Stability of Nonlinear Control Systems Under Intermittent Information with Applications to Multi-Agent Robotics

Transcription

1 Introduction Stability Optimal Intermittent Fdbk Summary Stability of Nonlinear Control Systems Under Intermittent Information with Applications to Multi-Agent Robotics Domagoj Tolić Fakultet Elektrotehnike i Računarstva Zagreb, Croatia ACROSS Colloquia D. Tolić, FER Stability Under Intermittent Information 1 / 45

2 Introduction Stability Optimal Intermittent Fdbk Summary Outline 1 Introduction Motivation Intermittent Information Recent Ideas 2 Stability Problem Formulation Approach Numerical Results 3 Optimal Intermittent Fdbk Motivation Approach Numerical Results 4 Summary D. Tolić, FER Stability Under Intermittent Information 2 / 45

3 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Intermittent Information Recent Ideas Outline 1 Introduction Motivation Intermittent Information Recent Ideas 2 Stability 3 Optimal Intermittent Fdbk 4 Summary D. Tolić, FER Stability Under Intermittent Information 3 / 45

4 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Intermittent Information Recent Ideas Multi-Agent Systems D. Tolić, FER Stability Under Intermittent Information 4 / 45

5 Systems of Systems (Heterogeneity) Xbee Link Ethernet 100Mbps Ethernet 100Mbps Embedded LL Controller Vicon MX Giganet Gigabyte Ethernet Vicon Machine DC Motor Ethernet 100Mbps Vicon Cameras Vicon Cameras Router 1 XBee Adapter RS-232 Ethernet 100Mbps CompactRIO Router 2 Ethernet 100Mbps ROS Machine LabView Machine XBee Radio module Xbee Link SSH Wireless Laser Range Finder Xbee Link Xbee Link Xbee Link Differential Drive Pioneer P3-AT Odometry Sonar Xbee Link Xbee Link Xbee Link

6 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Intermittent Information Recent Ideas Illustrative Example D. Tolić, FER Stability Under Intermittent Information 6 / 45

7 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Intermittent Information Recent Ideas What Do We Mean by Intermittent Information? How often should up-to-date information about systems be collected and sent to neighbors? Intrinsic Properties packet collisions, sampling period, network throughput, scheduling protocols, occlusions of sensors, a limited communication/sensing range, etc. D. Tolić, FER Stability Under Intermittent Information 7 / 45

8 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Intermittent Information Recent Ideas What Do We Mean by Intermittent Information? How often should up-to-date information about systems be collected and sent to neighbors? Intrinsic Properties packet collisions, sampling period, network throughput, scheduling protocols, occlusions of sensors, a limited communication/sensing range, etc. Intermittent Feedback a user-designed property of a system the goal is to decrease energy consumption as well as processing and sensing requirements flexibility and resource efficiency D. Tolić, FER Stability Under Intermittent Information 7 / 45

9 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Intermittent Information Recent Ideas Why Do We Care? When designing controllers we think of: D. Tolić, FER Stability Under Intermittent Information 8 / 45

10 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Intermittent Information Recent Ideas Why Do We Care? What we really have is: D. Tolić, FER Stability Under Intermittent Information 9 / 45

11 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Intermittent Information Recent Ideas Event-Triggering vs. Self-Triggering Event-Triggering sampling (i.e., transmission of up-to-date information) is triggered when a triggering event occurs D. Tolić, FER Stability Under Intermittent Information 10 / 45

12 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Intermittent Information Recent Ideas Event-Triggering vs. Self-Triggering Self-Triggering the current sampling instance is used to predict triggering events and preclude undesired performance D. Tolić, FER Stability Under Intermittent Information 11 / 45

13 Introduction Stability Optimal Intermittent Fdbk Summary Problem Formulation Approach Numerical Results Outline 1 Introduction 2 Stability Problem Formulation Approach Numerical Results 3 Optimal Intermittent Fdbk 4 Summary D. Tolić, FER Stability Under Intermittent Information 12 / 45

14 Introduction Stability Optimal Intermittent Fdbk Summary Problem Formulation Approach Numerical Results Related Work 1 Small-gain theorem approaches [Nešić and Teel, 2004], [Tabbara et al., 2007]; D. Tolić, FER Stability Under Intermittent Information 13 / 45

15 Introduction Stability Optimal Intermittent Fdbk Summary Problem Formulation Approach Numerical Results Related Work 1 Small-gain theorem approaches [Nešić and Teel, 2004], [Tabbara et al., 2007]; 2 Dissipativity or passivity-based approaches [Yu and Antsaklis, 2011a], [Yu and Antsaklis, 2011b]; D. Tolić, FER Stability Under Intermittent Information 13 / 45

16 Introduction Stability Optimal Intermittent Fdbk Summary Problem Formulation Approach Numerical Results Related Work 1 Small-gain theorem approaches [Nešić and Teel, 2004], [Tabbara et al., 2007]; 2 Dissipativity or passivity-based approaches [Yu and Antsaklis, 2011a], [Yu and Antsaklis, 2011b]; 3 Input-to-State Stability (ISS) approaches [Tabuada, 2007], [Anta and Tabuada, 2010], [Lemmon, 2010]; and D. Tolić, FER Stability Under Intermittent Information 13 / 45

17 Introduction Stability Optimal Intermittent Fdbk Summary Problem Formulation Approach Numerical Results Related Work 1 Small-gain theorem approaches [Nešić and Teel, 2004], [Tabbara et al., 2007]; 2 Dissipativity or passivity-based approaches [Yu and Antsaklis, 2011a], [Yu and Antsaklis, 2011b]; 3 Input-to-State Stability (ISS) approaches [Tabuada, 2007], [Anta and Tabuada, 2010], [Lemmon, 2010]; and 4 Other approaches [Estrada and Antsaklis, 2008], [Li et al., 2007], [Tallapragada and Chopra, 2011]. D. Tolić, FER Stability Under Intermittent Information 13 / 45

18 Introduction Stability Optimal Intermittent Fdbk Summary Problem Formulation Approach Numerical Results Closed-Loop Systems first design a closed-loop system without taking into account communication networks plant: controller: ẋ p = f p (t, x p, u, ω p ), y = g p (t, x p ), (1) ẋ c = f c (t, x c, y, ω c ), u = g c (t, x c ) (2) second introduce communication networks D. Tolić, FER Stability Under Intermittent Information 14 / 45

19 Introduction Stability Optimal Intermittent Fdbk Summary Problem Formulation Approach Numerical Results Introducing Communication Networks output error vector: e := [ŷ ] y =: û u [ ey e u ] (3) input error vector: e ω := ˆω p ω p (4) D. Tolić, FER Stability Under Intermittent Information 15 / 45

20 Introduction Stability Optimal Intermittent Fdbk Summary Problem Formulation Approach Numerical Results Problem Statement Input-Output Triggering Based on the value of ŷ and ˆω p received at t i, i N, find a time interval τ i until the next transmission instant of y and ω p yielding the closed-loop system (1)-(2) stable in some appropriate sense. D. Tolić, FER Stability Under Intermittent Information 16 / 45

21 Introduction Stability Optimal Intermittent Fdbk Summary Problem Formulation Approach Numerical Results Contributions 1 The design of an input-output triggered sampling policy employing the small-gain theorem; D. Tolić, FER Stability Under Intermittent Information 17 / 45

22 Introduction Stability Optimal Intermittent Fdbk Summary Problem Formulation Approach Numerical Results Contributions 1 The design of an input-output triggered sampling policy employing the small-gain theorem; 2 Consideration of external inputs, and realistic communication channels and sensors; D. Tolić, FER Stability Under Intermittent Information 17 / 45

23 Introduction Stability Optimal Intermittent Fdbk Summary Problem Formulation Approach Numerical Results Contributions 1 The design of an input-output triggered sampling policy employing the small-gain theorem; 2 Consideration of external inputs, and realistic communication channels and sensors; 3 The formulation of novel conditions for L p -stability of hybrid systems; and D. Tolić, FER Stability Under Intermittent Information 17 / 45

24 Introduction Stability Optimal Intermittent Fdbk Summary Problem Formulation Approach Numerical Results Contributions 1 The design of an input-output triggered sampling policy employing the small-gain theorem; 2 Consideration of external inputs, and realistic communication channels and sensors; 3 The formulation of novel conditions for L p -stability of hybrid systems; and 4 The design of a novel method for computing L p -gains over a finite horizon by resorting to convex programming. D. Tolić, FER Stability Under Intermittent Information 17 / 45

25 Introduction Stability Optimal Intermittent Fdbk Summary Problem Formulation Approach Numerical Results Contributions 1 The design of an input-output triggered sampling policy employing the small-gain theorem; 2 Consideration of external inputs, and realistic communication channels and sensors; 3 The formulation of novel conditions for L p -stability of hybrid systems; and 4 The design of a novel method for computing L p -gains over a finite horizon by resorting to convex programming. Our approach does not require construction of storage nor Lyapunov functions. D. Tolić, FER Stability Under Intermittent Information 17 / 45

26 Introduction Stability Optimal Intermittent Fdbk Summary Problem Formulation Approach Numerical Results Interconnecting Nominal and Error System I We write (1)-(2) as the following interconnected hybrid system [Sanfelice, 2007]: } ẋ = f δ (t, x, e, e ω ) ė = g δ t [t i, t i+1 ), (5a) (t, x, e, e ω ) i N 0 } x(t + ) = x(t) e(t + t T. (5b) ) = h(t, e(t)) where δ = ˆω p and x = (x p, x c ). D. Tolić, FER Stability Under Intermittent Information 18 / 45

27 Introduction Stability Optimal Intermittent Fdbk Summary Problem Formulation Approach Numerical Results Interconnecting Nominal and Error System II f (t, x, e, ˆω p, e ω ) := h(t i, e) := [ ] hy (t i, e(t i )), h u (t i, e(t i )) [ ] fp (t, x p, g c (t, x c ) + e u, ˆω p e ω ), f c (t, x c, g p (t, x p ) + e y, ˆω p ) g(t, x, e, ˆω p, e ω ) := ˆf p(t,x p,x c,g p(t,x p)+e y,g c(t,x c)+e u,ˆω p e ω) gp t (t,x p) gp }{{} 0 for zero-order-hold estimation strategy {}}{ ˆf c(t,x p,x c,g p(t,x p)+e y,g c(t,x c)+e u,ˆω p) gc t (t,x c) gc xc (t,xc)fc(t,xp,gp(t,xp)+ey,ˆωp) xp (t,xp)fp(t,xp,gc(t,xc)+eu,ˆωp eω) D. Tolić, FER Stability Under Intermittent Information 19 / 45

28 Introduction Stability Optimal Intermittent Fdbk Summary Problem Formulation Approach Numerical Results L p -stability of Switched Systems I Theorem Consider a hybrid system Σ δ. Let K 0 and p [1, ). If δ is such that (i) There exist constants K(τ δ i ), γ(τ δ i ) such that y[ti δ, t ] p K(τ i δ ) x(t δ+ ) + γ(τ i δ ) ω[ti δ, t ] p. (6) for all t [ti δ, ti+1 δ ] and all i N 0, where τi δ that exist, and i = t δ i+1 tδ i, and such K M := sup i N 0 K(τ δ i ), (7) γ M := sup i N 0 γ(τ δ i ), (8) D. Tolić, FER Stability Under Intermittent Information 20 / 45

29 Introduction Stability Optimal Intermittent Fdbk Summary Problem Formulation Approach Numerical Results L p -stability of Switched Systems II Theorem (ii) The condition holds, i=1 x(t δ+ i ) K x(t 0 ), (9) then Σ δ is L p -stable from ω to y with constant K M (K + 1) and gain γ M for the given δ. For p =, the same result holds with the constant K M K and gain γ M when (9) is replaced with sup i N x(ti δ+ ) K x(t 0 ). In addition, if the state x is L p to L p detectable, then conditions (i) and (ii) are both sufficient and necessary. D. Tolić, FER Stability Under Intermittent Information 21 / 45

30 Introduction Stability Optimal Intermittent Fdbk Summary Problem Formulation Approach Numerical Results L p -stability Over a Finite Horizon for Error System Theorem Assume δ = ˆω p is fixed to be constant. Suppose that there exist A A + n e such that A <, and a continuous function ỹ : R R nx R nω R nω R ne + such that the output error dynamics in (5a) satisfies ė = g(t, x, e, ˆω p, e ω ) Aē + ỹ(t, x, ˆω p, e ω ) (10) for all e R ne and all (t, x(t), ˆω p, e ω (t)) S provided that t [t 0, t 0 + τ], where S R R nx R nω R nω. Then, the output error system is L p -stable from ỹ to e over a finite horizon τ 0 for any p [1, ] with ( ) 1/p exp( A pτ) 1 K e (τ) =, p A (11) γ e (τ) = exp( A τ) 1. A (12) D. Tolić, FER Stability Under Intermittent Information 22 / 45

31 Introduction Stability Optimal Intermittent Fdbk Summary Problem Formulation Approach Numerical Results Input-Output Triggering stabilizing intersampling intervals are given by τi = 1 ( A δ i ln κ Aδ i ) γn δ + 1 γn δ γ e (τi ) = κ, (13) where κ (0, 1). D. Tolić, FER Stability Under Intermittent Information 23 / 45

32 Introduction Stability Optimal Intermittent Fdbk Summary Problem Formulation Approach Numerical Results Input-Output Triggering Theorem For p [1, ] assume that (i) There exists L 0 such that ω p L 1 nω, (ii) Theorem 3 holds, (iii) Σ δ n is L p -stable from (e ω, e) to ỹ for every δ c, c P, with gain γ δ n that has an upper- and lower-bound, (iv) Sampling policy (13) yields x that satisfies (9) for given δ : [t 0, ) P, and (v) x is L p to L p detectable from (ỹ, e ω, e). Then, the closed-loop system (1)-(2) is L p -stable from e ω to (x, e) for given δ. D. Tolić, FER Stability Under Intermittent Information 24 / 45

33 Introduction Stability Optimal Intermittent Fdbk Summary Problem Formulation Approach Numerical Results Trajectory Tracking tracking error: ω R1 x p2 v R1 + v R2 cos x p3 ẋ p = ω R1 x p1 + v R2 sin x p3. ω R2 ω R1 controller: v R1 = v R2 cos x p3 + k 1 x p1, ω R1 = ω R2 + k 2 v R2 sin x p3 x p3 x p2 + k 3 x p3. D. Tolić, FER Stability Under Intermittent Information 25 / 45

34 Introduction Stability Optimal Intermittent Fdbk Summary Problem Formulation Approach Numerical Results Trajectory Tracking - with Switched Modeling x 1 x x 3 80 states of the system x time[s] norm of (x,e) time[s] sampling period τ sampling period τ time[s] time[s] D. Tolić, FER Stability Under Intermittent Information 26 / 45

35 Introduction Stability Optimal Intermittent Fdbk Summary Problem Formulation Approach Numerical Results Trajectory Tracking - without Switched Modeling x 1 x 2 x states of the system x norm of (x,e) time[s] time[s] 0.01 sampling period τ time[s] D. Tolić, FER Stability Under Intermittent Information 27 / 45

36 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Approach Numerical Results Outline 1 Introduction 2 Stability 3 Optimal Intermittent Fdbk Motivation Approach Numerical Results 4 Summary D. Tolić, FER Stability Under Intermittent Information 28 / 45

37 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Approach Numerical Results What do we Mean by Optimal Intermittent Feedback? think of an airplane driven by an autopilot system designed to follow the shortest path between two points any deviation from the shortest path caused by intermittent feedback increases total fuel consumption this increase in fuel consumption is probably more costly than the cost of energy saved due to intermittent feedback D. Tolić, FER Stability Under Intermittent Information 29 / 45

38 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Approach Numerical Results What do we Mean by Optimal Intermittent Feedback? think of an airplane driven by an autopilot system designed to follow the shortest path between two points any deviation from the shortest path caused by intermittent feedback increases total fuel consumption this increase in fuel consumption is probably more costly than the cost of energy saved due to intermittent feedback we encode these energy consumption trade-offs in a cost function, and design an Approximate Dynamic Programming (ADP) approach that yields optimal intertransmission intervals with respect to the cost function D. Tolić, FER Stability Under Intermittent Information 29 / 45

39 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Approach Numerical Results Contributions 1 Formulation of the optimal self-triggering problem as a Dynamic Programming (DP) problem; D. Tolić, FER Stability Under Intermittent Information 30 / 45

40 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Approach Numerical Results Contributions 1 Formulation of the optimal self-triggering problem as a Dynamic Programming (DP) problem; 2 Employment of Particle Filters (PFs) fed by intermittent feedback to account for partially observable states; and D. Tolić, FER Stability Under Intermittent Information 30 / 45

41 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Approach Numerical Results Contributions 1 Formulation of the optimal self-triggering problem as a Dynamic Programming (DP) problem; 2 Employment of Particle Filters (PFs) fed by intermittent feedback to account for partially observable states; and 3 Formulation of properties that successful approximation architectures in ADP approaches satisfy. D. Tolić, FER Stability Under Intermittent Information 30 / 45

42 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Approach Numerical Results Contributions 1 Formulation of the optimal self-triggering problem as a Dynamic Programming (DP) problem; 2 Employment of Particle Filters (PFs) fed by intermittent feedback to account for partially observable states; and 3 Formulation of properties that successful approximation architectures in ADP approaches satisfy. Tolić, D.; Fierro, R.; Ferrari, S.;, Optimal Self-Triggering for Nonlinear Systems via Approximate Dynamic Programming, 2012 IEEE Multiconference on Systems and Control (MSC 2012), 2012, accepted for publication D. Tolić, FER Stability Under Intermittent Information 30 / 45

43 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Approach Numerical Results Performance Index - Dynamic Programming I Goal Minimize the following cost function V : R nx R { V τi (x 0 ) = E γ i[ eω i=1 ti t i 1 (x T p Qx p + u T Ru)dt + S }{{} l(x p,u,τ i ) over all sampling policies τ i and for all initial conditions x 0 R nx. ] } (14) D. Tolić, FER Stability Under Intermittent Information 31 / 45

44 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Approach Numerical Results Performance Index - Dynamic Programming I Goal Minimize the following cost function V : R nx R { V τi (x 0 ) = E γ i[ eω i=1 ti t i 1 (x T p Qx p + u T Ru)dt + S }{{} l(x p,u,τ i ) over all sampling policies τ i and for all initial conditions x 0 R nx. ] } (14) This cost function captures performance vs. energy trade-offs. D. Tolić, FER Stability Under Intermittent Information 31 / 45

45 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Approach Numerical Results Performance Index - Dynamic Programming II minimization of (14) is equivalent to Hamilton-Jacobi-Bellman equation ( ) V (z) = inf l(z, u, τ) + γ E {V (f (z, u, τ, ˆω p, e ω ))}. τ [0,τ max ] eω D. Tolić, FER Stability Under Intermittent Information 32 / 45

46 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Approach Numerical Results Performance Index - Dynamic Programming II minimization of (14) is equivalent to Hamilton-Jacobi-Bellman equation ( ) V (z) = inf l(z, u, τ) + γ E {V (f (z, u, τ, ˆω p, e ω ))}. τ [0,τ max ] eω V (z) is called the optimal value function (or optimal cost-to-go function) V (z) represents the cost incurred by an optimal policy τ when the initial condition in (14) is z. D. Tolić, FER Stability Under Intermittent Information 32 / 45

47 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Approach Numerical Results Performance Index - Dynamic Programming III introduce the Bellman operator M as ( ) Mg = (Mg)(z) = inf l(z, u, τ) + γ E {g(f (z, u, τ, ˆω p, e ω ))} eω for any g : R nx R τ [0,τ max ] D. Tolić, FER Stability Under Intermittent Information 33 / 45

48 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Approach Numerical Results Performance Index - Dynamic Programming III introduce the Bellman operator M as ( ) Mg = (Mg)(z) = inf l(z, u, τ) + γ E {g(f (z, u, τ, ˆω p, e ω ))} eω for any g : R nx R τ [0,τ max ] since γ (0, 1), therefore M is a contraction, i.e., where v s = sup z R np v(z) Mu Mv s γ u v s D. Tolić, FER Stability Under Intermittent Information 33 / 45

49 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Approach Numerical Results Performance Index - Dynamic Programming III introduce the Bellman operator M as ( ) Mg = (Mg)(z) = inf l(z, u, τ) + γ E {g(f (z, u, τ, ˆω p, e ω ))} eω for any g : R nx R τ [0,τ max ] since γ (0, 1), therefore M is a contraction, i.e., where v s = sup z R np v(z) Mu Mv s γ u v s the set B of all bounded, real valued functions with the norm s is a Banach space D. Tolić, FER Stability Under Intermittent Information 33 / 45

50 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Approach Numerical Results Performance Index - Dynamic Programming III introduce the Bellman operator M as ( ) Mg = (Mg)(z) = inf l(z, u, τ) + γ E {g(f (z, u, τ, ˆω p, e ω ))} eω for any g : R nx R τ [0,τ max ] since γ (0, 1), therefore M is a contraction, i.e., where v s = sup z R np v(z) Mu Mv s γ u v s the set B of all bounded, real valued functions with the norm s is a Banach space therefore, for each initial V 0 B, the sequence of value functions V n+1 = MV n = M n+1 V 0 converges to V D. Tolić, FER Stability Under Intermittent Information 33 / 45

51 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Approach Numerical Results Issues 1 Curses of dimensionality ( ) (Mg)(z) = inf l(z, u, τ) + γ E {g(f (z, u, τ, ˆω p, e ω ))} eω τ [0,τ max ] 2 Approximation architecture 3 Partially observable states D. Tolić, FER Stability Under Intermittent Information 34 / 45

52 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Approach Numerical Results Issues 1 Curses of dimensionality ( ) (Mg)(z) = inf l(z, u, τ) + γ E {g(f (z, u, τ, ˆω p, e ω ))} eω τ [0,τ max ] 2 Approximation architecture 3 Partially observable states This is why we use Approximate Dynamic Programming. D. Tolić, FER Stability Under Intermittent Information 34 / 45

53 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Approach Numerical Results Curses of Dimensionality (Mg)(z) = inf τ [0,τ max ] ( ) l(z, u, τ) + γ E {g(f (z, u, τ, ˆω p, e ω ))} eω 1 Uncountable and multi-dimensional state space > choose a finite set of points X C x R nx 2 Computing expectation > a sum of a quadrature approximation (e.g., Simpson formula) 3 Optimization > gradient search methods with constraints starting from different initial points D. Tolić, FER Stability Under Intermittent Information 35 / 45

54 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Approach Numerical Results Approximation Architecture I introduce an approximate value function ˆV i of V i compute ˆV i+1 only for the points in X generalize/interpolate for ˆV i+1 for C x \ X D. Tolić, FER Stability Under Intermittent Information 36 / 45

55 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Approach Numerical Results Approximation Architecture I introduce an approximate value function ˆV i of V i compute ˆV i+1 only for the points in X generalize/interpolate for ˆV i+1 for C x \ X Properties (i) ˆV i+1 (x ) = (MˆV i )(x ); (ii) supp(ˆv i+1 ˆV i ) = C i, where C i C x is a convex and compact neighborhood of x ; and (iii) for any c C i the following holds ˆV i+1 [S] [ˆV i+1 (c), ˆV i+1 (x )], where ˆV i+1 [S] is the image of the segment S connecting x and c. D. Tolić, FER Stability Under Intermittent Information 36 / 45

56 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Approach Numerical Results Approximation Architecture II online learning of Neural Networks (NNs) D. Tolić, FER Stability Under Intermittent Information 37 / 45

57 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Approach Numerical Results Approximation Architecture II online learning of Neural Networks (NNs) for batch learning, properties (i), (ii) and (iii) cannot be guaranteed since NNs are expansion approximators D. Tolić, FER Stability Under Intermittent Information 37 / 45

58 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Approach Numerical Results Approximation Architecture II online learning of Neural Networks (NNs) for batch learning, properties (i), (ii) and (iii) cannot be guaranteed since NNs are expansion approximators exceptions are kernel-based NNs and recurrent NNs in certain settings D. Tolić, FER Stability Under Intermittent Information 37 / 45

59 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Approach Numerical Results Approximation Architecture II online learning of Neural Networks (NNs) for batch learning, properties (i), (ii) and (iii) cannot be guaranteed since NNs are expansion approximators exceptions are kernel-based NNs and recurrent NNs in certain settings randomly pick points x i C x in each step we do not have to specify X we avoid the problem of exploration vs. exploitation (see [Powell, 2007] and [Sutton and Barto, 1998]) D. Tolić, FER Stability Under Intermittent Information 37 / 45

60 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Approach Numerical Results Partially Observable States we assume that the controller can access its state x c consequently, the controller can access u at any given time the controller does not have access to the state of the plant x p but merely to ˆω p and ŷ we propose a particle filter to extract ˆx p from ˆω p and ŷ and feeds the controller D. Tolić, FER Stability Under Intermittent Information 38 / 45

61 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Approach Numerical Results Trajectory Tracking cost to go V parametrized with estimate of x 3 = 0 [m] estimate of x 2 [m] estimate of x 1 [m] D. Tolić, FER Stability Under Intermittent Information 39 / 45

62 Introduction Stability Optimal Intermittent Fdbk Summary Motivation Approach Numerical Results Results - Trajectory Tracking x 1 x 2 x states of the system x norm of (x,e) time[s] time[s] sampling period τ sampling period τ time[s] time[s] D. Tolić, FER Stability Under Intermittent Information 40 / 45

63 Introduction Stability Optimal Intermittent Fdbk Summary Outline 1 Introduction 2 Stability 3 Optimal Intermittent Fdbk 4 Summary D. Tolić, FER Stability Under Intermittent Information 41 / 45

64 Introduction Stability Optimal Intermittent Fdbk Summary Summary input-output triggered control for nonlinear systems the small-gain theorem is employed to prove stability L p -gains over a finite horizon novel results regarding L p -stability of hybrid systems are presented optimal self-triggering via Approximate Dynamic Programming D. Tolić, FER Stability Under Intermittent Information 42 / 45

65 References I Anta, A. and Tabuada, P. (2010). To sample or not to sample: Self-triggered control for nonlinear systems. IEEE Transactions on Automatic Control, 55(9): Estrada, T. and Antsaklis, P. J. (2008). Stability of model-based networked control systems with intermittent feedback. In Proc. of the 17th IFAC World Congress on Automatic Control, pages Lemmon, M. (2010). Event-triggered Feedback in Control, Estimation, and Optimization, volume 405 of Lecture Notes in Control and Information Sciences. Springer Verlag. Li, C., Feng, G., and Liao, X. (2007). Stabilization of nonlinear systems via periodically intermittent control. IEEE Trans. on Circuits and Systems II: Express Briefs, 54(11): Nešić, D. and Teel, A. R. (2004). Input-output stability properties of Networked Control Systems. IEEE Transactions on Automatic Control, 49(10): Powell, W. B. (2007). Approximate Dynamic Programming: Solving the Curses of Dimensionality. Wiley Series in Probability and Statistics. John Wiley and Sons, Inc., Hoboken, NJ. Sanfelice, R. (2007). Robust Hybrid Control Systems. PhD thesis, University of California Santa Barbara. D. Tolić, FER Stability Under Intermittent Information 43 / 45

66 References II Sutton, R. and Barto, A. (1998). Reinforcement Learning. The MIT Press, Cambridge, Massachusetts. Tabbara, M., Nešić, D., and Teel, A. R. (2007). Stability of wireless and wireline networked control systems. IEEE Transactions on Automatic Control, 52(9): Tabuada, P. (2007). Event-triggered real-time scheduling of stabilizing control tasks. IEEE Transactions on Automatic Control, 52(9). Tallapragada, P. and Chopra, N. (2011). On event triggered trajectory tracking for control affine nonlinear systems. In Proceedings of the IEEE Conference on Decision and Control, pages Yu, H. and Antsaklis, P. (2011a). Event-triggered real-time scheduling for stabilization of passive and output feedback passive systems. In Proceedings of the American Control Conference, pages , San Francisco, CA. Yu, H. and Antsaklis, P. J. (2011b). Output Synchronization of Multi-Agent Systems with Event-Driven Communication: Communication Delay and Signal Quantization. Department of Electrical Engineering, University of Notre Dame. technical report. D. Tolić, FER Stability Under Intermittent Information 44 / 45

67 Questions? Comments? Suggestions? D. Tolić, FER Stability Under Intermittent Information 45 / 45

68 Questions? Comments? Suggestions? Thank You for Your attention!! D. Tolić, FER Stability Under Intermittent Information 45 / 45