce the terms including complex conjugate poles/zeros in the form \(\frac{\omega _n^2}{s^2+2\zeta \omega _n s + \omega _n^2}\), also having unit static gain, as they are the same as those included in Eq. (3.9).
- 8.
It should only be applied to stable systems [7], as in other cases the output is unbounded.
- 9.
With multiple roots \( y_h(t)=\sum _{i=1}^n c_i(t) e^{p_i t}\), where \(c_i(t)\) is a polynomial with degree less than the multiplicity of the root \(p_i\) [1].
- 10.
The step response describes the relationship between an input that changes from zero to a constant value abruptly (a step input) and the corresponding output [1].
- 11.
Value that the output of the system reaches after all elements of the transient response have declined [4].
- 12.
In the case of linearized systems, it is the ratio between the change experienced by the output and the amplitude of the step input, considering their steady-state values.
- 13.
As treated in Chap. 7, the equivalent transfer function of two systems in series \(G_1(s)\) and \(G_2(s)\) is the product \(G_1(s)G_2(s)\), as a consequence of applying the properties of the Laplace transform.
- 14.
In such case, \(Y(s)=k\omega _n^3/(s^2+\omega _n^2)^2\), providing \(y(t)=(k/2)(\sin {(\omega _n t)}-\omega _n t \cos {(\omega _n t)})\) from the application of the inverse Laplace transform (line 15 in Table 3.2).
- 15.
A concept that is discussed in Chap. 4 related to the speed of response of the (usually closed-loop) system (the higher the bandwidth, the faster the response).
- 16.
Subscript 1 is used both in the output and the transfer function because it facilitates the explanation.
- 17.
Property of derivation in t of the Laplace transform (Table 3.1).
- 18.
Subscript 2 is used both in the output and the transfer function because it facilitates the explanation.
- 19.
Other indices can be used as the integral of absolute error (IAE) defined by \(\int _{t_0}^{t_f} |e_m(t)| dt\) or the integral of time multiplied by absolute error (ITAE), given by \(\int _{t_0}^{t_f} t |e_m(t)| dt\). In practice, these indices are not only used for model identification purposes, but for measuring the performance of closed-loop systems. ISE discriminates between excessively overdamped and excessively underdamped systems. The minimum value of the integral occurs for a compromise value of damping. Therefore, this criterion gives a higher weighting when there is a large error and a soft weighting when the error is small. IAE is one of the easiest implementable indices. It provides optimal results when dealing with reasonable damping and satisfactory transient response. However, this performance index is not easy to evaluate by analytical means. ITSE has the characteristic that with a unit step input the response has a large initial error because it has a small weight, but as time progresses, the error is penalized more heavily.
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Guzmán, J.L., Costa-Castelló, R., Berenguel, M., Dormido, S. (2023). Time Response. In: Automatic Control with Interactive Tools. Springer, Cham. https://doi.org/10.1007/978-3-031-09920-5_3
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