Ellipsoidal Cones and Rendezvous of Multiple Agents

Transcription

1 Ellipsoidal Cones and Rendevous of Multiple gents Raktim Bhattacharya Jimmy Fung bhishek Tiari Richard M Murray College of Engineering and pplied Science California Institute of Technology Pasadena, C bstract In this paper e use ellipsoidal cones to achieve rendevous of multiple agents Rendevous of multiple agents is shon to be equivalent to ellipsoidal cone invariance and a controller synthesis frameork is presented We first demonstrate the methodology on first order LTI systems and then extend it to rendevous of mechanical systems, that is systems that are force driven I INTRODUCTION Invariant sets play an important role in many situations hen the behaviour of the closed-loop system is constrained in some ay Blanchini in ref [1] provides an excellent survey of set invariant control Invariant sets that are cones have found application in problems related to areas as diverse as industrial groth [2], ecological systems and symbiotic species [3], arms race [4] and compartmental system analysis [5], [6] In general, cone invariance is an essential component in problems involving competition or cooperation For those interested in cones and dynamical systems, the book by Berman et al [7] ill be useful In our earlier ork [8], e demonstrated that rendevous of multiple agents is equivalent to cone invariance Cones in general could be polyhedral or ellipsoidal and the rendevous problem can be cast as a cone invariance problem of either type In this paper e use ellipsoidal cones to develop a frameork for controller synthesis that achieves multi-agent rendevous In [9] e analye rendevous using polyhedral cones The paper is organied as follos We first present mathematical preliminaries that is fundamental to our research ork The equivalence of rendevous and cone invariance is established next We then Postdoctoral Scholar, raktim@cdscaltechedu Graduate Student, fung@caltechedu Graduate Student, atiari@cdscaltechedu Professor, murray@cdscaltechedu present the controller synthesis frameork for multiple vehicles modeled as first order linear time invariant (LTI systems This frameork is extended to agents ith higher order dynamics The synthesis algorithms presented in this paper are then verified and illustrated by theoretical results and simulations II MTHEMTICL PRELIMINRIES Ellipsoidal Cones n ellipsoidal cone in R n is the folloing, Γ n {ξ R n : K n (ξ, Q, ξ T u n }, (1 here K n (ξ, Q ξ T Qξ, Q R n,n is a symmetric nonsingular matrix ith a single negative eigen-value λ n and u n is the eigen-vector associated ith λ n The boundary of the cone Γ n is denoted by Γ n and is defined by ξ Γ n {ξ Γ n : K n (ξ, Q } The outard pointing normal is the vector Qξ for ξ Γ n Lemma 1 (27 in [1] If Γ n is an ellipsoidal cone, then there exists a nonsingular transformation matrix M R n,n such that [ ] (M 1 T QM 1 P Q 1 n here P R n 1,n 1, P > and P P T Let the transformed state be x Mξ The ellipsoidal cone in x is therefore, ( T [ P Γ n {x : } (2 1

2 Fig 1 Ellipsoidal cone in 3-dimension condition for exponential non-negativity of ellipsoidal cones Theorem 1 (35 in [1] necessary and sufficient condition for p(γ n is that there exists γ R such that, Q n + T Q n γq n here Q n is defined in Lemma 1 Proof Please refer to pg162 of [1] here x ( T, R n 1, R n ellipsoidal cone in three dimension is shon in Fig(1 The axis of the cone is the eigen-vector associated ith the axis B Ellipsoidal Cone Invariance Consider a linear autonomous system ξ ξ (3 cone Γ n is said to be invariant ith respect to the dynamics in eqn(3 if x(t Γ n x(t Γ n, t t, ie if the system starts inside the cone, it stays in the cone for all future time Such a condition is also knon as exponential non-negativity, ie e t Γ n Γ n It is ell knon that certain structure in the matrix imposes constraints on e t [11] The most ell knon result is the condition of non-negativity on hich states that if ij for i j, then nonnegative initial conditions yield non-negative solutions Schneider and Vidyasagar [12] introduced the notion of cross-positivity of on Γ n hich as shon to be equivalent to exponential non-negativity Meyer et al [13] extended cross-positivity to nonlinear fields Let us characterie p(γ n to be the set of matrices R n,n hich are exponentially non-negative on Γ n It is defined by the folloing lemma Lemma 2 (31 in [1] Let Γ n be an ellipsoidal cone as in eqn(2 Then, p(γ n { R n,n : < ξ, Qξ >, ξ Γ n } (4 Lemma 2 states that is such that the flo of the associated vector field is directed toards the interior of Γ n, ie the dot product of the outard normal of Γ n and the field is negative at the boundary of the cone This leads to the result on the necessary and sufficient III RENDEZVOUS OF MULTIPLE GENTS In our earlier ork [8] e had defined rendevous to be the problem of driving multiple agents to a desired point such they all arrive ithin a small time interval of each other It is also required that the trajectories of the agents be such that they arrive at the destination point only once In the rest of the paper, e refer to the destination point as the origin Rendevous Interpretation on the Phase Plane Consider to scalar systems V 1 and V 2 defined by V 1 : ẋ 1 f 1 (x 1, f 1 (, V 2 : ẋ 2 f 2 (x 2, f 2 ( If V 1 and V 2 are exponentially stable systems then they ill arrive at the origin at eventually as time tends to infinity The trajectories may also be such that V 1 arrives long before V 2 does, hich is undesirable Therefore, just exponential stability doesn t ensure rendevous To achieve rendevous in finite time e relaxed the definition of rendevous to be such that rendevous is achieved if the agents enter a certain neighborhood around the origin We defined this region to be the rendevous region We also demonstrated that rendevous is best visualied in the phase plane To interpret rendevous for V 1 and V 2, e defined the folloing regions in the phase plane, U 1 {(x 1, x 2 : δ x 1 δ}, U 1 {(x 1, x 2 : δ x 1 δ}, S U 1 U 2, F U 1 U 2 U 1 U 2, W R 2 U 1 U 2 The rendevous problem is ell posed if the initial condition of the to agents satisfy (x 1 (, x 2 ( W, ie they both start far aay from the destination point The set F is the set of all points here one agent enters 2

3 the rendevous region much before the other Therefore, trajectories must avoid F for a valid rendevous, ie x 1 (t, x 2 (t / F t (5 Invariant Region B Forbidden Regions Rendevous Square B Forbidden Regions Rendevous Square C (a pproximate rendevous Fig 3 (b Invariant edge, the region I Fig 2 Rendevous in Phase Plane In Fig 2, trajectory B starts from an invalid initial condition and trajectory C enters the rendevous region prior to the final entry Such trajectories are not valid rendevous trajectories Trajectory is an example of a valid rendevous trajectory With the constraint defined in eqn(5, the only ay trajectories can approach the origin is through the corners of S, ie through one of the points (δ, δ, (δ, δ, ( δ, δ or ( δ, δ This also restricts the trajectories to be confined to the quadrant they originate from Entering S from one of its corners also implies that the agents enter the rendevous region at precisely the same time In reality it may be acceptable to allo agents to arrive ithin T seconds of each other We distinguished beteen the to cases by referring to rendevous ith T as perfect rendevous and rendevous ith T small as real or approximate rendevous The notion of approximate rendevous led to a design parameter ρ des and a measure of rendevous ρ defined by ρ max( x 1(t a, x 2 (t a (6 δ here t a is arrival time of the first agent From the definition of ρ it is clear that for a given trajectory ρ 1 Therefore a specification of rendevous is meaningful if and only if ρ des 1 The notion of approximate rendevous is illustrated in Fig3(a pproximate rendevous allos trajectories in phase plane to enter U 1 U 2 as long as they are ithin the edge defined by the points (δ, δρ des, (, and (δρ des, δ In Fig 3(a, trajectory achieves approximate rendevous, but trajectory B does not Therefore the only admissible trajectories for approximate rendevous are those that arrive at the origin hile remaining in the edge-like region I as shon in Fig 3(b For n agents achieving rendevous, the region I becomes a cone in n-dimensional space Depending on the norm used to define ρ in eqn(6, the cone could be either polyhedral or ellipsoidal For -norm, as is in eqn(6, the cone is a polyhedral cone ith 2 N 2 sides The measure ρ can also be defined using 1-norm or 2-norm The edge in that case becomes a polyhedral cone ith n sides for 1-norm or an ellipsoidal cone for 2-norm This is shon in Fig4 Desired region of invariance Norm Complexity Fig 4 Polyhedral Cone ith n-sides Quadratic Cone in n-dimension Polyhedral Cone ith 2 n -2 sides n constraints 1 constraint 2 n -2 constraints Region I in 3 dimensional state space B Rendevous as Cone Invariance In our earlier ork [8] e shoed that agents achieve rendevous if they rendered the region I invariant s mentioned before, the region I in a higher dimensional phase space becomes a cone Therefore for n agents, rendevous is guaranteed if the cone in the n-dimensional state space is invariant ith respect to their dynamics 3

4 Cone invariance alone does not guarantee that the agents reach the origin Figure 5 shos trajectories, B and C Trajectory achieves cone invariance but does not reach the origin Trajectory B reaches the origin but escapes the cones Trajectory C is the only trajectory that reaches the origin and stays ithin the cone We are interested in trajectories such as C Forbidden Regions Rendevous Square Fig 5 Possible trajectories in I In this paper e assume that ρ is defined using the 2-norm and hence e are interested in the invariance of ellipsoidal cones C Problem Formulation Let us assume that there are n agents for hich rendevous is desired Let us also assume that the agents are modeled as first order LTI systems Collectively, they can be ritten as ξ ξ 1 ξ 2 ξ n a 1 a 2 C B a n b 1 b 2 + b n ξ + Bu ξ 1 ξ 2 ξ n u 1 u 2 u n We also assume that e are given an ellipsoidal cone Γ n as defined by eqn(2, here Q depends on the specified measure of rendevous ρ des Therefore, given a cone Γ n and n agents modeled as first order LTI systems, e are interested in determining control u(t such that the folloing are true, ξ(t Γ n ξ(t Γ n, t t, and ξ(t as t (7 D Controller Synthesis in the Lyapunov Function Frameork In this section e address ellipsoidal cone invariance in the frameork of Lyapunov functions Given a cone Γ n as in eqn(2, let us define to Lyapunov functions V ( and V ( as V ( T P (8 V ( 2 (9 Note that V is a valid candidate for Lyapunov function as P > Cone invariance in this context is defined by V ((t V ((t, t t This is guaranteed if and only if V < V hen V V (1 hich is the condition for exponentially non-negativity on Γ n For controller synthesis e transform Γ n as defined by lemma 1 Therefore, x Mξ ẋ MM 1 x + MBu If e consider a full state feedback control frameork, then u F ξ F M 1 x and the closed-loop system is therefore ẋ M( + BF M 1 x (11 With respect to the partition x ( T, the closedloop system in eqn(11 can be ritten as ( [ ẇ (12 ż Therefore, the inequality V < V in eqn(1 implies ( [ T T ] P + P P ( < T P T 2 Substituting T P 2 to impose the condition V 4

5 V, V < V at the boundary of the cone implies [ ] T P + P 2 P P < T P T (13 To ensure that V and V reaches ero as time goes to infinity, it is sufficient to constrain R < α des The parameter α des is a positive real number that governs the decay rate of (t Therefore, the controller synthesis problem in this frameork is the folloing LMI feasibility problem in the state feedback gain matrix F : T P + P 2 P P T P T (14 Therefore, if there exists an F such that the LMIP in eqn(14 is feasible, then the control la u(t F ξ(t solves the problem posed by eqn(7 < The constraint in eqn(13 is a necessary and sufficient condition for cone invariance and can be proved as follos Theorem 1 states that the necessary and sufficient condition for exponential non-negativity is the existence of γ R such that Q n + T Q n γq n This is equivalent to [ T P + P γp P T P T γ 2 hich is eqn(13 for γ 2 ] <, The next theorem states the Lyapunov certificate theorem for the rendevous problem defined by eqn(7 Theorem 2 The LMI in eqn(14 implies that V (, T P + 2 is a Lyapunov function for the closed-loop system in eqn(12 Proof For P >, V (, is a valid Lyapunov function V (, for the system in eqn(12 is, ( V (, T [ Equation (14 implies, ( T [ T P + P T P T T P + P P T P T P 2 < 2 T P for all (t, (t dding 2 2 to both sides gives us ( or T [ T P + P T P T P 2 V (, < 2 V (, T < 2 ( T P + 2 This implies that the largest exponent of the closedloop system is With <, e can conclude V (, <, hence the proof We next analye trajectories that start outside the cone Γ n It is interesting to note that the condition of cone invariance implies that all trajectories that start outside the cone enter the cone This is given by the folloing theorem Theorem 3 For a system ẋ x as in eqn(11, K n(x, Q n <, x Γ n K n(x, Q n <, x : K n(x, Q n > Proof Equation (14 implies, ( [ T T P + P P T P T < 2 T P for all (t, (t dding 2 2 to both sides gives us ( [ ] T T ( P + P P T T P T 2 < 2 ( T P 2 Recalling that < and K n (x, Q n > implies T P 2 >, e can conclude that ( [ ] T T ( P + P P T P T 2 <, x : K n(x, Q n >, hich is equivalent to K n (x, Q n < for x outside the cone K n (x, Q n This condition implies that all trajectories that start outside the cone ill arrive at the boundary of the cone Example 1 Figure 6 demonstrates rendevous for three agents modeled as first order open-loop unstable systems ith a i 1, b i 1 The trajectory of ξ 3 is orth noting The initial condition for ξ 3 is closer to the origin relative to that for ξ 1 and ξ 2 It is interesting to observe that ξ 3 initially moves aay from the origin before making the final entry, along ith ξ 1 and ξ 2 Therefore the control 5

6 formulation presented in this paper allos agents to procrastinate, as demonstrated by ξ 3, in order to achieve rendevous Figure 7(a shos that V < V for all times Figure 7(b shos the trajectory inside the cone Γ n ξ(t u(t Fig 6 State and control trajectories - solid(ξ 1, u 1, dash-dot(ξ 2, u 2, dash(ξ 3, u 3 V and V (a Solid(V, dash(v Fig 7 ξ ξ 2 2 (b Trajectory inside K n(ξ, Q < Cone invariance and asymptotic stability IV RENDEZVOUS OF GENTS WITH HIGHER ORDER DYNMICS In the previous section e formulated the control synthesis problem for agents that ere modeled as first order systems In reality agents ill have higher order dynamics We continue to restrict our interest to linear systems and consider the problem of multi-agent rendevous for agents that are mechanical systems ie agents hose dynamics can be represented by the linear second order differential equation m i ξi (t + d i ξi (t + k i ξ i (t u i (t here m i, d i and k i are mass, damping and stiffness respectively In matrix-vector notation the dynamics can ξ 1 be represented by ( [ d ξi dt ξ i 1 k m d m ξi ξ i + [ 1 m ] u i For n agents the collective dynamics can be represented by the equation ( [ [ ] ξ I N ξ + u (15 η ηξ ηη η B η and e assume that the system is controllable For dynamical systems given by eqn(15, the cone Γ n defined on position states ξ is not closed-loop holdable (pg65 [7] cone Γ n is said to be closed-loop holdable if there exists control u(t such that the condition of exponential non-negativity can be enforced, ie u(t : K n (ξ, Q <, ξ Γ For the system in eqn(15 and the cone in eqn(1, K n (ξ, Q ξ T Qξ + ξ T Q ξ η T Qξ + ξ T Qη hich is independent of u Therefore, the condition of exponential non-negativity cannot be enforced by any choice of u Controller Synthesis We propose to solve the rendevous problem by the folloing to-step controller synthesis algorithm Step 1 - We first consider the dynamical system ξ η In the first step e determine η(t such that ξ(t achieves rendevous We assume that η(t F ξ(t (16 That is, e treat η(t as a control variable and determine the state feedback gain F such that Γ n is invariant ith respect to the system ξ η F ξ This synthesis can be achieved by solving the LMIP in eqn(14 Step 2 - Once F is knon e treat η r (t F ξ(t as the reference signal and design a tracking controller so that η(t η r (t as t ny control design methodology can be used to design 6

7 this controller Note that the dynamics of η need not necessarily be linear In general, the methodology can be used to achieve rendevous for agents ith dynamics ξ η η f(ξ, η + g(ξ, ηu If f(ξ, η and g(ξ, η are nonlinear functions, one ould have to adopt a nonlinear control design frameork to obtain a tracking controller agents arrive at the origin, implying trajectories outside the cone enter the cone before arriving at the origin This is clearly visible in Fig9(b Figure 9(b also demonstrates that the reference η r is valid outside the cone The initial loop in the trajectory indicates that the agents ere initially heading toards the rong direction This is due to a large initial error in the velocity By reducing this error to ero, the tracking controller as able to achieve rendevous Equation (16 defines a desired η(t ith respect to position ξ(t Even if the agents start inside Γ n, it is likely that the initial condition of η(t ill not be the desired value Depending on ho large this initial offset is, it is possible that the agents escape the cone hile η r (t is being tracked Once the agents leave the cone, it is important to analye if η r (t F ξ(t is still a valid reference for rendevous Theorem 3 states that all trajectories starting outside Γ n ill intersect the surface of Γ n, ie the controller obtained by solving eqn(14 ill drive agents into the cone for all initial conditions outside the cone Hence, agents ith dynamics as in eqn(15 ill achieve rendevous if they track the reference η r (t F ξ(t Position v gent1 gent3 gent2 v 1 v Example 2 In this example e demonstrate the application of the proposed control methodology to rendevous of three double integrators Figure 8 shos the position and the velocity of the three systems In the subplots for velocities v 1, v 2 and v 3, the solid line is the desired velocity and the dashed line is the velocity of the agents The three agents started from position (35, 5, 2 ith velocity (414, 122, 215 The tracking controller for velocity reference as designed as a linear quadratic regulator using the formulation min K such that ( ξ η and [ ] (η η r T M(η η r + u T Nu dt, ([ ] [ I N + ηξ ηη Bη η r F ξ ] ( ξ K η The initial condition as deliberately chosen so that the agents started outside the cone ith large error in velocity Figure 9(a shos the V > V before the Fig 8 Position and tracked velocity V SUMMRY In this paper e presented a control synthesis frameork for multi-agent rendevous problem It as shon that the problem of rendevous of multiple agents can be cast as a cone invariance problem We restricted our attention to ellipsoidal cones The proposed synthesis algorithm is based on determining control such that the closed-loop system renders a given ellipsoidal cone invariant We first demonstrated this on agents modeled as first order LTI systems and extended it to agents that are mechanical systems ith second order dynamics The frameork presented in the paper is still restricted to rendevous on a line, ie it can achieve rendevous on a single state variable of the agent In reality, it is desired that rendevous be achieved on multiple state variables Extension of this frameork to such cases is a subject of our current research CKNOWLEDGEMENTS The authors are grateful for support through: FOSR F for bhishek Tiari; NSF-ITR-CI for Jimmy Fung; CIT Special Institute Fello- 7

8 V and V (a Solid(V, dash(v [7] Berman, M Neumann, and R J Stern Nonnegative Matrices in Dynamic Systems Wiley-Interscience Publication, Ne York, 1989 [8] Tiari, J Fung, J M Carson III, R Bhattacharya, and R M Murray Frameork for Lyapunov Certificates for Multi- Vehicle Rendevous Problems In Proceedings of the merican Control Conference, Boston, 24 [9] Tiari, J Fung, R Bhattacharya, and R M Murray Polyhedral Cone Invariance pplied to Rendevous of Multiple gents In IEEE Conference on Decision and Control (submitted, Bahamas, 24 [1] R J Stern and H Wolkoic Exponential Nonnegativity on the Ice Cream Cone Siam J Matrix nal ppl, 12:16 165, January 1991 [11] J Yorke Invariance for Ordinary Differential Equations Math Sys Theory, 1: , 1967 [12] H Schneider and M Vidyasagar Cross-Positive Matrices SIM J Numer nal, 7(4:58 519, 197 [13] D G Meyer, T L Piatt, H N G Wadley, and R Vancheesaran Nonlinear Invariance: Cross-Positive Vector Fields In Proceedings of the merican Control Conference, Philadelphia, 1998 (b Trajectory outside the cone enters the cone Γ n Fig 9 V > V close to the origin ship and CIT ESD Felloship for John Carson; and DRP F for Raktim Bhattacharya REFERENCES [1] F Blanchini Set Invariance in Control utomatica, 35: , 1999 [2] J Stiglit, editor Collected Scientific Papers of Paul Samuelson MIT Press, Cambridge, Massachusetts, 1966 [3] D G Luenberger Introduction to Dynamic Systems, chapter 6 Wiley, Ne York, 1979 [4] L F Richardson rms and Insecurity Boxood Press, Pittsburgh Quadrangle Books, Pittsburgh,Pennsylvania, 196 [5] J Jacque Compartmental nalysis in Biology and Medicine, volume 5 Elsevier, Ne York, 1972 [6] D H nderson Compartmental Modeling and Tracer Kinetics Lecture Notes in Biomathematics,