# Iterative Solution of Linear Matrix Inequalities for the Combined Control and Observer Design of Systems with Polytopic Parameter Uncertainty and Stochastic Noise

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## Abstract

**:**

## 1. Introduction

## 2. Observer-Based Control of Systems with Polytopic Uncertainty: A Cautionary Tale

**Remark**

**1.**

## 3. Methodological Background: LMI-Based Control and Observer Design

**Case****1:**- The control signal is defined as$$\mathbf{u}={\mathbf{u}}_{\mathrm{ff}}-\mathbf{K}\xb7\widehat{\mathbf{x}},$$$$\begin{array}{cc}\hfill \dot{\widehat{\mathbf{x}}}& ={\mathbf{A}}_{\mathrm{nom}}\xb7\widehat{\mathbf{x}}+{\mathbf{B}}_{\mathrm{nom}}\xb7\mathbf{u}+\mathbf{H}\xb7\left({\mathbf{y}}_{\mathrm{m}}-\widehat{\mathbf{y}}\right)\hfill \\ \hfill \widehat{\mathbf{y}}& ={\mathbf{C}}_{\mathrm{m}}\xb7\widehat{\mathbf{x}},\hfill \end{array}$$
**Case****2:**- The control signal is defined as$$\mathbf{u}={\mathbf{u}}_{\mathrm{ff}}-{\mathbf{K}}_{\mathrm{o}}\xb7\mathbf{C}\xb7\widehat{\mathbf{x}},$$
**Case****3:**- The control signal is defined as$$\mathbf{u}={\mathbf{u}}_{\mathrm{ff}}-{\mathbf{K}}_{\mathrm{y}}\xb7{\mathbf{C}}_{\mathrm{y}}\xb7{\widehat{\mathbf{y}}}_{\mathrm{f}},$$

**Remark**

**2.**

#### 3.1. Polytopic Uncertainty Modeling

#### 3.2. Robust State Feedback Control

**Theorem**

**1**

**.**Robust asymptotic stability of the closed-loop control system according to Case 1 for a noise- and error-free state feedback (i.e., $\mathbf{x}\equiv \widehat{\mathbf{x}}$) is ensured if the gain matrix $\mathbf{K}$ satisfies the bilinear matrix inequalities

**Proof.**

**Corollary**

**1.**

#### 3.3. Robust Output Feedback Control

**Theorem**

**2**

**.**Robust asymptotic stability of the closed-loop control system according to Case 2 for an error-free output feedback (i.e., $\mathbf{C}\mathbf{x}\equiv \mathbf{C}\widehat{\mathbf{x}}$) is ensured if the gain matrix ${\mathbf{K}}_{\mathrm{o}}$ satisfies the bilinear matrix inequalities

**Proof.**

**Corollary**

**2**

**.**For precisely known matrices $\mathbf{C}$, an LMI formulation of Theorem 2 is obtained by introducing a linearizing change of variables with $\mathbf{Q}={\mathbf{P}}^{-1}\succ 0$ and the equality constraints

**Remark**

**3.**

**Remark**

**4.**

#### 3.4. Robust State Observation

**Theorem**

**3**

**.**The error dynamics of a robust state observer according to Equation (11) with the exactly known output matrix ${\mathbf{C}}_{\mathrm{m}}$ are robustly asymptotically stable if the observer gain $\mathbf{H}$ satisfies the bilinear matrix inequalities

**Corollary**

**3**

**.**To allow for an efficient solution of the bilinear matrix inequalities in Theorem 3, after considering the duality between control and observer synthesis by evaluating the transposed of (27) according to

## 4. Main Results: Iterative LMI Solution for a Combined Control and Observer Synthesis in the Presence of Stochastic Noise

#### 4.1. Observer-Based Feedback Control in the Presence of Noise

#### 4.2. Optimality of the Noise Insensitive Observer-Based Control Design

**Theorem**

**4**

**.**The parameterization of the observer-based controller for the augmented system (36)–(40) is optimal in the sense of a minimization of the influence of noise if the controller and observer gains are chosen to minimize the cost function

**Proof.**

**Remark**

**5.**

#### 4.3. Summary of the Proposed Algorithm and Further Discussion

## 5. Simulation Results

#### 5.1. Modeling

**Remark**

**6.**

#### 5.2. Tracking the Unstable Branch of the Bifurcation Diagram

#### 5.3. Trajectory Tracking Control

**S1:**- ${\mathbf{G}}_{\mathrm{p}}={10}^{-2}\xb7\mathbf{I}$, ${g}_{u}=2\xb7{10}^{-2}\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$, ${g}_{\mathrm{m}}={10}^{-2}\phantom{\rule{0.166667em}{0ex}}\mathrm{rad}\approx 0.573\phantom{\rule{0.166667em}{0ex}}\mathrm{deg}$;
**S2:**- ${\mathbf{G}}_{\mathrm{p}}={10}^{-2}\xb7\mathbf{I}$, ${g}_{u}=0\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$, ${g}_{\mathrm{m}}={10}^{-2}\phantom{\rule{0.166667em}{0ex}}\mathrm{rad}\approx 0.573\phantom{\rule{0.166667em}{0ex}}\mathrm{deg}$;
**S3:**- ${\mathbf{G}}_{\mathrm{p}}={10}^{-2}\xb7\mathbf{I}$, ${g}_{u}=0\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$, ${g}_{\mathrm{m}}=5\xb7{10}^{-2}\phantom{\rule{0.166667em}{0ex}}\mathrm{rad}\approx 2.865\phantom{\rule{0.166667em}{0ex}}\mathrm{deg}$;
**S4:**- ${\mathbf{G}}_{\mathrm{p}}={10}^{-2}\xb7\mathbf{I}$, ${g}_{u}=0\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$, ${g}_{\mathrm{m}}=10\xb7{10}^{-2}\phantom{\rule{0.166667em}{0ex}}\mathrm{rad}\approx 5.730\phantom{\rule{0.166667em}{0ex}}\mathrm{deg}$.

## 6. Conclusions and Outlook on Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Stability analysis of the observer-based closed-loop control system (7) for different values of ${a}_{\mathrm{nom}}$ and $a\in \left[1\phantom{\rule{0.277778em}{0ex}};\phantom{\rule{0.277778em}{0ex}}3\right]$: The choice of ${a}_{\mathrm{nom}}\ne \overline{a}$ requires a significant increase of both k and h above the thresholds $k>3$ and $h>3$.

**Figure 4.**Guidance of the Zeeman catastrophe machine along the unstable branch of equilibria in the $({x}_{3},{x}_{1})$-plane; comparison of an observer-based full state feedback control parameterized by the LQG technique with the LMI-based parameterization of (12).

**Figure 5.**Comparison of the control effort for the LMI-based full state feedback controller (Case 1), the LMI-based output feedback controller (Case 2), and the LQG technique with the ideal noise-free control signal.

**Figure 6.**Comparison of the output tracking behavior as well as the state reconstruction accuracy for the proposed LMI-based output feedback controller (Case 2) after the 10th iteration.

**Figure 7.**Comparison of the control signal $u\left(t\right)$ for 10 iterations of the proposed LMI-based output feedback controller (Case 2).

**Figure 8.**Reduction of the output tracking error over 10 iterations for the proposed LMI-based output feedback controller (Case 2).

**Table 1.**Comparison of tracking errors and control effort of all controllers for the scenarios S1–S4 in terms of the root mean square deviations to an ideal noise-free trajectory tracking; the control improvement is quantified by a comparison of Case 2 with the LQG in terms of $\Delta {u}_{\mathrm{RMS}}$.

Scenario | Case 1 | Case 2 | LGQ | Control Improve- | |||
---|---|---|---|---|---|---|---|

$\mathbf{\Delta}{\mathit{y}}_{\mathbf{RMS}}/\mathbf{deg}$ | $\mathbf{\Delta}{\mathit{u}}_{\mathbf{RMS}}/\mathbf{m}$ | $\mathbf{\Delta}{\mathit{y}}_{\mathbf{RMS}}/\mathbf{deg}$ | $\mathbf{\Delta}{\mathit{u}}_{\mathbf{RMS}}/\mathbf{m}$ | $\mathbf{\Delta}{\mathit{y}}_{\mathbf{RMS}}/\mathbf{deg}$ | $\mathbf{\Delta}{\mathit{u}}_{\mathbf{RMS}}/\mathbf{m}$ | ment in % | |

S1 | $0.945$ | $0.0083$ | $1.332$ | $0.0027$ | $0.713$ | $0.0040$ | $33.25$ |

S2 | $0.091$ | $0.0030$ | $0.1275$ | $0.0009$ | $0.092$ | $0.0021$ | $56.51$ |

S3 | $0.339$ | $0.0246$ | $0.463$ | $0.0047$ | $0.364$ | $0.0106$ | $55.95$ |

S4 | $0.636$ | $0.0425$ | $0.925$ | $0.0090$ | $0.705$ | $0.0211$ | $57.59$ |

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**MDPI and ACS Style**

Rauh, A.; Dehnert, R.; Romig, S.; Lerch, S.; Tibken, B.
Iterative Solution of Linear Matrix Inequalities for the Combined Control and Observer Design of Systems with Polytopic Parameter Uncertainty and Stochastic Noise. *Algorithms* **2021**, *14*, 205.
https://doi.org/10.3390/a14070205

**AMA Style**

Rauh A, Dehnert R, Romig S, Lerch S, Tibken B.
Iterative Solution of Linear Matrix Inequalities for the Combined Control and Observer Design of Systems with Polytopic Parameter Uncertainty and Stochastic Noise. *Algorithms*. 2021; 14(7):205.
https://doi.org/10.3390/a14070205

**Chicago/Turabian Style**

Rauh, Andreas, Robert Dehnert, Swantje Romig, Sabine Lerch, and Bernd Tibken.
2021. "Iterative Solution of Linear Matrix Inequalities for the Combined Control and Observer Design of Systems with Polytopic Parameter Uncertainty and Stochastic Noise" *Algorithms* 14, no. 7: 205.
https://doi.org/10.3390/a14070205