Filtered Repetitive Control with Nonlinear Systems: An Actuator-Focused Design Method

Abstract

The internal model principle (IMP) was first proposed by Francis and Wonham [2, 3]. It states that if any exogenous signal can be regarded as the output of an autonomous system, then the inclusion of this signal model, namely, internal model, in a stable closed-loop system can assure asymptotic tracking or asymptotic rejection of the signal. Until now, to the best of the authors’ knowledge, there exist at least two viewpoints on IMP. In the early years, for linear time-invariant (LTI) systems, IMP implies that the internal model is to supply closed-loop transmission zeros which cancel the unstable poles of the disturbances and reference signals. This is called cancelation viewpoint here and only works for problems able to be formulated in terms of transfer functions. In the mid-1970s, Francis and Wonham proposed the geometric approach [4] to design an internal model controller [2, 3]. The purpose of internal models is to construct an invariant subspace for the closed-loop system and make the regulated output zero at each point of the invariant subspace. This is called geometrical viewpoint here.

9.6 Appendix

9.1.1 9.6.1 Proof of Theorem 9.3

Choose a Lyapunov functional as

$$ V\left( \mathbf {z}_{t}\right) =\frac{1}{2}\mathbf {z}^{\text {T}}\left( t\right) \left( \mathbf {PE}+\mathbf {E}^{\text {T}}\mathbf {P}\right) \mathbf {z}\left( t\right) + {\displaystyle \int \nolimits _{-T}^{0}} \mathbf {z}_{t}^{\text {T}}\left( \theta \right) \mathbf {Q\mathbf {z}} _{t}\left( \theta \right) \text {d}\theta . $$

Taking its derivative along (9.13) yields

$$\begin{aligned} \dot{V}\left( \mathbf {z}_{t}\right)&=\mathbf {z}^{\text {T}}\left( t\right) \left( \mathbf {PE}+\mathbf {E}^{\text {T}}\mathbf {P}\right) \dot{\mathbf {z}}\left( t\right) +\mathbf {z}^{\text {T}}\left( t\right) \mathbf {Qz}\left( t\right) -\mathbf {z}^{\text {T}}\left( t-T\right) \mathbf {Qz}\left( t-T\right) \nonumber \\&\le -\lambda _{1}\left\| \mathbf {z}\left( t\right) \right\| ^{2}-\lambda _{1}\left\| \mathbf {z}\left( t-T\right) \right\| ^{2}\le 0, \end{aligned}$$

(9.41)

where (9.18) is utilized. Based on (9.41), two conclusions are proven in the following:

(i) $\mathbf {z}\left( t\right) =\mathbf {0}$ in (9.13) is globally exponentially stable when $\mathbf {A}_{\varepsilon }>0$. For system (9.13), there exist $\gamma _{1},\gamma _{2},\rho >0$ such that

$$\begin{aligned} \gamma _{1}\left\| \mathbf {z}\left( t\right) \right\| ^{2}&\le V\left( \mathbf {z}_{t}\right) \le \gamma _{2}\left\| \mathbf {z}\left( t\right) \right\| ^{2}+\rho {\displaystyle \int \nolimits _{-T}^{0}} \left\| \mathbf {\mathbf {z}}_{t}\left( \theta \right) \right\| ^{2}\text {d}\theta \\ \dot{V}\left( \mathbf {z}_{t}\right)&\le -\lambda _{1}\left\| \mathbf {z}\left( t\right) \right\| ^{2}. \end{aligned}$$

According to [32, Theorem 1], (9.13) is globally exponential convergence. Furthermore, $\mathbf {z}\left( t\right) =\mathbf {0}$ is globally exponentially stable according to the stability definition.

(ii) If $\sup _{t\in \left[ 0,T\right] }\left\| \left( \mathbf {I}_{m}+\mathbf {L}_{1}\left( t\right) \mathbf {D}\left( t\right) \right) ^{-1}\right\| <1,$ then $\mathbf {z}\left( t\right) =\mathbf {0}$ in (9.13) is globally exponentially stable when $\mathbf {A}_{\varepsilon }=\mathbf {0}.$ For system (9.13), there exist $\gamma _{2},\rho >0$ such that

$$\begin{aligned} \lambda _{2}\left\| \mathbf {x}\left( t\right) \right\| ^{2}&\le V\left( \mathbf {z}_{t}\right) \le \gamma _{2}\left\| \mathbf {z}\left( t\right) \right\| ^{2}+\rho {\displaystyle \int \nolimits _{-T}^{0}} \left\| \mathbf {\mathbf {z}}_{t}\left( \theta \right) \right\| ^{2}\text {d}\theta \\ \dot{V}\left( \mathbf {z}_{t}\right)&\le -\lambda _{1}\left\| \mathbf {z}\left( t\right) \right\| ^{2}. \end{aligned}$$

Similar to the proof in [32, Theorem 1], $\mathbf {x}\left( t\right)$ is globally exponential convergence. Arranging (9.10) results in

$$\begin{aligned} \mathbf {v}\left( t\right)&=\left( \mathbf {I}_{m}+\mathbf {L}_{1}\left( t\right) \mathbf {D}\left( t\right) \right) ^{-1}\mathbf {v}\left( t-T\right) \nonumber \\&-\left( \mathbf {I}_{m}+\mathbf {L}_{1}\left( t\right) \mathbf {D}\left( t\right) \right) ^{-1}\mathbf {L}_{1}\left( t\right) \left( \mathbf {C} ^{\text {T}}\left( t\right) +\mathbf {D}\left( t\right) \mathbf {L} _{2}\left( t\right) \right) \mathbf {x}\left( t\right) . \end{aligned}$$

(9.42)

Since $\sup _{t\in \left[ 0,T\right] }\left\| \left( \mathbf {I} _{m}+\mathbf {L}\left( t\right) \mathbf {D}\left( t\right) \right) ^{-1}\right\| <1$ and $\mathbf {x}\left( t\right)$ is globally exponential convergence, then $\mathbf {v}\left( t\right) =\mathbf {0}$ is globally exponential convergence as well. Consequently, according to the stability definition, $\mathbf {z}=[\mathbf {v}^{\text {T}}$ $\mathbf {x} ^{\text {T}}]^{\text {T}}=\mathbf {0}$ is globally exponentially stable.

9.1.2 9.6.2 Uniformly Ultimate Boundedness Proof for Minimum-Phase Nonlinear System

Design a Lyapunov functional to be

$$ V\left( \mathbf {z}_{t}\right) =\frac{k}{2}\mathbf {x}^{\text {T}}\left( t\right) \mathbf {D}\left( \mathbf {q}\left( t\right) \right) \mathbf {x}\left( t\right) +\frac{\varepsilon }{2}\mathbf {v}^{\text {T}}\left( t\right) \mathbf {v}\left( t\right) +\frac{1}{2}\int _{-T}^{0}\mathbf {v} _{t}^{\text {T}}\left( \theta \right) \mathbf {v}_{t}\left( \theta \right) \text {d}\theta , $$

where (A1) is utilized. Taking the derivative of V along the solutions of (9.33) results in

$$\begin{aligned} \dot{V}\left( \mathbf {z}_{t}\right)&=-k\mathbf {x}^{\text {T}}\left( t\right) \mathbf {Mx}\left( t\right) +k\mathbf {x}^{\text {T}}\left( t\right) \mathbf {v}\left( t\right) +\frac{1}{2}\left( \mathbf {v} ^{\text {T}}\left( t\right) \mathbf {v}\left( t\right) -\mathbf {v} ^{\text {T}}\left( t-T\right) \mathbf {v}\left( t-T\right) \right) \\&\quad +\mathbf {v}^{\text {T}}\left( t\right) \left( -\mathbf {v} \left( t\right) +\left( 1-\alpha \varepsilon \right) \mathbf {v}\left( t-T\right) \right) +k\mathbf {v}^{\text {T}}\left( t\right) \left( \mathbf {x}_{\text {d}}\left( t\right) -\mathbf {x}\left( t\right) \right) \\&\le -k\mathbf {x}^{\text {T}}\left( t\right) \mathbf {Mx}\left( t\right) -\frac{{\alpha \varepsilon \left( {2-\alpha \varepsilon }\right) }}{2}\mathbf {v} ^{\text {T}}\left( t\right) \mathbf {v}\left( t\right) +k\mathbf {v} ^{\text {T}}\left( t\right) \mathbf {x}_{\text {d}}\left( t\right) , \end{aligned}$$

where (A2) is utilized. Since

$$ {\varepsilon }\mathbf {v}^{\text {T}}\mathbf {v}+2k\mathbf {v}^{\text {T}} \mathbf {x}_{\text {d}}+\frac{1}{{\varepsilon }}k^{2}\mathbf {x}_{\text {d} }^{\text {T}}\mathbf {x}_{\text {d}}\ge 0 $$

for any ${\varepsilon }>0,$ one has

$$\begin{aligned} \dot{V}\le -k\mathbf {x}^{\text {T}}\mathbf {Mx}-\left( \frac{{\alpha \varepsilon \left( {2-\alpha \varepsilon }\right) }}{2}-{\varepsilon }\right) \mathbf {v}^{\text {T}}\mathbf {v}+\frac{1}{{\varepsilon }}k^{2}\mathbf {x} _{\text {d}}^{\text {T}}\mathbf {x}_{\text {d}}. \end{aligned}$$

(9.43)

Here, ${\varepsilon }$ is chosen sufficiently small so that $\frac{{\alpha \varepsilon \left( {2-\alpha \varepsilon }\right) }}{2}-{\varepsilon >0.}$ Therefore, the given Lyapunov functional satisfies

$$\begin{aligned} \gamma _{0}\left\| \mathbf {z}{\left( t\right) }\right\| ^{2}&\le V\left( \mathbf {z}_{t}\right) \le \gamma _{1}\left\| \mathbf {z}{\left( t\right) }\right\| ^{2}+\frac{1}{2}\int _{-T}^{0}{\left\| \mathbf {z} {_{t}}\left( \theta \right) \right\| ^{2}}\text {d}\theta \\ \dot{V}\left( \mathbf {z}_{t}\right)&\le -\gamma _{2}\left\| \mathbf {z}{\left( t\right) }\right\| ^{2}+\chi \left( \sup _{t\in \left[ 0,T\right] }{\left\| \mathbf {x}_{\text {d}}\left( t\right) \right\| ^{2}}\right) , \end{aligned}$$

where $\gamma _{0}=\min \left( \frac{k\underline{\lambda }_{D}}{2} {,\frac{{\varepsilon }}{2}}\right) ,$ $\gamma _{1}=\max \left( \frac{k\bar{\lambda }_{D}}{2}{,\frac{\varepsilon }{2}}\right) ,$$\gamma _{2}=\min \left( k\lambda _{\min }\left( \mathbf {M}\right) {,}\frac{{\alpha \varepsilon \left( {2-\alpha \varepsilon }\right) }}{2}-{\varepsilon }\right) \$ and function $\chi \$belongs to class $\mathscr {K}$ [29, Definition 4.2, p. 144]. According to [32, Theorem 1], the solutions of (9.33) are uniformly bounded and uniformly ultimately bounded.

9.1.3 9.6.3 Uniformly Ultimate Boundedness Proof for Nonminimum-Phase Nonlinear System

Design a Lyapunov functional to be

$$\begin{aligned} V\left( \mathbf {z}_{t}\right) =\frac{1}{2}p_{1}\eta ^{2}\left( t\right) +\frac{1}{2}p_{2}z^{2}\left( t\right) +\frac{\varepsilon }{2}v^{2}\left( t\right) +\frac{1}{2}\int _{-T}^{0}v_{t}^{2}\left( \theta \right) \text {d}\theta ,\nonumber \end{aligned}$$

where $\mathbf {z\triangleq [}\eta$ z $v\mathbf {]}^{\text {T}}$ and $p_{1},p_{2},\varepsilon >0.$ Taking the derivative of V along the solutions of (9.37) results in

$$\begin{aligned} \dot{V}\left( \mathbf {z}_{t}\right)&=p_{1}\eta \left( t\right) \dot{\eta }\left( t\right) +p_{2}z\left( t\right) \dot{z}\left( t\right) +\varepsilon v\left( t\right) \dot{v}\left( t\right) +\frac{1}{2}\left( {v^{2}\left( t\right) -v^{2}}\left( t-T\right) \right) \\&=-p_{1}q_{1}\eta ^{2}\left( t\right) -p_{2}kz^{2}\left( t\right) -\frac{{\alpha \varepsilon \left( {2-\alpha \varepsilon }\right) }}{2}v^{2}\left( t\right) \\&\quad +\left( {p_{1}-p_{2}q_{2}}\right) \eta \left( t\right) z\left( t\right) +\left( {p_{2}-\rho }\right) z\left( t\right) v\left( t\right) +v\left( {q_{1}\eta \left( t\right) +\sin \eta \left( t\right) }\right) \\&\quad +p_{1}\eta {\left( t\right) }d_{\eta }{\left( t\right) }+p_{2}\left( {d_{\xi }\left( t\right) -d_{\eta }\left( t\right) \left( {q_{1}\left( t\right) +\cos \eta \left( t\right) }\right) }\right) z{\left( t\right) }+\rho v{\left( t\right) }y_{\text {d}}{\left( t\right) }. \end{aligned}$$

By fixing $p_{1},q_{1}$ and choosing $p_{2}=\rho$ and $0<{\alpha \varepsilon <1}$, if k is chosen sufficiently large, then

$$\begin{aligned} -p_{1}q_{1}\eta ^{2}-p_{2}kz^{2}-\frac{{\alpha \varepsilon \left( {2-\alpha \varepsilon }\right) }}{2}v^{2}+\left( {p_{1}-p_{2}q_{2}}\right) \eta z\nonumber \\ +\left( {p_{2}-\rho }\right) zv+v\left( {q_{1}\eta +\sin \eta }\right) \le -\theta _{1}\eta ^{2}-\theta _{2}z^{2}-\theta _{3}v^{2}, \end{aligned}$$

(9.44)

where $\theta _{1},\theta _{2},\theta _{3}>0$. Furthermore, there exists a class $\mathscr {K}$ function $\chi :\left[ 0,\infty \right) \rightarrow \left[ 0,\infty \right)$ [29, Definition 4.2, p. 144] such that

$$ \dot{V}\le -\theta _{1}^{\prime }\eta ^{2}-\theta _{2}^{\prime }z^{2}-\theta _{3}^{\prime }v^{2}+\chi \left( \sup _{t\in \left[ 0,T\right] }\left( {\left\| {y_{\text {d}}}\left( t\right) \right\| ^{2}+\left\| {d_{\eta }}\left( t\right) \right\| ^{2}+\left\| {d_{\xi }}\left( t\right) \right\| ^{2}}\right) \right) , $$

where $\theta _{1}^{\prime },\theta _{2}^{\prime },\theta _{3}^{\prime }>0$. Therefore, the given Lyapunov functional satisfies

$$\begin{aligned} \gamma _{0}\left\| \mathbf {z}{\left( t\right) }\right\| ^{2}&\le V\left( \mathbf {z}_{t}\right) \le \gamma _{1}\left\| \mathbf {z}{\left( t\right) }\right\| ^{2}+\frac{1}{2}\int _{-T}^{0}{\left\| \mathbf {z} {_{t}}\left( \theta \right) \right\| ^{2}}\text {d}\theta \\ \dot{V}\left( \mathbf {z}_{t}\right)&\le -\gamma _{2}\left\| \mathbf {z}{\left( t\right) }\right\| ^{2}+\chi \left( \sup _{t\in \left[ 0,T\right] }\left( {\left\| {y_{\text {d}}}\left( t\right) \right\| ^{2}+\left\| {d_{\eta }}\left( t\right) \right\| ^{2}+\left\| {d_{\xi }}\left( t\right) \right\| ^{2}}\right) \right) , \end{aligned}$$

where $\gamma _{0}=\min \left( {\frac{1}{2}p_{1},\frac{1}{2}p_{2} ,\frac{{\varepsilon }}{2}}\right) ,$ $\gamma _{1}=\max \left( {\frac{1}{2} p_{1},\frac{1}{2}p_{2},\frac{\varepsilon }{2}}\right) \$and $\gamma _{2} =\min \left( {\theta _{1}^{\prime },\theta _{2}^{\prime },\theta _{3}^{\prime } }\right) .$ According to [32, Theorem 1], the solutions of (9.37) are uniformly bounded and uniformly ultimately bounded. Furthermore, $\xi$ is also uniformly ultimately bounded by using the relationship $\xi =z-q_{1}\eta -\sin \eta$.