# Projection onto the Set of Rank-Constrained Structured Matrices for Reduced-Order Controller Design

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## Abstract

**:**

## 1. Introduction

#### Notation

## 2. Reduced-Order Controller Design Problems

#### 2.1. Reduced-Order Stabilizing Controllers

**Problem**

**1**

**.**Find ${X}_{1},{X}_{2}\in {\mathcal{S}}_{n}$ such that the rank constraint

#### 2.2. Reduced-Order ${H}^{\infty}$ Controllers

**Problem**

**2**

**.**Find ${X}_{1},{X}_{2}\in {\mathcal{S}}_{n}$ such that the rank constraint (6), the LMI (7), and the following LMIs:

## 3. Algorithms

#### 3.1. Proposed Algorithm

**Problem**

**3.**

**Lemma**

**1.**

**Remark**

**1.**

#### 3.2. Projection onto the Set ${\Omega}_{r}$ of Rank-Constrained Structured Matrices

#### 3.3. Projection onto the Set $\Lambda $ Described by LMIs

**Remark**

**2.**

Algorithm 1 Algorithm to solve Problem 3 |

Require:
Initial guess $({X}_{1}\left[0\right],{X}_{2}\left[0\right])\in {\mathcal{S}}_{n}$Ensure:$({X}_{1},{X}_{2})\in {\Omega}_{r}\cap {\Omega}_{\Lambda}$for $k=0,1,2,\dots ,N-1$do ${X}_{1}\leftarrow {X}_{1}\left[k\right]$ ▹ Projection onto ${\Omega}_{r}$ ${X}_{2}\leftarrow {X}_{2}\left[k\right]$ $\tilde{Z}\left[0\right]\leftarrow 0$ $W\left[0\right]\leftarrow 0$ for $i=0,1,2,\dots ,M-1$ do ${Z}_{1}[i+1]\leftarrow {\left(1+\frac{\rho}{2}\right)}^{-1}\left({X}_{1}+\frac{\rho}{2}{M}_{11}\left[i\right]\right)$ ${Z}_{2}[i+1]\leftarrow {\left(1+\frac{\rho}{2}\right)}^{-1}\left({X}_{2}+\frac{\rho}{2}{M}_{22}\left[i\right]\right)$ $\tilde{Z}[i+1]\leftarrow {\Pi}_{{\mathcal{C}}_{r}}\left(\left[\begin{array}{ccc}{Z}_{1}[i+1]& II& {X}_{2}[i+1]\end{array}\right]-W\left[i\right]\right)$ $[i+1]\leftarrow W\left[i\right]+\tilde{Z}[i+1]-\left[\begin{array}{ccc}{Z}_{1}[i+1]& II& {Z}_{2}[i+1]\end{array}\right]$ end for $X\leftarrow {[{Z}_{1}\left[M\right],{Z}_{2}\left[M\right]]}^{\top}$ ▹ Projection onto ${\Omega}_{\Lambda}$ $({Z}_{1},{Z}_{2},W)\leftarrow \underset{{Z}_{1},{Z}_{2},W\in {\mathcal{S}}_{n}}{\mathrm{arg}\phantom{\rule{0.166667em}{0ex}}\mathrm{min}}\mathrm{tr}\left(W\right)$ subject to $F({Z}_{1},{Z}_{2})\u2aaf0$, $\left[\begin{array}{cc}W& {(Z-X)}^{\top}\\ Z-X& I\end{array}\right]\u2ab0\u03f5I$, ${X}_{1}[k+1]\leftarrow {Z}_{1}$ ${X}_{2}[k+1]\leftarrow {Z}_{2}$ end for${X}_{1}\leftarrow {X}_{1}\left[N\right]$ ${X}_{2}\leftarrow {X}_{2}\left[N\right]$ |

## 4. Numerical Examples

#### 4.1. Stabilizing Static Controllers

- 1
- 2
- Cone complementarity linearization algorithm [12] (CCL)
- 3
- Nuclear norm minimization [15] (NNM)
- 4
- Alternating projection with approximate projection onto ${\Omega}_{r}$ [13] (GS96)
- 5
- Alternating projection with the proposed precise projection in Section 3 (Proposed)

`EB5`and

`JE1`) for which it takes a very long CPU time by the se methods. For example, GS96 takes 53,450 [s] and the proposed method takes 52,315 [s] for

`EB5`. However, there are models for which the proposed method successfully gives a stabilizing static controller while some of the other methods fail. We summarize the results in Table 2.

`TF1`. We emphasize our method is an effective method that may provide a solution to a reduced-order controller design problem that cannot be solved by standard methods as nonsmooth ${H}^{\infty}$, cone complementarity linearization, and nuclear norm minimization.

#### 4.2. Stabilizing Low-Order Controllers

`TF1`. In this model, the state-space matrices are given as follows:

#### 4.3. ${H}^{\infty}$ Static Controllers

`AC4`,

`NN1`,

`NN12`, and

`HE6`from COMPL${}_{e}$ib.

`AC4`is from a autopilot control problem for an air-to-air missile discussed in [37].

`NN1`and

`NN12`are academic test problems proposed in [38] and [39], respectively.

`HE6`is a helicopter model that has four inputs, 20 states, and six outputs [40]. For these plant models, we seek the ${H}^{\infty}$ static controller by using the bisection method on $\gamma $. Namely, we first give a sufficiently large upper bound $\overline{\gamma}$, for example $\overline{\gamma}=100$, and a sufficiently small lower bound $\underline{\gamma}$ (e.g., $\underline{\gamma}=0$). Then we set $\gamma =(\underline{\gamma}+\overline{\gamma})/2=50$ and solve Problem 2. If there is a feasible solution, then we update the upper bound to $\overline{\gamma}=\gamma =50$, otherwise we set the lower bound to $\underline{\gamma}=\gamma =50$. We note that the problem is assumed to be infeasible if a solution of Problem 2 is not found after 15 iterations of (16). We repeat this process until sufficient accuracy is achieved.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Proof of Lemma 1

## Appendix B. ADMM Algorithm

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NS${\mathit{H}}^{\mathit{\infty}}$ [36] | CCL [12] | NNM [15] | GS96 [13] | Proposed | |
---|---|---|---|---|---|

# success | 75 | 61 | 53 | 56 | 59 |

CPU time [s] | 0.365 | 56.5 | 4.58 | 663 | 666 |

**Table 2.**Stabilizing static controller results (the full list available at [28]).

NS${\mathit{H}}^{\mathit{\infty}}$ [36] | CCL [12] | NNM [15] | GS96 [13] | Proposed | |
---|---|---|---|---|---|

HE6 | unstable | stable | unstable | stable | stable |

HE7 | stable | stable | unstable | stable | stable |

REA3 | unstable | stable | unstable | unstable | stable |

DIS1 | stable | stable | unstable | unstable | stable |

PAS | unstable | stable | stable | stable | stable |

TF1 | unstable | unstable | unstable | unstable | stable |

TF2 | unstable | stable | stable | stable | stable |

NN1 | stable | unstable | unstable | unstable | stable |

NN11 | stable | stable | unstable | stable | stable |

NN12 | unstable | unstable | unstable | stable | stable |

FS | unstable | stable | stable | stable | stable |

NS${\mathit{H}}^{\mathit{\infty}}$ [36] | CCL [12] | NNM [15] | GS96 [13] | Proposed | |
---|---|---|---|---|---|

TF1 (${n}_{c}=1$) | unstable | stable | stable | stable | stable |

CPU time [s] | 0.016910 | 0.214947 | 0.177307 | 0.194252 | 1.836873 |

TF1 (${n}_{c}=2$) | unstable | stable | stable | stable | stable |

CPU time [s] | 0.016012 | 0.263448 | 0.205307 | 0.206729 | 4.250059 |

Model | AC4 | NN1 | NN12 | HE6 |
---|---|---|---|---|

$\gamma $ | 1.000 | 74.72 | 28.09 | 520.0 |

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**MDPI and ACS Style**

Nagahara, M.; Iwai, Y.; Sebe, N.
Projection onto the Set of Rank-Constrained Structured Matrices for Reduced-Order Controller Design. *Algorithms* **2022**, *15*, 322.
https://doi.org/10.3390/a15090322

**AMA Style**

Nagahara M, Iwai Y, Sebe N.
Projection onto the Set of Rank-Constrained Structured Matrices for Reduced-Order Controller Design. *Algorithms*. 2022; 15(9):322.
https://doi.org/10.3390/a15090322

**Chicago/Turabian Style**

Nagahara, Masaaki, Yu Iwai, and Noboru Sebe.
2022. "Projection onto the Set of Rank-Constrained Structured Matrices for Reduced-Order Controller Design" *Algorithms* 15, no. 9: 322.
https://doi.org/10.3390/a15090322