Homogeneous Dynamical Systems Theory Bijoy K

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462 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 3, MARCH 2002 Homogeneous Dynamical Systems Theory Bijoy K. Ghosh, Fellow, IEEE, and Clyde F. Martin, Fellow, IEEE

Abstract—In this paper, we study controlled homogeneous dy- where the scalar output may be considered to be the slope namical systems and derive conditions under which the system is of the line spanned by the vector and where perspective controllable. We also derive conditions under which the system is observable in the presence of a control over the complex base field. In the absence of any control input, we derive a necessary and sufficient condition for observability of a homogeneous (1.3) dynamical system over the real base field. The observability criterion obtained generalizes a well known Popov–Belevitch–Hautus (PBH) rank criterion to check the observability of a linear dynamical system. Finally, we introduce rational, exponential, interpola- For a dynamical system of the form (1.1), (1.2), one is tion problems as an important step toward solving the problem of interested in controlling only the direction of the state vector realizing homogeneous dynamical systems with minimum state di- and such problems are, therefore, of interest in mensions. gaze control. To generalize the control problem, we consider a Index Terms—Author, please supply your own keywords or send dynamical system with state variable , control variable a blank e-mail to [email protected] to receive a list of suggested and observation function , the projective keywords. space of homogeneous lines in , where we assume that . The dynamical system is described as I. INTRODUCTION HE class of problems we study in this paper has to do with (1.4) T the problem of controlling and observing the orientation of a state vector in . Of course, the problem of “orientation-con- and the projective valued observation function is defined as trol” has been a subject of study in the nonlinear control literature for at least the last two decades [1]. A typical example of such a control problem is “satellite orientation control” by (1.5) active means such as gas jets, magnetic torquing etc. see [2]. In recent years, an important example of the orientation con- where is the homogeneous line spanned by the nonzero trol problem arises in Biology and is known as the “gaze-con- vector . The set is defined as trol” problem. The “gaze-control” problem has close connection with the problem of “eye-movement” (see [3]–[5]), where the problem is to orient the eye so that a target is in the field of view. An example of the dynamical system we study in this The pair (1.4) and (1.5) is a linear dynamical system with a paper is described as follows: homogeneous observation function and has been introduced in [6]–[9] as an example of a perspective dynamical system. The following two problems would initiate two of the important (1.1) questions discussed in this paper. Problem 1.1 (Perspective Control Problem): Let and be two distinct elements of , does there exist a and (1.2) , such that

(1.6)

where is a nonzero vector in such that ? Problem 1.2 (Perspective Observation Problem): Let Manuscript received February 23, 2001; revised July 30, 2001 and August 21, 2001. Recommended by Associate Editor M. E. Valcher. This work was sup- and be two distinct elements of , does there exist ported in part by the National Science Foundation under Grants ECS 9720357 and , such that and ECS 9976174. B. K. Ghosh is with the Department of Systems Science and Math- ematics, Washington University, St. Louis, MO 63130 USA (e-mail: [email protected]). C. F. Martin is with the Department of Mathematics, Texas Tech University, Lubbock, TX 79409 USA (e-mail: [email protected]). (1.7) Publisher Item Identifier S 0018-9286(02)02823-4.

for all where , and where II. PERSPECTIVE CONTROL PROBLEM AND THE RICCATI FLOW and are two nonzero vectors in such that , We start this section with the following definition of perspec- ? tive controllability. The two problems 1.1 and 1.2 refer to the extent the state Definition 2: We shall say that the dynamical system (1.4) is vector can be controlled up to its direction using a control perspective controllable if problem 1.1(1.1) has an affirmative input and can be observed up to its direction using a pro- answer for almost every pair of points , jective valued observation . The proposed class of problems in the usual product topology. can be motivated from machine vision and robotics where the Remark 2.2: A property is said to be generically satisfied, if goal is to guide a robot, possibly a mobile robot, with charge it is satisfied for an open and dense set of points in a topological coupled device cameras as sensors. These tasks are typically space. The phrase almost every pair is used to mean every pair known as visual servoing and has been of interest for at least that belongs to a generic subset of . One choice the last two decades (see [10], [11] for an early and more re- of the generic set would be to choose all pairs of vectors cent reference). The problem 1.2 arises in computer vision as in such that , 2 where is defined in (2.1) the observability problem [12], [13] from visual motion, where as follows. the goal is to estimate the location of a moving object from the For a pair of nonzero vectors , in , let us define image data and may be the structure of the object and motion parameters as well. Often, the image data is produced by a line span (2.1) [14]. and The problem of visual servoing typically deals with the problem of controlling the movement of a robot arm or a column span (2.2) mobile platform guided only by a visual sensor, such as a charged coupled device camera. In the last decade, this problem for . The following theorem characterizes the main re- has been studied in great details and the associated literature sult in perspective control. is large. A major difference in the servoing scheme has been Theorem 2.3: The dynamical system (1.4) is perspective passive servoing wherein the camera is assumed to be held controllable iff for almost every pair of points in permanently fixed to the ceiling and active servoing [15], [16], , the subspaces and have the same where the camera moves with the manipulator. The problem dimension and there exists a real number such that of active camera manipulation is of particular importance in mobile and walking robots, see [17] and [18]. (2.3) The observability problem or perhaps the identifiability problem, if the parameters are changing in time, deals with the Remark 2.4: Note that the notion of controllability in the per- problem of ascertaining the location of a target at the very least spective setting is slightly different from the concept of control- and subsequently estimating the structure and motion parame- lability of a linear dynamical system. A linear system is control- ters, if possible (see [9], [19], and [20] for some new references lable if every pair of vectors in is controllable, whereas on this problem). What makes the perspective observation for perspective controllability, it is enough to have almost every problem interesting is that typically a single camera is unable to pair in to be controllable. This difference is observe the precise position of a target exactly. Thus it becomes important, indeed if and , then there imperative to observe the targets up to their directions with the is no way to reach a state outside of despite the fact that the hope that eventually multiple cameras can precisely locate the system may still be perspective controllable. position. This point of view is in sharp contrast with stereo Proof of Theorem 2.3: Initializing the dynamical system based algorithms, wherein multiple cameras are also used, but (1.4) at we have one requires feature correspondence between various cameras. In the last ten years, many excellent books, tutorials, and sur- (2.4) veys have been written on the topic of vision based control and observation (see [21]–[27]). To summarize the main content of It is well known (see [33]) that the set of all vectors in the right this paper, we analyze problem 1.1, and show that the dynamical hand side of (2.4) for various choice of , is given by all vec- system (1.1) is perspective controllable in between two states iff tors in the subspace . Assume that (1.4) is perspective control- the two states are in the same orbit of a Riccati flow. We also an- lable, it follows that for almost every pair : alyze problem 1.2, and show that the perspective observability of the dynamical system (1.1), (1.2) can be ascertained via a suitable generalization of the Popov–Belevitch–Hautus (PBH) (2.5) rank test, (see [28]). Such rank tests have already been derived in [29]–[31] assuming that the control input is not present. We for some scalars , and and for some , introduce perspective realizability problems and establish con- . Thus, we have nection between these realization problems and the well-known exponential interpolation problems [32]. span (2.6) 464 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 3, MARCH 2002 from which we infer that . Multiplying (2.5) by and let us represent by Grass , the space of all dimen- , we also have sional homogeneous planes in . Clearly the equation

span (2.7) (2.12)

Hence, we infer that and we have . defines a flow in Grass described by Thus, (2.3) is satisfied. Conversely, because and have the same dimension it Grass Grass follows that either and both belong to or they both do (2.13) not. In the former case, it is trivial to find , and such that (2.5) is satisfied. In the latter case it follows from (2.3) that The flow (2.12) can also be described via a Riccati equation as there exists a such that , i.e., follows. Let us denote

(2.8) span of (2.14) where . Note also that for otherwise it would where is a nonsingular matrix and where is a follow that violating the assumption that it is not. It now matrix. Writing follows easily from (2.8) that one can satisfy (2.5) by choosing , and an appropriate choice of . (Q.E.D.) (2.15) Note that Theorem 2.3 may be viewed as a criterion for checking perspective controllability of a homogeneous system (1.4), in between the two directions and . The following where is a matrix, and defining ,we is an important corollary for perspective control. have Corollary 2.5: If , the dynamical system (1.4) is perspective controllable iff (2.16) (2.9) It is trivial to verify that the Riccati flow (2.16) is an equiva- Proof of Corollary 2.5: Assume (2.9), it follows that for all lent representation of the homogeneous flow (2.13) up to time that are not in ,wehave and hence (2.3) is when is singular. trivially satisfied. Thus from Theorem 2.3 it follows that the dy- Remark 2.8: Regarding the proof of Corollary 2.5, for all namical system is perspective controllable for every pair of vec- the vectors that are not in , dimensions of and tors not in . These vectors would give rise to a generic are equal and is one greater than the dimension of .Soifthe pair of points in , hence, the dynamical system (1.4) is dimension of is or , it would follow that and perspective-controllable. are the same subspace and is equal to the entire space . If the Conversely, assume that .For there dimension of is less than , the subspaces and still exists two vectors not in and are such that have the same dimension but they are proper subspaces of .If we consider the flow described by (2.13), and initialize the flow, (2.10) say on , the problem is to ascertain if the other subspace lies on the orbit of the flow. Except for the special case , for all . Hence (1.4) is not perspective controllable as noted in Remark 2.6, it is always possible to choose a generic in between , and . Moreover there exists an open neigh- pair of vectors , such that (2.10) is satisfied. borhood of and , such that (2.10) is satisfied. Hence, a For many other properties of the phase portrait of a Riccati generic pair is not perspective controllable. (Q.E.D.) equation we refer to [34]. We now have the following restate- Remark 2.6: For , the above corollary 2.5 is not true. ment of Theorem 2.3 which we state without proof. If it would follow that for two nonzero vectors Theorem 2.9: The dynamical system (1.4) is perspective and in , we would have for . controllable in between the two directions and in Perspective controllability would depend upon, whether or not iff the subspaces and have the same dimension, (2.3) is satisfiable for some . For certain choice of and say , and are in the same orbit of the homogeneous flow (2.4) is satisfiable for every pair of nonzero vectors. One such (2.13). choice is when and when

III. PERSPECTIVE OBSERVABILITY IN THE ABSENCE OF A CONTROL Remark 2.7: In this remark we point out that the perspective As a special case of the perspective observability problem 1.2 controllability problem has connection with a flow on an asso- considered in the introduction, we now consider the dynamical ciated Grassmannian which is described by a Riccati equation. system (1.4) under the assumption that , i.e., there is no Let us assume that influence of the control. We continue to assume that the observation function is projective valued and is given by (1.5). (2.11) The perspective observation problem is described as follows. GHOSH AND MARTIN: HOMOGENEOUS DYNAMICAL SYSTEMS THEORY 465

Problem 3.1: Let and be two linearly independent vec- 1) For , the dynamical system (1.4) is perspective tors in , does there exist such that unobservable. 2) There exists a real number and a real decomposable vector , such that for all ? Note that the problem 3.1 is equivalent to problem 1.2 when (3.3) the control input is set to zero. The notion of perspective observability in this section would always refer to problem 3.1. 3) There exist two numbers , which are either both real Problem 3.1 already has a satisfactory answer over and has (may be the same) or complex conjugates of each other been studied in [29] and [30]. It has been shown that a necessary such that and sufficient condition for perspective observability over is that the rank of the matrix rank (3.4)

rank (3.1) Remark 3.4: The first two equivalent conditions of the theorem 3.3 basically says that the perspective unobservability of for all pairs of eigenvalues , (may be the same) of the (1.4), assuming , is equivalent to the regular unobserv- matrix .Over the rank condition (3.1) is only sufficient. ability (in the sense of a linear system) of the hat system (3.3) In order to state the main observability result of this section, we with the additional requirement that the unobseravbility sub- define two-fold wedge product of as follows. space of the hat system must contain a decomposable vector. Definition 3.2: Define as the vector space whose Decomposability of a vector in is hard to check. The vectors are linearly generated by the vectors third condition of the Theorem 3.3 provides a computationally feasible solution, which involves checking the rank of a matrix for every real or complex conjugate pairs of eigenvalues of the matrix . Addition in is multilinear and alternating in the compo- Remark 3.5: Theorem 3.3 essentially obtains a necessary nents. If is a basis of , then it follows from the and sufficient condition for perspective unobservability over multilinearity and the alternating property of the wedge product and the condition is described as a generalization of the well that: known PBH rank condition. Proof of Theorem 3.3 : ( ) Assume that there exists a vector such that (3.3) is satisfied. It follows is a basis of . If a vector in has a representation that: for some particular vectors , one says that is a decomposable vector. (3.5) We now introduce the following hat system that has already been considered in [29]. Define the vector space , [36] From (3.5), we conclude the following: and consider the linear map

given by

(3.6) We also consider the linear map Thus, as a vector in , and are linearly independent implying that (1.4) is perspective unobservable for . ( ) Assume that (1.4) is perspective unobservable for given by . It follows that there exists two vectors such that and:

(3.7) We now define the promised hat system as follows:

(3.2) In order to show (3.3), we need to show that there is a real decomposable eigenvector of in the kernel of . where and . The main result of this Define two complex conjugate vectors and as follows: section is summarized in the following theorem. Theorem 3.3: The following three conditions are equivalent over . (3.8) 466 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 3, MARCH 2002

It follows from (3.7) and (3.8) that: When and are complex conjugates of each other we write (3.9)

The relation (3.9) follows easily from the fact that: (3.15)

Let be as before. We then define We now proceed by expanding the vectors and in terms of the eigenvectors (generalized eigenvectors) of . Let (3.16) (3.10) It is easy to verify that be the self conjugate set of eigenvectors (generalized eigenvectors) of the matrix . Expanding in terms of the eigenvectors (generalized eigenvectors) of for we write (3.17)

Moreover where we can assume that and either , are both real or are complex conjugates of each other. We can also assume that the vectors in the set (3.10) have been ordered so that we since claim that (3.18)

It is an easy calculation to show that and when ,wehave

where is the multiplicity of the eigenvalue corresponding It would therefore follow that: to . Such an ordering is always possible and has been detailed in [29]. It now follows from (3.9) that: and (3.11) where is an eigenvector of corresponding to the eigenvalue . We define and and the proof is complete. If and are both real, then the proof is over. If they are (2 3) By assumption, we have such that (3.3) complex conjugates of each other, we can write is satisfied. By decomposing the vector in terms of the eigenvectors (generalized eigenvectors) of the matrix , we can (3.12) repeat the necessity proof of the first part of this theorem to show that there exist two eigenvectors (or generalized eigenvectors) It is easy to check that . It follows from and of such that for all .In (3.11) that: particular, we have . Define a scalar such that

(3.13) For it would follow that and Thus, we have completing the proof. (Q.E.D.)

(3.14) IV. PERSPECTIVE OBSERVABILITY IN THE PRESENCE OF A CONTROL Since is always real, this completes the proof. In this section, we would consider the perspective observa- Proof of Theorem 3.3 : ( ) When tion problem 1.2. However, the base field is assumed to be the and are both real, there is nothing else to prove. If is complex field , therefore appropriate changes have to be made a vector such that and , to the statement of problem 1.2. The essential question is that we define and . It is easy to see that if the state of the dynamical system (1.4), (1.5) is not ob- and which servable from the output function in (1.5), with , implies (3.3). can it be made observable by a proper choice of a control signal GHOSH AND MARTIN: HOMOGENEOUS DYNAMICAL SYSTEMS THEORY 467

? It is perhaps clear that if we have two nonzero vectors and such that for all , (1.7) can never be satisfied for any control input . In order to state the main result, we assume that these two nonzero vectors satisfy in are linearly dependent for any choice of control input and for every . Thus and are perspective (4.1) unobservable. (Q.E.D.) Before we state the main result on perspective observability, for , where . we consider the following definition. Definition 4.1: A pair of points , in are said Definition 4.3: We shall say that the dynamical system (1.4) to be perspective observable if problem 1.2 has an affirmative is perspective observable if problem 1.2 has an affirmative an- answer. swer for every pair of distinct points , We now prove the following lemma. . Lemma 4.2: Over the base field , consider the dynamical We are now in a position to state and prove one of the main system (1.4), (1.5) and the pair of vectors and satisfying theorems of this section which generalizes an earlier result (4.1). The pair of points and in are perspective reported in [29] and [30], wherein the control input was not unobservable iff present. The next theorem is about a PBH rank condition to test perspective observability of a dynamical system in the presence (4.2) of a control. for every in . Theorem 4.4: Assume that the matrix pair (C, A) is an ob- Proof of Lemma 4.2: Assume that the pair of points servable pair, i.e., and are perspective unobservable. It follows that for every , every , , and for (4.8) every control signal we have The dynamical system (1.4), (1.5) is perspective unobservable over the base field iff there exists a pair of eigenvalues of , such that for all pairs of complex numbers (may be the same) in the set one has (4.3) for some nonzero scalar , which may be dependent on .It rank (4.9) follows, by choosing in particular, that for every fixed , there must exist a homogeneous line , possibly dependent where is the set of eigenvalues of such that the on ,in such that subspace spanned by the corresponding eigenvectors or generalized eigenvectors of is . (4.4) Remark 4.5: Note that the existence of follows from the fact that is -invariant and the property in question Additionally, it follows that: is true for any such subspace. Note also that if the above observability rank condition (4.8) is not satisfied, then the system (1.4) (4.5) is perspective unobservable. Remark 4.6: In view of the results in Section III on the per- is a subset of for otherwise we can always choose , spective observability in the absence of any control input, it is and can conclude that under the restriction (4.1), there possible to state the results of Theorem 4.4 over the base field would always exist a control such that (4.2) is not satisfied . Such a generalization has not been attempted in this paper. for in some interval . Thus, we infer that Theorem 4.7: In linear systems theory, adding control does (4.6) not change observability. In the perspective setting, this is not the case. Sometimes, an unobservable pair of initial conditions Combining (4.4) and (4.6), we have can be rendered observable by a proper choice of control. Proof of Theorem 4.4: Before we sketch the formal proof, (4.7) we note the following. If we assume that (4.9) is satisfied it follows that there exist a nonzero vector such that: which implies (4.2). Conversely, assume that (4.2) is satisfied for every in .It follows that there exists a homogeneous line , possibly dependent on such that (4.7) is satisfied and that for every , (4.10) such that and we have (4.4) and (4.6) It may be remarked that the pair of matrices and satisfied for every . We conclude that the vectors commute, a fact that would be used subsequently in the proof. Let be the eigenspace spanned by the eigenvectors or generalized eigenvectors of corresponding to 468 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 3, MARCH 2002 eigenvalues it follows that is an element of and can where is the multiplicity of the eigenvalue . For details on be written as: the existence of this ordering see [29]. It follows from (4.14) that and are linearly dependent for all values (4.11) of . In particular, since and are linearly dependent, there would exist a scalar such that would be in the for some scalars and . Finally, we have null space of the matrix . It follows that:

rank (4.18) Hence, we conclude that The above argument can actually be repeated for each and every (4.12) pair of eigenvectors in indicating that for every pair of eigenvalues , in the set (4.16), the rank condition (4.9) Sufficiency: Assume that (4.9) is satisfied for a pair of eigen- would be satisfied. (Q.E.D.) values , of . It follows from (4.12) that if is the sub- Remark 4.8: To clarify an important point made in the proof space spanned by the eigenvectors or generalized eigenvectors of Theorem 4.4, if and are two eigenvalues of a matrix , of corresponding to eigenvalues ,wehave and if is a vector in the kernel of , such a must necessarily be spanned by a pair of eigenvectors, gen- (4.13) eralized eigenvectors of .If is the associated eigenspace, Let and be two linearly independent vectors in , it follows spanned by the pair, it follows that is at most one-di- from (4.13) that (4.2) would be satisfied. Thus, the pair mensional. The choice of and in (4.11) is not unique but cannot be observed. what is important in the proof is that the image of the subspace Necessity: Assume that (1.4), (1.5) is perspective unobserv- under the map is at most one-dimensional provided able. It follows that there exist two independent vectors , that the rank condition (4.9) is satisfied. In the case when is such that and cannot be observed by (1.4), (1.5). More- higher dimensional, it can be defined as the span of the eigenvec- over, (4.1) is automatically satisfied by , because of the tors, generalized eigenvectors, of the eigenvalues of in the set rank assumption (4.8). Using Lemma 4.2, we conclude that (4.2) . Assuming that (4.9) is satisfied for every is satisfied, for every in . It follows that there exists a homo- pair of eigenvalues , in the above set, it would follow once geneous line , possibly depending on such that again that the image of the subspace under the map is at most one dimensional. (4.14) To end this section, we write down the condition for perspec- and tive observability explicitly. Remark 4.9: Assuming the observability rank condition (4.15) (4.8) for the matrix pair , the dynamical system (1.4), In what follows we show that from (4.14) and (4.15) we can infer (1.5) is perspective observable, over the base field iff for the following. There exists a pair of eigenvectors (generalized every pair of eigenvalues of , there exist some pair eigenvectors) and of with associated eigenvalues and in the set such that such that for all pairs of complex numbers , in the set rank (4.19) (4.16) the rank condition (4.9) is satisfied, which would complete the Over the base field , the rank condition (4.19) is only sufficient. proof. The following corollary of the Theorem 4.4 is perhaps sur- Let be a set of eigenvectors or gener- prising. alized eigenvectors of with corresponding eigenvalues Corollary 4.10: Assume that the dynamical system (1.4) and such that (1.5) is such that rank (4.20)

(4.17) then the same dynamical system is perspective observable over where and and where we assume that the base field iff the observability rank condition (4.8) is sat- without any loss of generality (if not, replace by for isfied. some choice of ). One can order the eigenvalues in such a way Proof of Corollary 4.10: If the observability rank condi- that tion is not satisfied, the dynamical system is clearly perspective unobservable. Conversely, it follows from (4.8) and (4.20) that: and when ,wehave Thus, from Lemma 4.2, it would follow that every pair of vectors would be perspective observable. (Q.E.D.) GHOSH AND MARTIN: HOMOGENEOUS DYNAMICAL SYSTEMS THEORY 469

V. R EALIZABILITY VIA THE RATIONAL EXPONENTIAL We now obtain INTERPOLATION (5.8) So far in the previous sections we have considered the problem of perspective control and observation assuming that where is a square matrix defined as the parameters of the homogeneous dynamical system (1.4) are known. If on the other hand, the parameters of the system are unknown, it is important to be able to identify the parameters from a record of the observation over a certain interval of time. . . . . In particular, it is important to realize homogeneous dynamical . . . . (5.9) systems with minimum state dimension, if possible, and ascertain if the choice of the parameters is unique. If not, it is important to be able to classify the extent of the nonuniqueness. The interpolation problem considered in this section is a step in that direction. It is easy to see that Roughly speaking, the problem we consider is described as (5.10) follows. Assume and that the output of a dynamical system (1.4) has been observed over a finite interval of time. where The problem is to identify the parameters of the system from the observed output. We shall see that the parameter identifica- (5.11) tion problem is connected with a certain class of interpolation problem. In particular, for a linear dynamical system, one con- and is of the form siders an exponential interpolation problem. On the other hand, for a homogeneous dynamical system, of the kind described in (1.4) one obtains a rational exponential interpolation problem. Before we describe the “rational exponential interpolation” problem, let us consider the exponential interpolation problem (5.12) . . already described in [32]. We consider a linear autonomous . . system

(5.1) where solution of which can be expressed as follows, assuming distinct eigenvalues of : (5.13)

(5.2) To see (5.13), note that where

We assume , and that is given at discrete data points , ; where are given fixed constants. We obtain It now follows from (5.10) and (5.12) that:

(5.3) From the , one computes the coefficients and Let us define the variables . In principle, one can compute , , , . Using (5.8), one can also compute and (5.4) hence . and rewrite (5.3) as This is a very brief explanation of Prony’s method. In the form described here it is obvious that the technique is numerically unstable. However, it can be implemented in a more stable (5.5) fashion [32]. A standard reference for Prony’s method is the numerical analysis text by Hildebrand [35]. Finally, we write (5.5) in vector form and define two We now proceed to consider the exponential rational inter- matrices polation problem and consider the homogeneous dynamical system (1.4), (1.5). Writing the output in homogeneous (5.6) coordinates of as and (5.14) (5.7) 470 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 3, MARCH 2002

Abbreviating , a specific where choice of coordinates of is given by

In the notation of (5.14), the vector can be defined as a rational exponential function

. . . . where and , . Discretizing as before, we obtain We need to find conditions on such that (5.19) can be solved for a nonzero . In principle, this can be done by rewriting (5.19) as

(5.20)

Choosing the notations as follows: where

we obtain and where can be appropriately defined. Note that is an dimensional vector independent of . The matrix depends on (5.15) and is a function of and . In order to solve (5.20) for a nonzero , the matrix has to have a corank at least one, giving rise to a set of determinant conditions on . By choosing such where , , . For notational simplicity, determinants, one would solve for the variables . The rest of we assume at this point that . The general case can be the algorithm is same as the classical Prony’s algorithm. easily constructed and will be remarked later on. We would now Remark 5.1: When , one can easily write a set of like to write (5.15) in matrix notation by defining matrix equations of the form (5.20). The details can be easily constructed. Remark 5.2: The exponential rational interpolation algorithm described above depends on our ability to solve a set of at and as in (5.9) as follows: least polynomial equations in the variables . Unfortunately, this process is not numerically robust. The modified Prony’s (5.16) algorithm described in this section needs to be compared with the rescaling algorithm described in [8], wherein the where . It is easy to elimi- exponential rational interpolation problem has been rescaled to nate from (5.16) and obtain a regular exponential interpolation problem. This can then be (5.17) solved by the classical Prony’s algorithm.

Recalling the structure of from (5.12), we obtain VI. SUMMARY AND CONCLUSION To summarize, the important contributions of this paper are three fold. First of all, it introduces a new perspective control problem which is important in steering a vector in up to its direction. It is not totally surprising that these problems have a connection with the Riccati flow. What would be important in this context of gaze control is to introduce dynamical systems that are somewhat more general than (1.4) (see, for example, . . . . [37]) and possibly makes contact with the dynamics of mechanical systems. One would like to be able to visually actuate these systems up to direction. We would like to refer to [31] wherein (5.18) (1.4) has been replaced by a more general Riccati dynamics. We can rewrite (5.17) as Second of all, we revisit the perspective observability problem [29], [30] and obtain a necessary and sufficient condition over (5.19) the real base field, in the absence of any control. We also obtain GHOSH AND MARTIN: HOMOGENEOUS DYNAMICAL SYSTEMS THEORY 471

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Clyde F. Martin (M’74–SM’83–F’91) received the B.S. degree from Kansas State Teachers College, Emporia, and the M.Sc. and Ph.D. degrees from the University of Wyoming, Laramie, in 1965, 1967, and 1971, respectively. He has held regular academic positions at Utah State University, Logan, Case Western Reserve University, Cleveland, OH, and where he is presently, Texas Tech University, Lubbock. He has held postdoctoral or visiting positions at Harvard University, Cambridge, MA, and the Royal Institute of Technology, Stockholm, Sweden, and has held two postdoctoral positions at NASA Ames Research Center, Moffett Field, CA. Currently, he is the Paul Whitfield Horn Professor and the Ex-Students Distinguished Professor of Mathematics at Texas Tech University. He is the Director of the Institute for the Mathematics of the Life Sciences at Texas Tech University. His primary technical interests are in the application of mathematical and control theoretical techniques to problems in the life sciences. He has published numerous papers, monographs, and edited conference proceedings. He is Vice Chair of the SIAM Activity Group in Control Theory, and is Editor-in-Chief of the Journal of Mathematical Systems, Estimation, and Control.