# Robust Stabilization via Super-Stable Systems Techniques

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Parametrically Certain Systems

#### 2.1. The Elementary Control Problem

- (1)
- by assigning only eigenvalues in a closed system (3), it is not always possible to achieve the desired transients of the state variables;
- (2)
- in multidimensional systems with vector control, there are certain computational difficulties in synthesis, called the “curse of dimensionality”;
- (3)
- full parametric certainty of the matrices $A$ and $B$ is required.

**Definition**

**1.**

#### 2.2. Synthesis of Modal Control Based on a Regular Form

**Definition**

**2.**

**Procedure**

**1.**

- Nonsingular transformation of system (1) to the regular form (6).

**Procedure**

**2.**

**Remark**

**1.**

**Definition**

**3**

**.**Matrix $A=({a}_{ij})\in {R}^{n\times n}$ and, consequently, the system $\dot{x}=Ax$ are called super-stable if $A$ is a negative-diagonal-dominated matrix, i.e., all the elements of its main diagonal are negative numbers ${a}_{ii}<0,\hspace{0.17em}i=\overline{1,\hspace{0.17em}n}$, which are greater in absolute value than the sum of the modules of the non-diagonal elements in the row:

**Lemma**

**1.**

**Proof**

**1.**

**Lemma**

**2.**

**Proof**

**2.**

## 3. Parametrically Uncertain Systems

#### 3.1. Elementary Control Problem

**Remark**

**2.**

**Lemma**

**3.**

**Proof**

**3.**

#### 3.2. Formalisation of a Class of Acceptable Non-Elementary Systems

**Lemma**

**4.**

**Proof**

**4.**

## 4. Simulations

**Example**

**1.**

- 1.a.
- In the matrix $B$, the bottom two rows are linearly independent and form a basis. It is not necessary to rearrange the rows. We assume$${B}_{2}=\left(\begin{array}{cc}2& 1\\ 0& 1\end{array}\right),{T}_{p}=I,T={T}_{a},x=\tilde{x}.$$
- 1.b.
- Using the second Formula (10), we find the cancellation matrix$${B}_{2}^{\ast}={\tilde{B}}_{1}{B}_{2}^{-1}=\frac{1}{2}\left(\begin{array}{cc}1& 0\end{array}\right)\hspace{0.17em}\left(\begin{array}{cc}1& -1\\ 0& 2\end{array}\right)=\left(\begin{array}{cc}0.5& -0.5\end{array}\right)$$$$T=\left(\begin{array}{ccc}1& -0.5& 0.5\\ 0& 1& 0\\ 0& 0& 1\end{array}\right),\hspace{0.17em}{T}^{-1}=\left(\begin{array}{ccc}1& 0.5& -0.5\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$$$$\begin{array}{l}{\dot{x}}_{1}=1.5{x}_{1}+(1.25\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-0.25){x}_{2},\\ {\dot{x}}_{2}=\left(\begin{array}{l}0\\ 1\end{array}\right){x}_{1}+\left(\begin{array}{cc}1& 0\\ 0.5& 0.5\end{array}\right){x}_{2}+\left(\begin{array}{l}2\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1\\ 0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1\end{array}\right)u.\end{array}$$
- 2.a.
- (Procedure 2) In the first subsystem, we take a valid eigenvalue from the given spectrum (61) as the reference matrix: ${A}_{1}=-1$. The local feedback matrix ${x}_{2}={F}_{1(2\times 1)}{x}_{1}$, providing (13), has infinitely many realizations. The solution obtained is similar to the first equality (5):$${A}_{11}+{A}_{10}{F}_{1}={A}_{1}\Rightarrow {F}_{1}={A}_{10}^{+}({A}_{1}-{A}_{11})=\left(\begin{array}{l}{f}_{1}\\ {f}_{2}\end{array}\right)=\left(\begin{array}{l}-25/13\\ \hspace{0.17em}5/13\end{array}\right),\hspace{0.17em}{A}_{10}^{+}=\left(\begin{array}{l}10/13\\ -2/13\end{array}\right)$$$${A}_{11}+{A}_{10}{F}_{1}={A}_{1}\Rightarrow 1.5+\left(\begin{array}{cc}1.25& -0.25\end{array}\right)\left(\begin{array}{l}{f}_{1}\\ {f}_{2}\end{array}\right)=-1\iff {f}_{2}=10+5{f}_{1}.$$Let us assume, for example ${F}_{1}={(-1\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}5)}^{T}$. After performing the transformation (15), we obtain the RF closed by the local relation (16), in the form$${\dot{e}}_{1}=-{e}_{1}+(1.25\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-0.25){e}_{2},\hspace{0.17em}\hspace{0.17em}\phantom{\rule{0ex}{0ex}}{\dot{e}}_{2}=\left(\begin{array}{l}-2\\ \hspace{0.17em}\hspace{0.17em}8\end{array}\right){e}_{1}+\left(\begin{array}{cc}2.25& -0.25\\ -5.75& 1.75\end{array}\right){e}_{2}+\left(\begin{array}{l}2\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1\\ 0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1\end{array}\right)u.$$
- 2.b.
- For the remaining complex-conjugate pair from a given spectrum $\lambda =-1\pm 3j$, we make a reference matrix, e.g., in the form of a Jordanian cell ${A}_{2}=\left(\begin{array}{cc}-1& 3\\ -3& -1\end{array}\right)$, and generate the feedback from the second Formula (17) in the form of$$u=\left(\begin{array}{l}0.5\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-0.5\\ \hspace{0.17em}\hspace{0.17em}0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1\end{array}\right)\left(\left(\begin{array}{l}\hspace{0.17em}\hspace{0.17em}2\\ -8\end{array}\right){e}_{1}+\left(\begin{array}{cc}-2.25& \hspace{0.17em}\hspace{0.17em}0.25\\ -5.75& -1.75\end{array}\right){e}_{2}+\left(\begin{array}{cc}-1& 3\\ -3& -1\end{array}\right){e}_{2}\right),\phantom{\rule{0ex}{0ex}}u=Ke=\left(\begin{array}{ccc}5& -3& 3\\ -8& 2.75& -2.75\end{array}\right)e,$$$${\dot{e}}_{1}=-{e}_{1}+(1.25\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-0.25){e}_{2},\hspace{0.17em}{\dot{e}}_{2}=\left(\begin{array}{cc}-1& 3\\ -3& -1\end{array}\right){e}_{2}.$$
- 2.c.
- Considering the transformations performed, let us find the feedback matrix and form a modal state control law for the initial system in form (19)$${F}_{2\times 3}=K{T}_{e}T=\left(\begin{array}{ccc}5& -3& 3\\ -8& 2.75& -2.75\end{array}\right)\left(\begin{array}{ccc}1& 0& 0\\ 1& 1& 0\\ -5& 0& 1\end{array}\right)\left(\begin{array}{ccc}1& -0.5& 0.5\\ 0& 1& 0\\ 0& 0& 1\end{array}\right),\phantom{\rule{0ex}{0ex}}u=Fx=\left(\begin{array}{ccc}-13& 3.5& -3.5\\ 8.5& -1.5& 1.5\end{array}\right)x,$$

**Example**

**2.**

**Example**

**3.**

## 5. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Utkin, V.A.; Utkin, A.V. Problem of Tracking in Linear Systems with Parametric Uncertainties under Unstable Zero Dynamics. Autom. Remote Control
**2014**, 75, 1577–1592. [Google Scholar] [CrossRef] - Ljung, L. Pespectives on system identification. IFAC Annu. Rev.
**2010**, 34, 1–12. [Google Scholar] [CrossRef] [Green Version] - Krstic, M.; Kanellakopoulos, I.; Kokotovic, P. Nonlinear and Adaptive Control Design; Wiley: New York, NY, USA, 1995. [Google Scholar]
- Ackermann, J. Robust Control: The Parameter Space Approach; Springer: London, UK, 2002. [Google Scholar]
- Kharitonov, V.L. Asymptotic Stability of a Family of Systems of Linear Differential Equations. Differ. Uravn.
**1978**, 1, 2086–2088. [Google Scholar] - Lao, X.X. Necessary and sufficient conditions for stability of a class of interval matrices. Int. J. Control
**1987**, 45, 211–214. [Google Scholar] - Polyak, B.T.; Tsypkin, Y.Z. Frequency Criteria of Robust Stability and Aperiodicity of Linear Systems. Autom. Remote Control
**1990**, 9, 1192–1200. [Google Scholar] - Neimark, Y.I. Robust stability and D-partitioning. Autom. Remote Control
**1992**, 53, 957–965. [Google Scholar] - Gadewadikar, J.; Lewis, F.L.; Abu-Khalaf, M. Necessary and Sufficient Conditions for H-infinity Static Output-Feedback Control. J. Guid. Control Dyn.
**2006**, 29, 4. [Google Scholar] [CrossRef] [Green Version] - Edwards, C.; Shtessel, Y.B. Adaptive Continuous Higher Order Sliding Mode Control. Automatica
**2016**, 65, 183–190. [Google Scholar] [CrossRef] [Green Version] - Antipov, A.S.; Krasnova, S.A.; Utkin, V.A. Methods of Ensuring Invariance with Respect to External Disturbances: Overview and New Advances. Mathematics
**2021**, 9, 3140. [Google Scholar] [CrossRef] - Polyak, B.T.; Sznaider, M.; Shcherbakov, P.S.; Halpern, M. Super-stable control systems. In Proceedings of the 15th IFAC, Barcelona, Spain, 21–26 July 2002; pp. 799–805. [Google Scholar]
- Utkin, V.I.; Yang, K.D. Methods for construction of discontinuity planes in multidimensional variable structure systems. Autom. Remote Control
**1979**, 39, 1466–1470. [Google Scholar] - Drakunov, S.V.; Izosimov, D.B.; Luk’yanov, A.G.; Utkin, V.A.; Utkin, V.I. The Block Control Principle. Autom. Remote Control
**1990**, 5, 601–608. [Google Scholar] - Krasnova, S.A.; Utkin, V.A.; Utkin, A.V. Block approach to analysis and design of the invariant nonlinear tracking systems. Autom. Remote Control
**2017**, 78, 2120–2140. [Google Scholar] [CrossRef] - Krasnova, S.A.; Utkin, V.A.; Sirotina, T.G. A structural approach to robust control. Autom. Remote Control
**2011**, 72, 1639–1666. [Google Scholar] [CrossRef] - Gantmacher, F.R. Theory of Matrices; Chelsea Publishing Company, Inc.: New York, NY, USA, 1959. [Google Scholar]
- Mu, J.; Yan, X.-G.; Spurgeon, S.K.; Mao, Z. Generalized regular form based smc for nonlinear systems with application to a wmr. IEEE Trans. Ind. Electron.
**2017**, 64, 6714–6723. [Google Scholar] [CrossRef] [Green Version] - Wonham, W.F. Linear Multivariate Control: A Geometric Approach; Springer: New York, NY, USA, 1985. [Google Scholar]

**Figure 1.**(

**a**) Plots of ${x}_{1}(t),\hspace{0.17em}{x}_{2}(t),\hspace{0.17em}{x}_{3}(t)$; (

**b**) Plots of ${u}_{1}(t),\hspace{0.17em}{u}_{2}(t)$ in the closed-loop system (59), (62) with $x(0)={(0.5,\hspace{0.17em}0.5,\hspace{0.17em}0.5\hspace{0.17em})}^{\mathrm{T}}$.

**Figure 2.**(

**a**) Plots of ${x}_{1}(t),\hspace{0.17em}{x}_{2}(t),\hspace{0.17em}{x}_{3}(t)$; (

**b**) Plots of ${u}_{1}(t),\hspace{0.17em}{u}_{2}(t)$ in the closed-loop system (59), (63) with $x(0)={(0.5,\hspace{0.17em}0.5,\hspace{0.17em}0.5\hspace{0.17em})}^{\mathrm{T}}$.

**Figure 3.**(

**a**) Plots of ${x}_{1}(t),\hspace{0.17em}{x}_{2}(t),\hspace{0.17em}{x}_{3}(t)$; (

**b**) Plots of ${u}_{1}(t),\hspace{0.17em}{u}_{2}(t)$ in the closed-loop system $\dot{x}=(1.1A+0.9BF)x$, (59), (69) with $x(0)={(0.5,\hspace{0.17em}0.5,\hspace{0.17em}0.5\hspace{0.17em})}^{\mathrm{T}}$.

**Figure 4.**(

**a**) Plots of ${x}_{1}(t),\hspace{0.17em}{x}_{2}(t),\hspace{0.17em}{x}_{3}(t)$; (

**b**) Plots of ${\overline{u}}_{1}(t),\hspace{0.17em}{\overline{u}}_{2}(t)$ in the closed-loop system $\dot{x}=(1.1A+0.9BF)x$, (59), (69), (70).

**Figure 5.**(

**a**) Plots of ${x}_{1}(t),\hspace{0.17em}{x}_{2}(t),\hspace{0.17em}{x}_{3}(t)$; (

**b**) Plots of ${\overline{u}}_{1}(t),\hspace{0.17em}{\overline{u}}_{2}(t)$ in the closed-loop system $\dot{x}=((1+\alpha )A+(1+\beta )BF)x$, (59), (69), (70), $x(0)={(0.5,\hspace{0.17em}0.5,\hspace{0.17em}0.5\hspace{0.17em})}^{T}$.

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**MDPI and ACS Style**

Krasnova, S.A.; Kokunko, Y.G.; Utkin, V.A.; Utkin, A.V.
Robust Stabilization via Super-Stable Systems Techniques. *Mathematics* **2022**, *10*, 98.
https://doi.org/10.3390/math10010098

**AMA Style**

Krasnova SA, Kokunko YG, Utkin VA, Utkin AV.
Robust Stabilization via Super-Stable Systems Techniques. *Mathematics*. 2022; 10(1):98.
https://doi.org/10.3390/math10010098

**Chicago/Turabian Style**

Krasnova, Svetlana A., Yulia G. Kokunko, Victor A. Utkin, and Anton V. Utkin.
2022. "Robust Stabilization via Super-Stable Systems Techniques" *Mathematics* 10, no. 1: 98.
https://doi.org/10.3390/math10010098