Geometry of Cascade Feedback Linearizable Control Systems

Control theory, simply defined, is an area of research in mathematics and engineering that is fundamentally driven by the question: how can one control the result of some mechanical, electrical, biological, or other process in an acceptable way? For example, autonmous driving and parallel parking, optimal power consumption for super computers, blood sugar levels in humans, confined plasma devices etc. It is an incredibly vague and daunting question that is almost always dependent on the particular process being considered. Nevertheless, there have been many important mathematical and engineering insights that have led to general techniques, as well as the recognition of types of control systems where said techniques may apply. In the course of researching this question, more specific lines of inquiry begin to arise, such as: can one optimally control the process subject to some constraint? How sensitive is the process to the choice of controls? Can the process be controlled even in the presence of random noise? and many more. This thesis is essentially concerned with a particular variant of the question: what types of seemingly nonlinear processes are actually linear in some–potentially expanded–sense? When this is the case, the full array of well developed techniques in linear control theory are suddenly applicable. Even this question has already accumulated a deep literature, some of which we will introduce in the remainder of Chapter 1. For a reasonably detailed overview of the big ideas and history of control theory, the reader is encouraged to peruse [18].

This thesis is in the area of geometric control theory, specifically from the viewpoint of differential geometry. There are many papers from this perspective in control theory, and we will be particularly concerned with results that answer questions of equivalence between control systems; that is, questions related to classifying control systems. Chapter 2 will address the key concepts from differential geometry, as they apply to geometric control theory, that are relevant to this thesis. These concepts include topics such as: distributions, exterior differential systems, derived systems, jet bundles, generalized Goursat bundles, etc. Chapter 3 is specifically concerned with the role of symmetry from the perspective of exterior differential systems, and provides a key property needed to define cascade feedback linearizable systems. Chapter 4 opens with the definition of a cascade feeback linearizable control system, and then presents the primary results of this thesis, including a new necessary condition for cascade feedback linearizable control systems, as well as the presentation of a new explicit class of such systems.

For more on geometric control theory, good places to begin are [12],[31], and [19].

In this section we will define explicitly what we mean by a control system for the purposes of this thesis, as well as introduce several examples that will appear throughout.

As alluded to in the introduction, we mention that, in general, a control system could refer to many different types of differential equations or processes, e.g. PDEs, SDEs, discrete DEs, general stochastic processes, etc. The author is curious to know the extent to which the ideas in this thesis may be applied to other types of control systems, and will likely investigate this to some degree in the future. For now, however, we will be content with our above definition of a control system.

One important property of control systems is the question of controllability.

Studying controllability of control systems is of central importance in the overall field of control theory, and there are many different types of controllability and related notions. We will not explore this topic any further in this thesis, except briefly in Section 1.3 and in the following example. One can refer to [12] for more on controllability. We now list several examples of control systems that appear throughout this thesis.

Solutions for this control system passing through the point $\displaystyle(\textbf{x}_{0}=\textbf{x}(t_{0}))\in\textbf{X}(M)\cong\mathbb{R}^{4}$ are easily seen to be given by

where $\displaystyle C=x^{4}(t_{0})-x^{2}(t_{0})$. We then notice the algebraic constraint $\displaystyle x^{4}(t)=x^{2}(t)+C$ for any choice of $\displaystyle f_{2}(t)$ and for all $\displaystyle t$. Thus, this control system is not controllable. All remaining examples introduced in this section are controllable.

This example first appears in [16], and will have importance as an illustration of the main results in Chapter 4. It has the property of being cascade feedback linearizable, as shown in [16], and in particular, it serves as an example of Theorem 18 in Chapter 4.

Example 3 above is an example of a control system that appears to be a nonlinear system of underdetermined ODE. However, the control system is equivalent to a linear system in a precise way to be defined in Sections 1.2 and 3.7.

The above system in Example 4 will be referred to as the BC system, for Battilotti and Califano, who introduced the system in [7]. This system is also cascade feedback linearizable, as will be demonstrated in Chapter 4.

This relatively simple looking control system will be used only once, and will not reappear until Chapter 4. The control parameters are $\displaystyle z^{1}_{2}$ and $\displaystyle z^{2}_{2}$. It provides a nice demonstration of the necessary condition found in Theorem 17, which is one of the main results of this thesis.

The above family of control systems possesses a familiar set of symmetries–the affine transformations of the real plane. Specific choices of the functions $\displaystyle g^{i}$ and the constants $\displaystyle a_{i}$ will be used to demonstrate various theorems in Chapter 3 and Chapter 4.

One important example to mention, which will not be explored in this thesis, is that of the planar vertical take-off and landing vehicle, (PVTOL) control system, listed above. An in-depth analysis of the system regarding cascade feedback linearization will appear in a later work.

Finally, Example 8 is an 11-parameter family of control systems. Any choice of the parameters leads to a system that is not linearizable (see Definition 6). However, this system does have the property of being linearized when additional differential equations are imposed. This concept will be made precise in Section 1.3.

All of the examples presented in this section have very different properties, and in terms of classification, are all inequivalent to one another.

The most fundamental and well studied class of control systems are those that are linear.

In particular, Example 1 is a linear control system. We demonstrated that it was not controllable, which naturally makes one wonder about when a linear control system is controllable.

We will restrict ourselves to controllable systems from here on. Given a control system, it may be possible to change the state and control variables in such a way that the new system is a linear control system. In particular, a theorem of Brunovský [9] says that all controllable linear control systems may be put into the following form by a specific type of transformation.

Sometimes it is possible to transform a seemingly nonlinear control system to Brunovský normal form via a change of coordinates. Indeed, in [27] and [22] Example (3) was shown to be equivalent to a Brunovský normal form via a change of coordinates where the new state variables are of the form $\displaystyle\textbf{z}=\textbf{z}(\textbf{x})$ and the new control variables have the form $\displaystyle\textbf{v}=\textbf{v}(\textbf{x},\textbf{u})$.

Notice that the forms of the maps only take state variables to state variables, while the new controls are allowed to depend on the old controls and old state variables. This is what is meant by “feedback”. For example, when one is driving a car, the current position of the car is used to determine how to change the steering wheel or acceleration to stay on the road. However, a driver has no way of controlling the shape or orientation of the road in order to keep the car on the road. This property is important for having meaningful solutions for a control system.

The first results concerning when a given nonlinear control system is SFL were given by Krener in [35], as well as by Brockett in [8] and then Jakubczyk and Respondek in [30]. Constructing explicit maps for SFL systems is harder, and that work was started by Hunt, Su, and Meyer in [27], and then a more geometric approach based on symmetry was developed in [20] by Gardner, Shadwick, and Wilkens, and finally work of Gardner and Shadwick [23],[21], and [22] provided what is now known as the GS algorithm for static feedback linearization. The work of Vassiliou in [49] and [50] provides a way to construct the required maps for ESFL systems, as well as identifying when a given control system is ESFL. It is Vassiliou’s work that will be central to this thesis. The main results of the two previously mentioned papers will appear in Chapter 2.

A particularly desirable property for a control system is when solutions can be written purely in terms of arbitrary function and their derivatives. Much like the case of Example 1, determining a solution curve requires no integration, and involves only algebraic expressions of arbitrary functions and their derivatives. Solutions to Brunovský linear control systems always have this property. However, there are nonlinear systems that may also have this property and are not SFL.

It turns out that EI systems are related to another type of linearization called dynamic feedback linearization.

There has been considerable effort to understand DFL systems, more than we can exhaustively list here. The concept first appears in [47], and was subsequently studied in [28], [13], and [14]. A geometric necessary condition based on ruled submanifolds was presented in [48]. A method for producing a DFL, if it exists, was the subject of work by Battilotti and Califano in [5],[6], and [7]. However, a complete classification of DFL systems has yet to be achieved.

The 11-parameter family from Example 8 was shown to be DFL in [48] by differentiating twice along the control $\displaystyle u^{1}$. That is, if we augment (10) by

then the augmented system is SFL. We also have the following nonautonomous version of a dynamic feedback linearizable system.

In forthcoming work [16], it is shown that

Example 4 is DFL, and in fact, in Chapter 4 of this thesis, we will show that Example 4 is EI. In order to prove this, the theory of cascade feedback linearization (CFL) is introduced and applied in Chapter 4. The key idea is the existence of two particular kinds of ESFL systems whose trajectories may be “composed” in order to compute trajectories of Example 4. CFL systems are introduced in [51] where it is shown that such systems are EI. In light of this result and Proposition 1 from [16], we can say that any CFL system is EDFL. In particular, as is shown in [16], carrying out the CFL process tells one how to construct a simple augmented system that is ESFL, therefore demonstrating directly that a CFL system is EDFL.

We would also like to remark briefly on Example 7 mentioned in Section 2. The control system is shown to be DFL by Martin, Devasia, and Paden in [38], and in [51] it is shown to be CFL. However, the sizes of the augmented systems differ from the two constructions, namely the CFL construction in [51] requires a larger augmented system. In another forthcoming work by the author, Example 7 will be explored through the lens of CFL theory more closely, demonstrating, in particular, that there is an augmented system of the same size as that presented in [38].

In this section we provide some background on exterior differential systems (EDS) and derived systems. For a comprehensive account of EDS, refer to [10] and [29]. Throughout this thesis, we will assume that the ranks of all bundles that appear are constant on sufficiently small open sets unless otherwise specified.

We will always consider the case that an EDS is finitely generated as an ideal. We have two ways of generating an EDS: algebraically or differentially. That is,

where $\displaystyle k$ is positive integer, each $\displaystyle\theta^{a}$ is a differential form on $\displaystyle M$, and $\displaystyle\langle\theta^{a}\rangle_{\mathrm{diff}}:=\langle\theta^{a},d\theta^{a}\rangle_{\mathrm{alg}}$. For shorthand, we will often drop the “diff” subscript so that $\displaystyle\langle\theta^{a}\rangle=\langle\theta^{a}\rangle_{\mathrm{diff}}$. An important question about a given EDS is whether or not it admits integral manifolds.

A straightforward example is the case of integral curves of a nowhere vanishing vector field $\displaystyle X$ on a manifold $\displaystyle M^{n}$. Let $\displaystyle\{\theta^{a}\}_{a=1}^{n-1}$ span the space of all 1-forms $\displaystyle\psi$ on $\displaystyle M$ such that $\displaystyle\psi(X)=0$. Then the set of integral manifolds of the EDS $\displaystyle\mathcal{I}=\langle\theta^{a}\rangle$ contains the integral curves of $\displaystyle X$. If one considers the space $\displaystyle L(X)=\text{Span}_{C^{\infty}(M)}\{X\}$, then the set of all integral curves of vectors in $\displaystyle L(X)$ are in 1-1 correspondence with integral manifolds of $\displaystyle\langle\theta^{a}\rangle$.

Sometimes it is desirable to find an $\displaystyle m$-dimensional integral manifold $\displaystyle f:N\to M$ of $\displaystyle\mathcal{I}$ such that $\displaystyle f^{*}\Omega\neq 0$ for a given $\displaystyle m$-form $\displaystyle\Omega=dx^{1}\wedge\cdots\wedge dx^{m}$ on $\displaystyle M$, where $\displaystyle\{x^{i}\}_{i=1}^{m}$ form part of a local coordinate system given by $\displaystyle(x^{1},\ldots,x^{m},y^{m+1},\ldots,y^{n})$. When this is the case, $\displaystyle\Omega$ is called an independence condition, and it plays the role of establishing independent variables for integral manifolds of the EDS. The requirement that $\displaystyle f^{*}\Omega\neq 0$ is equivalent to claiming that $\displaystyle(x^{1},\ldots,x^{m})$ may be chosen as local coordinates for $\displaystyle N$. Then the integral manifold $\displaystyle f:N\to M$ may be thought of as a graph given by $\displaystyle(\textbf{x},\textbf{y}(\textbf{x}))$ where $\displaystyle\textbf{x}=(x^{1},\ldots,x^{m})$ and $\displaystyle\textbf{y}=(y^{m+1},\ldots,y^{n})$. Returning to the example of a vector field $\displaystyle X$, it may be possible to pick a 1-form $\displaystyle\Omega=dt$ such that integral curves to $\displaystyle X$ (and therefore its associated EDS) may be written locally as graphs $\displaystyle(t,x^{1}(t),\ldots,x^{n-1}(t))$, where $\displaystyle(t,x^{1},\ldots,x^{n-1})$ form coordinates for $\displaystyle M$.

The EDS in this thesis will be either Pfaffian or linear Pfaffian systems. Since our systems will be Pfaffian, we will often formulate results using the dual notion of distributions.

We will denote distributions by a set of sections of $\displaystyle TM$ that generate the distribution by $\displaystyle C^{\infty}$ linear combinations. That is, if $\displaystyle\mathcal{V}$ is a distribution of rank $\displaystyle s$, then

where the $\displaystyle X_{i}$, for $\displaystyle 1\leq i\leq s$, are linearly independent sections of $\displaystyle\mathcal{V}\subset TM$. In the case that we have a Pfaffian system, there is a natural distribution whose integral manifolds are the same as those of the Pfaffian system.

We now discuss two equivalent versions of an important theorem in the study of EDS and distributions.

The condition (28) may also be stated as

for all $\displaystyle 1\leq a\leq n-r$. Condition (28) is also equivalent to saying that $\displaystyle\mathcal{I}$ is algebraically generated by 1-forms, i.e.

Certainly, not all Pfaffian systems are Frobenius, and indeed, one might be interested in measuring how far a system deviates from being completely integrable. One can do this by removing all the forms in $\displaystyle\mathcal{I}$ that obstruct the EDS from being Frobenius. This is the idea of the derived system.

Notice that if $\displaystyle{I}^{(1)}=I$ then $\displaystyle\mathcal{I}$ generates a Frobenius Pfaffian system. Similarly, if $\displaystyle\mathcal{V}^{(1)}=\mathcal{V}$ then $\displaystyle\mathcal{V}$ is Frobenius. Furthermore, the derived system of a Pfaffian system is a diffeomorphism invariant since pullback commutes with exterior differentiation and the wedge product. Additionally, we mention that if a distribution and Pfaffian system are related by $\displaystyle\mathcal{V}=\mathrm{ann}\,I$, then $\displaystyle\mathcal{V}^{(1)}=\mathrm{ann}\,{I}^{(1)}$. This fact follows from the identity

for any $\displaystyle X,Y\in\Gamma(TM),\,\theta\in\Omega^{1}(M)$. As there is a first derived system, one can repeat the constructions in Definition 17 to generate a second derived system, and so on.

Additionally, we have the following integer invariants of a derived flag.

The last bundle in the derived flag is always Frobenius. We see that, for distributions, if $\displaystyle\mathcal{V}^{(k+1)}=\mathcal{V}^{(k)}$ then $\displaystyle[\mathcal{V}^{(k)},\mathcal{V}^{(k)}]\subseteq\mathcal{V}^{(k)}$, which is exactly the condition needed to apply the Frobenius theorem. In the case of a Pfaffian system, $\displaystyle{I}^{(k+1)}={I}^{(k)}$ means that $\displaystyle d\theta\equiv\,0\mod{I}^{(k)}$ for all $\displaystyle\theta\in{I}^{(k)}$. Hence the Frobenius condition is satisfied.

Consider a completely integrable Pfaffian system $\displaystyle\mathcal{I}$ of rank $\displaystyle n-r$ on a manifold $\displaystyle M^{n}$. Then the Frobenius theorem says there is a coordinate system $\displaystyle(y^{1},\ldots,y^{n-r},x^{1},\ldots,x^{r})$ where the coordinate functions $\displaystyle\{y^{1},\ldots,y^{n-r}\}$ can be chosen so that $\displaystyle\mathcal{I}$ is generated by $\displaystyle I=\{dy^{1},\ldots,dy^{n-r}\}\subseteq T^{*}M$. Hence the coordinate functions $\displaystyle y^{1},\ldots,y^{n-r}$ are first integrals of $\displaystyle\mathcal{I}$. Furthermore, in this coordinate system, any other first integral $\displaystyle F$ of $\displaystyle\mathcal{I}$ must be of the form $\displaystyle F(y^{1},\ldots,y^{n-r})$. Indeed, if $\displaystyle F(\textbf{x},\textbf{y})$ is a first integral, then $\displaystyle dF\wedge\Omega_{y}=0$, where $\displaystyle\Omega_{y}=dy^{1}\wedge\cdots\wedge dy^{n-r}\neq 0$. However, this means that $\displaystyle\displaystyle\frac{\partial F}{\partial x^{i}}dx^{i}\wedge\Omega_{y}=0$ for all $\displaystyle 1\leq i\leq r$, and thus $\displaystyle F$ has no dependence on $\displaystyle x^{i}$ for all $\displaystyle 1\leq i\leq r$.

A classic example of a completely non-integrable system is the EDS generated by a contact form on $\displaystyle\mathbb{R}^{3}$. Indeed, if

then it is clear that $\displaystyle d(dy-z\,dx)=-dz\wedge dx$. This 2-form is not zero modulo $\displaystyle dy-z\,dx$. Thus the first derived system is the zero ideal, and therefore this EDS has no first integrals. When we consider control systems as linear Pfaffian systems, we will also assume that such systems are completely non-integrable. This is a necessary condition for a control system to be controllable. In fact, the contact system on $\displaystyle\mathbb{R}^{3}$ is a simple example of a control system with 1 control and 1 state, where $\displaystyle x$ is the independent variable. The contact system has only curves as integral manifolds, and such curves are given in coordinates by $\displaystyle(x,f(x),f^{\prime}(x))$ where $\displaystyle f:\mathbb{R}\to\mathbb{R}$.

Another important bundle that is associated to a given EDS/distribution is the Cauchy bundle.

Note that $\displaystyle\mathrm{Char}\,{\mathcal{\mathcal{V}}}$ is integrable. This follows directly from the Jacobi identity on Lie brackets.

Let $\displaystyle X$ be any Cauchy characteristic for $\displaystyle\mathcal{I}$ and let $\displaystyle N$ be any $\displaystyle m$-dimensional integral manifold of $\displaystyle\mathcal{I}$ on $\displaystyle M^{n}$ that is transverse to $\displaystyle X$. Consider the family of submanifolds $\displaystyle N_{s}=\varphi_{s}(N)$, where $\displaystyle\varphi_{s}$ is the 1-parameter family of diffeomorphisms generated by the flow of $\displaystyle X$. Each submanifold in the family $\displaystyle N_{s}$ is a an integral manifold of $\displaystyle\mathcal{I}$, and moreover, the manifold $\displaystyle\bigcup_{s}N_{s}$ is an $\displaystyle m+1$ dimensional integral manifold of $\displaystyle\mathcal{I}$. Hence, knowledge of the Cauchy bundle of an EDS can be used to construct more integral manifolds to the EDS. Additionally, we will always assume that the rank of any Cauchy bundle that appears in this thesis is constant.

Cauchy characteristic vector fields turn out to be a special type of infinitesimal symmetry of an EDS. In general, the flow generated by an infinitesimal symmetry of an EDS will take integral manifolds to integral manifolds. However, we may not be able to construct higher dimensional integral manifolds as in the case of transverse Cauchy characteristics. In this thesis, symmetry plays a particularly important role and will be the main subject in Chapter 2.

We will frequently use a diffeomorphism invariant from [49] to identify particular types of distributions.

A particularly special structure that appears in the study of generalized Goursat bundles (see section 2.6 below) is that of the resolvent bundle. In this section we present the definition of a resolvent bundle as well as important theorems from [49]. Let $\displaystyle\mathcal{V}$ be a subbundle of $\displaystyle TM$ and consider the map

The kernel of this map is exactly the Cauchy bundle of $\displaystyle\mathcal{V}$.

Given some $\displaystyle[X]\in\mathbb{P}\mathcal{V}$, the map $\displaystyle\text{deg}_{\mathcal{V}}:\mathbb{P}\mathcal{V}\to\mathbb{N}$ is called the degree of $\displaystyle[X]$ and is defined by

Note that for $\displaystyle X\in\mathrm{Char}\mathcal{V}$, $\displaystyle\text{deg}_{\mathcal{V}}\left([X]\right)=0$. For this reason, we consider the quotient $\displaystyle\pi:TM\to TM/\mathrm{Char}\mathcal{V}$ and denote all quotient objects by an overbar, so that $\displaystyle\overline{TM}=TM/\mathrm{Char}\mathcal{V}$.