#
Multiple Estimation Architecture in Discrete-Time Adaptive Mixing Control^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation: Uncertain Plant

- P1.
- The degree n of ${D}_{0}(z,{\theta}^{*})$ is known.
- P2.
- The plant is strictly proper, i.e., $m\le n-1$.
- P3.
- ${M}_{\Delta}\left(z\right)$ is proper, rational, and analytic in $\left|z\right|\ge \sqrt{{\delta}_{0}}$ for some known $0<{\delta}_{0}\le 1$.
- P4.
- ${\theta}^{*}\in \Omega $ for some known compact convex set $\Omega \subset {\mathbb{R}}^{n+m+1}$.

- P5.
- ${Q}_{m}\left(z\right)$, ${N}_{0}\left(z\right)$ are coprime.

## 3. Multicontroller and Mixer

- M1.
- $\beta (\xb7)$ is Lipschitz in Ω.
- C1.
- The elements ${p}_{0}(\xb7)$, $\overline{p}(\xb7)$, and $\overline{l}(\xb7)$ are Lipschitz in ${\mathbb{R}}^{N}$.
- C2.
- For all ${\theta}^{*}\in \Omega $, let ${\beta}^{*}:=\beta \left({\theta}^{*}\right)$; then $C(z;{\beta}^{*})$ internally stabilizes the plant ${G}_{0}(z;{\theta}^{*})$.

## 4. Multiple Parallel Estimators and Switching Logic

- A1.
- For each $\sigma (\xb7)\in \mathcal{S}$ and $i\in \mathcal{I}$, ${\mathcal{J}}_{i}\left(t\right)$ admits a limit (which may be infinite) as $t\to \infty $;
- A2.
- For each $\sigma (\xb7)\in \mathcal{S}$, there exist integers $\mu \in \mathcal{I}$ such that ${\mathcal{J}}_{\mu}(\xb7)$ is bounded.

**HSL Lemma**[17] Let σ be the switching sequence resulting from (16) and (17). Then, if A1 and A2 hold, there is a finite time ${t}^{*}\in {\mathbb{Z}}_{+}$, after which no more switching occurs. Moreover, ${\mathcal{J}}_{\sigma \left({t}^{*}\right)}(\xb7)$ is bounded.

**Theorem 1**Consider the parallel robust adaptive laws given by (8)–(11) and hysteresis switching logic (16) and (17) with performance signals (14) and (15). Then, the resulting multiple estimator resulting from the interconnection of the parallel adaptive laws with the hysteresis switching logic satisfies

- E1.
- ${\u03f5}_{\sigma \left(t\right)}$, ${\u03f5}_{\sigma \left(t\right)}{m}_{s}$, $\Delta {\theta}_{\sigma \left(t\right)}\in {l}_{2}$ if ${M}_{\Delta}$, $d=0$.
- E2.
- ${\u03f5}_{\sigma \left(t\right)}$, ${\u03f5}_{\sigma \left(t\right)}{m}_{s}$, $\Delta {\theta}_{\sigma \left(t\right)}\in \mathcal{S}\left(\frac{{\eta}^{2}}{{m}_{s}^{2}}\right)\cap {l}_{\infty}$ if ${M}_{\Delta}$, $d\ne 0$.

## 5. Stability and Stability Robustness of Multiple-Estimator Adaptive Mixing Control

**Theorem 2**Let the unknown plant given by (1)–(3) satisfy the plant assumptions P1–P5. Consider the adaptive mixing controller with the multicontroller given by (6) and satisfying assumptions C1–C2; the mixer (7) satisfying M1; the parallel robust adaptive laws given by (8)–(11) and hysteresis switching logic (16) and (17). Then the following results hold:

- If ${M}_{\Delta}$, $d=0$, then all the closed-loop signals are bounded, i.e., ϕ, u, $y\in {l}_{\infty}$; furthermore ${e}_{1}\left(t\right):=y\left(t\right)-{y}_{m}\left(t\right)\to 0$ as $k\to \infty $.
- If ${M}_{\Delta}$, $d\ne 0$, then there exists ${\mu}^{*}>0$ such that, if $c{\Xi}_{1}^{2}<{\mu}^{*}$ where ${\Xi}_{1}={\u2225\frac{{N}_{0}{M}_{\Delta}}{{\Lambda}_{p}}\u2225}_{2{\delta}_{0}}$, ${\u2225\xb7\u2225}_{2\delta}$ is the system norm defined as ${\u2225H\u2225}_{2\delta}:=\frac{1}{\sqrt{2\pi}}{\left\{{\int}_{-\infty}^{\infty}{\left|H\left(\sqrt{\delta}{e}^{j\omega}\right)\right|}^{2}\phantom{\rule{0.277778em}{0ex}}\mathrm{d}\omega \right\}}^{1/2}$, and $c>0$ a finite constant, then the adaptive mixing control scheme guarantees ϕ, u, y, ${e}_{1}\in {l}_{\infty}$ and$$\frac{1}{T}\sum _{t}^{t+T-1}{\left|{e}_{1}\left(\tau \right)\right|}^{2}\le {c}_{0}{\mu}^{2}+\frac{{c}_{1}}{T},\phantom{\rule{1.em}{0ex}}\forall t,T\ge 0$$

**Remark 1**The stability and robustness results are conceptually similar to those in robust adaptive control [9,10,18]. The advantage of AMC in comparison with conventional robust adaptive control is that for the proposed scheme the stabilizability of the estimated plant is no longer an issue in calculating on-line the controller parameters. In addition, it allows well developed results from robust control to be incorporated in the design. In fact, thanks to the modularity of the design, the analysis of the overall system relies on certain properties of its individual parts. Furthermore, the Multi-AMC can handle controllers that are not directly parameterized by the unknown plant parameters, e.g., ${\mathcal{H}}_{\infty}$ or μ-synthesized robust controllers, extending the class of feedback adaptive control systems.

**Remark 2**Equation (19) is a mean square condition: it does not guarantee that the tracking error will be bounded from above by the modeling error bound at all times. As in conventional adaptive control, a phenomenon known as “bursting” [19], where the tracking error assumes large values over short intervals of time, cannot be excluded by the mean square bound. One way to get rid of bursting is to use a dead zone to switch-off adaptation when convergence to steady state values is achieved [20].

## 6. Numerical Example

${\mathit{s}}_{\mathit{i}\mathbf{1}}$ | ${\mathit{s}}_{\mathit{i}\mathbf{2}}$ | ${\mathit{s}}_{\mathit{i}\mathbf{3}}$ | ${\mathit{s}}_{\mathit{i}\mathbf{4}}$ | ${\mathit{r}}_{\mathit{i}\mathbf{1}}$ | ${\mathit{r}}_{\mathit{i}\mathbf{2}}$ | ${\mathit{r}}_{\mathit{i}\mathbf{3}}$ | ${\mathbf{\Theta}}_{\mathit{i}}$ | |
---|---|---|---|---|---|---|---|---|

${C}_{1}$ | 146.257 | −428.616 | 419.360 | −136.999 | −2.464 | 2.038 | −0.563 | $[0.04,\phantom{\rule{0.277778em}{0ex}}0.12)$ |

${C}_{2}$ | 79.779 | −233.668 | 228.558 | −74.668 | −2.457 | 2.026 | −0.559 | $(0.07,\phantom{\rule{0.277778em}{0ex}}0.23)$ |

${C}_{3}$ | 43.739 | −128.102 | 125.361 | −40.997 | −2.446 | 2.010 | −0.553 | $(0.16,\phantom{\rule{0.277778em}{0ex}}0.42)$ |

${C}_{4}$ | 31.680 | −92.798 | 90.877 | −29.757 | −2.435 | 1.994 | −0.547 | $(0.23,\phantom{\rule{0.277778em}{0ex}}0.603)$ |

${C}_{5}$ | 22.632 | −66.341 | 65.092 | −21.380 | −2.411 | 1.958 | −0.533 | $(0.37,\phantom{\rule{0.277778em}{0ex}}0.95)$ |

${C}_{6}$ | 19.309 | −56.746 | 56.013 | −18.571 | −2.323 | 1.830 | −0.485 | $(0.65,\phantom{\rule{0.277778em}{0ex}}2.05)$ |

${C}_{7}$ | 17.744 | −54.669 | 57.372 | −20.436 | −1.865 | 1.187 | −0.252 | $(1.07,\phantom{\rule{0.277778em}{0ex}}3.5]$ |

- The root mean square (RMS) tracking error over the first 50 seconds. This criterion is used to evaluate transient and tracking error performance.
- The average time it takes for the scheme to converge to the final controller without any further switching (used in the comparison of the single estimator based AMC with the Multi-AMC).

${\mathit{\theta}}^{*}$ | Tracking error RMS | Multi-AMC RMS Improvement | Convergence to final controller | |
---|---|---|---|---|

Multi-AMC | 2.395 | 7.5 sec. | ||

AMC | 0.05 | 6.567 | 63.5 % | 22.0 sec. |

APPC | 20.393 | 88.3 % | ||

Multi-AMC | 1.852 | 7.1 sec. | ||

AMC | 0.1 | 4.331 | 57.2 % | 16.5 sec. |

APPC | 15.488 | 88.0 % | ||

Multi-AMC | 1.411 | 7.2 sec. | ||

AMC | 0.2 | 3.120 | 54.8 % | 12.7 sec. |

APPC | 11.721 | 88.0 % | ||

Multi-AMC | 0.989 | 5.9 sec. | ||

AMC | 0.3 | 1.815 | 45.5 % | 11.3 sec. |

APPC | 6.816 | 85.5 % | ||

Multi-AMC | 0.641 | 3.7 sec. | ||

AMC | 0.5 | 1.045 | 38.7 % | 9.5 sec. |

APPC | 3.304 | 80.6 % | ||

Multi-AMC | 0.458 | 4.1 sec. | ||

AMC | 1.0 | 0.606 | 24.4 % | 4.9 sec. |

APPC | 1.164 | 60.7 % | ||

Multi-AMC | 0.584 | 0 sec. | ||

AMC | 2.0 | 0.584 | 0 % | 0 sec. |

APPC | 0.761 | 22.7 % |

${\mathit{\theta}}^{*}$ | Tracking error RMS | Multi-AMC RMS Improvement | Convergence to final controller | |
---|---|---|---|---|

Multi-AMC | 1.881 | 7.0 sec. | ||

AMC | 0.05 | 4.390 | 57.2 % | 22.9 sec. |

APPC | 15.215 | 87.6 % | ||

Multi-AMC | 1.746 | 7.0 sec. | ||

AMC | 0.1 | 3.927 | 55.5 % | 13.8 sec. |

APPC | 14.047 | 87.6 % | ||

Multi-AMC | 1.669 | 7.2 sec. | ||

AMC | 0.2 | 3.192 | 47.7 % | 12.8 sec. |

APPC | 11.939 | 86.0 % | ||

Multi-AMC | 1.340 | 5.1 sec. | ||

AMC | 0.3 | 2.083 | 35.7 % | 12.7 sec. |

APPC | 7.739 | 82.7 % | ||

Multi-AMC | 0.744 | 4.6 sec. | ||

AMC | 0.5 | 1.014 | 26.6 % | 10.3 sec. |

APPC | 3.890 | 80.9 % | ||

Multi-AMC | 0.508 | 4.2 sec. | ||

AMC | 1.0 | 0.619 | 17.9 % | 4.8 sec. |

APPC | 1.415 | 64.1 % | ||

Multi-AMC | 0.586 | 0 sec. | ||

AMC | 2.0 | 0.586 | 0 % | 0 sec. |

APPC | 0.803 | 27.0 % |

**Figure 4.**Ideal case, ${\theta}^{*}=0.05$: Multi-AMC (solid), AMC (dash-dotted), APPC (dashed). (

**a**) output response $y\left(t\right)$: Multi-AMC (solid), AMC (dash-dotted), APPC (dashed); (

**b**) Parameter estimate $\theta \left(t\right)$: Multi-AMC (solid), AMC (dash-dotted), APPC (dashed).

**Figure 5.**Unmodeled dynamics, ${\theta}^{*}=0.5$: Multi-AMC (solid), AMC (dash-dotted), APPC (dashed). (

**a**) output response $y\left(t\right)$: Multi-AMC (solid), AMC (dash-dotted), APPC (dashed); (

**b**) Parameter estimate $\theta \left(t\right)$: Multi-AMC (solid), AMC (dash-dotted), APPC (dashed).

**Figure 6.**Ideal case, ${\theta}^{*}=0.05$: AMC controller weights $\beta \left(t\right)$: Multi-AMC (solid), AMC (dash-dotted). (

**a**) ${\beta}_{1}\left(t\right)$, ${\beta}_{2}\left(t\right)$, ${\beta}_{3}\left(t\right)$, ${\beta}_{4}\left(t\right)$; (

**b**) ${\beta}_{5}\left(t\right)$, ${\beta}_{6}\left(t\right)$, ${\beta}_{7}\left(t\right)$.

**Figure 7.**Unmodeled dynamics, ${\theta}^{*}=0.5$: AMC controller weights $\beta \left(t\right)$: Multi-AMC (solid), AMC (dash-dotted). (Note that, for both schemes, all the controller weights from ${\beta}_{1}$ to ${\beta}_{4}$ are zero for all $t\ge 0$). (

**a**) ${\beta}_{1}\left(t\right)$, ${\beta}_{2}\left(t\right)$, ${\beta}_{3}\left(t\right)$, ${\beta}_{4}\left(t\right)$; (

**b**) ${\beta}_{5}\left(t\right)$, ${\beta}_{6}\left(t\right)$, ${\beta}_{7}\left(t\right)$.

**Figure 8.**Unmodeled dynamics, ${\theta}^{*}=0.18$: Multi-AMC (solid), AMC (dash-dotted), APPC (dashed). (

**a**) output response $y\left(t\right)$: Multi-AMC (solid), AMC (dash-dotted), APPC (dashed); (

**b**) Parameter estimate $\theta \left(t\right)$: Multi-AMC (solid), AMC (dash-dotted), APPC (dashed).

**Figure 9.**Unmodeled dynamics, ${\theta}^{*}=0.18$: AMC controller weights $\beta \left(t\right)$: Multi-AMC (solid), AMC (dash-dotted). (

**a**) ${\beta}_{1}\left(t\right)$, ${\beta}_{2}\left(t\right)$, ${\beta}_{3}\left(t\right)$, ${\beta}_{4}\left(t\right)$; (

**b**) ${\beta}_{5}\left(t\right)$, ${\beta}_{6}\left(t\right)$, ${\beta}_{7}\left(t\right)$.

## 7. Conclusions

## Appendix: Proof of Theorem 1

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^{*}This work has been supported by the European Commission FP7-ICT-5-3.5, Engineering of Networked Monitoring and Control Systems, under the contract #257806 AGILE.

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Baldi, S. Multiple Estimation Architecture in Discrete-Time Adaptive Mixing Control. *Machines* **2013**, *1*, 33-49.
https://doi.org/10.3390/machines1010033

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Baldi S. Multiple Estimation Architecture in Discrete-Time Adaptive Mixing Control. *Machines*. 2013; 1(1):33-49.
https://doi.org/10.3390/machines1010033

**Chicago/Turabian Style**

Baldi, Simone. 2013. "Multiple Estimation Architecture in Discrete-Time Adaptive Mixing Control" *Machines* 1, no. 1: 33-49.
https://doi.org/10.3390/machines1010033