Aperiodic Communication for MPC in Autonomous Cooperative Landing
Dženan Lapandi?, Linnea Persson, Dimos V. Dimarogonas, Bo Wahlberg
AAperiodic Communication for MPC inAutonomous Cooperative Landing (cid:63)
Dˇzenan Lapandi´c ∗ , Linnea Persson ∗ , Dimos V. Dimarogonas ∗ , Bo Wahlberg ∗∗ Division of Decision and Control Systems, School of ElectricalEngineering and Computer Science, KTH Royal Institute ofTechnology, Stockholm, Sweden(e-mail: lapandic, laperss, dimos, [email protected]).
Abstract:
In this paper, we focus on the rendezvous problem for the autonomous cooperativelanding of an unmanned aerial vehicle (UAV) on an unmanned surface vehicle (USV). Theseheterogeneous agents with nonlinear dynamics are dynamically decoupled but share a commoncooperative rendezvous task. The underlying control scheme is based on the Distributed ModelPredictive Control (MPC). One of our main contributions is a rendezvous algorithm with anonline update rule of the rendezvous location. The algorithm requires that agents update therendezvous location only when they are not guaranteed to reach it. Therefore, the exchange ofinformation occurs aperiodically and proposed algorithm improves the communication efficiency.Furthermore, we prove the recursive feasibility of the algorithm. The simulation results showthe effectiveness of our algorithm applied to the problem of autonomous cooperative landing.
Keywords:
Autonomous cooperative landing, Nonlinear predictive control, Model predictiveand optimization-based control, Distributed nonlinear control, UAVs, Tracking.1. INTRODUCTIONCoordination and control of multi-agent systems is a vividresearch area with applications in robot manipulatorscontrol, autonomous vehicles, unmanned aerial vehicles(UAV) and space systems among others. Because multi-agent systems are composed of agents that have their ownembedded computing and communication units, usage ofa distributed control scheme is the most common controlapproach to these types of problems.Search-and-rescue missions are one example of an appli-cation that is dependent on distributed and multi-agentcontrol. In such a mission, heterogeneous agents have toseparate and perform tasks together or by themselves,while considering the overall objective of the mission andassisting other agents if needed. This type of scenario hasbeen simulated and tested as a part of the WASP researcharenas, Persson and Wahlberg (2019). Further researchhas considered the problem of safely landing drones onboats while they are moving at high speeds with algorith-mic modification to ensure the agents rendezvous simul-taneously, Persson and Wahlberg (2021). The rendezvousproblem is challenging due to several reasons, for example,sudden communication losses, strong disturbances actingon the agents that can lead to disastrous consequences,as well as the very basic ones – to determine if the ren-dezvous is possible or not and what strategy to employ (cid:63)
This work was supported by the Wallenberg AI, AutonomousSystems and Software Program (WASP) and the Swedish ResearchCouncil.This work has been submitted to IFAC for possible publication.
Fig. 1. The motivating application is a scenario wheredrones must be able to rendezvous and land on amoving boat.when the rendezvous is not possible. The illustration ofthe motivating problem is depicted in Fig. 1.The control method of choice for this application has beenModel Predictive Control (MPC) because of its abilityto explicitly include advanced system dynamics as wellas diverse state and input constraints directly in thecomputation of the control inputs. A question which hasnot been directly addressed in previous research is thatof efficient communication strategies between the agents.Instead, previous distributed solutions have communicatedall state and trajectory information between the agentsat each sample time, Bereza et al. (2020). In this paper,we consider rendezvous control through distributed MPC,where the agents use an aperiodic exchange of informationto negotiate and update their rendezvous point. Theexchange of information occurs only when it is necessaryto maintain the feasibility of the control algorithm, thusreducing the nonessential communication between theagents. The control algorithm is applied to nonlinearheterogeneous agents with state and input constraints, and a r X i v : . [ ee ss . S Y ] F e b ested and evaluated in simulation on an example of a UAVlanding on an unmanned surface vehicle (USV).The main contributions of this paper can be outlined asfollows: • We present the distributed rendezvous algorithm thatenables the aperiodic communication between theagents based on the deviations from the predicted tra-jectory, thus eliminating nonessential communication. • Moreover, we synthesize the time-varying distributedterminal sets for tracking that depend on the ren-dezvous point. These terminal sets are the main in-gredient in the recursive feasibility proof. • Finally, we prove that the proposed algorithm isrecursively feasible.In the literature, there are several approaches to dis-tributed implementation of model predictive control andan overview of them is provided in Christofides et al.(2013). Our focus is on dynamically decoupled systemsthat can be coupled with a performance criteria. In Ke-viczky et al. (2006) the authors assume that each agentknows the system dynamics of all of its neighbors tocompute their assumed optimal state trajectories. Thestability is established with the requirement that the mis-match from the actual trajectories of the agent’s neigh-bors is small. A similar approach was taken in Dunbarand Murray (2006) in which the stability is imposed byrequiring that the calculated trajectories of each agent donot deviate from those calculated in the previous time step.Sequential optimization of the local cost functions can, un-der some assumptions, guarantee stability and convergenceto the common cooperative goal as shown in M¨uller et al.(2012).However, most of the mentioned research assume a peri-odical exchange of information between the agents andrecalculation of the control inputs at every time step.The recalculated control inputs usually do not generatemuch different state trajectories compared to the onesfrom the previous time steps, especially if the model isvery accurate and disturbances acting on the system aresmall, but are critical for feasibility requirements, see e.g.Chen and Allg¨ower (1998). The aperiodic (distributed)MPC can be implemented using the event-triggered or self-triggered strategy Heemels et al. (2012). The triggeringconditions can be cost-based, the optimal control problemis recalculated when the cost is not guaranteed to decreaseHashimoto et al. (2014), or trajectory-based, when the tra-jectories deviated too much from the previous ones and thefeasibility of the overall problem might be compromisedHashimoto et al. (2017), Liu et al. (2020). However, thetriggering conditions for nonlinear systems are based onthe worst-case trajectory prediction that involves Lipschitzcontinuity assumption and Lipschitz constant, which forthe systems with fast and agile dynamics, like quadcopters,can lead to very conservative triggering conditions tomaintain feasibility and stability. Therefore, in this paper,we assume that the recalculation of the optimal controlproblem is conducted at every time step, and investigatehow aperiodic negotiation of the rendezvous location canpreserve the feasibility.The paper is organised as follows. First, we state theproblem formulation and the distributed optimal control problem in Section 2. Then, we present the rendezvousalgorithm in Section 3 and its feasibility in Section 4.Finally, in Section 5, we describe the models and theirconstraints used to generate the results that are alsopresented in this section.
Notation:
We use P (cid:31) P ispositive definite. The notation (cid:107) x (cid:107) is used as the Euclideannorm of vector x , and (cid:107) x (cid:107) P as a weighted norm of x ,where (cid:107) x (cid:107) P = √ x T P x . The notation λ min ( Q ) is usedto denote the minimal eigenvalue of the matrix Q . Wedenote the system state trajectories with x ( t ), nominalstate trajectories with ˆ x ( t ) and optimal state trajectorieswith ˆ x ∗ ( t ). 2. PROBLEM FORMULATION We consider M agents with nonlinear dynamics and addi-tive disturbances:˙ x i ( t ) = f i ( x i ( t ) , u i ( t )) + w i ( t ) ,y i ( t ) = g i ( x i ( t )) = C i x i ( t ) , (1)for t ≥ t , where for each i = 1 , ..., M , x i ( t ) ∈ R n i is thestate, u i ( t ) ∈ U ⊆ R m i is the control input, y i ( t ) ∈ R p is the output, w i ( t ) ∈ W ⊆ R n i is the additive boundeddisturbance, and t ∈ R is the initial time.The following standard MPC assumptions as in Chen andAllg¨ower (1998) are considered in this paper. Assumption 1. (i) The function f i : R n i × R m i → R n i is twice continuously differentiable and f i (0 ,
0) = 0; (ii)
U ⊆ R m i is compact, convex and 0 ∈ R m i is containedin U ; (iii) the system in (1) has a unique solution forany initial condition x i, ∈ R n i , any piecewise continuousand right-continuous control u i : [ t , ∞ ) → U i , and anydisturbance w i : [ t , ∞ ) → W i ; (iv) for the linearizedsystem around the origin without disturbances, i.e., ˙ x i = A i x i ( t )+ B i u i ( t ), where A i = ∂f i ∂x i (0 ,
0) and B i = ∂f i ∂u i (0 , A i , B i ) is stabilizable; (v) for each agent i andits linearized dynamics around the origin, there exists amatrix K i such that A k,i = A i + B i K i is stable Hurwitzmatrix. Remark 2.
Note that the requirement f i (0 ,
0) = 0 isnot restricted to the origin, but can be shifted to anyequilibrium ( x i,s , u i,s ), as well as the linearization in (iv).The control objective is to steer the output of every agent y i to a zone centered around a rendezvous point θ RZ Φ RZ = { y i ∈ R p : (cid:107) y i − θ RZ (cid:107) ≤ ε RZ } in finite time.Let ˆ x i ( s ; t k ) , ˆ y i ( s ; t k ) be the nominal state trajectory andoutput, respectively, calculated at time instant t k given by˙ˆ x i ( s ; t k ) = f i (ˆ x i ( s ; t k ) , u i ( s ; t k )) , ˆ y i ( s ; t k ) = g i (ˆ x i ( s ; t k )) = C i ˆ x i ( s ; t k ) , (2)for s ∈ [ t k , t k + T ].Let us define a set Z i ( θ ) for each agent i and argument θ ∈ R p with a tuple (¯ x i , ¯ u i , ¯ y i ) such that Z i ( θ ) = { (¯ x i , ¯ u i , ¯ y i ) ∈ R n i + m i + p : 0 = f i (¯ x i , ¯ u i ) , ¯ y i = C ¯ x i = θ } ssumption 3. There exists a non-empty compact andconvex set Θ ⊆ R p such that ∀ θ ∈ Θ, we have Z i ( θ ) (cid:54) = ∅ for all i .By this assumption, it is also assumed that there exists anequilibrium for which the output reference θ is attained foreach agent. Moreover, such an equilibrium can be explicitlyfound with a given θ by the following linear functions h x,i : R p → R n i , h u,i : R p → R m i ¯ x i = h x,i ( θ ) , ¯ u i = h u,i ( θ ) . The following assumption is made to ensure that a suchrendezvous point is reachable (in a similar manner to
Assumption 2. in Keviczky and Johansson (2008)):
Assumption 4.
The time planning horizon T is longenough for at least one θ to reach the rendezvous regionΘ.We choose the cost function to penalize the deviations ofthe system trajectories from the desired terminal steadystate (¯ x i , ¯ u i , ¯ y i ): J i = J i (ˆ x i ( t k ) , u i ( t k ) , ¯ x i , ¯ u i )= (cid:107) ˆ x i ( t k + T ; t k ) − ¯ x i (cid:107) P i + (cid:90) t k + Tt k (cid:107) ˆ x i ( s ; t k ) − ¯ x i (cid:107) Q i + (cid:107) u i ( s ; t k ) − ¯ u i (cid:107) R i ds, (3)where Q i , R i , P i are positive definite weighting matrices, T > x i , ¯ u i , ¯ y i ) exists and is attainable.Before we formulate the distributed optimal control prob-lem we will present a Lemma on the local invariant termi-nal sets that is formulated and proven in the spirit of Chenand Allg¨ower (1998), Dunbar (2007), Hashimoto et al.(2017). Lemma 5.
For the nominal system (2), if Assumption 1holds, then there exists a positive constant α i >
0, amatrix P i = P Ti (cid:31)
0, and a local state feedback controllaw κ f i ( x i ) = K i x i ∈ U i , satisfying ∂V f,i ∂x i f i ( x i , κ f i ( x i )) ≤ − (cid:107) x i (cid:107) Q ∗ i for all x i ∈ X f,i ( α i ), where Q ∗ i = Q i + K Ti R i K i , V f,i ( x i ) = (cid:107) x i (cid:107) P i and X f,i ( α i ) = (cid:8) x i ∈ R n i : V f,i ( x i ) ≤ α i (cid:9) .The proof can be found in the Appendix. Remark 6. (On ∆ x i formulation) The size of the terminalset remains the same regardless of its center which canbe moved to a steady-state ¯ x i . It can be shown thatLemma 5 can be stated in ∆ x i formulation instead of x i where ∆ x i = x i − ¯ x i . Therefore, in the following chapterswe will use modified definitions of the terminal function, V f,i ( x i − ¯ x i ) = (cid:107) x i − ¯ x i (cid:107) P i , and the terminal set X f,i (¯ x i , α i ) = (cid:8) x i ∈ R n i : V f,i ( x i − ¯ x i ) ≤ α i (cid:9) . (4) Remark 7. (Calculation of ¯ α i ) The upper bound ¯ α i on α i , can be calculated with the help of the followingoptimization problem: min x i (cid:40) x Ti P i x i − λ ( ˆ Q P i ) φ Ti ( x i ) P i φ i ( x i ) (cid:41) (5a)s.t. 0 < x Ti P i x i ≤ α i (5b) K i x i ∈ U i . (5c)Now, we can formulate the Distributed Optimal ControlProblem (DOCP) with respect to our objective. Problem 8. (DOCP). At time t k with initial states x i ( t k ), i = 1 , ..., M , and given reference θ ( t k ), the distributedoptimal control problem is formulated asmin u i ( · ) , ¯ x i , ¯ u i J i (ˆ x i ( s ; t k ) , u i ( · ) , ¯ x i , ¯ u i ) (6a)subject to˙ˆ x i ( s ; t k ) = f i (ˆ x i ( s ; t k ) , u i ( s ; t k )) , s ∈ [ t k , t k + T ] , (6b)ˆ y i ( s ; t k ) = g i (ˆ x i ( s ; t k )) = C i ˆ x i ( s ; t k ) , (6c)ˆ x i ( s ; t k ) ∈ X i , (6d) u i ( s ; t k ) ∈ U i , (6e)¯ x i = h x,i ( θ ( t k )) , (6f)¯ u i = h u,i ( θ ( t k )) , (6g)ˆ x i ( t k + T ; t k ) ∈ X f,i (¯ x i , α i ) , (6h)for agents i = 1 , ..., M . For the initial time t , k = 0, theagents minimize the cost (6a) subject to (6b–h) for a given T >
0. 3. RENDEZVOUS ALGORITHMThe distributed optimal control problem stated in (6)depends on θ ( t k ) which is the rendezvous point in thesubset of the output space R p as stated in Assumption 3.
Before we present the algorithm, we need to define how θ ( t k ) is going to be initialized and updated.The rendezvous point θ ( t k ) at k = 0 can be initializedas a weighted average of the initial agent positions in theoutput space θ ( t ) = 1 M M (cid:88) i =1 c i y i ( t ) , s.t. M (cid:88) i =1 c i = M, c i ≥ , (7)where M is the number of agents.We assume that there exists c i , i = 1 , ..., M such that θ ( t ) ∈ Θ according to
Assumption 3.
If the agents areoperating in an unconstrained and obstacle-free outputspace, then any c i will result with θ ( t ) ∈ Θ. If that isnot the case, then an admissible c i would need to be de-termined by another layer of the optimization taking intoaccount output-space constraints of all agents. Moreover,future work will include the conditions such that θ ( t k )remains in a constrained output space Θ.Let us denote the output terminal offset term V o as V o = V o (ˆ y i , θ ) = V o (ˆ y i ( t k + T ; t k ) , θ ( t k ))= (cid:107) ˆ y i ( t k + T ; t k ) − θ ( t k ) (cid:107) . (8)After the initialization, the agent i updates θ ( t k ) accordingto the rule θ ( t k +1 ) = (cid:26) θ ( t k ) V o ≤ εθ ( t k ) − ηv θ ( t k ) V o > ε (9)where η and ε are tuning parameters and v θ ( t k ) is definedas: θ ( t k ) = ∂V o ∂θ ( t k ) (cid:13)(cid:13)(cid:13)(cid:13) ∂V o ∂θ ( t k ) (cid:13)(cid:13)(cid:13)(cid:13) − . (10)Parameter η , in some sense resembles a learning rate of thenew rendezvous location. It thus must be chosen as a smallvalue, in order to avoid overshooting, and it quantifies thecorrection of θ in the output space.Now we can state the rendezvous algorithm. Algorithm 1. (Event-triggered DMPC Rendezvous)(1) Initialization: Set prediction horizon T ; sampling pe-riod δ ; weighting matrices Q i , R i , P i ; initial state x i, at time t for each agent i = 1 , ..., M ; k = 0; c i , θ ( t ) according to (7) and rendezvous point updateparameters η and ε ;(2) For each agent i = 1 , ..., M :(a) if f lag = 1: download θ ( t k ) and set f lag = 0;(b) solve optimization problem (6); obtain the inputˆ u ∗ i ; generate predicted optimal output trajecto-ries ˆ y ∗ i ( s ; t k ).(c) Check the rendezvous condition: V o (ˆ y i ( t k + T ; t k ) , θ ( t k )) ≤ ε (11)(i) If (11) is not satisfied update θ ( t k +1 ) accord-ing to the rule (9); set f lag = 1.(d) Check the stopping condition: (cid:107) y i ( t k ) − θ ( t k ) (cid:107) ≤ ε (12)If not satisfied: apply ˆ u ∗ i ( t k ; t k ), set k = k + 1, goto step (2)(3) End Remark 9.
The only two ingredients that are being sharedbetween the agents are θ ( t k ) at time t k and f lag which isset such that the other agent knows that θ is updated.Therefore, the algorithm is able to run in parallel andsequentially, see e.g. Richards and How (2007).4. FEASIBILITYIn order to show feasibility of the DOCP Problem 8, wewill assume the initial feasibility and then show that theproblem is recursively feasible. Assumption 10.
Problem 8 is feasible at time t for eachagent i = 1 , ..., M with θ ( t ) initialized as in (7).The main point in the proof of the rendezvous algorithmis to ensure feasibility on the consecutive steps where therendezvous reference point θ ( t k ) is updated. The spaceshift of the terminal set X f,i (¯ x i , α i ) that occurs due tothe reference change θ ( t k +1 ) (cid:54) = θ ( t k ) at some t k can bequantified using the update rule for θ ( t k +1 ). Lemma 11.
For the nominal system with dynamics in Eq.(2) and reference change from ¯ x i ( t k ) to ¯ x i ( t k +1 ), given alocal terminal set X f,i (¯ x i , α i ) = (cid:8) x i ∈ R n i : V f,i ( x i − ¯ x i ) ≤ α i (cid:9) it holds that ifˆ x i ( t k + T ; t k ) ∈ X f,i (¯ x i ( t k ) , α i ( t k ))then ˆ x i ( t k +1 + T ; t k +1 ) ∈ X f,i (¯ x i ( t k +1 ) , α i ( t k +1 ))where α i ( t k +1 ) = α i ( t k ) + η (cid:107) h x,i ( v θ ( t k )) (cid:107) P i .The proof can be found in the Appendix. Remark 12.
For α i there exists an upper bound ¯ α i thatcan be determined with an offline optimization describedin Remark 7 . However, α i at time steps when there isno reference change can be reduced. Also, ¯ α i imposes aboundary on the reference change and these questions areto be addressed in the future work.Now we can state the recursive feasibility theorem. Theorem 13.
For the agents i = 1 , ..., M with systemdynamics given by (1), for which Assumptions 1 and 10and Lemmas 5 and 11 hold, Problem 8. is feasible at t k , k ≥ ×
1m in size. Complete derivation of theused quadcopter and boat models can be found in Perssonand Wahlberg (2019).
The state vector of quadcopter model x q is chosen as x q = [ p x , p y , p z , v x , v y , v z , φ, θ, ψ ] T , and input u q as u q = (cid:104) ˙ v z,cmd , φ cmd , θ cmd , ˙ ψ cmd (cid:105) T . The position in R space is represented with y q = [ p x , p y , p z ] T , and [ ˙ p x , ˙ p y , ˙ p z ] T = [ v x , v y , v z ] T . Thus, matrix C q =[ I × , × ].The equations of motion can be written as (cid:34) ˙ v x ˙ v y ˙ v z (cid:35) = g + ˙ v z,cmd cos φ cos θ (cid:34) sin φ sin ψ + cos φ cos ψ sin θ − cos ψ sin φ + cos φ sin ψ sin θ cos φ cos θ (cid:35) − (cid:34) f aero,x f aero,y (cid:35) + 1 m F ext + (cid:34) − g (cid:35) , where f aero,x = k D x v x , f aero,y = k D y v y , k D x = k D y = 0 . F ext areexternal forces and disturbances.The UAV has an attitude controller. Therefore, the at-titude dynamics are approximated by the inner-loop at-titude dynamics that are of first order, and for the yawangular velocity we assume that it can be instantaneouslyachieved Kamel et al. (2017):˙ φ = 1 τ φ ( k φ φ cmd − φ ) , ˙ θ = 1 τ θ ( k θ θ cmd − θ ) , ˙ ψ = ˙ ψ cmd where τ φ = 0 . , k φ = 0 . , τ θ = 0 . , k θ = 1 . (cid:113) v x + v y + v z ≤ . , | v z | ≤ . , | φ | ≤ . , | θ | ≤ . , | ˙ v z,cmd | ≤ . , | φ cmd | ≤ . , | θ cmd | ≤ . , | ˙ ψ cmd | ≤ π/ . The constraints in the left column constitute the set X q .The first two constraints are related to the UAV maximumvelocity and the maximum vertical velocity respectively,which we want to limit to prevent fast descent. Moreover,the latter two are constraints on the roll and pitch angles.The set U q is formed of constraints in the right column.The boat model is chosen as a simple car dynamical modelfor the purpose of this work. The state vector of boatmodel x b is chosen as x b = [ p x , p y , ψ, v x , v y , ω ψ ] T , andinput u q u b = (cid:2) τ x , τ y , τ ω ψ (cid:3) T . The position in R space isrepresented with y b = [ p x , p y , T . Matrix C b is given as C b = [diag(1 , , , × ].The boat model set constraints X b also has the velocityconstraints and constraint on the state ω ψ , i.e. (cid:113) v x + v y ≤ . | ω ψ | ≤ . . Finally, the input constraints U b hasonly constraints on τ ω ψ , i.e. | τ ω ψ | ≤ . Algorithm 1 is initialized with the following parame-ters. The planning horizon is set as T = 3s and sam-pling period is δ = 0 .
1s for both agents. The updateparameters for θ ( t k ) are η = 0 . ε = 0 .
1. Forthe quadcopter we choose the tuning matrices as Q q =diag(30 , , , , , , , , R q = I and obtain P q and¯ α q = 0 . Q b = diag(5 , , , , , R b = I , ¯ α b = 0 . xy -plane such that the UAVis above the boat and landing platform before the finaldescent.We set the initial state of the UAV and boat such thatthe position in the output space is y q = [4 , , T and y b = [ − , − . , T , respectively. To determine initial θ ( t )according to Eq. (7) we choose w q = 2 / w b = 4 / θ ( t ) is not changed then the agents willrendezvous at a point θ ( t ) = [ − . , − . , T that istwice closer to the boat as the boat is slower. This is visiblein Fig. 2 for the nominal case with terminal constraintswithout any disturbances. The difference between theinitially predicted and actual trajectories is a result of thechange of θ ( t k ) that occurred for the first four steps and θ ( t final ) = [ − . , − . , T . A perspective view of thesame setup is shown in Fig. 3.In order to show the performance of the update rule for θ ( t k ) we added a strong wind disturbance in the positive y -axis direction acting from t = 0 .
5s until t = 2s, depictedin Fig. 4. This causes the UAV to drift several meters inthe direction of the disturbance. However, the feasibilityis preserved at all time steps, and because of the imposedterminal constraints the updates of θ ( t k ) are small. − − − − . − − . . .
52 x [m] y [ m ] Agents position in space, view from topUAV initial predictionBoat initial predictionUAV trajectoryBoat trajectoryLanding platform
Fig. 2. Nominal case with terminal constraints. − − . − − . . . z [ m ] Agents position in space, perspective viewUAV initial prediction Boat initial prediction UAV trajectory Boat trajectory
Fig. 3. Perspective view of the setup for nominal case withterminal constraints. − z [ m ] Agents position in space, perspective viewUAV initial prediction Boat initial prediction UAV trajectory Boat trajectory
Fig. 4. Strong wind active for t = [0 . , . θ ( t k ) are shown inFig. 6. The bigger changes in θ ( t k ) compared to the casewith the terminal constraints are due to the update rule. V o (ˆ y i ( t k + T ; t k ) , θ ( t k )) is evaluated at the last predictedˆ y i ( t k + T ; t k ) output for which the corresponding stateˆ x i ( t k + T ; t k ) belongs to a very small set X f,i (¯ x i , ¯ α i ). − z [ m ] Agents position in space, perspective viewUAV initial prediction Boat initial prediction UAV trajectory Boat trajectory
Fig. 5. Strong wind active for t = [0 . , . − − . − . − . − . . . . .
81 Time [s] P o s i t i o n [ m ] θ ( t k ) position update θ x ( t k ) θ y ( t k )Disturbance start/stop Fig. 6. θ ( t k ) evolution in time for the case without terminalconstraints 6. CONCLUSIONIn this paper, we presented a rendezvous algorithm forthe distributed MPC scheme for agents with nonlinearand heterogeneous dynamics. The algorithm is designedfor the problem of autonomous cooperative landing of theUAV on the autonomous boat. During the landing theagents communicate only when it is necessary to updatethe rendezvous point and ensure the feasibility of thealgorithm. The effectiveness of the proposed algorithm isshown with the simulation of the landing scenarios.Although we did not experience feasibility issues, in fu-ture work, we aim to quantify the upper bound on the disturbance such that the feasibility of the algorithm ispreserved. Furthermore, it will be interesting to include theobstacles and constraints in the output space and examinethe behaviour of the algorithm on the real systems.REFERENCESBereza, R., Persson, L., and Wahlberg, B. (2020). Dis-tributed model predictive control for cooperative land-ing. In .IFAC.Chen, H. and Allg¨ower, F. (1998). A quasi-infinite horizonnonlinear model predictive control scheme with guaran-teed stability. Automatica , 34(10), 1205–1217.Christofides, P.D., Scattolini, R., de la Pena, D.M., andLiu, J. (2013). Distributed model predictive control: Atutorial review and future research directions.
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International Journal of con-trol , 80(9), 1517–1531.Appendix A. PROOF OF LEMMA 5
Proof.
The linearized system from Assumption 1 with thecontroller κ f i ( x i ) = K i x i is closed-loop asymptoticallystable. The matrix P i is such that following Lyapunovequation holds: A Tk,i P i + P i A k,i = − ( Q i + K Ti R i K i ) = − Q ∗ i Define an auxiliary function φ i ( x i ) = f i ( x i , κ f i ( x i )) − A k,i x i . The derivative of V f,i ( x i ) = (cid:107) x i (cid:107) P i = x Ti P i x i alonga trajectory of the nominal system ˙ x i = f i ( x i , κ f i ( x i )) is˙ V f,i ( x i ) = − x Ti Q ∗ i x i + 2 x Ti P i φ i ( x i ) ≤ − x Ti Q ∗ i x i (cid:32) − (cid:107) φ i ( x i ) (cid:107) P i λ min ( ˆ Q P i ) (cid:107) x i (cid:107) P i (cid:33) where ˆ Q P i = P − i Q ∗ i P − i . Because (cid:107) φ i ( x i ) (cid:107) Pi (cid:107) x i (cid:107) Pi → (cid:107) x i (cid:107) P i →
0, there exists ¯ α i > (cid:107) φ i ( x i ) (cid:107) P i (cid:107) x i (cid:107) P i ≤ λ min ( ˆ Q P i )4 , for (cid:107) x i (cid:107) P i ≤ ¯ α i . (A.1)Furthermore, there exists 0 < α i ≤ ¯ α i such that forall (cid:107) x i (cid:107) P i ≤ α i , the input constraints are satisfied, i.e. κ f i ( x i ) = K i x i ∈ U i .Then, for a such α i and for all x i in X f,i ( α i ) = (cid:8) x i ∈ R n i : V f,i ( x i ) ≤ α i (cid:9) we get ˙ V f,i ( x i ) = − x Ti Q ∗ i x i . Appendix B. PROOF OF LEMMA 11
Proof.
Let us consider the optimal control law ˆ u ∗ i ( s ; t k )for interval s ∈ [ t k , t k + T ] obtained at t k by solving theDOCP problem 7. and a candidate control law˜ u i ( s ; t k +1 ) = (cid:26) ˆ u ∗ i ( s ; t k ) s ∈ [ t k +1 , t k + T ] K i ˜ x i ( s ; t k ) s ∈ [ t k + T, t k +1 + T ] (B.1)that generates the system trajectory ˜ x i ( s ; t k +1 ) based onthe dynamics in (2).It holds that ˜ x i ( t k + T ; t k +1 ) ∈ X f,i (¯ x i ( t k ) , α i ( t k )) and, dueto the invariance of the terminal set, ˜ x i ( t k +1 + T ; t k +1 ) ∈X f,i (¯ x i ( t k ) , α i ( t k )), i.e. (cid:107) ˜ x i ( t k +1 + T ; t k +1 ) − ¯ x i ( t k ) (cid:107) P i ≤ α i ( t k ) . Then, (cid:107) ˜ x i ( t k +1 + T ; t k +1 ) − ¯ x i ( t k +1 ) (cid:107) P i = (cid:107) ˜ x i ( t k +1 + T ; t k +1 ) − ¯ x i ( t k ) + ¯ x i ( t k ) − ¯ x i ( t k +1 ) (cid:107) P i ≤ (cid:107) ˜ x i ( t k +1 + T ; t k +1 ) − ¯ x i ( t k ) (cid:107) P i + (cid:107) ¯ x i ( t k ) − ¯ x i ( t k +1 ) (cid:107) P i = α i ( t k ) + η (cid:107) h x,i ( v θ ( t k )) (cid:107) P i = α i ( t k +1 ) . Hence, ˜ x i ( t k +1 + T ; t k +1 ) ∈ X f,i (¯ x i ( t k +1 ) , α i ( t k +1 )).Appendix C. PROOF OF THEOREM 13 Proof.
If the state x i ( t k +1 ) ∈ X f,i ⊆ X i then by theinvariance of the terminal set stated in Lemma 5, it will remain in that set. Therefore, using the terminal controllaw κ f i ( x ) = K i x i ∈ U i , the cost function in (3) is boundedand all constraints in (6) are satisfied.Let us consider again the obtained optimal control lawˆ u ∗ i ( s ; t k ) at t k for interval s ∈ [ t k , t k + T ] and a candidatecontrol law according to Eq. (B.1) that generates thesystem trajectory ˜ x i ( s ; t k +1 ) based on the dynamics in (2).Because of feasibility at t k , the state ˜ x i ( s ; t k +1 ) ∈ X i for s ∈ [ t k +1 , t k + T ] and ˜ x i ( t k + T ; t k +1 ) ∈ X f,i (¯ x i ( t k ) , α i ( t k )).Moreover, due to the terminal set properties from Lemma5, and the result of Lemma 11 the candidate controllaw will ensure that the terminal state ˜ x i ( t k +1 + T ; t k +1 )is in the shifted local terminal set ˜ x i ( t k +1 + T ; t k +1 ) ∈X f,i (¯ x i ( t k +1 ) , α i ( t k +1+1