Structurally Stable Properties of Control Systems

Shiva Shankar

arXiv:1812.08992·math.OC·November 14, 2019

Structurally Stable Properties of Control Systems

Shiva Shankar

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TL;DR

This paper explores the properties of control systems that remain stable under structural changes, contributing to the theoretical understanding of system robustness and stability.

Contribution

It introduces new theoretical insights into the stability properties of control systems, extending classical results with modern mathematical techniques.

Findings

01

Identification of conditions for structural stability

02

New criteria for robustness of control systems

03

Theoretical framework for analyzing stability under perturbations

Abstract

Talk dedicated to the 110th anniversary of L.S.Pontryagin. Steklov Mathematical Institute of Russian Academy of Sciences, Moscow.

Equations6

\begin{array}[]{lccc}P(\partial):&(\mathcal{C}^{\infty})^{k}&\longrightarrow&(\mathcal{C}^{\infty})^{\ell}\\ &f&\mapsto&P(\partial)f\end{array}

\begin{array}[]{lccc}P(\partial):&(\mathcal{C}^{\infty})^{k}&\longrightarrow&(\mathcal{C}^{\infty})^{\ell}\\ &f&\mapsto&P(\partial)f\end{array}

P (σ, σ^{- 1}) : (C^{k})^{Z^{n}} ⟶ (C^{ℓ})^{Z^{n}}

P (σ, σ^{- 1}) : (C^{k})^{Z^{n}} ⟶ (C^{ℓ})^{Z^{n}}

(\frac{d}{d t} I_{ℓ \times ℓ} - X, - U) : (C^{\infty} (R))^{ℓ + m} ⟶ (C^{\infty} (R))^{ℓ}

(\frac{d}{d t} I_{ℓ \times ℓ} - X, - U) : (C^{\infty} (R))^{ℓ + m} ⟶ (C^{\infty} (R))^{ℓ}

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Taxonomy

TopicsElasticity and Wave Propagation · Mathematical Control Systems and Analysis · Geotechnical and Geomechanical Engineering

Full text

Structurally Stable Properties of Control Systems

Shiva Shankar111Chennai Mathematical Institute, Chennai (Madras), India

Abstract of talk dedicated to the 110th anniversary of L.S.Pontryagin,

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow.

We study properties of control systems that are stable with respect to perturbations. These ideas go back to the notion of structural stability of autonomous systems due to Andronov and Pontryagin. In contrast, control systems usually admit inputs, and are therefore non-autonomous.

We study linear systems defined by partial differential or difference equations. These are systems defined over the ring $A=\mathbb{C}[\partial_{1},\ldots,\partial_{n}]$ of partial differential operators, or the ring $B=\mathbb{C}[\sigma_{1},\sigma_{1}^{-1},\ldots,\sigma_{n},\sigma_{n}^{-1}]$ of Laurent polynomials, respectively. More precisely:

(i) Let $P\subset A^{k}$ be an $A$ -submodule, and suppose it is generated by $p_{1},\ldots,p_{\ell}$ . Let $p_{i}=(p_{i1}(\partial),\ldots,p_{ik}(\partial))$ , and let $P(\partial)$ be the $\ell\times k$ matrix whose rows are the $p_{i}$ . Let $\mathcal{C}^{\infty}$ be the space of smooth functions on $\mathbb{R}^{n}$ . Then

[TABLE]

is an $A$ -module map. The distributed system $\mathcal{B}(P)$ defined by $P$ is the kernel $\{f|P(\partial)f=0\}$ of the above map (it does not depend on the choice of generators for $P$ which defined the matrix $P(\partial)$ , indeed $\mathcal{B}(P)\simeq\mathop{\sf Hom}_{A}(A^{k}/P,~{}\mathcal{C}^{\infty})$ ).

Remark: More generally, we can replace $\mathcal{C}^{\infty}$ by any $A$ -submodule $\mathcal{F}$ of the space $\mathcal{D}^{\prime}$ of distributions on $\mathbb{R}^{n}$ , and study the system $\mathop{\sf Hom}_{A}(A^{k}/P,~{}\mathcal{F})$ . Examples include the spaces $\mathcal{S}^{\prime}$ of tempered distributions, $\mathcal{S}$ of rapidly decreasing functions, the inverse limit $\sf\overleftarrow{\mathcal{H}}$ of the Sobolev spaces, etc. The answers to the questions we address depend on the choice of $\mathcal{F}$ (for instance [5]). We confine ourselves here to the space $\mathcal{C}^{\infty}$ , and the classical result of Malgrange and Palamodov that it is an injective, cogenerating, $A$ -module is of crucial importance [3].

(ii) Let $\mathbb{Z}^{n}$ be the integer lattice, and let $(\mathbb{C})^{\mathbb{Z}^{n}}$ be the set of all functions $w:\mathbb{Z}^{n}\rightarrow\mathbb{C}$ . The term $\sigma_{i}\in B$ acts on $w$ by shift: $\sigma_{i}(w)(m_{1},\ldots m_{n})=w(m_{1},\ldots,m_{i}+1,\ldots,m_{n})$ . Composition then defines the action of a monomial, and by linearity extends to an action of $B$ on $(\mathbb{C})^{\mathbb{Z}^{n}}$ . The $n$ -D system $\mathcal{B}(P)$ defined by a submodule $P\subset B^{k}$ is the kernel of

[TABLE]

where the $\ell$ rows of the matrix $P(\sigma,\sigma^{-1})$ generate $P$ . Again, $\mathcal{B}(P)\simeq\mathop{\sf Hom}_{B}(B^{k}/P,~{}(\mathbb{C})^{\mathbb{Z}^{n}})$ . Here it is elementary that $(\mathbb{C})^{\mathbb{Z}^{n}}$ is an injective cogenerating $B$ -module.

We interpret the above formulation: let $R$ denote either the ring $A$ or $B$ , and $\mathcal{F}$ denote $\mathcal{C}^{\infty}$ or $\mathbb{C}^{\mathbb{Z}^{n}}$ . A system is described by some $k$ ( $\mathbb{C}$ -valued) attributes at various points of $\mathbb{R}^{n}$ or $\mathbb{Z}^{n}$ , each such description is an evolution of the system. A priori, perhaps the system could evolve according to any $f\in\mathcal{F}^{k}$ , but the laws governing the system restrict the possible evolutions $f$ to a subset of $\mathcal{F}^{k}$ . Here the set of laws governing the system is a submodule $P$ of $R^{k}$ , and to say that $f$ must satisfy these laws is to say that it lies in the kernel of equation (1) or (2) above.

Example: State space systems: the state $x:\mathbb{R}\rightarrow\mathbb{R}^{\ell}$ of the system evolves according to $\frac{d}{dt}x=Xx+Uu$ , where $u:\mathbb{R}\rightarrow\mathbb{R}^{m}$ is the input, $X$ and $U$ are $\ell\times\ell$ and $\ell\times m$ matrices with entries in $\mathbb{R}$ . The possible evolutions $f=(x,u):\mathbb{R}\rightarrow\mathbb{R}^{\ell+m}$ that can occur is the kernel of

[TABLE]

$\square$

We wish to study perturbations of such systems, hence we need to topologise the set of all systems. In the context of (i), we need to topologize the set of all submodules $P\subset A^{k}$ , and towards this, we need to first topologize the set $\mathcal{M}_{k}$ of all matrices with $k$ columns and entries from the ring $A$ . We consider structured perturbations - here it means that we consider matrices with a fixed number $\ell$ of rows. Denote this subset by $\mathcal{M}_{\ell,k}$ .

Let $\mathcal{M}_{\ell,k}(d)$ be the subset of those matrices in $\mathcal{M}_{\ell,k}$ whose entries are all bounded in degree by $d$ . There are ${n+d\choose n}$ monomials of degree at most $d$ in $n$ indeterminates, hence we identify $\mathcal{M}_{\ell,k}(d)$ with the $\mathbb{C}$ -affine space of dimension $\ell k{n+d\choose n}$ with the Zariski topology. For $d_{1}<d_{2}$ , $\mathcal{M}_{\ell,k}(d_{1})\hookrightarrow\mathcal{M}_{\ell,k}(d_{2})$ as a Zariski closed subspace, and as $d$ tends to infinity, the direct limit $\mathcal{M}_{\ell,k}$ is equipped with the direct limit topology. A similar construction equips the set of all $\ell\times k$ matrices with entries from the Laurent polynomial ring $B$ with the Zariski topology. These topologies descend to submodules of $A^{k}$ and $B^{k}$ , and hence to distributed and $n$ -D systems respectively (details appear in [6, 7]).

We can now ask if a certain property of a distributed or $n$ -D system is generic with respect to the above topology. In other words, we ask if the property holds for an open dense set of systems.

In this talk, I ask whether the property of being controllable is generic for distributed systems [6]. The question whether the degree of autonomy of an $n$ -D system is generic is answered in [7].

Definition[8]: The distributed system $\mathcal{B}(P)$ defined by a submodule $P\subset A^{k}$ is controllable if for any two subsets $U_{1}$ and $U_{2}$ of $\mathbb{R}^{n}$ whose closures do not intersect, and any two elements $f_{1}$ and $f_{2}$ of $\mathcal{B}(P)$ , there is an element $f$ in $\mathcal{B}(P)$ such that $f=f_{1}$ on some neighbourhood of $U_{1}$ and $f=f_{2}$ on some neighbourhood of $U_{2}$ .

This definition generalizes the definition of a controllable state space system (of Example 1) introduced by Kalman [1] in 1960, here in Moscow!

Theorem 1.1.

[4]** The distributed system $\mathcal{B}(P)$ is controllable if and only if $A^{k}/P$ is torsion free.

Definition The distributed system $\mathcal{B}(P)$ is strictly underdetermined if the submodule $P\subset A^{k}$ can be generated by fewer than $k$ elements (i.e. $\ell<k$ in the notation of (i)). Otherwise, it is overdetermined.

Theorem 1.2.

[6]** A generic strictly underdetermined system is controllable, for this set of systems contains a Zariski open set. Conversely, a generic overdetermined system is uncontrollable.

This follows from the following characterisation of controllability:

Theorem 1.3.

[6]** Let $P$ be a submodule of $A^{k}$ , and let $P(\partial)$ be any $\ell\times k$ matrix whose $\ell$ rows generate $P$ . Suppose the ideal $\mathfrak{i}_{\ell}$ of $\ell\times\ell$ minors of $P(\partial)$ is nonzero (so that $\ell\leqslant k$ ). Then the system $\mathcal{B}(P)$ is controllable if and only if the codimension of the variety of $\mathfrak{i}_{\ell}$ is greater than or equal to 2.

The paper [7] considers the important notion of the degree of autonomy of an $n$ -D system, and shows that this degree is a generic property with respect to the Zariski topology. To prove this, it is first shown that while it is difficult to calculate the dimension of the variety of a specific ideal of the ring $A$ , generically a variety is a complete intersection.

Bibliography8

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] R.E.Kalman, On the general theory of control systems, Proceedings, 1st World Congress of the International Federation of Automatic Control, 1960, Moscow, 481-493.
2[2] U.Oberst, Multidimensional constant linear systems, Acta Applicandae Mathematicae , 1990, 20:1-175.
3[3] V.P.Palamodov, A remark on exponential representation of solutions of differential equations with constant coefficients, Math. USSR Sbornik, 1968, 5: 401-416.
4[4] H.Pillai and S.Shankar, A behavioural approach to control of distributed systems, SIAM jl. Control and Optimization, 1998, 37: 388-408.
5[5] S.Shankar, On the dual of a flat module in TOP, Linear Algebra and its Applications, 2010, 433:1077-1081.
6[6] S.Shankar, The Hautus test and genericity results for controllable and uncontrollable behaviors, SIAM jl. Control and Optimization, 2014, 52:32-51.
7[7] S.Shankar and P.Rocha, The generic degree of autonomy, SIAM jl. Applied Algebra Geometry, 2018, 2:410-427.
8[8] J.C.Willems, The behavioral approach to open and interconnected systems, IEEE Control Systems Magazine, 2007, 27:46-99.

TL;DR

Contribution

Findings

Abstract

Peer Reviews

Videos

Taxonomy

Structurally Stable Properties of Control Systems

Theorem 1.1**.**

Theorem 1.2**.**

Theorem 1.3**.**

Theorem 1.1.

Theorem 1.2.

Theorem 1.3.