Global Controllability for General Nonlinear Systems

Controllability is one of the central issue in modern control theory. For linear systems, it has been fully studied and it turns out that the controllability of a linear system is completely characterized by the rank of its controllability matrix. As for nonlinear systems, things are very different. First, the reachability of linear systems is symmetric (which means that $x_{0}$ is reachable by $x_{1}$ if and only if $x_{1}$ is reachable by $x_{0}$) while in general it is not the case for nonlinear systems. Second, there is a difference between local and global controllability for nonlinear systems.

For global controllability, existing researches mainly focus on a special class of nonlinear systems known as affine systems, that is, control systems of the following form

where $x$ is the state and $u$ is the control. (Rigorous definitions will be given in Section 2.) Although, as far as we know, a complete characterization of global controllability for system (1) is still lacking, many insightful results have been achieved. For example, the controllability problem of switched linear systems has been completely solved in [1] and the criterion is very similar to that of linear ones. For codimension-$1$ affine systems, i.e., that with $n$ state variables and $n-1$ independent inputs, necessary and sufficient conditions are given (separately) in [2]. However, for the $2$-dimensional case, a simple and easily verifiable condition which is both necessary and sufficient is obtained in [3]. The author proved an assertion which is essential for the sufficiency of the condition, that is, any control curve separates the plane into two disjoint components. The result was then extended to $3$-dimensional case under some mild assumptions (see [4]). To our best knowledge, the closest result to a complete solution to this problem was achieved in [5] which, to some extent, covers all the aforementioned results.

In the study of controllability of nonlinear systems, Chow’s theorem (see [6]) plays a key role. It gives a sufficient condition for distributions to be globally controllable, but the condition needs not to be necessary. Indeed, counterexamples are given in [7] along with a condition which is both sufficient and necessary. The ideal is to extend the given distribution to a new and “bigger” one. This ideal is essential. However, the controllability of system (1) is different from that of a distribution. Since the reachability of the latter is symmetric while generally it is not the case for the former. To overcome this difficulty, the concept of foliation was introduced (see [8, 9]). It decomposes the controllability of a nonlinear system into two directions, that is, directions along and traverse the leaves of a foliation. This concept turns out to be powerful in analysis of nonlinear systems, see [10, 11, 12, 13] for some applications of it. For a comprehensive discussion in the global controllability of nonlinear systems, we refer to the books [14, 15, 16].

This note concerns the global controllability of a general nonlinear system, i.e. control system of the following form,

where $x$ is the state and $u$ is the control. (Rigorous definitions will be given in Section 2.) We are going to prove the equivalence between the global controllability of system (2) and a specific affine system using some of the aforementioned techniques.

The organization of the note is as follows. In Section 2, we review some basic preliminaries and fix some notations for statement ease. Our main result along with some illustrative examples are presented in Section 3. The proof of the main result is given in Section 4. Finally, conclusions are given in Section 5.

By a general nonlinear control system, we mean system of the form (2), where the state $x$ belongs to a paracompact manifold $M$, called the state space of the system. Throughout the note the word “manifold ” means smooth, finite-dimensional, second countable, connected and Hausdorff. We use $n$ to denote the dimension of $M$. The control $u$ is a function valued in $\mathcal{C}$, called the control space and, each component $u_{i}$ of $u$ belongs to $\mathcal{U}$, called the set of feasible controls. In the note, $\mathcal{C}$ is always assumed to be the Euclidean space $\mathbb{R}^{m}$ (except in Corollary 2 and its proof) and $\mathcal{U}$ the set of piecewise constant functions. (In the literature, $\mathcal{U}$ is usually assumed to be the set of measurable or piecewise continuous functions, etc. But the choice of $\mathcal{U}$ doesn’t affect the controllability much, see [17, 18].) Moreover, $f:M\times\mathbb{R}^{m}\rightarrow TM$ is a smooth mapping which assigns, to each $x\in M$ and $u\in\mathbb{R}^{m}$ a tangent vector $f(x,u)$ in $T_{x}M$. Here, $TM$ denotes the tangent bundle of $M$ and $T_{x}M$ the tangent space of $M$ at $x$.

Affine control system is a special nonlinear system which is of the form (1), where $x$ and $u$ are as previously. In addition, $f(\cdot)$ and $g_{1}(\cdot),\cdots,g_{m}(\cdot)$ are smooth vector fields on $M$. Here, $g_{i},i=1,\cdots,m$ denotes the $i$-th component of $g$.

An absolutely continuous curve $x:[0,T]\rightarrow M$ is said to be a trajectory of system (2), if there exists a feasible control $u:[0,T]\rightarrow\mathbb{R}^{m}$ such that $\dot{x}(t)=f(x(t),u(t))$ for almost every $t\in[0,T]$. In this case, we call it the trajectory of system (2) associated with control $u$. Given two points $x_{0},x_{1}\in M$, we say that $x_{1}$ is reachable by $x_{0}$, if there exists a feasible control $u:[0,T]\rightarrow\mathbb{R}^{m}$ such that the associated trajectory $x:[0,T]\rightarrow M$ satisfies $x(0)=x_{0}$ and $x(T)=x_{1}$. The set of all points in $M$ that are reachable by $x_{0}$ is called the reachable set of $x_{0}$ and is denoted by $R(x_{0})$. The system (2) is said to be globally controllable, if for each $x\in M$, $R(x)=M$. In the rest of the note, the word “controllability” always means “global controllability” and “controllable” always means “globally controllable”, unless otherwise specified.

In the note, we shall use $\mathbb{R}_{+}$ to denote the subset of $\mathbb{R}$ that contains all the positive real numbers and $\mathbb{R}^{l}_{+}$ to denote its $l$-fold Cartesian product $\underbrace{\mathbb{R}_{+}\times\cdots\times\mathbb{R}_{+}}_{l}$. We shall use $I_{\epsilon}^{l}$ to denote the open subset $\underbrace{(-\epsilon,\epsilon)\times\cdots\times(-\epsilon,\epsilon)}_{l}$ of $\mathbb{R}^{l}$. We simply use $I^{l}$ when the specific value of $\epsilon>0$ is of no great importance.

Let $F$ be a family of vector fields on $M$, then $X$ belongs to $F^{l}$, denoted by $X\in F^{l}$, if there exist $l$ elements $X_{1},\cdots,X_{l}\in F$ such that $X=(X_{1},\cdots,X_{l})$. Let $t=(t_{1},\cdots,t_{l})\in\mathbb{R}^{l}$, we use $e^{tX}$ to denote $e^{t_{1}X_{1}}\circ\cdots\circ e^{t_{l}X_{l}}$ and $e^{tX}_{\ast}$ to denote its tangent map. A mapping $e^{tX}:M\times\mathbb{R}\rightarrow M$ is called physically realizable by system (2) if $t\geq 0$ and

To formulate our main result, we extend system (2) to the following affine system by adding an integrator,

Here $(x,y)^{T}\in M\times\mathbb{R}^{m}$ is regarded as the state and $v\in\mathcal{U}^{m}$ the control of the extended system (3). We shall prove, in the next section, the equivalence between controllability of system (2) and (3), that is,

We shall also prove that if $M$ and $\mathcal{C}$ are compact, then

Here, we give examples to illustrate some applications of our main result.

The previous two examples show the necessary of Theorem 1, the following example which is given in [9] will show the sufficiency of Theorem 1.

To prove Theorem 1, we study the controllability of system (3). Note that the state space $M\times\mathbb{R}^{m}$ of it admits a nature foliation $\mathcal{F}=\left\{M\times\left\{y\right\}|y\in\mathbb{R}^{m}\right\}$, and the vector field $(f(\cdot,\cdot),0)$ is tangent to the leaf $L_{y}\coloneqq M\times\left\{y\right\}$ for any $y\in\mathbb{R}^{m}$. Therefore, we can restrict the vector field $(f(\cdot,\cdot),0)$ to the submanifold $L_{y}$ of $M\times\mathbb{R}^{m}$ to obtain a vector field $(f(\cdot,y),0)$ on $L_{y}$. Moreover, let $g_{i},i=1,\cdots,m$ be the vector field associated with the $i$-th input $v_{i}$, that is $g_{i}=\partial_{y_{i}}$, and $G=\left\{g_{1},\cdots,g_{m}\right\}$, then clearly the mapping $\varphi:L_{y}\rightarrow L_{y_{0}}$ defined by

where $g=(g_{1},\cdots,g_{m})\in G^{m}$ and $t=y_{0}-y\in\mathbb{R}^{m}$, is a diffeomorphism. Hence, we can push-forward the vector fields $(f(\cdot,y),0)$ on all leaves $L_{y}$ of $\mathcal{F}$ to a single leaf $L_{y_{0}}$, resulting in a family of vector fields

on $L_{y_{0}}$ for each $y_{0}\in\mathbb{R}^{m}$.

Note that the family of vector fields $\bar{F}$ on $L_{y_{0}}$ can be identified with the family of vector fields $F\coloneqq\left\{f(\cdot,u)|u=\text{const}\right\}$ on $M$. Indeed, let $\pi$ be the projection mapping from $L_{y_{0}}$ to $M$, i.e.,

It is clear that $\pi$ is a diffeomorphism. Moreover, the tangent mapping

induced by $\pi$ defines a one-to-one correspondence between $\bar{F}$ and $F$.

In addition, we need the following lemma which follows directly from the proof of Lemma 1 in [19].

In this note, we have proved that the global controllability of a general nonlinear system is preserved when an integrator is added. Also, we have extended this result to the case with bounded inputs. By making use of our result, the controllability problem of a general nonlinear system can be converted to that of an affine system which has been studied more thoroughly. The power of our result has been demonstrated by several examples. In addition, the differences between our approach and existing ones have been illustrated.