Robust Stability of Positive Continuous Timesystems

Abstract

This paper investigates 2D mixed continuous–discrete-time systems whose coefficients are polynomial functions of an uncertain vector constrained into a semialgebraic set. It is shown that a nonconservative linear matrix inequality (LMI) condition for ensuring robust stability can be obtained by introducing complex Lyapunov functions depending polynomially on the uncertain vector and a frequency. Moreover, it is shown that nonconservative LMI conditions for establishing upper bounds of the robust ${\mathcal{H}}_{\infty }$ and ${\mathcal{H}}_2$ norms can be obtained by introducing analogous Lyapunov functions depending rationally on the frequency. Some numerical examples illustrate the proposed methodology.

Introduction

The study of 2D mixed continuous–discrete-time systems has a long history, with some early works such as Fornasini and Marchesini (1978) and Roesser (1975) introducing basic models, systems theory and stability properties. Applications of these systems can be found in repetitive processes (Rogers & H, 1992), disturbance propagation in vehicle platoons (Fornasini & Valcher, 1997), and irrigation channels (Knorn & Middleton, 2013).

Researchers have investigated several fundamental properties of 2D mixed continuous–discrete-time systems, in particular stability, for which key contributions include Bouagada and Van Dooren (2013), Chesi and Middleton (2014a), Galkowski, Paszke, Rogers, Xu, and Lam (2003), Kar and Singh (2003), and Rogers and H (1992). Other fundamental properties that have been investigated in 2D mixed continuous–discrete-time systems are the ${\mathcal{H}}_{\infty }$ and ${\mathcal{H}}_2$ norms, for which the contributions include Chesi and Middleton (2015), Paszke, Galkowski, Rogers, and Lam (2008) and Paszke, Rogers, and Galkowski (2011) where conditions based on linear matrix inequalities (LMIs) have been provided. The reader is also referred to Li, Gao, and Wang (2012) and Li, Lam, Gao, and Gu (2015) for related contributions in other areas of 2D systems.

However, these conditions cannot be used whenever the matrices of the model are affected by uncertainty. In fact, in such a case, one should repeat the existing conditions addressing the uncertainty-free case for all the admissible values of the uncertainty. Clearly, this is impossible since the number of values in a continuous set is infinite and one cannot just consider a finite subset of values such as the vertices in the case this set is a polytope. It should be mentioned that various methods have been proposed in the literature for stability and performance analysis of 1D systems affected by uncertainty, such as Aguirre, Garcia-Sosa, Leyva, Solis-Daun, and Carrillo (2015), Aguirre, Ibarra, and Suarez (2002), Bliman (2004), Chesi, 2005, Chesi, 2013, Oliveira and Peres (2007) and Scherer and Hol (2006).

This paper investigates 2D mixed continuous–discrete-time systems affected by uncertainty. It is supposed that the coefficients of the systems are polynomial functions of an uncertain vector constrained into a semialgebraic set. It is shown that an LMI condition for ensuring robust stability can be obtained by introducing complex Lyapunov functions depending polynomially on the uncertain vector and a frequency. Moreover, it is shown that LMI conditions for establishing upper bounds of the robust ${\mathcal{H}}_{\infty }$ and ${\mathcal{H}}_2$ norms can be obtained by introducing analogous Lyapunov functions depending rationally on the frequency. These LMI conditions are sufficient for any chosen degree of the complex Lyapunov functions, and also necessary for a sufficiently large degree of these functions under some mild assumptions on the set of admissible uncertainties. The LMI conditions proposed in this paper exploit Putinar's Positivstellensatz (Putinar, 1993), which allows one to investigate positivity of a polynomial over a semialgebraic set by using polynomials that can be written as sums of squares of polynomials (SOS). Some numerical examples illustrate the proposed methodology.

This paper extends the preliminary conference papers (Chesi, 2014, Chesi and Middleton, 2014b) by showing that the LMI condition for determining the robust ${\mathcal{H}}_{\infty }$ norm is nonconservative (Theorem 3) and by adding the investigation of the robust ${\mathcal{H}}_2$ norm (Section  5).

The paper is organized as follows. Section  2 provides the problem formulation and some preliminaries about SOS matrix polynomials. Section  3 investigates the robust exponential stability. Section  4 addresses the robust ${\mathcal{H}}_{\infty }$ norm. Section  5 addresses the robust ${\mathcal{H}}_2$ norm. Section  6 presents some illustrative examples. Section  7 concludes the paper with some final remarks. Lastly, the appendices report some additional results.

Section snippets

Problem formulation

The notation is as follows. The real and complex number sets are denoted by $\mathbb{R}$ and $\mathbb{C}$ . The imaginary unit is $j$ . The symbol $I$ denotes the identity matrix (of size specified by the context). The notations $Re(\cdot ),Im(\cdot )$ and $|\cdot |$ denote the real part, imaginary part and magnitude. The Euclidean norm and the ${\mathcal{L}}_2$ norm are denoted by ${\Vert \cdot \Vert }_2$ and ${\Vert \cdot \Vert }_{{\mathcal{L}}_2}$ . The adjoint, determinant, null space and trace are denoted by $adj(\cdot ),det(\cdot ),ker(\cdot )$ and $trace(\cdot )$ . The sign is denoted by $sgn(\cdot )$ . The notation $A\otimes B$ denotes the

Robust exponential stability

Let us start by observing that, for the case of 2D mixed continuous–discrete-time systems without uncertainty, a necessary condition for exponential stability is that the matrix multiplying $x_c(t,k)$ in the expression of $\frac{d}{dt}{x}_c(t,k)$ is Hurwitz and the matrix multiplying $x_d(t,k)$ in the expression of $x_d(t,k+1)$ is Schur. In particular, we say that $M\in {\mathbb{C}}^{n\times n}$ is Hurwitz if $Re\left({\lambda }_i\left(M\right)\right)<0\forall i=1,\dots ,n$ and we say that $M\in {\mathbb{C}}^{n\times n}$ is Schur if $\left|{\lambda }_i\left(M\right)\right|<1\forall i=1,\dots ,n\text{.}$ This means that, without loss of generality, we can

Robust ${\mathcal{H}}_{\infty }$ norm

Let us denote with $U_L(s,k)$ and $Y_L(s,k)$ the Laplace transforms of $u(t,k)$ and $y(t,k)$ , where $s\in \mathbb{C}$ . Let us denote with $U_{LZ}(s,z)$ and $Y_{LZ}(s,z)$ the Z-transforms of $U_L(s,k)$ and $Y_L(s,k)$ , where $z\in \mathbb{C}$ . The transfer function from $u(t,k)$ and $y(t,k)$ is denoted by $F(s,z,p)$ and satisfies $Y_{LZ}(s,z)=F(s,z,p)U_{LZ}(s,z)\text{.}$ Standard manipulations show that $F(s,z,p)=F_3(s,p){(zI-F_1(s,p))}^{-1}{F}_2(s,p)+F_4(s,p)$ where $F_1(s,p)$ is given by (22) and $\{\begin{array}{ccc}\hfill F_2(s,p)& \hfill =\hfill & A_{dc}\left(p\right){(sI-A_{cc}\left(p\right))}^{-1}{B}_c\left(p\right)+B_d\left(p\right)\hfill \\ \hfill F_3(s,p)& \hfill =\hfill & C_c\left(p\right){(sI-A_{cc}\left(p\right))}^{-1}{A}_{cd}\left(p\right)+C_d\left(p\right)\hfill \\ \hfill F_4(s,p)& \hfill =\hfill & C_c(p\hfill \end{array}$

Robust ${\mathcal{H}}_2$ norm

The quantity ${\gamma }_2\left(p\right)$ in (10) can be written as ${\gamma }_2\left(p\right)={\Vert F(\cdot ,\cdot ,p)\Vert }_{LZ-{\mathcal{H}}_2}$ where ${\Vert F(\cdot ,\cdot ,p)\Vert }_{LZ-{\mathcal{H}}_2}$ is the Laplace-Z ${\mathcal{H}}_2$ norm of $F(s,z,p)$ defined as $\begin{array}{c}\hfill {\Vert F(\cdot ,\cdot ,p)\Vert }_{LZ-{\mathcal{H}}_2}=\frac{1}{2\pi }\sqrt{{\int }_{-\infty }^{\infty }{\int }_{-\pi }^{\pi }trace\left(F{(j\omega ,e^{j\theta },p)}^{H}F(j\omega ,e^{j\theta },p)\right)d\theta d\omega }\text{.}\hfill \end{array}$ Hence, it follows that ${\gamma }_2\left(p\right)=\sqrt{\frac{1}{2\pi }{\int }_{-\infty }^{\infty }{\Vert F(j\omega ,\cdot ,p)\Vert }_{Z-{\mathcal{H}}_2}^{2}d\omega }$ where ${\Vert F(j\omega ,\cdot ,p)\Vert }_{Z-{\mathcal{H}}_2}$ is the $Z{\mathcal{H}}_2$ norm of $F(j\omega ,\cdot ,p)$ defined as $\begin{array}{c}\hfill {\Vert F(j\omega ,\cdot ,p)\Vert }_{Z-{\mathcal{H}}_2}=\sqrt{\frac{1}{2\pi }{\int }_{-\pi }^{\pi }trace\left(F{(j\omega ,e^{j\theta },p)}^{H}F(j\omega ,e^{j\theta },p)\right)d\theta }\text{.}\hfill \end{array}$ Therefore, a necessary condition for ${\gamma }_2\left(p\right)$ to be finite is $\underset{\omega \to \infty }{lim}F(j\omega ,e^{j\theta },p)=0\text{.}$ The idea to construct upper bounds on ${\gamma }_2$ is

Examples

In this section we present some illustrative examples of the proposed results. The SDPs (34), (54), (76) are solved with the toolbox SeDuMi (Sturm, 1999) for Matlab on a personal computer with Windows 8, Intel Core i7, 3.4 GHz, 8 GB RAM. The SDP (34) is solved with the choice ${\omega }_0=1$ and $p_0$ equal to the center of $\mathcal{P}$ .

Conclusion

This paper has proposed LMI conditions for establishing robust exponential stability and for determining the robust ${\mathcal{H}}_{\infty }$ and ${\mathcal{H}}_2$ norms of 2D mixed continuous–discrete-time systems whose coefficients are polynomial functions of an uncertain vector constrained into a semialgebraic set. The proposed LMI conditions are based on the introduction of complex Lyapunov functions depending polynomially or rationally on a frequency and polynomially on the uncertainty. It has been shown that these LMI

Acknowledgments

The author would like to thank the Associate Editor and the Reviewers for their useful comments.

Graziano Chesi received the Laurea from the University of Florence and the Ph.D. from the University of Bologna. He served as Associate Editor for Automatica, the European Journal of Control, the IEEE Transactions on Automatic Control, the IEEE Transactions on Computational Biology and Bioinformatics, and Systems and Control Letters. He founded and served as chair of the Technical Committee on Systems with Uncertainty of the IEEE Control Systems Society. He also served as chair of the Best

References (31)

• Exponential stability of 2-D systems

  Systems & Control Letters

  (1984)

• G. Chesi

  Establishing stability and instability of matrix hypercubes

  Systems & Control Letters

  (2005)

• B. Aguirre et al.

  Sufficient algebraic conditions for stability of cones of polynomials

  Systems & Control Letters

  (2002)

• B. Aguirre et al.

  Conditions for the schur stability of segments of polynomials of the same degree

  Boletin de la Sociedad Matematica Mexicana

  (2015)

• P.-A. Bliman

  A convex approach to robust stability for linear systems with uncertain scalar parameters

  SIAM Journal on Control and Optimization

  (2004)

• D. Bouagada et al.

  On the stability of 2D state-space models

  Numerical Linear Algebra with Applications

  (2013)

• G. Chesi

  LMI techniques for optimization over polynomials in control: a survey

  IEEE Transactions on Automatic Control

  (2010)

• G. Chesi

  Sufficient and necessary LMI conditions for robust stability of rationally time-varying uncertain systems

  IEEE Transactions on Automatic Control

  (2013)

• Chesi, G. (2014). On the robust H∞ norm of 2D mixed continuous-discrete-time systems with uncertainty. In IEEE...
• G. Chesi et al.

  Homogeneous polynomial forms for robustness analysis of uncertain systems

  (2009)

G. Chesi et al.

Necessary and sufficient LMI conditions for stability and performance analysis of 2D mixed continuous-discrete-time systems

IEEE Transactions on Automatic Control

(2014)

Chesi, G., & Middleton, R.H. (2014b). On the robust stability of 2D mixed continuous-discrete-time systems with...

G. Chesi et al.

H-infinity and H-two norms of 2D mixed continuous-discrete-time systems via rationally-dependent complex Lyapunov functions

IEEE Transactions on Automatic Control

(2015)

E. Fornasini et al.

Doubly-indexed dynamical systems: State-space models and structural properties

Mathematical Systems Theory

(1978)

E. Fornasini et al.

Recent developments in 2D positive system theory

International Journal of Applied Mathematics and Computer Science

(1997)

Cited by (40)

Recommended articles (6)

Graziano Chesi received the Laurea from the University of Florence and the Ph.D. from the University of Bologna. He served as Associate Editor for Automatica, the European Journal of Control, the IEEE Transactions on Automatic Control, the IEEE Transactions on Computational Biology and Bioinformatics, and Systems and Control Letters. He founded and served as chair of the Technical Committee on Systems with Uncertainty of the IEEE Control Systems Society. He also served as chair of the Best Student Paper Award Committees for the IEEE Conference on Decision and Control and the IEEE Multi-Conference on Systems and Control. He is author of the books "Homogeneous Polynomial Forms for Robustness Analysis of Uncertain Systems" and "Domain of Attraction: Analysis and Control via SOS Programming". He is first author in more than 100 publications.

Richard H. Middleton completed his Ph.D. (1987) from the University of Newcastle, Australia. He was a Research Professor at the Hamilton Institute, The National University of Ireland, Maynooth from May 2007 till 2011 and is currently Professor at the University of Newcastle and Head of the School of Electrical Engineering and Computer Science. He has served as Program Chair (CDC 2006), CSS Vice President Membership Activities, and Vice President Conference Activities. In 2011, he was President of the IEEE Control Systems Society. He is a Fellow of IEEE and of IFAC, and his research interests include a broad range of Control Systems Theory and Applications, including Robotics, control of distributed systems and Systems Biology with applications to Parkinson's Disease and HIV Dynamics.

Copyright © 2016 Elsevier Ltd. Published by Elsevier Ltd. All rights reserved.

scottthame1939.blogspot.com

Source: https://www.sciencedirect.com/science/article/abs/pii/S0005109816000431

Share this post