Switched Systems: Mixing Logic with Differential Equations

Controlo 2002 5th Portuguese Conference on Automatic Control University of Aveiro, September 5-7, 2002 Switched Systems:Mixing Logic withDifferential Equations João P. Hespanha University of Californiaat Santa Barbara slides will be made available at http://www.ece.ucsb.edu/~hespanha

Outline Logic-based switched systems framework Congestion control in data networks Impactmaps Lyapunov tools Vision-based control application areas analytical tools Adaptive control Interconnection of systems

Logic-based switched systems discrete modes hybrid automaton representation transitions

Logic-based switched systems hybrid automaton representation 2 1

Logic-based switched systems hybrid automaton representation switching times s(t)´switching signal 2 1 differential equation dynamical system representation discrete transition

Congestion control in data networks sources destinations B Congestion control problem: How to adjust the sending rates of the data sources to make sure that the bandwidth B of the bottleneck link is not exceeded? B is unknown to the data sources and possibly time-varying

Congestion control in data networks queue (temporary storage for data) r1 bps r2 bps rate ·B bps q(t) ´ queue size r3 bps When åi ri exceeds B the queue fills and data is lost (drops) )drop (discrete event) Event-based control: The sources adjust their rates based on the detection of drops

Window-based rate adjustment wi (window size) ´ number of packets that can remain unacknowledged for by the destination source i destination i e.g., wi = 3 round-triptime(RTT) t0 1st packet sent t1 2nd packet sent t2 t0 3rd packet sent 1st packet received & ack. sent t1 2nd packet received & ack. sent t2 t3 1st ack. received4th packet can be sent 3rd packet received & ack. sent t t wi effectively determines the sending rate ri: round-trip time

Window-based rate adjustment queuegets full longerRTT ratedecreases queuegets empty negative feedback wi (window size) ´ number of packets that can remain unacknowledged for by the destination per-packet transmission time ´ sending rate total round-trip time propagationdelay time in queueuntil transmission This mechanism is still not sufficient to prevent a catastrophic collapse of the network if the sources set the wi too large

TCP Reno congestion control • While there are no drops, increase wi by 1 on each RTT • When a drop occurs, divide wi by 2 (congestion controller constantly probe the network for more bandwidth) Network/queue dynamics Reno controllers drop detected(one RTT after occurred) ß ß drop occurs disclaimer: this is a simplified version of Reno that ignores several interesting phenomena…

Switched system model for TCP queue-not-full transition enabling condition (drop occurs) queue-full (drop detected) state reset

Switched system model for TCP queue-not-full s = 1 (drop occurs) queue-full (drop detected) s = 2 alternatively… continuous dynamics discrete dynamics s2 {1, 2}

Impact maps x1 x2 T t0 t1 t2 t3 t4 t5 t6 queue-not-full queue full queue-not-full queue full queue-not-full queue full ´kth time the system enters the queue-not-full mode queue-not-full x1 x2 impact map queue-full state space

Impact maps x1 x2 T t0 t1 t2 t3 t4 t5 t6 queue-not-full queue full queue-not-full queue full queue-not-full queue full ´kth time the system enters the queue-not-full mode Theorem [1]: The function T is a contraction. In particular, • Therefore • xk!x1 as k!1x1´constant • x(t) !x1 (t) as t!1x1( t ) ´ periodic limit cycle

NS-2 simulation results N1 Flow 1 S1 N2 Flow 2 S2 Bottleneck link TCP Sinks TCP Sources Router R2 Router R1 20Mbps/20ms Flow 7 S7 N7 window size w1 S8 N8 Flow 8 window size w2 window size w3 window size w4 window size w5 window size w6 window size w7 window size w8 queue size q 500 400 300 Window and Queue Size (packets) 200 100 0 0 10 20 30 40 50 time (seconds)

Random early detection (RED) Performance could be improved if the congestion controllers were notified of congestion before a drop occured queue-not-full (notification of congestion) Ni´ notification counter (incremented whenever a notification of congestion arrives) function to be adjusted In RED, N is a random variable with Stochastic switched system

Impact maps queue-not-full xk xk+1 impact map queue-full state space Impact maps are difficult to compute because their computation requires: Solving the differential equations on each mode (in general only possible for linear dynamics) Intersecting the continuous trajectories with a surface(often transcendental equations) It is often possible to prove that T is a contraction without an explicit formula for T…

Outline Logic-based switched systems framework Congestion control in data networks Impact maps Lyapunov tools Vision-based control application areas analytical tools Adaptive control Interconnection of systems

Vision-based control of a flexible manipulator kflex mtip mtip bflex  qtip qtip bbase bbase qbase qbase u u Ibase Ibase flexible manipulator very lightly damped 4th dimensional small-bending approximation Control objective: drive qtip to zero, using feedback from qbase! encoder at the base qtip! machine vision (essential to increase the damping of the flexible modes in the presence of noise)

Vision-based control of a flexible manipulator mtip To achieve high accuracy in the measurement of qtip the camera must have a small field of view qtip bbase qbase u Ibase feedback output: Control objective: drive qtip to zero, using feedback from qbase! encoder at the base qtip ! machine vision (essential to increase the damping of the flexible modes in the presence of noise)

Switched process manipulator y u

Switched process controller 1 manipulator y u controller 2 controller 1 optimized for feedback from qbase and qtip and controller 2 optimized for feedback only from qbase E.g., LQG controllers that minimize

Switched system feedback connectionwith controller 1 (qbase and qtip available) feedback connection with controller 2 (only qbase available) How does one check if the overall system is stable and that eventually converges to zero ? controller’s state

Common Lyapunov functions V(x)´common Lyapunov function V(x) Suppose that there exists a cont. diff. function V(x) such that V(x) provides a measure of the size of x: V(x) decreases along trajectories of both systems: • positive definite • radially unbounded switched system is stable and x ! 0 as t !1 independently of how switching takes place

Common Lyapunov functions How to find a common Lyapunov function V(x) ? V(x) • Algebraic conditions for the existence of a common Lyapunov function • The matrices commute, i.e., A1A2 = A2A1 • The Lie Algebra generated by {A1, A2} is solvable • For all l2[0,1] the matrices lA1+(1– l)A2 and lA1+(1– l)A2 -1 are asymptotically stable(only for 2£2 matrices) [S1,S2] But, all these conditions fail for the problem at hand …

Multiple Lyapunov functions Common Lyapunov function… same Lyapunov function must decrease for every controller V(x) one Lyapunov function for each controller (more flexibility) Multiple Lyapunov functions… V1(x) V1(x) V2(x) V2(x) [2,3, etc.]

Multiple Lyapunov functions V1(x) V1(x) V2(x) V2(x) Suppose that exist positive definite, radially unbounded cont. diff. functions V1(x), V2(x) such that Vi(x) decreases along trajectories of Ai: Vi(x) does not increase during transitions: at points z where a switching from i to j can occur switched system is stable and x ! 0 as t!1

Multiple Lyapunov functions Multiple Lyapunovfunctions can be foundfor the problem at hand !!! V1(x) V1(x) V2(x) V2(x) Suppose that exist positive definite, radially unbounded cont. diff. functions V1(x), V2(x) such that Vi(x) decreases along trajectories of Ai: Vi(x) does not increase during transitions: at points z where a switching from i to j can occur switched system is stable and x ! 0 as t!1 [S2]

Closed-loop response 5 0 -5 0 10 20 30 40 50 60 5 0 -5 0 10 20 30 40 50 60 no switching(feedback only fromqbase) qtip(t) with switching (close to 0 feedback also from qtip ) qtip(t) qmax = 1 [S2]

Robustness When will a small perturbation in the dynamics result in a small perturbation in the switched system’s trajectory? For purely continuous systems (difference or differential equations) Lyapunov stability automatically provides some degree of robustness This is not necessarily true for switched systems: When is this a problem?Is there a notion of stability that automatically provides robustness? • Important for … • numerical simulation of switched systems • digital implementation of switched controllers • analysis and design based on numerical methods

Prototype adaptive control problem process can either be: process u y Control objective: Stabilize process (keep state of the process bounded)

Prototype adaptive control problem process can either be: controller 1 process u y controller 2 controller 1 stabilizes P1 and controller 2 stabilizes P2

Prototype adaptive control problem 2 1  How to choose online which controller to use?  controller 1 process u y controller 2  switching signal taking values on the set {1,2} 

Estimator-based architecture e1 e2 output estimation errors logic-based supervisor P1observer P2observer   controller 1 process u y controller 2 “likelihood” ofprocess beingP1 is high should usecontroller 1 e1 small Certainty equivalence inspired

Estimator-based architecture e1 e2 output estimation errors logic-based supervisor P1observer P2observer   controller 1 process u y controller 2 switched system

Scale-independent hysteresis switching performance signals (measure the “size” of the estimation errors) hysteresis constant (positive) switching signal  estimation errors e1 e2 switched system

Scale-independent hysteresis switching How does one verify if x remains bounded along solutions to the hybrid system? Analyzing the system as a whole is too difficult. We need to: switching signal 1.abstract the complex behavior of each subsystem (supervisor & switched) to a small set of properties 2.infer properties of the overall system from the properties of the interconnected subsystems  e1 e2 switched system

One-diagram analysis outline hysteresisswitching logic H1 switching signal (discrete)  estimation errors (continuous signals) switched system H2 One can show [S3] that… H1 has the property that finite “L2-induced gain” from smallest error to the switched error H2 has the property that vice-versa(“detectability”through es) (with d = 0 when there is no unmodeled dynamics)

One-diagram analysis outline hysteresisswitching logic H1 switching signal (discrete)  The system is stable provided that g . d < 1 estimation errors (continuous signals) switched system H2 One can show [S3] that… H1 has the property that finite “L2-induced gain” from smallest error to the switched error H2 has the property that vice-versa(“detectability”through es) (with d = 0 when there is no unmodeled dynamics)

Interconnections of switched system H1 H1 H1 H2 H2 H2 For interconnections through continuous signals, existing tools can be extended to hybrid systems (small gain, passivity, integral quadratic constrains, ISS, etc.) [6,7] H1 H1 H1 H2 H2 H2 For mixed interconnections new tools need to be developed… continuous signal discrete signal

Conclusion Impactmaps Congestion control in data networks Lyapunov tools Vision-based control application areas analytical tools Adaptive control Interconnection of systems Switched systems are ubiquitous and of significant practical application A unified theory ofswitched systems is barely starting to become available

References Background surveys & tutorials: [S1] D. Liberzon, A. S. Morse, Basic problems in stability and design of switched systems, In IEEE Control Systems Magazine, vol. 19, no. 5, pp. 59-70, Oct. 1999. [S2] J. Hespanha. Chapter Stabilization Through Hybrid Control. In Encyclopedia of Life Support Systems, 2002. To appear. [S3] J. Hespanha. Tutorial on Supervisory Control. Lecture Notes for the workshop Control using Logic and Switching for the 40th Conf. on Decision and Contr., Orlando, Florida, Dec. 2001. Papers referenced specifically in this talk: [1] S. Bohacek, J. Hespanha, J. Lee, K. Obraczka. Analysis of a TCP hybrid model. In Proc. of the 39th Annual Allerton Conference on Communication, Control, and Computing, Oct. 2001. [2] P. Peleties P., R. A. DeCarlo. Asymptotic stability of m-switched systems using Lyapunov-like functions. In Proc. of the 1991 Amer. Contr. Conf., pp. 1679–1684, 1991. [3] M. S.Branicky. Studies in Hybrid Systems: Modeling, Analysis, and Control. Ph.D. thesis, MIT, Cambridge, MA, 1995. [4] J. Hespanha. Extending LaSalle's Invariance Principle to Linear Switched Systems. In Proc. of the 40th Conf. on Decision and Contr., Dec. 2001. [5] J. Hespanha, A. S. Morse. Certainty Equivalence Implies Detectability. Syst. & Contr. Lett., 36(1):1-13, Jan. 1999. [6] M. Zefran, F. Bullo, M. Stein, “A notion of passivity for hybrid systems.” In Proc. of the 40th Conf. on Decision and Contr., Dec. 2001. [7] J. Hespanha. Root-Mean-Square Gains of Switched Linear Systems. To appear in Trans. of Autom. Contr. See also http://www.ece.ucsb.edu/~hespanha/published.html

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Logic-based switched systems discrete modes hybrid automaton representation transitions

Logic-based switched systems hybrid automaton representation 2 1

Switched Systems: Mixing Logic with Differential Equations