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Sampled-Data Filtered Repetitive Control With Nonlinear Systems: An Additive-State-Decomposition Method

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Filtered Repetitive Control with Nonlinear Systems

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Abstract

In the previous chapter, continuous-time filtered repetitive control (FRC, or filtered repetitive controller, which is also designated as FRC), was proposed using an additive-state-decomposition-based method for nonlinear systems. In this chapter, the sampled-data FRC is proposed for the same nonlinear system, but with a different problem under the additive-state-decomposition-based tracking control framework [1].

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Notes

  1. 1.

    The neutral-type system in the critical case is in the form \(\dot{x}\left( t\right) \) \(-\dot{x}\left( t-T\right) \) \(=A_{1}x\left( t\right) \) \(+A_{2}x\left( t-T\right) ,\) where \(x\left( t\right) \in \mathbb {R} ^{n},\) \(T>0,\) and \(A_{1},A_{2}\in \mathbb {R} ^{n\times n}\) [8, 9].

  2. 2.

    If \(\mathbf {A}\) is unstable, then, because of the observability of \(\left( \mathbf {A},\mathbf {c}^{\text {T}}\right) \) in Assumption 8.1, there always exists a vector \(\mathbf {p}\in \mathbb {R} ^{n}\) such that \(\mathbf {A}+\mathbf {pc}^{\text {T}}\) is stable, whose eigenvalues can be assigned freely. Then, (8.1) can be rewritten as \({\dot{\mathbf x}}=\left( \mathbf {A}+\mathbf {pc}^{\text {T}}\right) \mathbf {x}+\mathbf {b}u+\left( \varvec{\phi }\left( y\right) -\mathbf {p}y\right) +\mathbf {d}.\) Therefore, without loss of generality, \(\mathbf {A}\) is assumed to be stable.

  3. 3.

    Given that (8.16) and (8.12) are the only models existing in the design, the initial values \(\mathbf {x}_{s}\left( 0\right) \) and \({\hat{\mathbf x}}_{s}\left( 0\right) \) are both assigned by the designer and are all determinate.

  4. 4.

    Using (8.15) and (8.16), \(\hat{y}_{\text {p}}\) and \({\hat{\mathbf x}}_\mathrm{s}\ \)are obtained instead of the true state \(\mathbf {x}.\) Meanwhile, \(\mathbf {x}\) or \(\mathbf {x}_{\text {p}}\) is still unknown. We avoid solving \(\mathbf {x}\) using additive state decomposition.

  5. 5.

    Here, \(\mathscr {B}\left( \delta \right) \triangleq \left\{ \xi \in \mathbb {R} \left| \left| \xi \right| \le \delta \right. \right\} ,\) \(\delta >0;\) the notation \(x\left( t\right) \rightarrow \mathscr {B}\left( \delta \right) \) denotes \(\underset{y\in \mathscr {B}\left( \delta \right) }{\min }\) \(\left| x\left( t\right) -y\right| \rightarrow 0\ \)as \(t\rightarrow \infty .\)

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Authors and Affiliations

  1. School of Automation Science and Electrical Engineering, Beijing University of Aeronautics and Astronautics, Beijing, China

    Quan Quan & Kai-Yuan Cai

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  1. Quan Quan
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  2. Kai-Yuan Cai
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Correspondence to Quan Quan .

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Quan, Q., Cai, KY. (2020). Sampled-Data Filtered Repetitive Control With Nonlinear Systems: An Additive-State-Decomposition Method. In: Filtered Repetitive Control with Nonlinear Systems. Springer, Singapore. https://doi.org/10.1007/978-981-15-1454-8_8

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