## Abstract

This paper analyses deviated linear cyclic pursuit in which an agent pursues its leader with an angle of deviation in both the continuous- and discrete-time domains, while admitting heterogeneous gains and deviations for the agents. Sufficient conditions for the stability of such systems, in both the domains, are presented in this paper along with the derivation of the reachable set, which is a set of points where the agents may converge asymptotically. The stability conditions are derived based on Gershgorin's theorem. Simulations validating the theoretical results presented in this paper are provided.

## 1. Introduction

Cyclic pursuit has attracted the interest of researchers for a long time [1–9]. However, there are many variants of cyclic pursuit that are still to be analysed and, if harnessed gainfully, can overcome some of the existing limitations of conventional cyclic pursuit. In cyclic pursuit, each agent *i* pursues its leader *i*+1 modulo *n*, where *n* is the total number of agents. Several results on cyclic pursuit, involving the set of reachable points where such agents may rendezvous, are available in the literature [6]. Stability of various formations for agents, in cyclic pursuit, have also been analysed [7]. In conventional cyclic pursuit, the velocity of agent *i* is proportional to the distance separating the agent from its leader *i*+1 and is in a direction along the vector joining the two of them. The constant of proportionality, or gain, is generally chosen to be the same for every agent [7]. However, in [6] it has been shown that the gains can be heterogeneous and at most one of the gains can be negative, with a lower bound, for stability. This negative gain expands the reachable set and allows rendezvous at any point in the state space, save for some pathological cases [6]. Besides, several other aspects of the rendezvous problem, for agents in cyclic pursuit, have received much attention in the literature [10].

It has been shown in [11] that by using deviated cyclic pursuit, where each agent pursues its leader along a direction deviated from its line of sight, as in figure 1 (with *δ*_{i}=*δ*, ∀ *i*), interesting trajectory patterns can be obtained. However, the reachable set has not been investigated for such cases, though a limit on the deviation (based on stability considerations) has been obtained [12], for homogeneous gain and deviation of each agent.

In this paper, the agents are allowed to have heterogeneous deviations and gains. Different combinations of homogeneous and heterogeneous gains and deviations are analysed. Gershgorin's discs are used to arrive at conditions for stability of deviated cyclic pursuit and the analysis is extended to cases that have not been investigated in the literature. Some regions in the state space, which cannot be reached using heterogeneous cyclic pursuit, as in [6], are shown to be reachable using the proposed deviated cyclic pursuit scheme. Some preliminary results have been reported in [13]. This paper also deals with discrete-time deviated cyclic pursuit. The discrete-time counterparts of some of the proposed continuous-time cyclic pursuit strategies are analysed in the following sections. Results on stability and reachability of the same are presented. Gershgorin's theorem [14] has been used to derive conditions on the stability of various discrete-time cyclic pursuit schemes.

The paper is organized in the following manner. In §2, the existing framework is discussed and the problem formulated. Section 3 discusses different types of deviated cyclic pursuit and analyses the stability conditions and reachability for such cases. Stability and reachability of discrete-time deviated cyclic pursuit are presented in §4. In §5, expansion of the reachability set is analysed for deviated cyclic pursuit. An illustrative example is discussed which shows the limitation of conventional heterogeneous linear cyclic pursuit, vis-

## 2. Existing framework and analysis

Consider the case of conventional linear cyclic pursuit where the position of agent *i* is given by (*x*_{i},*y*_{i}). The kinematics are given by
*i* now pursues its leader, agent *i*+1 (modulo *n*), along a direction deviated by an angle *δ* from its line of sight, the equation of motion can be represented as
*p*_{r}, are given by
*δ*<*π*/*n*, the real parts of all the roots (except zero) are negative and hence the system is stable. This result is the same as that obtained in [11,12]. However, when both the deviation and the gain for each agent are allowed to be different as in this paper, the system equation can be expressed as

## 3. Continuous-time deviated cyclic pursuit

Three types of deviated cyclic pursuit schemes are considered in this paper. The circulant property of the system matrix, used in [12] to derive stability conditions, is lost in these cases. So new techniques are required. First, the case of homogeneous gains with heterogeneous deviations is considered. Next, a case of heterogeneous gains with homogeneous deviations is considered. Finally, a case of heterogeneous gains and heterogeneous deviations is considered. In each of these cases, the reachable set is derived. The following lemma provides an important result pertaining to the stability of the deviated cyclic pursuit system.

### Lemma 3.1

*Consider the system given by* (2.5). *If the number of agents is greater than two, with* *k*_{i}>0, ∀ *i* *and* |*δ*_{i}|<*π*/*n* ∀ *i*, *then*

### Proof.

It can be observed that *δ*_{i}|<*π*/*n* ∀ *i*. Let *p*_{i}. Then, the argument of *p*_{i} is given by *ϕ*_{i}=*γ*−*δ*_{i}. Since, *k*_{i}>0, ∀ *i*, the magnitude of *p*_{i} is *p*_{i} and *p*_{j} for any pair (*i*, *j*) is given by *ϕ*_{i}−*ϕ*_{j}=*δ*_{j}−*δ*_{i}. The maximum value for this quantity can be 2*π*/*n* since |*δ*_{i}|<*π*/*n*, ∀ *i*. Since *n*>2, the maximum angle of the sector over which the phasors *p*_{i} may be spread is less than *π* because no two phasors *p*_{i} and *p*_{j} can be separated by an angle greater than *π*. It is clear that the sum of the phasors, confined within a sector spread over an angle less than *π*, cannot be zero. Hence,

The term *s* in the characteristic equation of (2.5). Thus, the consequence of lemma 3.1 is that the characteristic equation of the most general deviated cyclic pursuit system (subject to |*δ*_{i}|<*π*/*n*, ∀ *i*) cannot have more than one root at the origin. This is because lemma 3.1 considers the most general case of heterogeneous gains and heterogeneous deviations and therefore any result applicable to this case is, in general, true. This result is critical to the stability of the system because a repeated root at the origin will imply instability of the system.

### (a) Homogeneous gains with heterogeneous deviations

The equation of motion of the agents in (2.5), whose deviations are *δ*_{i}, *i*∈{1,2,…,*n*}, and gains *k*_{i}=*k* (>0), is given by
*q*=*s*/*k*

For a point, say *q*, to be a root of equation (3.3) one requires that *a*. The radial vectors, with their arrowhead directions reversed, represent the complex variable [0−(−*e*^{jδi})]. Each of them has an argument equal to *δ*_{i}. This explains why the origin is a root of the characteristic equation. Now, designating *θ*_{i}∈(−*π*,*π*), one of the conditions for a point *q* to be a root of (3.2) is that *π*/*n*. This same condition is imposed for all the cases considered in this paper. However, this is done only to aid the analysis and towards the end of the paper, it will be shown for a specific case (of deviation by one agent only) that this is not a necessary condition for stability. Upon imposing the same condition, that is, −*π*/*n*<*δ*_{m}≤*δ*_{i}≤*δ*_{M}<*π*/*n*, ∀ *i*, it follows that *θ*_{i}<*δ*_{i}, ∀ *i*, or *θ*_{i}>*δ*_{i}, ∀ *i*, for all candidate roots in a particular region of the complex plane and *x* and *y*, belonging to the same open interval of length *L* will satisfy the condition *x* (mod *L*)≠*y* (mod *L*).

Figure 2*b* shows the intersection with the RHP of the union of the Gershgorin discs (corresponding to each row of the system matrix), through hatched lines, for the system in (3.1). The centres of these discs lie on the dotted circle, of unit radius, shown in figure 2*b*. For analysis purposes, let the deviations be bounded by 0≤|*δ*_{i}|≤|*δ*_{M}|<*π*/*n*, ∀ *i* (that is, |*δ*_{M}|>|*δ*_{m}|). If it can be ensured that no point in the RHP, other than the origin (which is a non-repeating root due to the assumption that *b*, it is clear that of all the points within the hatched region, the point *P* subtends the maximum angle (in terms of absolute value), at the centre of any Gershgorin circle, with the horizontal. This angle *θ*_{iR} (corresponding to the centre of the Gershgorin disc for the *i*th agent), bounded by |*θ*_{iR}|≤*θ*_{M}, is given by

### Theorem 3.2

*Consider a system of agents in deviated linear cyclic pursuit given by (*3.1*) with k>0. If the deviations are bounded by −π/n<δ*_{m}*≤δ*_{i}*≤δ*_{M}*<π/n, then the system will be stable if
**and
*

### Proof.

The maximum value of |*θ*_{i}| within the hatched portion of figure 2*b* is given by
*θ*_{i}=*δ*_{i}, ∀ *i*. For any other candidate root of (3.2) within the hatched portion of figure 2*b*, *θ*_{i}<*δ*_{i}, ∀ *i*. Now, if *δ*_{m}<0 and |*δ*_{M}|<|*δ*_{m}|, then the hatched portion would extend to the first quadrant besides the fourth that is shown in figure 2*b* and *θ*_{i}>*δ*_{i}, ∀ *i* for a point *q*, in the first quadrant, resulting in *δ*_{m}>−*π*/*n*. By a similar reasoning as before, *δ*_{M} by *δ*_{m} in (3.5a). Since these hatched portions are the only parts of the RHP where roots of the characteristic equation may lie (Gershgorin's theorem), stability is ensured when *δ*_{M}|>|*δ*_{m}| or *δ*_{M}|<|*δ*_{m}|. ▪

Clearly, in (3.5a) or (3.5b) equality instead of inequality does not imply loss of stability unless *δ*_{i}=*δ*, ∀ *i*. Hence, the inequalities only provide sufficient conditions for stability. Several observations follow from the above results.

### Corollary 3.3

*If all the agents have the same deviations, that is* *δ*_{i}= *δ*, ∀ *i*, *then* *δ*_{m}=*δ*_{M}=*δ* *and* (3.5) *reduces to* *δ*<*π*/*n*.

### Proof.

Follows from substituting *δ*_{i}=*δ*, ∀ *i* and *δ*_{M}=*δ*_{m}=*δ* in (3.5). ▪

### Corollary 3.4

*For* *n*≥4 (*so that* *δ*_{i}<*π*/4, ∀ *i*), (3.5) *can be further rewritten as* *for* |*δ*_{M}|>|*δ*_{m}| *and* *for* |*δ*_{m}|>|*δ*_{M}|.

### Proof.

Along the dotted arc AB (moving from *A* to *B*) in figure 2*b*, the angle subtended at any point on the arc (which is a candidate for the centre of a Gershgorin disc) by the horizontal line and a vector joining the point with *P* (which can be *θ*_{i} for some *i*) decreases monotonically from *A* to *B* for *δ*_{M}<*π*/4. This is explained as follows. Consider the function *π*/4<*δ*_{i}< *π*/4 ∀ *i*, for *n*≥4. Now, *δ*_{i}<*δ*_{M}<*π*/4, ∀ *i*, then *δ*_{i} if *δ*_{M}<*π*/4. The proof follows from noting the fact that *π*/2,*π*/2). Similar arguments hold for |*δ*_{m}|>|*δ*_{M}|. ▪

Corollary 3.4 provides an easier verification for stability because for *n*≥4, only the highest and lowest angles of deviations are considered instead of searching across all the angles of deviation.

### Remark 3.5

For *n*<4, the condition *θ*_{iR} is as defined in (3.4) or by replacing *δ*_{M} with *δ*_{m} therein, for |*δ*_{m}|>|*δ*_{M}|. However, it cannot be explicitly represented in terms of *δ*_{m} and *δ*_{M} as in corollary 3.4.

### Remark 3.6

To investigate the sector of permissible uncertain deviations of homogeneous linear cyclic pursuit, *δ*_{i}=−*δ*_{M} can be imposed in (3.5) to obtain *δ*_{i}=0 is substituted in (3.5) to obtain the bound as

The point of convergence of the agents following deviated cyclic pursuit as in (3.1) is now obtained. Choosing the transformation *f*’ and ‘0’ denote the final and initial values, respectively. Now, for rendezvous *z*_{if}=*z*_{jf}, ∀ *i*,*j*∈{1,2,…,*n*}. This implies that the rendezvous point *Z*_{f} is given by
*Z*_{f} need not belong to the convex hull of the initial positions of the agents.

### (b) Heterogeneous gains with homogeneous deviations

The equation of motion of the agents, in this case, can be described as
*δ* is the deviation of each agent and *k*_{i} (>0, ∀ *i*) are the respective gains. The stability of this system is equivalent to the confinement of the roots of a real polynomial to a sector. This is because the eigenvalues (or roots of the characteristic equation) of the undeviated system matrix are rotated by an angle *δ* to yield the eigenvalues of the system matrix in (3.9). It needs to be determined if the roots of the undeviated system are confined to the sector indicated by the hatched portion of figure 3*a*. If such conditions are met, then the system is stable for a deviation of *δ* in each agent. There are several Routh-like arrays to test for the confinement of roots within a sector, such as the one outlined in [15]. In order to avoid complications arising out of a complex Routh array, a 2*n*-order real polynomial may be tested to check the stability of an *n*th-order complex polynomial whose roots are the eigenvalues of the system in (3.9). In general, the following theorem may serve as a guideline for ensuring stability.

### Theorem 3.7

*Consider the system in (*3.9*) with gains k*_{i}*>0 restricted by 0<k*_{m}*≤k*_{i}*≤k*_{M}*, ∀ i. The system will be stable for a uniform deviation of δ if*

### Proof.

Consider figure 3*b*. Here, as before, among all points within the hatched region (which is the only part of the RHP where roots of the characteristic polynomial corresponding to the system given by (3.9) may lie) the point *P* subtends the maximum angle between the centres of the Gershgorin discs and the horizontal. Among all such angles, the angle subtended with the centre of the smallest disc (whose radius is *k*_{m}) is the greatest. This angle *τ* is given by

Upon substituting *k*_{m}=*k*_{M}=*k*_{i}, ∀ *i*, one arrives at the condition *δ*<*π*/*n*. The rendezvous point is unaffected by the deviation *δ* and is given by the same expression as in [6], which is

### (c) Heterogeneous gains with heterogeneous deviations

This is the most general case. Both the gains *k*_{i} (>0, ∀ *i*) as well as the deviations *δ*_{i} are different for the agents in this case. This is thus a generalization of all the cases discussed in the literature as well as in this paper. The main interest in studying this case would be to investigate the stability of the system and also to obtain the reachable set. The governing equation is given by (2.5) where, it is considered that |*δ*_{i}|<*π*/*n*, ∀ *i*, as before. The hatched portion of figure 4 shows the intersection of the union of the Gershgorin discs with the RHP. The characteristic equation for the system can be obtained as

Clearly, the phase condition to be met by a root of equation (3.11) is similar to the one in (3.2). For a point within the hatched region of figure 4 to satisfy (3.11), one must have *q* is a candidate root of the characteristic equation (3.11). As before, if the origin is the only point within the hatched region that satisfies this condition, the system is stable because, by lemma 3.1, there cannot be repeated roots at the origin. It can be seen from figure 4 that at any candidate root, *q*, of (3.11) within the hatched region, either *θ*_{i}<*δ*_{i}, ∀ *i* or *θ*_{i}>*δ*_{i}, ∀ *i*, for *n*≥4. Thus, *A* corresponds to a positive angle of deviation *δ*_{p}, then,
*B* should be defined as

### Theorem 3.8

*Consider the system of n (≥4) agents in deviated linear cyclic pursuit as described by (*2.5*) with k*_{i}*>0, ∀ i. If the deviations are restricted by |δ*_{i}*|<π/n, ∀ i, and A and B are as defined by (*3.12*) and (*3.14*), then the system will be stable if*

### Proof.

As before, within the hatched portion of figure 4, either the point *P* or the point *Q* (in figure 4) subtends the greatest angle *θ*_{i} (magnitude-wise) at the centre of the *i*th Gershgorin circle. For example, in figure 4, either *θ*_{p} or *θ*_{q} is the greatest value of |*θ*_{i}| within the hatched portion. Figure 4 also shows that the term *A* in (3.12) is either equal to the length of the segment OP or OQ. Subsequently, *B* results in the greatest value of

This theorem, however, provides a sufficient condition for stability and is thus conservative. But this conservatism is the price to be paid in order to deal with heterogeneous gains as well as deviations for each agent. There are 2*n* decision variables that can be chosen and so it offers a considerable amount of flexibility in design. It can be seen that by putting *k*_{i}=*k*>0, ∀ *i*, in this setting, one arrives at the same condition as obtained in (3.5).

A similar analysis as before, with

## 4. Discrete-time deviated cyclic pursuit

As with the continuous-time case, the discrete-time cyclic pursuit law essentially implies that each of the *n* agents, in cyclic pursuit, decides its position at the next sampling instant based on the length and direction of the vector joining it to its leader, at the current sampling instant. This implies that the kinematic equation of the system can be expressed as
*z*(*t*) denotes the positions of the *n* agents at the sampling instant *t* and circ(.,…,.) denotes a circulant matrix represented by its first row. The term 0<*ρ*<1 is the constant of proportionality, also called the gain of the system. It may be observed that for the system described above, the gain is the same for all the agents. An essential feature of discrete-time deviated cyclic pursuit is that the limits on the deviations depend on the values of the gains even when the gain is homogeneous, unlike in the continuous-time case. Thus, the results are not straightforward extensions of the continuous-time versions. As described in (4.1) and (4.2), the discrete-time cyclic pursuit law with homogeneous gains will be stable if the roots of the circulant matrix *A* lie within the disc of unit radius, centred at the origin of the complex plane. This follows directly from the stability of discrete-time systems. It can be readily verified that the characteristic equation of the matrix *A* in (4.1) is given by
*ρ*) and radius equal to *ρ* as shown in figure 5*a*. They can be expressed as
*A* corresponding to the eigenvalue 1 is [1 1 …1]^{T}, which would lead to positional consensus if the system is stable. Furthermore, from figure 5*a*, it is also evident that if 0<*ρ*<1 then the eigenvalues are all inside the unit circle, shown with dotted lines. This would guarantee stability of the system given by (4.1) and (4.2).

As discussed earlier, depending on the different kinds of deviations and gains, the deviated cyclic pursuit system may be broadly classified into three categories. The first comprises agents having different gains, but the same deviation. The second consists of agents having the same gain but different deviations. The third and the most general category includes agents having different gains and deviations. The equation of motion for this general case may be written as
*s*−1 by λ, the constant term in the characteristic equation vanishes. The coefficient of λ is given by *δ*_{i}|<*π*/*n*, ∀ *i*. The following result then holds.

### Lemma 4.1

*For the system described by* (4.5), *if* |*δ*_{i}|<*π*/*n*, ∀ *i*, *and* 0<*ρ*_{i}<1, ∀ *i*, *then the coefficient of* (*s*−1) *in the characteristic equation* (4.6), *given by* *does not vanish if the number of agents is greater than two*.

### Proof.

This proof is in the same lines as lemma 3.1 for the continuous-time case. Once again the spread of the vectors may be used to see that the coefficient of (*s*−1) does not vanish in the characteristic equation (4.6). ▪

The above lemma implies that the characteristic equation for deviated cyclic pursuit cannot have multiple roots at *s*=1 for any of the three types mentioned above. It should be noted that unlike the continuous-time deviated cyclic pursuit, in the discrete time, no result on the stability of deviated cyclic pursuit with homogeneous gains and deviations is available in the literature. Hence, the analysis for this case is first presented in the following subsection. Thereafter, the other three cases are considered as in the case of continuous time.

### (a) Homogeneous gains and deviations

Suppose the agents pursue each other with some angle of deviation *δ*, then the equation of motion is given by
*δ*_{c} is the critical limit on *δ* for stability, that is, |*δ*|<*δ*_{c}. Under this condition, the agents will execute circular motion about a point instead of converging to it, which is similar to continuous-time cyclic pursuit [12]. However, it may be noted that the critical value of *δ* obtained here is dependent on the value of *ρ*. This is an interesting feature of discrete-time cyclic pursuit. It may be deduced that in the case of discrete-time cyclic pursuit, the margin for stability (measured in terms of homogeneous deviation) is less by

From (4.10), the limit on the homogeneous deviation has been derived to be *δ*_{c} will also be derived based on the same geometric principle, in order to present a unified approach.

Consider a deviation of *δ* for each agent, as in (4.7) and (4.8). The eigenvalues of the system will be distributed as shown in figure 5*b* for a certain value of *δ*=*δ*_{c}. Clearly, when this condition is met, there will be two roots of unit magnitude. One of them is unity and will lead to consensus, whereas, the other, at *Q*, will cause the agents to move in a circle around a point. The second eigenvalue of unit magnitude will result in an undamped oscillatory response with respect to time, both along the *x*- and *y*-axes, thereby resulting in a circular motion. This is similar to the result obtained for the continuous-time case in [11]. If *δ*>*δ*_{c}, one of the eigenvalues lies outside the unit circle, leading to instability. It is thus important to obtain the value of *δ*_{c}. Consider figure 5*b*. C is the centre of the circle of radius *ρ*. The triangles OCP and OCQ are congruent because OC is the common side, OP and OQ are of the same length (unity) and both CP and CQ are of length *ρ*. Thus, the line OC bisects the angle *θ* and the angle *θ*/2+*δ*_{c}=*π*/*n*. Considering the triangle OPQ, an application of the rule of cosines implies *d*. Similarly, in triangle PCQ, *d* from these expressions and using the relation *θ*/2+*δ*_{c}=*π*/*n*, it readily follows that

### (b) Homogeneous gains with heterogeneous deviations

The equation of motion of the agents, whose deviations are *δ*_{i}, *i*∈{1,2,…,*n*}, and gains equal 0<*ρ*<1, is given by
*q*=(*s*−1). From (4.12), it may be concluded that the sum of the arguments of the complex numbers joining the points −*e*^{jδi} (*i*=1,2,…,*n*) to a candidate root of the equation, say *q*, must be *π*/*n*. In all the three cases discussed in the present section, the same condition is imposed on the angle of deviation, that is *δ*_{i}<*π*/*n*, ∀ *i*. This aids analysis because it implies that

Suppose that *s* to be a root of (4.12), it is required that *θ*_{i}<*δ*_{i}, ∀ *i*, or *θ*_{i}>*δ*_{i}, ∀ *i*, for all candidate roots in a particular region of the complex plane and *π*,*π*). Hence, when both *π*, they cannot be equal either. The observation is based on the fact that for two distinct real numbers to be equal in modulo *L* (another real number), they must be separated by an integral multiple of *L*.

Figure 6*a* shows the intersection with the region outside the unit circle of the union of the Gershgorin discs (corresponding to each row of the system matrix, or agent), by hatched lines, for the system in (4.11). As suggested above, let the deviations be bounded by 0≤|*δ*_{i}|≤|*δ*_{M}|<*π*/*n*, ∀ *i*. If it can be ensured that no point outside the unit circle within the hatched area of figure 6*a*, except (1,0) (which is on the periphery of the unit circle) is a root of equation (4.12), then system (4.11) is stable. Even the root at 1 is non-repeating due to lemma 3.1. It may be observed from figure 6*a* that of all the points within the hatched region, either the point *M* or the point *M*′ subtends the maximum angle at the centre of any of the Gershgorin discs with the horizontal. Thus, each of the *θ*_{i}'s obtains a maximum value at *M* or *M*′ within the hatched portion. Suppose this angle is designated as *θ*_{Mi}. Let

In order to evaluate *θ*_{Mi}, the points of intersection of each of the Gershgorin discs with the unit circle must be obtained. One of the points is (1,0) while the second point of intersection is either *M* or *M*′ for one of the discs. Now, some of the agents have positive angle of deviation while others may have a negative angle of deviation. If the absolute value of the maximum positive deviation is greater than that of the maximum negative deviation, then *M* would contribute to *θ*_{Mi}, else *M*′ needs to be considered. For any point above (1,0) within the hatched region, *θ*_{i}>*δ*_{i}, ∀ *i*, and for a point below (1,0) in the hatched portion, *θ*_{i}<*δ*_{i}, ∀ *i*. Hence, if *i*th Gershgorin disc by the point M or *M*′ with the horizontal, then *a*, either *δ*_{i}|<*π*/*n* as well. Thus, under these conditions

Consider the point *M* or the point *M*′ in figure 6*a*. The point *M*, or *M*′ lies on the unit circle, and is given by *i*th Gershgorin circle is given by *M* or *M*′ corresponds to that Gershgorin disc which has the maximum |*δ*_{i}|, given by *δ*_{M}. Note that, by definition *δ*_{M}>0. Hence, the intersections of the circle given by *M*, or the intersections of the circle given by *M*′, depending on whether

The point of intersection at *M* or *M*′ given by *ψ*∈(0,*π*):
*M* or *M*′) subtends an angle (absolute value) less than *π*/*n* at the centre of any Gershgorin disc

### Theorem 4.2

*Consider system (*4.11*) with 0<ρ<1 and |δ*_{i}*|<δ*_{M}*< π/n ∀ i. The system will be stable if* *where ψ∈(0,π) is given by (*4.14*) and x=1 if* *corresponds to a positive deviation while x=−1 if* *corresponds to a negative deviation.*

### Proof.

The proof can be split into two cases.

*Case 1*. Suppose *a*, it may be seen that of all the points within the hatched portion, the point *M* subtends the maximum angle at the centre of each Gershgorin disc, in terms of its absolute value. Of these, *ψ*∈(0,*π*) as given by (4.14), represents the maximum contribution of any point within the hatched region towards *π*/*n*, then, *θ*_{i}>*δ*_{i}, ∀ *i*, or *θ*_{i}<*δ*_{i}, ∀ *i*, implying that *a* can be a root of (4.12) because the arguments of the two phasors, *π*, thus ensuring the stability of the system.

*Case 2*. Suppose *ρ*<1 and |*δ*_{i}|<*δ*_{M}<*π*/*n*, ∀ *i*. In this case, the point *M*′ subtends the maximum angle at the centre of any Gershgorin disc instead of *M*, among all the points in the hatched portion. It suffices to ensure that

Using the same reasoning as employed to arrive at (3.8) (since the system matrix in (4.11) has an eigenvalue at 1—equivalent to a continuous time system having an eigenvalue at the origin—and the corresponding eigenvector is [1 1…1]^{T}), it follows that the set of reachable points will be given by

### (c) Heterogeneous gains with homogeneous deviations

The equation of motion for agents, in this case, is given by (4.11) with all *δ*_{i}=*δ*. In this case, all the agents have different gains 0<*ρ*_{i}<1, ∀ *i*, but the same deviation *δ*. The characteristic equation is given by
*r*, in the complex plane to satisfy the characteristic equation (4.16), it is obvious that the sum of the arguments (modulo 2*π*) of the phasors [(*r*−1+*ρ*_{i}*e*^{jδ})], given by *θ*_{i}, must equal *nδ*(modulo 2*π*). Moreover, this point must lie within the intersection of the Gershgorin discs as shown in figure 6*b*. Now, the system in this case will be stable if no point within the hatched region satisfies this condition and hence does not satisfy (4.16). This is a sufficiency condition for stability. Clearly, the phasors [(*r*−1+*ρ*_{i}*e*^{jδ})] are nothing but phasors joining the centres of the Gershgorin discs to the candidate root of (4.16). Among all the points in the hatched area of figure 6*b*, the point *M* subtends the maximum angle between the horizontal and the centre of any Gershgorin disc. Since *M* lies on the intersection of the unit circle with the largest Gershgorin disc, corresponding to *θ*_{Mi} subtended by the point *M* at the centre of the Gershgorin disc corresponding to *ρ*_{i} is given by
*ψ* is as defined earlier. Clearly, if *θ*_{i} is the angle subtended at the centre of the *i*th Gershgorin disc by any candidate root of (4.16) within the hatched portion of figure 6*b*. Moreover, *θ*_{i}<*δ*, ∀ *i*, as is evident from figure 6*b*. Thus, if *n*|*δ*|<*π*, then it can be guaranteed that *b*, thereby ruling out the possibility of containing any roots of (4.16) in the same region. Hence, the stability of system (4.11) with *δ*_{i}=*δ*, ∀ *i*, is guaranteed. It should be noted that if *δ*<0, similar arguments can be presented with *θ*_{i}>*δ*_{i}, ∀ *i*, and the coordinate of *M* being given by

### Theorem 4.3

*Consider the system given by (*4.11*) with* *and |δ*_{i}*|=δ< π/n, ∀ i. the system will be stable if* *where,* *with ψ∈(0,π).*

### Proof.

Follows from the discussion above. ▪

The reachable set does not expand any further than in case of heterogeneous cyclic pursuit. It may be verified, using the same technique as used to obtain (3.8), that the rendezvous point in this case is given by

### (d) Heterogeneous gains with heterogeneous deviations

The equation of motion in this case is given by (4.5). This is the most general case as both the gains, 0<*ρ*_{i}<1, and *δ*_{i} are heterogeneous. Consider figure 6*c*, where intersection of the region outside the unit circle with the union of the Gershgorin discs is shown by hatched lines. If any root of the characteristic equation for this system, (4.6) (other than the non-repeating root at 1), lies within the hatched portion of figure 6*c*, the system will lose stability. It is, therefore, sufficient, from stability considerations, to ensure that no root of (4.6) lies within the hatched portion. Consider any candidate root, *p* of (4.6). Let the phasor *θ*_{i}. Now, for a point *p* to be a root of (4.6), *p*, both *π*,*π*) and *p* cannot be a root of (4.6). Moreover, if it can be ensured that for any point within the hatched portion, the above inequality holds and both *π*/*n*, (implying that *p* with the centre of the *i*th Gershgorin disc. Thus, for any point other than (1,0), either *θ*_{i}>*δ*_{i}, ∀ *i*, or *θ*_{i}<*δ*_{i}, ∀ *i*. Hence, the inequality *π*/*n* is sufficient to rule out the possibility of having any root of (4.6) within the hatched portion of figure 6*c*. The following theorem summarizes the sufficient conditions for stability of the system described by (4.5).

### Theorem 4.4

*Consider the system described by (*4.5*) with |δ*_{i}*|<π/n, ∀ i. Let L={i:|ψ*_{i}*|≥|ψ*_{j}*| ∀ j∈{1,2,…,n}∖{i}} where, ψ*_{i} *is given by
**The system is guaranteed to be stable if*

### Proof.

Consider the point *M*, or *M*′, in figure 6*c*. Because it lies on the unit circle, its coordinate can be given by *l* corresponds to the agent whose Gershgorin disc intersects with the unit circle and subtends the maximum absolute angle at the origin, with the horizontal. Thus, if there is more than one agent that intersect the unit circle at the point *M* or *M*′, compose a set *L*={*i*:|*ψ*_{i}|≥|*ψ*_{j}| ∀ *j*∈{1,2,…,*n*}∖{*i*}} comprising the indices of all such agents. Clearly, of all the points within the hatched portion, the point *M* (or *M*′) subtends the maximum angle at the centre of any Gershgorin disc, with the horizontal. This angle is given by *ψ*_{i} is obtained, as in the previous subsections, by solving for the intersection of the *i*th Gershgorin disc with the unit circle, which gives *δ*_{i}|<*π*/*n*, ∀ *i*, according to the conditions of the present theorem. Thus, the stability of the system is guaranteed. ▪

The set of reachable points in this case can be derived, using similar techniques as employed earlier, to be

In all the three theorems pertaining to deviated cyclic pursuit systems in this section, only the point (1,0) within the closed hatched region qualifies as a root of the characteristic equation of the system. This guarantees stability. However, a repeated root at (1,0) will not retain the stability of the system. But, lemma 3.1 ensures that there can be no repeated roots at (1,0).

## 5. Analysis of reachable set using deviated linear cyclic pursuit

In this section, the reachable set is analysed for agents in deviated cyclic pursuit (continuous time). Because the set is essentially the same for both continuous- and discrete-time domains, the analysis for either is similar. It will be shown that by using deviated pursuit, it is possible to reach certain points that could not be reached even with heterogeneous gains (with one gain negative) but with no deviations. It is known that [6] without using deviations and using only positive gains, it is possible to reach any point in the interior of the convex hull of the initial positions of the agents. In the following, it is assumed that all the agents start from a vertex of the convex hull, that is, none of the agents are initially in the interior of the convex hull. Then, even by using one negative gain as in [6], some regions are still not reachable. This is illustrated in the following example. Although a specific initial configuration of the agents is considered below, it is sufficient to illustrate the general principle.

Consider 4 agents at the vertices of a square of side *d*, [(0,0), (*d*,0), (*d*,*d*),(0,*d*)]. Suppose the desired rendezvous point is (*d*+*ϵ*,*βd*), where *ϵ*>0 is a small quantity and 0<*β*<1. Note that this point is outside the convex hull of the initial positions of the agents.

### Proposition 5.1

*For the initial configuration with four agents at the vertices of a square, as given above, it is not possible to achieve the desired rendezvous point, located outside the convex hull of the initial positions, as given above.*

### Proof.

Let the gains for the agents (without any deviation) be *k*_{i}. The following equation may then be written for rendezvous using (3.10):
*v*_{1}=[−1 1 −1 1]^{T} and *v*_{2}=[−(1+*ϵ*/*βd*)(1+*ϵ*/*d*)/*β* 0 1]^{T}. The reciprocals of the gains are given by
*c*_{1} and *c*_{2} are some real (non-zero) constants. If sgn(*c*_{1})=sgn(*c*_{2})=+1, it follows that both *k*_{1} and *k*_{3} are negative. So, the system will not be stable due to multiple negative gains [6]. Similarly, if sgn(*c*_{1})=sgn(*c*_{2})=−1, both *k*_{2} and *k*_{4} become negative and, by the same reasoning, lead to instability. If sgn(*c*_{1})=+1 and sgn(*c*_{2})=−1, then *k*_{3}<0. Two cases may arise, depending on whether |*c*_{1}|<|*c*_{2}| or |*c*_{1}|>|*c*_{2}|. In the former case, *k*_{4}<0, thereby leading to instability. In the latter, the condition for stability, that is, (1+*ϵ*/*d*)/*β*<|*c*_{1}|/|*c*_{2}|<1+*ϵ*/*βd*, where the lower bound is greater than the upper bound on |*c*_{1}|/|*c*_{2}|, cannot be met. Finally, the case with sgn(*c*_{1})=−1 and sgn(*c*_{2})=+1, is considered. From [6], it is known that, for stability, *k*_{1}*k*_{2}*k*_{3}+*k*_{2}*k*_{3}*k*_{4}+*k*_{3}*k*_{4}*k*_{1}+*k*_{4}*k*_{1}*k*_{2}>0. As before, it can be seen that if |*c*_{1}|>|*c*_{2}|, *k*_{4}<0 and if |*c*_{1}|<|*c*_{2}|, then *k*_{1}<0. If any other gain is negative in either case, then, by a similar reasoning, stability is lost. Even if all the remaining three gains in either case were positive, the product *k*_{1}*k*_{2}*k*_{3}*k*_{4}<0. But,
*k*_{1}*k*_{2}*k*_{3}*k*_{4}, leads to *k*_{1}*k*_{2}*k*_{3}+*k*_{2}*k*_{3}*k*_{4}+*k*_{3}*k*_{4}*k*_{1}+*k*_{4}*k*_{1}*k*_{2}<0. This also implies loss of stability. Thus, it is impossible to rendezvous at a point (*d*+*ϵ*,*βd*), 0<*β*<1, *ϵ*>0, using heterogeneous gains in cyclic pursuit for the given configuration. ▪

Now, consider the case of deviated cyclic pursuit with all gains positive. Without loss of generality, the gains *k*_{i} may be chosen so that

### Theorem 5.2

*Using the deviated cyclic pursuit law, described by (*2.5*), agents in cyclic pursuit can rendezvous at certain points outside the convex hull of their initial coordinates, which are otherwise unreachable using any known cyclic pursuit law, when all the agents start from the vertices of a convex polygon.*

### Proof.

First, let the gains be chosen to reach a point *Z*_{f} which is arbitrarily close to an edge of the convex hull of the initial positions, but inside it. Consider, further, that a deviation is given to any one of the two agents at the ends of the edge of the hull that *Z*_{f} (the point of convergence under no deviation) is closest to. The effect of this deviation is analysed as follows. If all the agents were to deviate, the resultant effective displacement of *Z*_{f}, given by Δ*Z*_{f}, may be approximated as the sum of the individual contributions, given by *δ*_{p}=*δ*_{p} (assuming small angle of deviation *δ*_{p}). Here, Δ*δ*_{p}=*δ*_{p} essentially implies that the deviation on agent *p* is like a small perturbation to the nominal heterogeneous cyclic pursuit system without any deviation. Consider the case shown in figure 7. The gains (*k*_{i}>0, ∀ *i*) have been designed to reach the point *Z*_{f} inside the convex hull of the initial positions of the agents. Now, if only agent *p* uses a non-zero deviation, the sensitivity of *Z*_{f} to that deviation is given by

The displacement of *Z*_{f} due to a deviation *δ*_{p} may thus be expressed as
*δ*_{p} is positive then the resulting Δ*Z*_{f} leads to convergence to a point outside the convex hull of initial positions, for the case shown in figure 7. This is because a counterclockwise rotation is considered to be positive and the interior of the convex hull lies to the right of the edge in figure 7. If the original point, *Z*_{f}, without any deviation, is very close to an edge of the convex hull as in figure 7 and inside it, then it can be expressed as *Z*_{f}=*λz*_{p0}+(1−λ)*z*_{q0}+*ϵ*, where, *ϵ* (|*ϵ*|>0) is a complex number, with a small magnitude, orthogonal to the edge joining agents *p* and *q* as shown in figure 7. Note this is just one such choice of gains. Of course, other choices of gains may result in the same point of convergence. But this is the one chosen for analysis purposes here and will be a guiding principle for designing gains and deviations. With this choice in mind, it is clear that if the point *Z*_{f} is closer to the edge *pq* than to any other agent, the gains *k*_{p} and *k*_{q} are the lowest among all *k*_{i}'s. This is also in accordance with (3.10), from where it may be logical to conclude that the agent/agents which starts closest to the desired rendezvous point have the lowest gain/gains. Now let *ϵ*=−*jζ*(*z*_{q0}−*z*_{p0}), where *ζ*>0 is a small real number and as *ϵ*|. Substituting these for *Z*_{f} in (5.5), it can be seen that
*Z*_{f}, which is dependent on *ζ*, causes a shift towards the interior of the convex hull rather than away from it for *δ*_{p}>0. This is due to the term *e*^{−jδp} present in the numerator which rotates *ζ*[*z*_{q0}−*z*_{p0}] by an angle *δ*_{p} clockwise. Thus, to maximize the distance of the final point of convergence from the aforementioned convex hull, perpendicular to the edge *pq* (that is, to expand the reachable set outside the hull), *ζ* must be decreased. In the limiting case, as *k*_{p}=1/λ and *k*_{q}=1/(1−λ). Thus, for two sets of *Z*_{f} and |*ϵ*|, (*Z*_{f1},|*ϵ*_{1}|) and (*Z*_{f2},|*ϵ*_{2}|), if λ and *δ*_{p} are the same, then |*ϵ*_{1}|<|*ϵ*_{2}| implies the component of Δ*Z*_{f2} which is perpendicular to [*z*_{q0}−*z*_{p0}], is less than that of Δ*Z*_{f1}.

If the point *Z*_{f} is closer to *z*_{p0} than *z*_{q0}, then *k*_{p}<*k*_{q}. The conditions for stability in (3.12) and (3.13) and theorem 3.8 now dictate a bound on *δ*_{p}. From (3.12), it follows that *k*_{p} and *k*_{q} are by far the two lowest gains among all *k*_{i}'s, it means that *k*_{j} in (3.13) may either correspond to *k*_{q} or *k*_{p}. In other words, the gains can certainly be chosen to satisfy this condition. In the former case, the condition for stability demands that *δ*_{p}|<*π*/*n*. In general, *Z*_{f} is closer to *z*_{q0} than *z*_{p0}, with *δ*_{q}<0 and *δ*_{i}=0, ∀ *i*≠*q*, for moving out of the convex hull. Thus, the outward displacement from an edge of the convex hull (orthogonal to the edge), denoted as |Δ*Z*_{f⊥}|, due to a deviation in the agent closer to *Z*_{f} (and part of the edge), is bounded as *π*/*n* or *δ*_{p} is a small angle then the deviation is nearly perpendicular to the edge of the convex hull and is thus effective for convergence at a point away from the edge of the convex hull, but the magnitude of *δ*_{p} results in a greater magnitude of *δ*_{p}. The function |Δ*Z*_{fmax}(λ,*δ*_{p})| is continuous in the domain *δ*_{p}. ▪

It should be noted that a non-zero deviation is made by that agent (of the two that comprise the edge) which is closer to the desired rendezvous point. The number of decision variables in this case are *n*+1 (*n* gains and one deviation).

It may be pointed out that in case it is known that there is a deviation in one agent (say agent *p*) only, the limit of stability can be pushed further. In such a case, all the centres of the Gershgorin discs, except one, will lie on the real axis. Thus, the angle *θ*_{i}, subtended by the point *P* in figure 4 at the centre of the *i*th circle (where agent *i* has no deviation), is given by *δ*_{p}. As before, it is argued that the point *P* contributes the greatest angle *θ*_{i} (in terms of absolute value) for the agent *i* among all points within the hatched region. Hence, if *P* cannot meet the argument condition of (3.11), no other point within the hatched region can. For the characteristic equation in (3.11), *P* satisfies *θ*_{p}=−*δ*_{p}. Hence, for *δ*_{p}>0, this inequality can be ensured by the following condition (for *δ*_{p}>0):
*δ*_{p}<0. This is, in general, a more relaxed condition than the one obtained earlier in the paper for multiple number of agents with deviations.

## 6. Simulation results

This section provides simulation examples to illustrate the analytical results related to stability and reachable sets. The following examples show that the set of reachable points expands by use of deviated cyclic pursuit, to include points that could not be reached using heterogeneous cyclic pursuit with one negative gain as in [6].

### Example 6.1

Consider four agents starting from (0, 0), (3, 0), (3, 3) and (0, 3). Let the gains be chosen (without deviation) to reach a point (2.95, 1.8), close to the edge joining agents 2 and 3 so that λ=0.6 for 3 and |*ϵ*|=0.05. One such set of gains is [300, 2.5210, 1.7045, 75]. Let the deviations be *δ*_{i}=0, ∀ *i*≠3, and *δ*_{3}=0.70 rad). Here, *A*=2.20 and *a* where the convex hull of the initial positions is shown with thicker lines. This convergence would not be possible using any known linear cyclic pursuit scheme in the literature. From (5.7), the value of |Δ*Z*_{f max}|=0.574 and direct computations from simulated results yield |Δ*Z*_{f}|=0.510 and |Δ*Z*_{f⊥}|=0.46. Figure 8*b* shows the connection invariance property (implying that the sequence of pursuit does not affect the point of convergence) of the deviated cyclic pursuit law. This is to be expected in accordance with (3.16).

### Example 6.2

In this example, six agents start from the vertices of a hexagon given by [(0,0), (−1,10), (4,−2), (5,8), (−2,3), (8,4)]. Their convex hull (the hexagon) is shown with a thicker line in figure 8*c*. Suppose the rendezvous is desired at a point just outside the convex hull (the hexagon) of initial coordinates and near the mid-point of the edge joining agents 1 and 3. An initial choice of gains to rendezvous inside the hull at (2.1,−0.9) is given by [2.151, 200, 1.946, 200, 156.25, 217.39], so that λ=0.51 for agent 3. If the deviations of all agents, other than agent 3, are chosen to be zero, the magnitude of the deviation is restricted by stability constraints. This implies, as before, that *δ*_{3}=0.30 rad for meeting the above condition, the rendezvous occurs at (1.930,−1.196) as shown in figure 8*c*.

For this particular example, the function Δ*Z*_{f max}(λ,*δ*_{3}) is given by
*δ*_{3} for three different values of λ. The graphs in figure 8*d* indicate that the function is mostly increasing in the domain of interest. However, at larger values of λ (when the rendezvous is much closer to one vertex than the others), the peak value of *π*/6=0.523. In such cases, for optimal benefit (in terms of reachability), a value below the highest permissible *δ*_{3} may be used.

### Example 6.3

Two agents start from [(5,15), (10,2)]. Using any known cyclic pursuit scheme, rendezvous can occur only at a point on the straight line segment joining the initial coordinates of the agents, or its extension on either side. However, using deviated cyclic pursuit with gains given by [2.1, 1.91] and a non-zero deviation, *δ*_{2}=*π*/4, for agent 2, rendezvous can be achieved at (4.95, 7.11), as shown in figure 8*e*. Thus, even for smaller number of agents the expansion in reachable set is significant. It is also easy to verify the exact conditions for stability using (3.11) for *n*≤3. In a similar manner, the two eigenvalues of the system matrix in this case are 0 and −3.4506−*j*1.3506, respectively, thereby ensuring stability.

### Example 6.4

Two agents are considered as in example 6.3 and the deviation of agent 1 is chosen as *π*/4, while that of agent 2 is zero. The gains for both agents are chosen to be 5. It may be seen from figure 8*f* that the agents rendezvous at a point (10.19, 9.53), which does not lie on the straight line joining the initial positions of the agents. Thus, heterogeneous deviations enable rendezvous outside the convex hull of the initial positions of the agents.

### Example 6.5

Here four agents start from the vertices of a square as in example 6.1, and their gains are given by [1, 1.5, 2, 2.5], while the homogeneous deviation is 0.2 rad. This deviation is within the limit proposed in theorem 3.7. The agents rendezvous at the point (1.364, 1.052), as shown in figure 8*g*, which is the same point where they would have converged to even without any deviation. Hence, as indicated in (3.10), homogeneous deviations do not affect the point of convergence.

It is easy to check that

### Example 6.6

Four agents start from the initial positions (0,0), (5,0), (3,3), and (0,8), respectively, with a uniform gain of 0.5 and deviation of 0.4241 rad., which is equal to the critical deviation for stability as given by *h*.

### Example 6.7

Here, two agents with same gain are considered. Thus, *ρ*=0.5. They start from (0,0) and (8,10), respectively, with agent 1 having a deviation of *π*/6 while agent 2 is undeviated. This results in a rendezvous at the point (2.66, 6.07) which is in accordance with (4.15). The resulting trajectories are shown in figure 8*i*. It is clear that the point of convergence does not lie on the straight line joining the initial positions of the two agents, or on the same line extended in either direction. Using any known cyclic pursuit scheme such as homogeneous or heterogeneous, rendezvous at this point would not have been possible. This result demonstrates expansion in reachability set. It may be verified that the stability of the system is ensured, as per theorem 3.2, with the choice of gains and deviations.

### Example 6.8

Consider four agents as in example 6.1 with gains given by [0.1 0.2 0.25 0.3]. The uniform deviation for each of them is *π*/30, which ensures stability according to theorem 3.7. Without this deviation, the agents would rendezvous at (1.21,0.98). It may be shown from figure 8*j* that even with this uniform deviation the point of convergence does not change as the agents converge to (1.21,0.98). Thus, homogeneous deviation does not have any effect on the point of convergence as seen from (4.18).

### Example 6.9

The agents start from the vertices of a square (as in example 6.1). The gains are given by [0.021 0.6 0.6 0.021]. The deviation of the first agent is *π*/6 while those of the others is zero. From theorem 3.8, this choice of gains and deviations ensures stability. According to (4.20), the point of convergence is given by (−0.29,1.53), as shown in figure 8*k*.

The above examples demonstrate the advantages of using deviated linear cyclic pursuit in expanding the reachable set, illustrate the sufficiency of the stability conditions derived in this paper, indicate the relationship between the angle of deviation and the expansion of the reachability set, and show the efficacy of the new scheme for both smaller (*n*<4) and slightly greater number of agents (*n*≥4). Besides, they also indicate a method for designing gains and deviation(s) according to the desired point of convergence outside the convex hull of initial coordinates.

## 7. Conclusion

In this paper, the deviated cyclic pursuit scheme has been analysed and conditions for stability have been derived for both the continuous-time and discrete-time versions of the same. It has been shown that even though Gershgorin discs are used for the stability analysis of both the continuous- and discrete-time versions of deviated cyclic pursuit, yet the results on the stability of discrete-time cyclic pursuit are not straightforward extensions of the continuous-time pursuit. The expansion of the reachable set has been analysed and illustrated through simulation results. This shows that use of deviation by one or more agents is effective in expanding the reachable set. The scheme is particularly useful when all the agents start from the vertices of a convex polygon. The examples also indicate that significant expansion in a reachable set is attainable through deviation by one agent only.

## Data accessibility

The paper contains no data other than that already present in the body of the paper.

## Conflict of interests

We declare we have no competing interests.

## Funding

This research was partly supported by the NPMicAV Programme and the AOARD.

## Disclaimer

Both the authors contributed equally to the ideas in the paper. D.M. is responsible for carrying out the simulations. Both authors gave final approval for publication.

## Acknowledgements

There were no contributors other than the authors enlisted.

- Received September 28, 2015.
- Accepted November 18, 2015.

- © 2015 The Author(s)