Decentralized Control with Moving-Horizon Linear Switched Systems: Synthesis and Testbed Implementation

Transcription

1 17 American Control Conference Sheraton Seattle Hotel May 4 6, 17, Seattle, USA Decentralized Control with Moving-Horizon Linear Switched Systems: Synthesis and Testbed Implementation Joao P Jansch-Porto and Geir E Dllerd1 Department of Mechanical Science and Engineering University of Illinois, Urbana, Illinois {janschp,dllerd}@illinoised systems was presented in [9 In this paper we present an extension to [9 considering receding horizon modal information, as it was shown for the centralized case in [8 Since we are interested in seeing how the switched controller performs in a real-world system, we have implemented the decentralized controller to mltiple robots We tilize testbed comprised of the HoTDeC hovercrafts developed at the University of Illinois This testbed was previosly sed in [1 for a persit-evasion problem In this paper we consider a vehicle formation scenario, where a leader follows a desired reference signals, and the following vehicles trail a certain distance behind We analized the performance of the system both in simlation and experimentation Abstract In this paper, we improve recent reslts on the decentralized switched control problem to inclde the moving horizon case and apply it to a testbed system Using known derivations for a centralized controller with look-ahead, we were able to extend the decentralized problem with finite memory to inclde receding horizon modal information We then compare the performance of a switched controller with finite memory and look-ahead horizon to that of a linear time independent (LTI) controller sing a simlation The decentralized controller is frther tested with a real-world system comprised of mltiple model-sized hovercrafts I I NTRODUCTION In this paper, we are interested in the application of decentralized control to nested systems with switching dynamics Nested systems represent a hierarchy of sbsystems with a nidirectional flow of information amongst them, sch as the configration presented in Fig 1 This setp can be encontered in many real-world applications, sch as economic theory, power systems, and vehicle formations For the controller synthesis, we will consider the H -type cost criteria This criteria is also referred to as distrbance attenation or root-mean sqare gain The main idea behind this performance measre is to minimize the effect of the worst-case distrbance onto the energy of the system Explicit state-space soltions to H optimal control of nested systems has only recently been developed rior work on distribted systems sing state-space techniqes inclde [1 and [, which provide sfficient conditions for controller existence and then explicit constrction conditionally In [3, the athor considered the decentralized control of continos-time time-invariant systems with nested interconnection strctre, presenting tight conditions for controller synthesis The discrete-time time-varying version of the H optimal control problem of interconnected systems was considered in [4 Other athors have discssed the decentralized control of nested systems in the past ([5 and [6), however considering different performance criteria For or application, we have to consider a nested discrete time switching system Synthesis conditions for a finitepath dependent centralized controller were given in [7 The reslts in [7 were frther extended in [8 to allow the controller access to a finite nmber of ftre parameters The synthesis reslts for a finite-path controller for nested Fig 1: Hovercraft Experimental setp The rightmost vehicle follows a reference signal, while each other vehicle maintains a distance ri away from the robot directly in front II S WITCHING S YSTEMS A Notation We denote the space of n-dimensional symmetric, positive-definite, and positive-semidefinite matrices by Sn, Sn+, and S n+ For any matrix W, W denotes fll colmn rank matrices satisfying Im(W ) ker(w ), W > W I We let `n be the space of infinite indexed seqence of elements x (x(), x(1), x(), ) with x(t) Rn for t N A n sbspace of `P is the Hilbert space `n (or simply ` ), with norm kxk t x(t) < To denote the nmber of decentralized sbsytstems in the nested setp, we se M The space of block-lower trianglar matrices takes the form H 11 H1 H HM 1 HM H M M by S (m1,, mm ), (k1,, km ) so that Hij Rmi kj and Hij for i < j Additionally, the notation J {1,, M } and J {,, M } is tilized For partitioned 1 The athors were partially spported by AFOSR Grant FA , and NSF Grant /$31 17 AACC 851

2 [ X1 X symmetric matrices, say X X, we will sppress repeated [ 3 X1 X sb-blocks as X 3 We will also write ineqalities of the form W GW as [ GW, for some matrix G B Preliminaries 1) Mode Dependent Switched Systems: A switched system is defined to be a mlti-model system that allows transitions among operation models, where each mode corresponds to a distinct state-space model ([11) The system dynamics are given by x(t + 1) A θ(t) x(t) + B θ(t) w(t) z(t) C θ(t) x(t) + D θ(t) w(t) where the system matrices depend on the switching signal θ(t) We assme that or switching signal takes vales from a discrete and finite set N {1,, n s }, and that switching between vales in time is governed by a finite-state atomata The set of admissible seqences of length r N generated by sch an atomaton is denoted as A r For the decentralized control problem, we consider the following mode-dependent switched plant: x(t + 1) A θ(t) x(t) + B w θ(t) w(t) + B θ(t) (t) (1) z(t) Cθ(t) z x(t) + Dzw θ(t) w(t) + Dz θ(t) (t) () y(t) C y θ(t) x(t) + Dyw θ(t) w(t) Here w(t) R nw is the distrbance inpt, z(t) R nz is the performance otpt, (t) R n is the control inpt, and y(t) R ny is the measrement available to the controller The states, inpts, and otpts are partitioned as x 1(t) 1(t) y 1(t) x(t), (t), y(t) x M (t) M (t) y M (t) where x i (t) R ni, i (t) R n i, and yi (t) R ny i The dimensions satisfy n M i1 n i, n M i1 n i, and ny M i1 ny i We also introdce the tple n (n 1,, n M ) and similarly define n and n y Since we are interested in nested systems with nidirectional flow of information, as in Fig 1, we want to enforce a certain strctre to or system matrices To this end, we make the following assmption: Assmption 1 We assme that A φ S( n, n), B φ S( n, n ), and C y φ S( ny, n) for all φ N This assmption restricts or controller synthesis problem to decentralized systems ) Path Dependent Systems: Now consider the switched system x(t + 1) A Ω(t) x(t) + B Ω(t) w(t) (3) z(t) C Ω(t) x(t) + D Ω(t) w(t) whose system matrices at time t depend on a switching path Ω(t) (θ(t L),, θ(t)) A L+1 consisting of L + 1 recent vales of the switching parameters We refer to these types of systems as finite-path dependent systems 1 M G θ(t) θ(t) K Θ(t) Θ(t) y 1 y M Fig : Interconnection diagram of a controller, K Θ(t), with plant, G θ(t), where Θ(t) is the switching seqence with memory of length L We can modify sch systems to be mode-dependent by introdcing indced atomata to reflect the path dependence This is done by assming the indced atomata state-space to be Ñ A L+r, for admissible seqences of length r > For r > we define the seqences Φ, A r+l, Φ Φ Ñ A L+1, and Φ N as Φ : (β 1,, β r+l ), Φ : (β,, β r+l 1 ), Φ : (β r,, β r+l ), Φ : β r+l 3) Switching control: For the plant (), or goal is to design a finite-dimensional, finite-path dependent linear controller with block lower trianglar sparsity strctre We se the following state space representation for or controller x K (t + 1) A K Ω(t) xk (t) + B K Ω(t) y(t) (t) C K Ω(t) xk (t) + D K Ω(t) y(t) (4) For a controller with memory L, the switching path corresponds to Ω(t) (θ(t L),, θ(t)) A L+1 The controller state x K (t) R nk is partitioned as [ (x K 1 (t)) (x K M (t)) with x K i (t) RnK i, ths satisfying + + n K M Or goal is then to design the above n K n K 1 controller with the following associated strctred controller matrices for every admissible seqence Ψ A L+1, A K Ψ S( n K, n K ), B K Ψ S( n K, n y ) C K Ψ S( n, n K ), D K Ψ S( n, n y ) where n K (n K 1,, n K M ) Ths, the reslting controller has a y to mapping with a lower trianglar sparsity strctre We can define the following kernel space matrices for i J, φ N, [ N N y y,x N N y,w [ N,x N,z [ (E y i ) C y φ (Ey i ) D yw φ [ (Ē i ) (B φ ) (Ē i ) (D z φ ) Additionally, consider the following constrction which will be sefl for the controller synthesis, [ N N N y (6) (5) 85

3 The closed loop scaling matrices are denoted by XΨ C S n+nk +, defined for each Ψ A L These matrices are partitioned into plant and controller sections as [ [ XΨ C XΨ XΨ GK (XΨ GK ) XΨ K, (XΨ C ) 1 Y Ψ YΨ GK (YΨ GK ) YΨ K (7) with X Ψ, Y Ψ S n +, XΨ GK, Y Ψ GK R n nk, and XΨ K, Y Ψ K S nk + We frther define the following for i J, { Z i,ψ : X Ψ XΨ GK ĒK i Y Ψ YΨ GK Ei K ( ( Ē K i ) X K Ψ ĒK i ( (E K i ) Y K Ψ E K i We then define the following associated matrices ) 1(X GK i Ēi K ) } 1 ) 1(Y GK ψ Ei K ) (8) Z a i,ψ :(E i Z i,ψ E i ) 1, Z b i,ψ : Za i,ψ(e i Z i,ψ Ē i ) Z c i,ψ :Ē i Z i,ψ Ē i (E i Z i,ψ Ē i ) (E i Z i,ψ E i ) 1 E i Z i,ψ Ē i (9) The matrices above give s the following factorization with [ Zi,ψ l Z i,ψ Z l i,ψ i,ψ) 1 i,ψ) l i,ψ) (1) I b i,ψ) Z c i,ψ [, Zi,ψ Z a i,ψ Z b i,ψ I (11) We are now ready to state the main reslt presented in [9: Theorem 1 Consider the mode-dependent systems () along with Assmption 1 There exists a synthesis of a finite-path dependent controller (4) which (i) is strctred as (5) (ii) has dimensions {n K i }M i1 (iii) achieves closed loop performance w z < 1 if and only if there exists an L N and matrices {Z a i,ψ, Zb i,ψ, Zc i,ψ } i J,Ψ A L satisfying the following Zi,Ψ a, Zi,Ψ c for all i J, Ψ A L (1a) i, [ Φ ) Z l i, Φ i, Φ ) A Φ Zi, Φ l i, Φ ) BΦ w I C z Φ Zi, Φ l DΦ zw N i, Φ) Zi, Φ l i,φ I for all i J, Φ A L+1, (1b) [ i 1,Ψ) Zi,Ψ l i 1,Ψ) Zi 1,Ψ l (1c) rank[ i 1,Ψ) Zi,Ψ l i 1,Ψ) Zi 1,Ψ l n + n K i (1d) Remark 1 If a closed loop performance of w z < γ is desired, Theorem 1 can be pdated to have Cφ z, Cy φ, Dzw φ, Dφ z, and Dyw φ scaled by γ 1 for all φ N The controller obtained for this modified system sing the procedre above can be sed to find the desired controller by scaling BΨ K and DΨ K with γ 1 for all Ψ C Receding Horizon Now we want to extend the previos reslt so that or nested switching system also acconts for ftre modes, as presented for the centralized case in [8 For this end, we define Φ +, Φ A r+l+h, Φ A r+l+h, and Φ N as Φ + : (β 1,, β r+l+h ), Φ : (β,, β r+l+h 1 ), Φ : (β,, β r+l+h ), Φ : β r+l Note that the above definitions are eqivalent to the case withot any look-ahead horizon Now, consider the switched system x(t + 1) A Θ(t) x(t) + B Θ(t) w(t) z(t) C Θ(t) x(t) + D Θ(t) w(t) (13) For a controller with memory L and look-ahead horizon H, the above system matrices at time t depend on a switching path given by Θ(t) ( θ(t L),, θ(t),, θ(t + H) ) A L+H+1 We can modify Lemma 4 in [9 to inclde a look-ahead horizon, as it was shown in [8 Lemma 1 The finite-path dependent system (13) with memory L N and look-ahead horizon H N is exponentially stable and satisfies w z < 1 if and only if there exists an r N and a set of positive-definite matrices {X Ψ } Ψ Ar+L+H satisfying [ [ [ [ XΦ AΦ B I Φ XΦ + AΦ B Φ C Φ D Φ I C Φ D Φ for all Φ A r+l+h+1 Now we can retrace the steps in [9 to obtain a set of LMIs that give necessary and sfficient conditions for the existence of the path dependent controller Theorem Consider the mode-dependent systems () along with Assmption 1 There exists a synthesis of a finite-path dependent controller (4) which (i) is strctred as (5) (ii) has dimensions {n K i }M i1 (iii) achieves closed loop performance w z < 1 if and only if there exists L, H N and matrices {Z a i,ψ, Zb i,ψ, Zc i,ψ } i J,Ψ A L+H satisfying the following Zi,Ψ a, Zi,Ψ c for all i J, Ψ A L+H (14a) i,φ [ + ) Z l i,φ + i,φ + ) A Φ Z l i,φ i,φ + ) BΦ w I C Φ z Z l i,φ DΦ zw N i,φ ) Z l i,φ i,φ I for all i J, Φ A L+H+1, (14b) [ i 1,Ψ) Zi,Ψ l i 1,Ψ) Zi 1,Ψ l (14c) rank[ i 1,Ψ) Zi,Ψ l i 1,Ψ) Zi 1,Ψ l n + n K i (14d) Remark Note that the soltion to the above ineqalities depend both on the switching path and the crrent mode of the system Hence the controller matrices will change based on the choice of L and H Remark 3 If one desires to obtain a decentralized controller 853

4 for a linear time invariant (LTI) plant, the reslts above can still be sed by setting N {1} That is, the atomaton only generates one admissible seqence As a reslt, for any L and H, there exists only one seqence in A L+H+1, and we can then drop the sbscripts in Theorem For controller constrction, refer to Theorem 16 in [9 D Example Fig 3: Switching atomata sed in the Example Let s consider the example shown in [9 in order to demosntrate the benefits of the look-ahead controller The two player system switches between 3 different modes, with a switching seqence generated by the atomaton in Fig 3 The corresponding system matrices are chosen as [ [ 14 7 A 1 A, A [ [ B1 B 1, B3 1 C y 1 Cy [ 1, C y 3 [ 1 1 D z 1 D z [ 1, D z 3 [4 and the following defined for φ {1,, 3} [ [ Bφ w 1 1, D yw φ 1, Cφ z [5, Dφ zw 5 Here we have chosen dimensions n 1 n n 1 n n y 1 ny nz n w 1 For different memory lengths, the above system was examined with n K i for i J The optimal bond w z < γ was fond sing a bisection algorithm, and the vales for different memory lengths are tablated below TABLE I: Optimal performance bond for the controller L/H For H L, the system is not stabilizable, reslting in infinite norms As one wold intitively expect, having a preview of ftre modes gives mch better H bonds than jst relying on previos modes III HARDWARE The hardware consists of the HoTDeC (Hovercraft Testbed for Decentralized Control) vehicle, which was developed at the University of Illinois at Urbana-Champaign, and a Vision system Detailed information abot the HoTDeC and earlier versions of the system can be fond in [1 and [13 A HoTDeC There are two different types of HoTDeC bodies: 3D printed and precision machined Each vehicle has five thrsters, each generating p to 1 N of thrst: one generates lift, and the others generate forces in the x- and y-direction De to their positioning, we can also se a combination of them to generate a moment and rotate the hovercraft body The HoTDeC consists of varios electronic boards The Powerboard reglates the voltage and power distribtion for the hovercraft The Main board is in charge of the integration of peripherals sch the Gmstix and the Digital Signal Processor (DSP) The Gmstix is the main processor of each vehicle, rnning on a real-time embedded Linx OS, and the DSP sed is a Texas Instrments TMS3F8335 Each motor board incldes a hall effect sensor, sed to measre the thrster anglar velocity, and an H-bridge B Dynamical Model The dynamic model of the HoTDeC is defined by ẋ(t) Ax(t) + B(t) + Gw(t) y(t) Cx(t) + Dv(t) (15) where the x(t) states are the positions and velocities in Cartesian coordinates, with the following matrices: 1 βx m 1 A 1 m, B G m m 1 β θ m 1 J [ [ 1 1 C 1, V Where β trans N s/m is the coefficient of translational friction, and β θ N m s/rad is the coefficient of rotational friction Detailed information abot the actator distrbance, w(t), and the measrement noise, v(t), are presented in [14 TABLE II: Mechanical parameters of the HoTDeC Body Type Mass, m (kg) M of Inertia, J (kg m ) Styrofoam D Printed The DSP board rns 5 independent proportional-integral (PI) loops to control the speed of each thrster We assme that the relationship between force F and thrster anglar velocity ω is ω 56 F Each thrster is modeled as ẋ 86x + w (16) 86876x with F { 81518x if x > 46 otherwise (17) 854

5 C Vision System & Network To determine the position of the hovercrafts, the HoTDeC lab has a setp of six cameras covering a total area of approximately 3m by m One compter processes the information from each camera, while another compter merges the information to remove any dplicate data from the overlapping areas and broadcasts the data The Vision system differentiates between different hovercrafts and determines angle of each by sing the patterns on their tops, as seen in Fig 1 We se the ZeroMQ messaging system to commnicate with the vehicles and the Vision system A Controller IV CONTROLLER DESIGN Or goal is to have the hovercrafts follow each other in a platoon of vehicles, as with the system depicted by Fig 1 Each vehicle will try to keep a set distance from the one in front of it, with the first vehicle following a ser defined path First, to set p or controller, we have to generate the state space system which describes the nested loop Let ε i R ni be the measred position vector of the i- th robot that is available to its followers Here we assme that we get fll state information via an estimator Then, if we desire that each hovercraft maintain a distance r i R ni away from its leader, we can choose the performance otpt e i y i ε i 1 r i, where that of the first hovercraft is e 1 y 1 r 1 Synthesis of the decentralized switching controller will be based on the n-weighted generalized plant of the nested strctre z e A B w B [ z C z D zw D z d (18) y C y D yw where z e [ e 1,, e M is the position performance otpt, z [ 1,, M is the controller performance otpt, y [ y 1,, y M is the measrement otpt, d [ r 1,, r M is the distrbance inpt, and [ 1,, M is the controller inpt The dynamical model was discretized with a sampling time of T s 1 seconds For or experiment, the hovercraft will operate near obstacles in either the x- or y-direction To avoid any collisions with the environment, the performance otpt switches between three modes given by z 1 [ 4x ẋ 4y ẏ 5θ θ 5 x 5 y 6 θ z [ 6x ẋ 3y ẏ 5θ θ 5 x 5 y 6 θ z 3 [ 3x ẋ 6y ẏ 5θ θ 5 x 5 y 6 θ (19) where mode 1 represents nobstrcted operation, and modes and 3 represent operation near an obstacle in the x- or y- direction respectively The weights were chosen based on the limitations of the hovercrafts, so as to not satrate the system All admissible seqences in or system can be generated by the atomaton in Fig Note that the weighting vectors above are for a single hovercraft For the nested system, these vectors will be expanded accordingly, with all hovercrafts sbjected to the same weights CVX, a MATLAB-based toolbox for convex optimization ([15), was sed to solve the LMIs in Theorem For or problem, we choose the crrent and ftre mode based on the position and velocities of the robots If the plant switching signal is not available, the controller switching signal can be generated by nmeros methods, sch as the one proposed in [16 B Estimator To deal with the noisy inpt from the Vision system, we have added a Kalman Filter sing the dynamics from Section III-B rnning at 1Hz While the H controller is able to estimate the system states, it tends to give very conservative estimates, so we decided to implement an Kalman filter which is widely sed in robotics applications To generate the filter gain we sed the covariance matrices, Q E [ [ w n wn σ F σf, στ R E [ [ v n vn σ () x σy σθ where σ x 5 mm, σ F 34 N, σ y 3 mm, σ τ 37 N, and σ θ 6 rad C Simlation In order to qickly test different controller weights before deploying the code to the hovercrafts, we have created a Simlink simlation where we modeled the HoTDeC dynamics and the Vision system Here, a nested strctre of for vehicles was considered The simlation acconted for the dynamics of the hovercrafts and the thrsters themselves, simlated the code rnning in the DSP, and modeled the Vision system inclding noise and delay We tested how well or hovercrafts followed a circlar trajectory nder different controller memory and horizon lengths In all test cases the leader followed a 6 m wide circlar path at a 3 Hz freqency A LTI case was generated sing only the first set of controller weights in Eqation 19 All controllers were generated at their optimal bonds For the case of the LTI controller in Fig 4, we can see that while the first robot (far right, red dashed line) does a good job of following the reference signal (far right, ble solid line), small deviations qickly increase as they propagate down the platoon In the controller with memory length and the controller with look-ahead horizon, the trailing robots do a better job at following the one in front For example, the average root mean sqare position error between a follower and the robot directly in front of it is approximately 8% for the LTI case, while it is redced to approximately 5% for the other two cases Under the look-ahead controller, the first robot is slightly worse at following the reference signal than the other cases, likely de to noise in the system affecting the predictions of ftre modes 855

6 y coordinate (m) y coordinate (m) y coordinate (m) x coordinate (m) Fig 4: Simlated paths of for hovercrafts sing the decentralized controller with varios switching parameters The rightmost ble solid line represents the reference signal Test cases: (a) LTI, (b) L, (c) H V HARDWARE IMPLEMENTATION We tested the controller with two hovercrafts The leader was told to move in a circle of radis 3 m, with its follower being told to trail behind at a distance of 1 m in its x- coordinate position The reference vales for the trajectory were generated via sinsoidal fnctions with freqency of 3 Hz So as to not be prely dependent pon predictions of ftre modes, we have tested a controller with memory L 1 and look-ahead horizon H 1, sing the same setp as described in the previos Section Main reslts from the hovercraft experiment are presented in the plots below It shold be noted that in Fig 5, the hovercrafts power on at the 4 second mark They immediately follow a circlar path, and the reference signal stops at 3 seconds In both plots, the x- and y-positions are the estimated otpt from the Kalman Filter x Position (m) y Position (m) Reference Leader Follower Time (s) Fig 5: Estimated position of both hovercrafts with a decentralized controller L1, H1 (a) x position, (b) y position We can see that while the leader did not follow the reference signal as well as the simlations wold indicate, the follower did a very good job at keeping a constant distance behind the leader Althogh the Vision System lost track (a) (b) (a) (b) (c) of the follower at approximately the second mark, the hovercraft qickly recovered to the desired distance VI CONCLUSION In this paper, we have presented the decentralized switched control problem and exact conditions for synthesis of a controller with finite memory We then developed an extension of the synthesis condition for the case of a controller with receding horizon modal information The derived controller was tested sing a system of hovercrafts: first with a MATLAB simlation, and then with a real-system experiment The simlations presented show how error can be redced sing this controller in comparison to a simply LTI system The real hovercraft system experiment shows that the controller is fnctional, althogh there is mch room for improvement in the experimental setp to better the continity between the experimental and simlation reslts Overall, the reslts of this paper have proven that a discrete-time linear nested system controller with finite memory and look-ahead horizon can be a sefl tool REFERENCES [1 R D Andrea and G E Dllerd Distribted control design for spatially interconnected systems IEEE Transactions on Atomatic Control, 48(9): , Sept 3 [ M Farhood, D Zhe, and G E Dllerd Distribted control of linear time-varying systems interconnected over arbitrary graphs International Jornal of Robst and Nonlinear Control, 5():179 6, 15 [3 C Scherer Strctred H -optimal control for nested interconnections: A state-space soltion Systems & Control Letters, 6(1): , 13 [4 A Mishra, C Langbort, and G E Dllerd Decentralized control of linear time-varying nested systems with H -type performance In 14 American Control Conference, pages , Jne 14 [5 P G Volgaris Control of nested systems In American Control Conference, Proceedings of the, volme 6, pages vol6, [6 L Lessard and S Lall Optimal controller synthesis for the decentralized two-player problem with otpt feedback In 1 American Control Conference (ACC), pages , Jne 1 [7 J W Lee and G E Dllerd Optimal distrbance attenation for discretetime switched and markovian jmp linear systems SIAM Jornal on Control and Optimization, 45(4): , 6 [8 R Essick, J W Lee, and G E Dllerd Control of linear switched systems with receding horizon modal information IEEE Transactions on Atomatic Control, 59(9):34 35, Sept 14 [9 A Mishra, C Langbort, and G E Dllerd Decentralized control of linear switched nested systems with l -indced norm performance IEEE Transactions on Control of Network Systems, (4):4 43, Dec 15 [1 S Lee, G E Dllerd, and E Polak On the real-time receding horizon control in harbor defense In 15 American Control Conference (ACC), pages , Jly 15 [11 D Liberzon Switching in Systems and Control Systems & Control: Fondations & Applications Birkhäser Boston, 1 [1 S Lee A Model Predictive Control Approach to a Class of Mltiplayer Minmax Differential Games PhD thesis, University of Illinois at Urbana-Champaign, 15 [13 J P Jansch-Porto Decentralized control of linear switched systems with receding horizon Master s thesis, University of Illinois at Urbana-Champaign, 16 [14 A Stbbs, V Vladimero, A T Flford, D King, J Strick, and G E Dllerd Mltivehicle systems control over networks: a hovercraft testbed for networked and decentralized control IEEE Control Systems, 6(3):56 69, Jne 6 [15 M Grant and S Boyd CVX: Matlab software for disciplined convex programming, version 1 March 14 [16 G Battistelli On stabilization of switching linear systems Atomatica, 49(5): ,