Stabilization of Control-Affine Systems by Local Approximations of Trajectories111A preliminary version  of this work has appeared in the Proceedings of the 20th IFAC World Congress, 2017.
We study convergence and stability properties of control-affine systems. Our considerations are motivated by the problem of stabilizing a control-affine system by means of output feedback for states in which the output function attains an extreme value. Following a recently introduced approach to extremum seeking control, we obtain access to the ascent and descent directions of the output function by approximating Lie brackets of suitably defined vector fields. This approximation is based on convergence properties of trajectories of control-affine systems under certain sequences of open-loop controls. We show that a suitable notion of convergence for the sequences of open-loop controls ensures local uniform convergence of the corresponding sequences of flows. We also show how boundedness properties of the control vector fields influence the quality of the approximation. Our convergence results are sufficiently strong to derive conditions under which the proposed output feedback control law induces exponential stability. We also apply this control law to the problem of purely distance-based formation control for nonholonomic multi-agent systems.
Consider a driftless control-affine system
on a smooth manifold with a real-valued output . We assume that the control vector fields are smooth, and that the output is given by a smooth function on . Suppose that we are interested in an output feedback control law for the real-valued input channels that stabilizes the system around states where attains an extreme value. Without loss of generality, we restrict our attention to the minima of . In this case, can be interpreted as a cost function that assigns every system state to a real number. Because we deal with a control-affine system, a natural attempt is to choose each input in such a way that, at every point of , the tangent vector points into a descent direction of . The system is then constantly driven into a descent direction of . This can be accomplished by defining each input at every point of as the negative directional derivative of along . In other words, the approach is to choose as the negative Lie derivative of the function along the vector field . We denote the Lie derivative of along at any point of by . However, this control law is not output feedback because its implementation requires information about the Lie derivatives in any state of the system. In our setup, both and are treated as unknown quantities.
The problem of stabilizing creftype 1 at minima of by means of output feedback has been studied extensively in the literature on extremum seeking control. In its most general form, the purpose of extremum seeking control is not restricted to control-affine systems with real-valued outputs, but is aimed at optimizing the output of a general nonlinear control system without any information about the model or the current system state. For an overview of different attempts and strategies on obtaining extremum seeking control, the reader is referred to [45, 21, 38]. A universal solution to this problem is not known.
The content of the present paper is motivated by an approach to extremum seeking control that was first introduced in , and then extended and improved in various subsequent works such as [7, 8, 6, 35, 37, 36]. As explained above, the negative Lie derivatives are promising candidates for the inputs to steer creftype 1 into a descent direction of , but they do not fall into the class of output feedback. Nonetheless, there are ways to obtain access to descent directions of by means of output feedback. For this purpose, we consider vector fields as derivations on the algebra of smooth functions. By doing so, one can define the Lie bracket of two smooth vector fields and as the vector field that acts on any smooth function on as the derivation . It is now easy to check that the vector field is the same as the Lie bracket , where denotes the vector field . Note that this choice of Lie bracket, which is due to , is not the only possible way to obtain access to the vector field . Another option, which is introduced in , is the Lie bracket . In , the nonsmooth approach with fixed is studied. A systematic investigation of suitable Lie brackets can be found in . As a common feature, one can recognize that these Lie brackets are of the form with suitably chosen functions .
At this point, we have merely rewritten the vector fields in terms of Lie brackets. However, it is not yet clear how to relate these Lie brackets to creftype 1 through output feedback, which is a crucial step in the procedure. The strategy is to find time-varying output feedback for creftype 1 such that the trajectories behave at least approximately like the trajectories of creftype 1 under the state feedback . For this purpose, one can exploit known convergence properties of control-affine systems under sequences of open-loop controls, which are extensively studied in [16, 17, 18, 42, 23, 24]. In the following, we only indicate this convergence theory through the example of system creftype 1. The general formalism is explained in the main part of the paper. Suppose that we have chosen suitable functions such that the Lie brackets coincide with the vector fields of interest. For we denote the vector fields and by and , respectively. This defines vector fields on . Our goal is to approximate the solutions of the differential equation
For each , we choose a sequence of measurable and bounded functions . Then, for every sequence index , we consider the differential equation
Note that each can be interpreted as a control-affine system with control vector fields under the open-loop controls . In this situation, we can apply the general convergence theory, which leads to the following result: If the sequences satisfy certain convergence conditions in terms of iterated integrals, then the solutions of the converge to the solutions of as the sequence index tends to . Clearly, the notion of convergence for the sequences has to depend somehow on the limit equation , but we do not go into details at this point. On the other hand, we have the notion of convergence for the trajectories. Roughly speaking, this means that for a fixed initial state and on a fixed compact time-interval, the sequence of solutions of the converges uniformly to the solution of .
The general convergence theory in [18, 24] is not restricted to the example above, but can be applied for arbitrary smooth vector fields in , and for the case in which iterated Lie brackets of the with possibly time-varying coefficient functions appear on the right-hand side of . A system of this form is then usually referred to as an extended system. It is shown in  that, for every extended system, one can find sequences of inputs such that the trajectories of the corresponding sequence of open-loop systems converge to the trajectories of the desired extended system. This result can be applied in the context of motion planning for nonholonomic systems, as in . Another application, which is also studied in the present paper, is the stabilization of control systems. The idea is that stability properties of the extended system carry over to members of the approximating sequence of systems with sufficiently large sequence index. It turns out that this strategy works quite well when the are homogeneous vector fields in the Euclidean space. For this situation, the following implication is known from [31, 30]: If the extended system is asymptotically stable, then the convergence of trajectories ensures that the same is true for members of the approximating sequence of systems with sufficiently large sequence index. However, this statement is not valid when the assumption of homogeneity is dropped. One obstacle is that convergence for a fixed initial state and a fixed compact time interval is in general not strong enough to transfer stability. Instead, the convergence of the trajectories must be uniform with respect to the initial states within compact sets, and the initial time. Under the assumption that this type of convergence is present, the authors of  prove the following implication: If the extended system is locally uniformly asymptotically stable (LUAS), then the approximating sequence of systems is practically locally uniformly asymptotically stable (PLUAS). Here, uniform means uniform with respect to the initial time because the systems are allowed to have possibly nonperiodic time dependencies. Moreover, practical means that we can only expect practical stability for members of the approximating sequence with sufficiently large sequence index.
The strategy of using convergence of trajectories to transfer stability from an extended system to an approximating sequence of systems has also been applied in papers on extremum seeking control, which we cited above. To see this, we insert the sequence of output feedback control laws for into creftype 1, where the functions and the sequences are chosen as before. We then obtain the sequence of differential equations . The results in  show that, if the sequences satisfy certain convergence conditions in terms of iterated integrals, which are slightly stronger than the conditions in [16, 24], then the convergence of trajectories is sufficiently strong to transfer stability. In particular, this leads to the following implication: If a point of is locally asymptotically stable for the autonomous , then this point is PLUAS for the sequence . Note that a sufficient condition to ensure that is locally asymptotically stable for is that attains a strict local minimum at , and that there exists a punctured neighborhood of in which has no critical point and the vector fields span the tangent bundle. In this case, one can solve the problem of stabilizing creftype 1 at a minimum of by means of output feedback, at least approximately.
The present paper contributes to the above research fields in various ways. For instance, we generalize the convergence results in , which are valid for extended systems with Lie brackets of two vector fields, toward extended systems with arbitrary iterated Lie brackets on the right-hand side, in the sense of [16, 24]. We treat the general case of a control-affine system with drift, and allow a possible time dependence of the vector fields. Our convergence results are sufficiently strong to ensure that the approximating sequence of systems is PLUAS whenever the extended system is LUAS. Moreover, we study how the quality of the approximation is influenced by boundedness assumptions on the vector fields of the control-affine system. Under suitable boundedness assumptions, we prove the following new implication: If an extended system is locally uniformly exponentially stable (LUES), then the members of the approximating sequence of systems with sufficiently large sequence index are LUES as well. We also study the problem of stabilizing a general control-affine system at minima of its output function by means of output feedback. Under the additional assumption that at least the minimum value of the output function is known, we present an output feedback control law that not only can induce practical stability but in fact exponential stability. Moreover, we show by the example of a nonholonomic system that our approach has the ability to obtain access to descent directions of the output function along Lie brackets of the control vector fields.
The remainder of this paper is organized as follows. In Section 2, we recall and extend the algebraic formalism from , which is needed to define a suitable notion of convergence for sequences of open-loop controls. The output feedback control law, which we present in Section 6, is not smooth but at least locally Lipschitz continuous. For this reason, we have to define Lie derivatives and Lie brackets in a suitable way. These and other some other definitions from differential geometry are summarized in Section 3. In Section 4, we recall some known notions of stability and present convergence conditions for trajectories of sequences of time-varying systems that are sufficient for the transfer of stability. Then, in Section 5, we present the main convergence results for control-affine systems. In Section 6, we apply these results to the problem of output optimization and distanced-based formation control.
2 Algebraic preliminaries
We begin this section by reviewing some aspects of the algebraic formalism in , which is used throughout this paper. Let be a finite set of noncommuting variables (also called indeterminates). A monomial in is a finite sequence of elements of , which is usually written as a product of the form , where is a finite multi-index with . The length of such a multi-index is denoted by . If and are two multi-indices, then we define their concatenation product as the multi-index by first writing the elements of , and then those of . The concatenation product for multi-indices naturally leads to the product of the associated monomials. In particular, for the empty multi-index of length , we obtain the empty product in , which is the unit element in the monoid of monomials. A noncommutative polynomial in over (or simply a polynomial) is a linear combination over of monomials in . If is a polynomial, it can then be written as with unique real-valued coefficients , where the sum over all multi-indices is finite because all but finitely many of the are zero. The set of all noncommutative polynomials in over is denoted by , which has a natural structure of an associative -algebra. Because of its universal property among all associative -algebra with generators (cf. ), the algebra is called the free associative algebra generated by over . For any two noncommutative polynomials , we define their Lie bracket as usual by . This turns into a Lie algebra. Let be the Lie subalgebra of generated by the variables with respect to the bracket product. The elements of are called Lie polynomials. One can show that is spanned by all Lie brackets of the form
with being a nonempty multi-index with . Because of its universal property among all Lie algebras with generators (cf. ), the Lie algebra is called the free Lie algebra generated by over .
Let be a map defined on with values in . We can then write with unique coefficient functions . The map is said to be bounded (Lebesgue measurable, continuous, locally absolutely continuous etc.) if all of its coefficient functions are bounded (Lebesgue measurable, continuous, locally absolutely continuous, etc.). For every positive integer , let denote the subspace of polynomials whose coefficients with multi-indices of length and are zero. As in , we introduce the following types of -valued maps.
Let be a positive integer. A polynomial input of order is a Lebesgue measurable and bounded -valued map defined on ; we write . An extended input of order is a polynomial input of order , which takes values in . An ordinary input is a polynomial input of order ; here, we write .
When it is not important to name the indeterminates explicitly, we simply write instead of for a polynomial input of order , and write instead of for an ordinary input.
The next definition is important ingredient to obtain a suitable notion of convergence for sequences of ordinary inputs. Again, we refer the reader to  for further explanations.
An th-order generalized difference of an ordinary input and a polynomial input of order is a locally absolutely continuous -valued map defined on that satisfies the differential equation
up to order , i.e., if we write , the differential equations
are satisfied almost everywhere on for and multi-indices of length .
As in [24, 23], we use the notation for an th-order generalized difference of an ordinary input and a polynomial input of order . Note that for any given initial value and any initial time , the unique global solution of creftype 2 with initial condition can be computed recursively by
with the usual convention for the Lebesgue integral of a locally Lebesgue integrable function if .
Now, we define the notion of convergence for sequences of ordinary inputs, which is used in Section 5 to prove convergence of trajectories of control systems. The conditions are slightly stronger than in . This is the price that we have to pay to obtain stronger convergence results for the trajectories.
Let be a sequence of ordinary inputs . Let be a polynomial input of order . We say that GD()-converges uniformly to if there exists a sequence of polynomial inputs of order , and a sequence of th-order generalized differences of and such that for all multi-indices , the following conditions are satisfied:
the sequence of converges to uniformly on as ;
the sequence of converges to uniformly on as ; and
if , then, for , the sequence of converges to uniformly on as ,
For later reference, we cite the following result from .
For some applications, it is important to know how to find a sequence of ordinary inputs that GD()-converges uniformly to a prescribed extended input of order . This problem is solved in . A careful analysis of the proof of the main result in  shows that the sequence constructed therein satisfies the above conditions of uniform GD()-convergence.
3 Time-varying functions, vector fields, and systems
We assume that the reader is familiar with basic concepts of differential geometry, as presented in [1, 19], for example. Let be a smooth manifold, i.e., a paracompact Hausdorff space endowed with an -dimensional smooth structure. Throughout this paper, smooth means , and the term function is reserved for real-valued maps. The algebra of smooth functions on is denoted by . A pseudo-distance function on is a nonnegative function with the following three properties for all : (i) implies , (ii) , and (iii) . If is a pseudo-distance function on , then for every and every nonempty subset of , we let denote the pseudo-distance between and with respect to . A distance function on is a pseudo-distance function on such that implies . We use the term distance function rather than metric to avoid any confusion with the notion of a Riemannian metric. A distance function on is said to be locally Euclidean if for every point there exist a smooth chart for at and constants such that for all . Here and in the following, denotes the Euclidean norm. Note that every distance function that arises from a Riemannian metric is locally Euclidean.
Next, we recall some definitions from . A time-varying function on is a function with domain . A Carathéodory function on is a time-varying function on such that (i) for every , the function is Lebesgue measurable, and (ii) for every , the function is continuous. Let be a time-varying function on , and let be a nonnegative function on . For a given subset of , we say that is uniformly bounded by a multiple of on if there is a constant such that holds for every . We say that is locally uniformly bounded by a multiple of if for every there is a neighborhood of in such that is uniformly bounded by a multiple of on . Moreover, we simply say that is locally uniformly bounded if it is locally uniformly bounded by . Finally, we say that is locally uniformly Lipschitz continuous if for every point there exist a smooth chart for at and a constant such that holds for every and all . Equivalently, using a locally Euclidean distance function on , the function is locally uniformly Lipschitz continuous if and only if for every point , there exist a neighborhood of in and a constant such that for every and all . Note that all of the above definitions also apply to functions on because every function on can be considered as the time-varying function . In this case, we usually omit the adjective “uniform”, which indicates uniformity with respect to time.
A time-varying vector field on is a map defined on that assigns every to a tangent vector to at . We consider the tangent space at any point as the vector space of all derivations of at . In this sense, every time-varying vector field on is a time-varying differential operator on . For every and every , we let denote the Lie derivative of along at , i.e., the real number obtained by applying the tangent vector to . Using Lie derivatives along smooth functions on , all definitions for time-varying functions can be made for time-varying vector fields. For instance, is a Carathéodory vector field if for every , the time-varying function is a Carathéodory function on . Using the same convention as for functions on , we associate every vector field on with the corresponding time-varying vector field .
A curve on is a continuous map from an open, nonempty interval into . A curve is said to be locally absolutely continuous if for every , the function is locally absolutely continuous. If is locally absolutely continuous, then it is differentiable at almost every , and we denote by the tangent vector to at that maps every to . Let be a time-varying vector field on . For given and , an integral curve of with initial condition is a locally absolutely continuous curve with and such that holds for almost every . We call the formal expression a time-varying system on . A solution (or also trajectory) of is an integral curve of . The system is said to have the existence property of solutions if for every and there is an integral curve of with the initial condition . A maximal solution of is a solution that cannot be extended to a solution of defined on a strictly larger interval. The system has the uniqueness property of solutions if, whenever and are two solutions of for which there exists with , then holds on . Suppose that has the existence and uniqueness property of solutions. Then we define for every triple with , , and in the domain of the unique maximal integral curve of with initial condition . The corresponding map is called the flow of (or also the flow of ).
In this paper, we are also interested in Lie derivatives of not necessarily differentiable functions. We define the Lie derivative for the following situation. Suppose that is a time-varying vector field on with the existence and uniqueness property of integral curves, and let be the flow of . Let be a time-varying function on , and let . If the limit
exists, then we call it the Lie derivative of along at . Note that this definition is consistent with the notion of Lie derivatives of smooth functions in terms of derivations (see, e.g., ). Moreover, if the limit
exists, then we call it the time derivative of at . Suppose that are time-varying vector fields on with the existence and uniqueness property of integral curves. Let . If for every , the Lie derivatives and exist, then
defines a tangent vector to at , which is called the Lie bracket of at .
Let be a smooth map from the smooth manifold to another smooth manifold . A function on is said to be constant on the fibers of if for all with we have . A time-varying vector field on is said to be -related to a time-varying vector field on if holds for all and , where denotes the tangent map of at . We say that a time-varying vector field on is tangent to the fibers of if it is -related to the zero vector field on . The map is said to be proper if the preimage of any compact subset of under is a compact subset of . The map is said to be a submersion if at every , the tangent map is surjective. Under the assumption that is a smooth surjective submersion, it is known (see ) that is constant on the fibers of if and only if there exists such that .
4 Stability of time-varying systems
Let be a smooth manifold, and let be a pseudo-distance function on . Throughout this section, we let be a sequence of time-varying systems on . We assume that each has the existence and uniqueness property of solutions. For every sequence index , let be the flow of . On the other hand, let be another time-varying system on with the existence and uniqueness property of solutions, which will play the role of a “limit system” with respect to the sequence . Let be the flow of .
The choice of a sequence index as parameter is motivated by the notation in [16, 42, 24, 23]. We note that all results in the present paper remain valid when the notion of a sequence is replaced by a net, as it is done in . In particular, the sequence index may be replaced by a small parameter that tends to as in [18, 31, 27, 30], or by a frequency parameter that tends to as in [7, 8, 35, 36].
4.1 Notions of stability
For a single system like , we first recall well-known notions of asymptotic and exponential stability with respect to a nonempty subset of ., see, e.g., .
The set is said to be locally uniformly asymptotically stable (abbreviated LUAS) for if
is uniformly stable for , i.e., for every there is such that for every and every with , the trajectory exists on with for every ; and if
is locally uniformly attractive for , i.e., there is some with the following property: for every there is such that for every and every with , the trajectory exists on with for every .
The set is said to be locally uniformly exponentially stable (abbreviated LUES) for if there are such that for every and every with , the trajectory exists on with for every .
For sequences of time-varying systems there is a weaker notions of asymptotic stability, which is due to .
The set is said to be practically locally uniformly asymptotically stable (abbreviated PLUAS) for if
is practically uniformly stable for , i.e., for every there are and a sequence index such that for every , every , and every with , the trajectory exists on with for every ; and if
is practically locally uniformly attractive for , i.e., there is some with the following property: for every there are and a sequence index such that for every , every , and every with , the trajectory exists on with for every .
The set is PLUAS for , then this ensures that the trajectories of are locally attracted into a prescribed -neighborhood of as long as is sufficiently large. However, in general, it is not known how large has to be chosen for a given . In the present paper, we prove a stronger notion of stability for a sequence of systems, namely the following.
For a given sequence index , we say that the set is LUES for if there are such that for every , every , and every with , the trajectory exists on with for every .
4.2 Convergence-based transfer of stability
Our aim is to carry over stability properties of to members of the sequence . It is shown in several earlier works, such as [31, 30, 7], that this can be done by means of convergence of trajectories. For this reason, we introduce the following notion of convergence.
Let be a subset of , and let be a nonnegative function on . We say that the trajectories of converge uniformly in on compact time intervals with respect to to the trajectories of if for every and every there exists a sequence index such that the following approximation property holds: whenever and are such that exists on with for every , then, for every , also exists on and
holds for every .
In particular, when the bound is identically equal to on , then Definition 4.6 coincides with the notion of convergence in [27, 29, 7]. If the bound is chosen as the pseudo-distance to a subset of , then Definition 4.6 corresponds to the approximation property in [28, 30]. Using the same strategy as in these papers, it is easy to derive the subsequent two propositions. We omit the proofs here. In the following, let be a nonempty subset of .
Suppose that is LUAS for . Define for every . Suppose that the trajectories of converge uniformly in some -neighborhood of on compact time intervals with respect to to the trajectories of . Then is PLUAS for .
Suppose that is LUES for . Define for every . Suppose that the trajectories of converge uniformly in some -neighborhood of on compact time intervals with respect to to the trajectories of . Then there exists a sequence index such that is LUES for .
5 Convergence results for control-affine systems
In this section, we present our main convergence results for trajectories of control-affine systems under sequences of open loop controls. They are strong enough to ensure that the convergence properties in Propositions 4.8 and 4.7 are satisfied. As in the previous sections, is any smooth manifold.
5.1 A suitable class of control-affine systems
As an abbreviation for several technical assumptions, we introduce the subsequent class of control-affine systems. A similar class is also considered in . Using the terminology and notation from Section 3, we make the following definition.
For given positive integers , we denote by the set of all -tuples of time-varying vector fields on such that the following properties hold for every .
The time-varying function is locally uniformly bounded, locally uniformly Lipschitz continuous, and, for every , the function is measurable.
For every multi-index of length with , the iterated Lie derivative
exists as a locally uniformly bounded and locally uniformly Lipschitz continuous time-varying function on , and its time derivative
exists as a locally uniformly bounded time-varying function on and is continuous as a function on .
For every multi-index of length with and every , the Lie derivative
exists as a locally uniformly bounded Carathéodory function on .
If , then for every multi-index of length with , we define the iterated Lie bracket
If , then we call the drift vector field and the control vector fields.
If is a vector field without time dependence, and if are vector fields without time dependence, then .
Our goal in this section is to derive a convergence result for control-affine systems under open-loop controls. The proof of this result is based on repeated integration by parts along trajectories of the system. For this purpose, we need the subsequent lemma. The notions of polynomial and ordinary inputs, which appear in the following statement, are introduced in Section 2.
For positive integers , let . Then, for every polynomial input of order , the time-varying system
has the existence and uniqueness property of solutions. Moreover, for any ordinary input , every solution of
every multi-index of length , and every , we have that
the function exists almost everywhere on and is locally Lebesgue integrable;
the function is locally absolutely continuous on and its derivative is given by
at almost every .
The proof is given in Section A.1.
5.2 An open-loop convergence result
In this subsection, let be positive integers and let . As in Definition 2.1, let be a sequence of ordinary inputs , and let be a polynomial input of order . For every sequence index , we consider the time-varying system
and we consider the time-varying system