On the finiteness of accessibility test for nonlinear discrete-time systems

TL;DR

This paper establishes finite tests for accessibility in certain classes of nonlinear discrete-time systems by determining a minimal input sequence length, with algorithms to compute this length and identify non-accessible points.

Contribution

It introduces algorithms to compute the accessibility index and its upper bound for analytic and rational systems, enabling finite accessibility testing and analysis.

Findings

Finite accessibility index can be computed for analytic and rational systems.

Algorithms provide a method to determine non-accessible points.

Relations between generic and pointwise accessibility are established.

Abstract

It is shown that for two large subclasses of discrete-time nonlinear systems - analytic systems defined on a compact state space and rational systems - the minimum length $r^*$ for input sequences, called here accessibility index of the system, can be found, such that from any point $x$, system is accessible iff it is accessible for input sequences of length $r^*$. Algorithms are presented to compute $r^*$, as well as an upper bound for it, which can be computed easier, and hence provide finite tests for determination of accessibility. The algorithms also show how to construct the set of points from which the system is not accessible in any finite number of steps. Finally, some relations between generic accessibility of the system and accessibility of individual points in finite steps are given.

Equations

$$x^{⟨1⟩}(t)=\Phi (x(t),u(t)),x(t_0)=x_0$$

$$\mathrm{rank}\left[\frac{\partial \Phi (x,u)}{\partial (x,u)}\right]=n$$

$${\phi }^{⟨1⟩}(x,u,u^{⟨1⟩},\dots ,u^{⟨k+1⟩}):=\phi (\Phi (x,u),u^{⟨1⟩},\dots ,u^{⟨k+1⟩}).$$

$${\Phi }_{x_0}^k(\text{u}):=\Phi (\Phi (\dots (\Phi (x_0,u_0),u_1)\dots ),u_{k-1}).$$

$${\mathcal{A}}^{+}(x_0)=\bigcup _{k\in \mathbb{N}}{\mathcal{A}}_k^{+}(x_0)$$

$$\sup \{\mathrm{rank}\frac{\partial }{\partial \mathbf{u}}{\Phi }_{x_0}^k(\mathbf{u}),\mathbf{u}\in U^k\}=n$$

$$\sup \{\mathrm{rank}\frac{\partial }{\partial \mathbf{u}}{\Phi }_{x_0}^k(\mathbf{u}),\mathbf{u}\in U^k,k\ge 1\}=n.$$

$${\Phi }_{x_0}^{k+1}(U^{k+1})=\{\Phi (y,u):y\in {\Phi }_{x_0}^k(U^k)\text{and}u\in U\}.$$

$$M_1(x,u)$$

$$M_k(x,\mathbf{u})$$

$$\mathrm{rank}(M_k(x_0,\mathbf{u}))<n\text{forall}\mathbf{u}\in U^k.$$

$$B^{⟨n-1⟩}=A^{⟨n-1⟩}{M}_{n-1}P$$

$$B^{⟨n⟩}=A^{⟨n⟩}{M}_{n-1}^{⟨1⟩}{P}^{⟨1⟩}.$$

$$A^{⟨n⟩}{B}^{⟨n-1⟩}=A^{⟨n⟩}{A}^{⟨n-1⟩}{M}_{n-1}P.$$

$$S_n\supseteq S_{n+1}\supseteq S_{n+2}\supseteq \dots \supseteq S_{\infty }.$$

$$p_{k,i}(x,\mathbf{u})=\sum _{j=1}^{j_{k,i}}{a}_{k,i,j}(x)b_{k,i,j}(\mathbf{u})$$

$$I_n⊊I_{n+1}⊊\dots ⊊I_{r^{\ast }}=I_{r^{\ast }+1}=\cdots$$

$${\overline{I}}_n\subseteq {\overline{I}}_{n+1}\subseteq \dots$$

$$x_1^{⟨1⟩}=x_1+T{x}_2,x_2^{⟨1⟩}=x_2+T(a{x}_{1}u-b{x}_2)$$

$$x_1^{⟨1⟩}=\frac{x_2}{u+x_1},x_2^{⟨1⟩}=x_1+x_2.$$

$$x_1^{⟨-1⟩}=\frac{1}{\Gamma }[(b{x}_1+x_2)T-x_1],x_2^{⟨-1⟩}=\frac{1}{\Gamma }[u^{⟨-1⟩}{x}_{1}aT-x_2]$$

$$z_1^{⟨1⟩}=\frac{1}{\Lambda }[(b{z}_1+z_2)T-z_1],z_2^{⟨1⟩}=\frac{1}{\Lambda }[v{z}_{1}aT-z_2]$$

$$x^{⟨1⟩}=\frac{x}{2}+u\sin \pi x,x\in \mathbb{R},u\in \mathbb{R}.$$

Full text

I Introduction

Different variants of accessibility property, being closely related to controllability, are useful concepts in nonlinear control theory, whereas in the linear case all of them coincide. Roughly speaking, forward accessibility from a point $x_{0}$ concerns the ability of input sequences to move the end-point state in all directions of some open subset of state space, and backward accessibility from a point $x_{0}$ concerns the ability of input sequences to move the state from all points of some open subset of the state space toward $x_{0}$. Accessibility is determined by the rank of the accessibility distribution (see [9] for discrete-time systems) computed at the initial state. In almost all control methods, it is assumed that the initial state is a regular point of accessibility distribution, meaning that the accessibility distribution has constant rank in a neighborhood of the initial state. But, checking the validity of this assumption is not a trivial task. Specific for nonlinear systems, it is necessary to distinguish between pointwise accessibility (or shortly accessibility), and generic accessibility (i.e. accessibility from almost any point). Various finite criteria exist for generic accessibility [3], [14], that require no more than $n$ steps, where $n$ is the state dimension. However, for pointwise accessibility, in general, it is not clear a priori how many steps are needed to distinguish between the accessibility singular points (from which the system is not accessible in any finite number of steps) and the points from which the system is forward accessible in a finite number of steps. Moreover, it is even unknown whether an upper bound exists for the number of steps needed. The similar problem exists in the continuous-time case too [22]. While the weaker property of accessibility in $k$ steps can be checked by a finite test [15], [21], the nonlinear system may become accessible from a given point $x$, only when the length of input sequence is larger than some unknown integer $r^{*}$ that can be far larger than the state space dimension. This is why the accessibility rank condition is usually expressed in terms of an infinite set of conditions. [9], [14].

Example 1**.**

Consider the nonlinear discrete-time system described by $x_{1}^{\langle 1\rangle}(t)=x_{2}(t)$ , and $x_{2}^{\langle 1\rangle}(t)=-x_{1}(t)+x_{2}(t)+u(t)x_{2}^{2}(t)-ux_{2}(t)$, where the notation ${\langle 1\rangle}$ denotes the value of the variable in the next time instance. By the results of [3] this system is generically accessible, meaning that starting from almost any initial state, the set of possible end point states in 2 steps for all input pairs $(u(0),u(1))$ contains non-empty open subsets of $\mathbb{R}^{2}$. However, starting from the initial state $x_{0}=(0,1)^{T}$ one can verify that, in spite of the chosen input pairs, the trajectory is $x(1)=(1,1)^{T},x(2)=(1,0)^{T}$. Hence the system is not forward accessible in one or two steps from $x_{0}$. Further computation of $x(3)=(0,-1)^{T}$, $x(4)=(-1,-1+2u(3))^{T}$, shows that in 4 steps, the input sequence affects the end point state only in one direction, hence the system is not forward accessible up to 4 steps. Computing further, we obtain $x(5)=(-1+2u(3),2u(3)+u(4)(2u(3)-1)(2u(3)-2))^{T}$, which shows that $x(5)$ can be moved in two independent directions. To conclude, the system is not forward accessible from $x_{0}$ in four steps or less, but eventually it becomes forward accessible in five steps.

The goal of this paper is to provide a finite test for deciding accessibility (in any number of steps), by finding a certain integer $r^{*}$, called accessibility index of the system, such that for all points of state space, examination of accessibility with respect to input sequences of length $r^{*}$ determines accessibility for any input sequence. To this aim, instead of assessing every individual point of state space separately, we look at the ”big picture” of the entire set of accessibility singular points, by assigning certain algebraic sets $S_{k}$ to the points that are not accessible for inputs of length $k$. Using the language of ideals and varieties, and some geometric properties of the system relying on analyticity and submersivity, we find the smallest $k$ such that $S_{k}$ is an invariant set. This $k$ is the accessibility index of the system. As a byproduct, our method also gives the entire set of accessibility singular points, using a single chain of strictly ascending chain of ideals.

The ideas, based on ideals and algebraic geometry were already successfully used in [4], [5], [10] for characterizing different obervability properties, in [11], [17], [18] for controllability and accessibility of polynomial systems. In [16] a sufficient condition was given for a polynomial system to be accessible from all points under some assumptions. In [12] similar problem was studied and a finite criterion for accessibility from an equilibrium point was introduced. However, there appears to be a gap in their results on accessibility (see Remark 21).

To the best of our knowledge, our results provide the first finite test for accessibility in any number of steps, for rational systems and analytic systems on compact semianalytic sets.

The paper is organized as follows. Preliminaries and definitions are given in Section II. Accessibility criterion, and relation between generic accessibility and pointwise accessibility are obtained in Section III. The main results of the paper are given in Section IV that investigates separately the cases of rational systems and analytic systems with compact state space. Appendix presents algorithms that are used in examples.

II Preliminaries

Consider the discrete-time nonlinear system

[TABLE]

where $x^{\langle 1\rangle}(t):=x(t+1)$ denotes the first order forward shift of the state vector $x(t)$ for any $t\in\mathbb{Z}$, $x(t)\in X\subseteq\mathbb{R}^{n}$, $u(t)\in U\subseteq\mathbb{R}^{m}$, and $X,U$ are open subsets, with the standard topology on $\mathbb{R}^{n}$. Throughout the paper, we assume that the state transition map $\Phi:X\times U\rightarrow X$ is an analytic function. For a matrix with analytic entries, we will use the terms “generic rank” and “maximal rank” interchangeably, since for such matrices these two are equal. The system is assumed to be generically submersive 111The assumption (2) is not restrictive, it is a necessary condition for accessibility [8], which means that

[TABLE]

holds almost everywhere (i.e. except on a zero-measure set) on $X\times U$.

Let $\cal K$ be the field of analytic functions in a finite number of independent variables from the infinite set $\mathcal{C}=\{x,u^{\langle k\rangle},k\geq 0\}$. For a function $\varphi\in\cal K$, the forward shift of $\varphi(x,u,\ldots,u^{\langle k\rangle})$ is defined as

[TABLE]

Let $U^{k}=\underbrace{U\times U\times\ldots\times U}_{k}$ denote the space of control sequences $\textbf{u}=\{u(0),\ldots,u(k-1)\}$. For initial point $x_{0}\in X$ and $k\geq 1$ we define the map $\Phi_{x_{0}}^{k}:U^{k}\rightarrow X$ as

[TABLE]

The image of the map $\Phi_{x_{0}}^{k}$, denoted by $\mathcal{A}_{k}^{+}(x_{0})$ and called the forward-attainable set in $k$ steps from $x_{0}$, describes the set of states that can be obtained by evaluating the solution of the system (1) in $k$ steps from the initial state $x_{0}$ using all control sequence $\textbf{u}\in U^{k}$. By

[TABLE]

we will denote the forward-attainable set in finite number of steps from $x_{0}$ for the system (1). Additionally, if $W\subseteq U^{k}$, then $\Phi_{x_{0}}^{k}(W):=\bigcup_{\textbf{u}\in W}\Phi_{x_{0}}^{k}(\textbf{u})$.

Definition 2**.**

The system (1) is said to be (forward) accessible in $k$ steps from the state $x_{0}\in X$, if the interior of the forward-attainable set $\mathcal{A}_{k}^{+}(x_{0})$ is nonempty. Otherwise it is non-accessible in $k$ steps from $x_{0}$. The system (1) is said to be (forward) accessible from $x_{0}$ (in finite number of steps), if ${\rm int}\mathcal{A}^{+}(x_{0})\neq\emptyset$. This system is said to be non-accessible from $x_{0}$ if it is not accessible from $x_{0}$.

Definition 3**.**

The system (1) is called accessible (pointwise accessible) from a set $R\subset X$ if it is accessible from any point of this set. System (1) is called generically accessible if there is an open and dense subset $D$ of $X$ such that the system is accessible from $D$.

Definition 4**.**

The (finite) integer $r^{*}$ is called the accessibility index of the system (1) over $X$ if $r^{*}$ is the maximum integer for which there exists at least one point $x_{0}\in X$ such that the system remains non-accessible from $x_{0}$ up to $r^{*}-1$ steps, but becomes accessible in $r^{*}$ steps. If there is no such finite integer $r^{*}$, then $r^{*}=\infty$.

III Accessibility Criteria

Various accessibility criteria can be found in literature (for example see [9], [20], [21]), which are more or less similar to the following:

Proposition 5**.**

[9]** For any fixed $x_{0}\in X$ the interior of the forward-attainable set $\mathcal{A}_{k}^{+}(x_{0})$ is nonempty if and only if

[TABLE]

and thus the necessary and sufficient condition for accessibility of system (1) from $x_{0}$ is

[TABLE]

The first step toward the simplification of checking the infinite set of conditions (5) is to show that, under the submersivity assumption (2), the non-accessibility in (exactly) $k$ steps is equivalent to non-accessibility up to $k$ steps .

Lemma 6**.**

Assume that the system (1) is generically submersive. If it is accessible from the state $x_{0}$ in $k$ steps, then it is accessible in $r$ steps too, for any integer $r>k$.

Proof.

Suppose that the system (1) is accessible from $x_{0}$ in $k$ steps. Thus, by definition, $\Phi_{x_{0}}^{k}(U^{k})$ contains an open subset of $X$. First we show that the set $\Phi_{x_{0}}^{k+1}(U^{k+1})$ contains an open subset of $X$, too. The set $\Phi_{x_{0}}^{k+1}(U^{k+1})$ can be written as

[TABLE]

Since, by assumption, $\Phi_{x_{0}}^{k}(U^{k})$ and $U$ contain open subsets of $X$ and $U$, respectively, then from the generic submersivity assumption and from local surjectivity theorem [1], there exists $y_{0}\in{\rm int}(\Phi_{x_{0}}^{k}(U^{k}))$ and an input $u_{0}\in U$ such that the map $\Phi$ is locally surjective around the point $(y_{0},u_{0})$. This means that, there exists an open neighborhood of $y_{0}$ denoted as $\beta(y_{0})\subseteq{\rm int}(\Phi_{x_{0}}^{k}(U^{k}))$, such that $\{\Phi_{y^{*}}(U),~{}y^{*}\in\beta(y_{0})\}$ contains an open subset of $X$. Hence from (6) we conclude that the set $\Phi_{x_{0}}^{k+1}(U^{k+1})$ contains an open subset of $X$. This means that the system (1) is accessible from $x_{0}$ in $k+1$ steps. A simple induction establishes the assertion. ∎

The criteria (5) can be stated in the form of rank of a series of matrices, similar to the controllability rank condition for the time-varying linear discrete-time systems [23].

Let $A(x,{\rm u}):=\frac{\partial\Phi(x,u)}{\partial x}$, $B(x,u):=\frac{\partial\Phi(x,u)}{\partial u}$ and put

[TABLE]

$k=2,3,4,\ldots$, where forward shifts of respective matrices are defined elementwise according to the rule given by (3).

Lemma 7**.**

The system (1) is not accessible up to $k$ steps from a point $x_{0}\in X$ if and only if

[TABLE]

Proof.

From Proposition 5 it follows that the system (1) is accessible from $x_{0}$ in $k$ steps if and only if the matrix $\frac{\partial}{\partial{u}}\Phi_{x_{0}}^{k}(\textbf{u})$ has generic rank $n$ over all $\textbf{u}\in U^{k}$. Differentiating the map $\Phi_{x_{0}}^{k}(\textbf{u})$ and using the chain rule, we obtain $\frac{\partial}{\partial\mathbf{u}}\Phi_{x_{0}}^{k}(\mathbf{u})=[A^{\langle k-1\rangle}A^{\langle k-2\rangle}\ldots A^{\langle 1\rangle}B|A^{\langle k-1\rangle}A^{\langle k-2\rangle}\ldots A^{\langle 2\rangle}B^{\langle 1\rangle}|\ldots\\ \ldots|A^{\langle k-1\rangle}B^{\langle k-2\rangle}|B^{\langle k-1\rangle}](x_{0},\mathbf{u})=M_{k}(x_{0},\mathbf{u})$, which establishes the claim. ∎

Before proceeding to the main result of the paper, we single out systems that are not generically accessible, by showing that such systems are non-accessible anywhere.

Theorem 8**.**

The system (1) is generically accessible if and only if $M_{n}(x,\mathbf{u})$ has generic rank $n$ on $X\times U^{n}$. Moreover, if the system (1) is not generically accessible, then it is non-accessible from any $x$ and for any number of steps.

Proof.

For simplicity, we prove the theorem for the single input case. If the generic rank of $M_{n}$ becomes $n$, then the system (1) is accessible from almost all points, in $n$ steps, and more, via Lemma 6.

For the converse, we use a deduction similar to the proof of the Cayley-Hamilton theorem. Assume that $M_{n}$ has generic rank less than $n$. Then, considering (III), there exists a vector $P=[p_{1},…,p_{n}]^{T}\in{\cal K}^{n}$ such that

[TABLE]

which, after forward shifting, becomes

[TABLE]

Multiplying both sides of (8) by $A^{\langle n\rangle}$ gives

[TABLE]

Now, note that the left hand sides of (9) and (10) are the last two columns of $M_{n+1}$, while the right hand sides of (9) and (10) are linear combinations of the first $n-1$ columns of $M_{n+1}$, thus the generic rank of $M_{n+1}$ is less than $n$. A simple induction proves that also all $M_{k}$ for $k\geq n$ has generic rank less than $n$. Because the generic rank of a matrix with analytic entries is its maximum rank, from Proposition 5 it follows that $\mathcal{A}_{k}^{+}(x)$ has empty interior for all $k$ and all $x\in X$. In this case, from (4) using Baire category theorem [1] we conclude that $\mathcal{A}^{+}(x)$ has empty interior for every $x\in X$. ∎

Theorem 8 shows that generic accessibility of the system is necessary for accessibility from any point $x_{0}$. This property can be checked by a (finite) rank test on the matrix $M_{n}(x,\mathbf{u})$. Therefore, in the rest of the paper, we assume that the system is generically accessible.

Denote by $S_{0}:=X$ and by $S_{k}\subseteq X$ the sets of all states from which the system (1) is not accessible up to $k$ steps. It means that for any $x_{0}\in S_{k}$, sets $\Phi_{x_{0}}^{i}(U^{i})$ for $1\leq i\leq k$ do not contain an open subset of $X$. Additionally, by $S_{\infty}$ we denote the set of points from which system is not accessible in any finite number of steps. These points are called accessibility singular points. Thanks to Lemma 6, we have the following descending chain of subsets:

[TABLE]

By construction, we obtain that $\Phi(S_{k+1},U)\subseteq S_{k}$.

Based on Theorem 8, for an analytic system (1) that is not generically accessible it holds that $S_{\infty}=X$, i.e., the system (1) is non-accessible from all points of $X$.

IV Finiteness conditions of accessibility

Let us recall some basic facts from ideal theory. An ideal $I$ of a commutative ring $\mathcal{P}$ is a subset of $\mathcal{P}$ with the properties that if $p\in\mathcal{P}$ and $a,b\in I$ then $a+b\in I$ and $ap\in I$. The radical of an ideal $I$ of $\mathcal{P}$, denoted by $\sqrt{I}$, is the set $\{p\in\mathcal{P}:\;p^{n}\in I\;\text{for\;some}\;n\in\mathbb{N}\}$. If $I$ coincides with its own radical, then $I$ is called a radical ideal. The real radical of $I$, denoted by $\sqrt[\mathbb{R}]{I}$, is the set of all $p\in\mathcal{P}$ for which there are natural $m,k$ and $q_{1},\ldots,q_{k}\in\mathcal{P}$ such that $p^{2m}+\sum_{i}^{k}q_{i}^{2}\in I$. If $I,J$ are ideals of $\mathcal{P}$, then $(i)$ the real radical of $I$ is an ideal of $\mathcal{P}$, $(ii)$ $I\subseteq\sqrt{I}\subseteq\sqrt[\mathbb{R}]{I}$, $(iii)$ if $I\subset J$ then $\sqrt[\mathbb{R}]{I}\subseteq\sqrt[\mathbb{R}]{J}$, see [6]. A semialgebraic (respectively semianalytic) set $X$ is a set such that for every $x\in X$ there is an open neighborhood $V$ of $x$ with property that $X\cap V$ is a finite Boolean combination of sets $\{\overline{x}\in V:\varphi(\overline{x})=0\}$ and $\{\overline{x}\in V:\psi(\overline{x})>0\}$ where $\varphi,\psi:V\rightarrow\mathbb{R}$ are polynomial functions (respectively analytic functions). For a set $A\subset X$, the Zariski closure of $A$ is defined as the smallest algebraic variety containing A, and is denoted by $\overline{A}$.

To study further forward accessibility of system (1) we need additional assumptions. We will address two cases:

A. function $\Phi$ is rational defined on $X\times U$

B. function $\Phi$ is analytic defined on $X\times U$ with a compact semianalytic $X$.

IV-A Rational case

Let $\mathbb{R}[x]=\mathbb{R}[x_{1},\ldots,x_{n}]$ be the commutative ring of polynomials in $x$. Recall that for an ideal $I$ of $\mathbb{R}[x]$ its zero-set is defined as $\mathcal{Z}(I):=\{x\in\mathbb{R}^{n}:p(x)=0\;\text{for\;all}\;p\in I\},$ i.e. $\mathcal{Z}(I)$ is the set of common zeroes of all polynomials in $I$. If $\mathfrak{Z}\subset\mathbb{R}^{n}$, then the zero-ideal of $\mathfrak{Z}$, denoted by $\mathcal{I}(\mathfrak{Z})$, is defined as the set $\mathcal{I}(\mathfrak{Z})=\{p\in\mathbb{R}[x]:p(x)=0\;\text{for\;all}\;x\in\mathfrak{Z}\}.$ If $I$ is an ideal of $\mathbb{R}[x]$ and $\mathfrak{Z}\in\mathbb{R}^{n}$, then $(i)$ $\mathcal{I}(\mathcal{Z}(I))=\sqrt[\mathbb{R}]{I}$, $(ii)$ $\mathfrak{Z}\subseteq\mathcal{Z}(\mathcal{I}(\mathfrak{Z}))$, $(iii)$ $I\subseteq\mathcal{I}(\mathcal{Z}(I))$, $(iv)$ $\mathcal{Z}(\mathcal{I}(\mathcal{Z}(I)))=\mathcal{Z}(I)$ and $(v)$ $\mathcal{I}(\mathcal{Z}(\mathcal{I}(\mathfrak{Z})))=\mathcal{I}(\mathfrak{Z})$, see [6].

Theorem 9**.**

Assume that the system (1) is rational and generically accessible. Then under submersivity assumption (2)

1. (i)

The descending chain of sets (11) eventually stabilizes. 2. (ii)

The inclusion relation in the descending chain of sets (11) before stabilization is strict. 3. (iii)

The system has a finite accessibility index $r^{*}$, where $r^{*}$ is the smallest integer such that $S_{r^{*}}=S_{r^{*}+1}$.

Proof.

Let $\mathbb{R}_{x}[\mathbf{u}]$ be the commutative ring of polynomials in $\mathbf{u}=\{u_{0},\ldots,u_{k-1}\}$ with coefficients in the ring $\mathbb{R}[x]$. For a fixed $k>n$, let $i_{k}$ be the number of all $n\times n$ submatrices of $M_{k}(x,\mathbf{u})$. Denote by $p_{k,1},\ldots,,p_{k,i_{k}}$ the numerator of determinants of these submatrices as polynomials in $R_{x}[\mathbf{u}]$. So, each $p_{k,i}$ can be written as

[TABLE]

where each $b_{k,i,j}(\mathbf{u})$ is a monomial in $\mathbf{u}$ with coefficient $a_{k,i,j}(x)$ in $\mathbb{R}[x]$. The index $j_{k,i}$ in (12) denotes the number of distinct monomials in $\mathbf{u}$.

By analyticity and using Lemma 6, a point $x_{0}\in X$ belongs to the set $S_{k}$ if and only if $a_{k,i,j}(x_{0})=0$ for all $1\leq i\leq i_{k}$ and $1\leq j\leq j_{k,i}$. Thus every set $S_{k}$ is an algebraic set in $X$. Since the dimension of $X$ is finite, the descending chain of algebraic sets (11) stabilizes after finitely many steps [13], i.e. there exists an integer $r^{*}$ such that $S_{n}\supseteq S_{n+1}\supseteq\ldots\supseteq S_{r^{*}}=S_{r^{*}+1}=\ldots=S_{\infty}.$ Hence the part (i) is proven.

Assume that the integer $l\geq n$ is the smallest integer such that $S_{l}=S_{l+1}$. Since $\Phi(S_{l+1},U)\subseteq S_{l}$ for any $\mathbf{u}\in U^{k}$, equality $S_{l+1}=S_{l}$ implies that $\Phi(S_{l},U)\subseteq S_{l}$, i.e. $S_{l}$ is closed with respect to the multivalued map $\Phi(\cdot,U)$. Now, by assuming generic accessibility, $S_{l}$ is a zero measure set. Hence the image of a point $x\in S_{l}$ under any sequence of inputs from $U^{k}$, belongs to a set of measure zero. Thus all points $x\in S_{l}$ are among accessibility singular points, or say, $S_{l}\subset S_{\infty}$. Obviously, from Lemma 6 we have $S_{\infty}\subset S_{l}$. Thus $S_{l}=S_{r^{*}}=S_{\infty}$ and we have the following strict inclusion property $S_{n}\supsetneq S_{n+1}\supsetneq\ldots\supsetneq S_{r^{*}}=S_{r^{*}+1}=\ldots=S_{\infty}$, which proves the part (ii) of the theorem. Part (iii) is a trivial conclusion of (ii). ∎

As a consequence of Theorem 9 we have the ascending chain

[TABLE]

of ideals in $\mathbb{R}[x]$ where $I_{k}=\mathcal{I}(S_{k})$. Let $I_{M_{k}}:=\langle a_{k,1,1},\ldots,a_{k,i_{k},j_{k,{i_{k}}}}\rangle$ where $\langle a_{k,1,1},\ldots,a_{k,i_{k},j_{k,{i_{k}}}}\rangle$ denotes the ideal generated by $a_{k,1,1},\ldots,a_{k,i_{k},j_{k,{i_{k}}}}$, and $a_{k,i_{k},j_{k,i_{k}}}$ are as in (12). Property $\mathcal{I}(\mathcal{Z}(I))=\sqrt[\mathbb{R}]{I}$ and Lemma 6 assures that $I_{k}=\sqrt[\mathbb{R}]{I_{M_{k}}}$. This allows the usage of computer algebra tools for obtaining the $S_{\infty}$ and respectively the accessibility index $r^{*}$, by checking equality of ideals. Namely, $r^{*}$ is the smallest integer such that $I_{r^{*}}=I_{r^{*}+1}$. Algorithm 1 in Appendix is based on Theorem 9.

Computation of ideals $I_{k}$ involves computation of real radical ideals, which is computationally challenging. So we introduce an alternative approach.

Consider another chain of ideals associated with the system (1) that satisfies conditions of Theorem 9. Namely, consider the ideals $\overline{I}_{k}:=\bigcup_{l=n}^{k}I_{M_{l}}$ for any $k\geq n$. By construction, we have the ascending chain

[TABLE]

of ideals in $\mathbb{R}[x]$.

Theorem 10**.**

Assume that the system (1) is rational and generically accessible. Under submersivity assumption (2), there exists a finite integer $\kappa$ such that $\overline{I}_{\kappa}=\overline{I}_{\kappa+1}$. Moreover, $r^{*}\leq\kappa$ and $\mathcal{Z}(\overline{I}_{\kappa})=S_{\infty}$.

Proof.

Since $\mathbb{R}[x]$ is a Noetherian ring, from the Hilbert Basis Theorem [7] it follows that there exists some $\kappa$ such that $\overline{I}_{\kappa}=\overline{I}_{\kappa+1}$. Now, from the proof of Theorem 9 $S_{k}=\mathcal{Z}\langle a_{k,1,1},\ldots,a_{k,i_{k},j_{k,{i_{k}}}}\rangle$. From the fact that $\mathcal{Z}(\bigcup_{i}I_{i})=\bigcap_{i}\mathcal{Z}(I_{i})$ (see [7]), definition of sets $S_{k}$ and Theorem 9 it follows that $\mathcal{Z}(\overline{I}_{k})=\bigcap_{l=k^{*}}^{k}S_{l}=S_{k}$. Hence, $\mathcal{Z}(\overline{I}_{k})=S_{k}=\mathcal{Z}(I_{k})$. Then from $\overline{I}_{\kappa}=\overline{I}_{\kappa+1}$ we obtain $S_{\kappa}=S_{\kappa+1}$, which using Theorem 9 gives $\mathcal{Z}(\overline{I}_{\kappa})=S_{\infty}$. Since $r^{*}$ is the smallest integer such that $S_{r^{*}}=S_{r^{*}+1}$, therefore from $S_{\kappa}=S_{\kappa+1}$ we have $r^{*}\leq\kappa$. ∎

The (finite) integer $\kappa$ in Theorem 10 serves as an upper bound for the accessibility index $r^{*}$, while its computation does not involve the real radical ideals, and therefore it may be easier to compute. Algorithm 2 in Appendix is based on Theorem 10.

Example 11**.**

Let us consider a current-controlled coil, rotating in a homogeneous magnetic field, as described in [15]. The system equations after Euler’s discretization with step $T$ is

[TABLE]

where $x_{1}$ and $x_{2}$ denote the rotational angle and the angular velocity of the coil, respectively. It is easy to see that system (14) is generically accessible (in 2 steps), and $I_{M_{2}}=\langle x_{1}(x_{1}+Tx_{2})\rangle$, shows that from the points of $\mathcal{V}(I_{M_{2}})$ the system (14) is not accessible in 2 steps. Using Algorithm 2, we obtain $\bar{I}_{2}=I_{M_{2}}=\langle x_{1}(x_{1}+Tx_{2})\rangle,~{}\bar{I}_{3}=\langle x_{1},x_{2}\rangle$, and $\bar{I}_{4}=\bar{I}_{3}$. Therefore, according to the Algorithm 2, $\mathcal{V}(\bar{I}_{3})=(0,0)$ is the only non-accessibility singular point, and accessibility index of the system is $r^{*}=3$.

The following example shows how to compute accessibility index and the set of accessibility singular points for a rational system.

Example 12**.**

Consider the following rational system

[TABLE]

One can easily verify that this system is generically accessible. If we go through the steps of Algorithm 1, firstly we obtain $\sqrt[\mathbb{R}]{I_{M_{2}}}=\langle x_{2}(x_{1}+x_{2})\rangle$. Thus $S_{2}=\{x\in\mathbb{R}^{2}:x_{2}=0\;\text{or}\;\;x_{1}+x_{2}=0\}$ is the set of points that are non-accessible up to two steps. Next, $I_{3}=\sqrt[\mathbb{R}]{I_{M_{3}}}=\langle x_{1},x_{2}\rangle$, so $S_{3}=\{x\in\mathbb{R}^{2}:x_{2}=0,\;x_{1}=0\}=\{(0,0)\}$ is the set of points that are non-accessible up to three steps. By computing $I_{4}=\sqrt[\mathbb{R}]{I_{M_{4}}}=\langle x_{1},x_{2}\rangle$ we obtain that $I_{4}=I_{3}$. So, according to Theorem 10, $S_{3}=S_{4}=S_{\infty}=\{(0,0)\}$, and accessibility index is $r^{*}=3$.

Remark 13*.*

The problem of backward accessibility can be tackled using an approach similar to that taken in this paper for the forward accessibility. System (1) is called * backward accessible* to a point $x_{0}\in X$ if the set of points in $X$ from which the state can be steered toward $x_{0}$ in a finite number of steps, contains a nonempty open subset of $X$. In case of reversible systems, including those that arise from sampling of continuous-time systems, it is known that the original system is backward accessible toward $x_{0}$ if and only if the time inverse system is forward accessible from $x_{0}$. As shown in [3], reversibility assumption may be relaxed and replaced by submersivity assumption.

Example 14**.**

Consider again the system (14) for investigation of backward accessibility. The backward-shift equations of (14) are

[TABLE]

with $\Gamma\coloneqq u^{\langle-1\rangle}aT^{2}+bT-1$. From (16) the inverse (in time) system is obtained as

[TABLE]

with $v$ as input and $\Lambda\coloneqq vaT^{2}+bT-1$. Based on Remark 13, we obtain the singular points of (forward) accessibility of (17). The system (17) is generically forward accessible (in 2 steps), with $I_{M_{2}}=\langle z_{1}(z_{1}-T(bz_{1}+z_{2}))\rangle$. By use of Algorithm 3, we obtain $\bar{I}_{2}:=I_{M_{2}}=\langle z_{1}(z_{1}-T(bz_{1}+z_{2})),~{}\bar{I}_{3}=\langle z_{1},z_{2}\rangle,~{}\bar{I}_{4}=\bar{I}_{3}$. In conclusion, the inverse system (17) is forward accessible from every point in at most 3 steps, except for the point $(0,0)$ which is singular point of forward accessibility for (17). As a result, the original system (14) is backward accessible to every point of the state space in at most 3 steps, except for the point $(0,0)$ which is singular point of backward accessibility for (14).

IV-B Analytic case

Since the ring of analytic functions on $\mathbb{R}^{n}$, unlike $\mathbb{R}[x]$, is not Noetherian, the accessibility index may fail to be finite, as in Example 15 below.

Example 15**.**

Consider the following system

[TABLE]

In this case, the set of points from which the system (18) is not accessible in one step is $S_{1}=\mathbb{Z}$. Going forward and assuming that $x\in S_{1}$, we obtain that $x^{\langle 1\rangle}=\frac{x}{2}$ and $x^{\langle 2\rangle}=\frac{x}{4}+u^{\langle 1\rangle}{\rm sin}\frac{\pi x}{2}.$ Since $x^{\langle 2\rangle}$ is independent on $u$ and $u^{\langle 1\rangle}$ if and only if $x\in 2\mathbb{Z}$, so $S_{2}=2\mathbb{Z}$. Inductively we can see that $S_{k}=2^{k-1}\mathbb{Z}$, $k\in\mathbb{N}$. So, we obtain infinite sequence $S_{1}\supset S_{2}\supset\ldots$ that will never stabilize.

Fortunately, the results of the previous subsection can be extended to the analytic case when the state space $X$ is assumed to be compact, since the ring of real analytic functions $A_{X}$ on a compact semianalytic set $X$ is Noetherian [2]. If $I$ is an ideal of $A_{X}$, then as before $\mathcal{Z}(I)$ denotes the zero-set of $I$ and $\mathcal{I}(\mathfrak{Z})$ denotes the zero-ideal of $\mathfrak{Z}\subset X$, which is now an ideal of $A_{X}$.

Theorem 16**.**

[19]** Let $I$ be an ideal of $A_{X}$. Then $\mathcal{I}(\mathcal{Z}(I))=\sqrt[\mathbb{R}]{I}$.

For a fixed $k>n$ and $\textbf{u}\in U^{k}$ let $q^{i}_{k,\textbf{u}}$, $i=1,\ldots,i_{k}$, denotes determinants of all $n\times n$ submatrices of $M_{k}(x,\textbf{u})$, where $i_{k}$ is the number of minors. For any control sequence u the function $q^{i}_{k,\textbf{u}}(x)$ is an analytic function parameterized by u. Therefore from the definition of $S_{k}$ and using Lemma 6 and analyticity, $S_{k}=\mathcal{Z}(I_{M_{k}})$, where $I_{M_{k}}\coloneqq\langle q^{i}_{k,\textbf{u}}:i=1,\ldots,i_{k},\textbf{u}\in U^{k}\rangle$ is the ideal of analytic functions $q^{i}_{k,\textbf{u}}$. Define $I_{k}=\mathcal{I}(S_{k})$. Theorem 16 assures that $I_{k}=\sqrt[\mathbb{R}]{I_{M_{k}}}$. Consider the ideals $\overline{I}_{k}:=\bigcup_{l=n}^{k}I_{M_{l}}$ for any $k\geq n$. By construction, we have the ascending chain $\overline{I}_{n}\subseteq\overline{I}_{n+1}\subseteq\ldots$ of ideals in $A_{X}$.

Theorem 17**.**

Assume that the system (1) is analytic on a compact semianalytic set $X$ and generically accessible. Then under submersivity assumption (2)

1. (i)

Properties $(i)$ - $(iii)$ of Theorem 9 are valid. 2. (ii)

There exists a finite integer $\kappa$ such that $\overline{I}_{\kappa}=\overline{I}_{\kappa+1}$. Moreover, $r^{*}\leq\kappa$ and $\mathcal{Z}(\overline{I}_{\kappa})=S_{\infty}$.

Proof.

Using the Noetherian property of ${A_{X}}$ on compact semianalytic set $X$ and Theorem 16, the proofs of the parts $(i)$ and $(ii)$ are similar to the proofs of Theorems 9 and 10, respectively. ∎

As a consequence of the part $(i)$ of Theorem 17 we have the ascending chain $I_{n}\subsetneq I_{n+1}\subsetneq\ldots\subsetneq I_{r^{*}}=I_{r^{*}+1}=\cdots$ of ideals of $A_{X}$ with $I_{k}=\mathcal{I}(S_{k})=\sqrt[\mathbb{R}]{I_{M_{k}}}$, $k>n$, which means that the accessibility index $r^{*}$ of the system is the first $k$ for which we have $I_{k}=I_{k+1}$. Part $(ii)$ ot Theorem 17 provides an alternative approach for obtaining an upper bound on $r^{*}$ along with the set of accessibility singular points in the following way: compute the chain of ideals $\overline{I}_{k}$ until for some $\kappa$ we reach that $\overline{I}_{\kappa}=\overline{I}_{\kappa+1}$. Then $\kappa$ serves as an upper bound on the accessibility index $r^{*}$, and $S_{\infty}=\mathcal{Z}(\overline{I}_{\kappa})$. .

Example 18**.**

Let us come back to system (18), but now take $(x,u)\in[0,2]\times[-1,1]$. Using the same reasoning as in Example 15 we obtain $S_{1}=\{0;2\}$, $S_{2}=\{0\}=S_{3}=\ldots$. So, now the sequence $S_{1}\supset S_{2}\supset\ldots$ stabilizes.

IV-C * Properties of the sets $S_{k}$*

The sets $S_{k}$ are important on their own, as in practice states on the sets $S_{k}$ for large enough $k$’s may be considered as singular points of accessibility. Also invariance property of $S_{\infty}$ allows to reduce the dimension of the system on $S_{\infty}$, and intuitively in a neighborhood around it.

Theorem 19**.**

For a generically accessible system, there exists an integer $k^{*}\leq n$ such that for $k\geq k^{*}$ the set $S_{k}$ is a zero-measure set, and for $k<k^{*}$ the set $S_{k}$ is the entire $X$.

Proof.

From Theorem 8 we know that for some $k^{*}\leq n$ the matrix $M_{k^{*}}$ becomes full-rank, which considering the smallest such $k^{*}$, the matrices $M_{k}$ for all $k<k^{*}$ has rank less than $n$ for all $x\in X$ and therefore for every $k<k^{*}$ the set $S_{k}$ is the entire $X$. Since $M_{k^{*}}$ is generically full rank, the set $S_{k^{*}}$ is a zero-measure set, and by (11) for every $k\geq k^{*}$ the set $S_{k}$ is a zero-measure set. ∎

With our notations, a set $S\subseteq X$ is a forward invariant set for the system (1) if for every $x\in S$ it holds that $\Phi_{x}^{i}(U^{i})\in S$ for all $i\in\mathbb{N}$.

Theorem 20**.**

Assume that the system (1) is rational or is analytic on a compact semianalytic set and generically accessible. The set $S_{r^{*}}$ is a forward invariant set for the system. In addition, any other forward invariant set of measure zero is contained in $S_{r^{*}}$.

Proof.

The first part was proved in Theorems 9 and 17 respectively. For the second part, assume that a set $\mathcal{B}\subset X$ is a zero measure set that is also forward invariant for the system (1). Let $y\in\mathcal{\mathcal{B}}$. Since $\mathcal{B}$ is forward invariant, then $\mathcal{A}^{+}(y)\subset\mathcal{B}$, and since $\mathcal{B}$ is a zero measure set, we conclude that ${\rm int}\mathcal{A}^{+}(y)=\emptyset$. So, the system (1) is non-accessible from any $y\in\mathcal{B}$ and hence $\mathcal{B}\subset S_{r^{*}}$. ∎

Remark 21*.*

It was correctly shown in [12] that for any $x_{0}\in X$ and fixed time step $k$, the set $\mathcal{A}_{k}^{+}(x_{0})$ is a semialgebraic set. Unfortunately, $\mathcal{A}^{+}(x_{0})$ which is the union of all sets $\mathcal{A}_{k}^{+}(x_{0})$ for $k\in\mathbb{Z}$, is not necessarily so, because an infinite union of semialgebraic sets in not necessarily a semialgebraic set. Therefore properties of the set $\mathcal{A}^{+}(x_{0})$ may not locally coincide with the properties of its Zariski closure. This point was overlooked in the proof of Lemma 4.1 of [12], and therefore this Lemma is not correct. For example, consider the linear discrete-time system $x_{1}^{\langle 1\rangle}=u,~{}~{}x_{2}^{\langle 1\rangle}=x_{2}+1$. For the initial state $x_{0}=(0,0)$, the set $\mathcal{A}_{k}^{+}(x_{0})$ for $k\geq 0$ is the line $x_{2}=k$, and $\mathcal{A}^{+}(x_{0})$ is the union of all lines $x_{2}=k$, and $\overline{\mathcal{A}^{+}(x_{0})}=\mathbb{R}^{2}$. Based on Lemma 4.1 of [12] the system must be accessible from $x_{0}$. However, the dynamics of $x_{2}$ is completely independent of $u$, and obviously the system is not accessible anywhere.

V Conclusion

The paper suggests minimum number of steps needed to decide (non-)accessibility for two large subclasses of nonlinear systems - rational systems, and analytic systems defined on a compact state space. This number may be far larger than the dimension of the state, unlike in the case of generic accessibility. As a byproduct, the set of singular points $S_{\infty}$ of accessibility have been characterized for these subclasses. From such points the system never becomes accessible. The set $S_{\infty}$ can be found as a limit of non-increasing algebraic subsets $S_{k}$ of state space. For the subclasses addressed, the sequence $S_{k}$ stabilizes in a finite number of steps, thanks to the fact that the rings of functions defined by such systems are Noetherian. Two algorithms are given. One of them allows to compute the accessibility index, i.e. the minimal number of steps to decide accessibility. Another algorithm is easier to apply but provides only an upper bound for accessibility index and also computes $S_{\infty}$.

Acknowledgment

The works of Zbigniew Bartosiewicz and Ewa Pawluszewicz have been supported by grants No. S/WI/1/2016 and S/WM/1/2016, respectively, of Bialystok University of Technology, financed by Polish Ministry of Science and Higher Education.

VI Appendix

Algorithm 1 gives the accessibility index $r^{*}$ and $I_{r^{*}}$ (and therefore $S_{\infty}$).

Algorithm 2 gives $\kappa$ (an upper bound for the accessibility index $r^{*}$), as well as the set $S_{\infty}$.