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**CONTROL of SWITCHED SYSTEMS with LIMITED INFORMATION**

Daniel Liberzon Coordinated Science Laboratory Electrical and Computer Engineering Univ. of Illinois, Urbana-Champaign, USA Will talk about a control problem that has two aspects: switching (not controlled) and limited information for feedback Semi-plenary lecture, MTNS, Groningen, July 2014 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAA

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**PROBLEM FORMULATION Switched system: Information structure:**

are (stabilizable) modes, (can be state-dependent, realizing discrete state in hybrid system) is a switching signal is a (finite) index set, Information structure: Sampling: state is measured at times sampling period ( ) Quantization: each is encoded by an integer from 0 to and sent to the controller, along with Even if we stabilize all individual modes (which is already nontrivial) it is not enough Between sampling times, system runs open-loop Alphabet size: N is number of symbols per state dimension, 0 is overflow symbol Data rate: Objective: design an encoding & control strategy s.t. based on this limited information about and

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**MOTIVATION Switching: Quantization:**

ubiquitous in realistic system models lots of research on stability & stabilization under switching tools used: common & multiple Lyapunov functions, slow switching assumptions Quantization: coarse sensing (low cost, limited power, hard-to-reach areas) limited communication (shared network resources, security) theoretical interest (how much info is needed for a control task) tools used: Lyapunov analysis, data-rate / MATI bounds MJLS: stochastic model, discrete time, \sigma is known Commonality of tools is encouraging Almost no prior work on quantized control of switched systems (except quantized MJLS [Nair et. al. 2003, Dullerud et. al. 2009])

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**NON - SWITCHED CASE Quantized control of a single LTI system:**

[Baillieul, Brockett-L, Hespanha, Matveev-Savkin, Nair-Evans, Tatikonda] System can be stabilized if error reduction factor at growth factor on e.g. Won’t go into details now for a single LTI system – more later K is a stabilizing gain (meaning u=Kx would have been stabilizing), e=x-\hat x goes to 0 => done Control: Crucial step: obtaining a reachable set over-approximation at next sampling instant How to do this for switched systems?

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**REACHABLE SET ALGORITHMS**

Many computational (on-line) methods for hybrid systems Puri–Varaiya–Borkar (1996): approximation by piecewise-constant differential inclusions; unions of polyhedra Henzinger–Preußig–Stursberg–et. al. (1998, 1999): approximation by rectangular automata; tools: HyTech, also PHAVer by Frehse (2005) Asarin–Dang–Maler (2000, 2002): linear dynamics; rectangular polyhedra; tool: d/dt Mitchell–Tomlin–et. al. (2000, 2003): nonlinear dynamics; level sets of value functions for HJB equations Kurzhanski–Varaiya (2002, 2005): affine open-loop dynamics; ellipsoids Chutinan–Krogh (2003): nonlinear dynamics; polyhedra; tool: CheckMate Girard–Le Guernic–et. al. (2005, 2008, 2009, 2011): linear dynamics; zonotopes and support functions; tool: SpaceEx Hybrid systems = state-dependent switching Results classified according to type of dynamics and shapes of the sets

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**OUR APPROACH We develop a method for propagating reachable set**

over-approximations for switched systems which is: Analytical (off-line) Leads to an a priori data-rate bound for stabilization (may be more conservative than on-line methods) Works with linear dynamics and hypercubes (with moving center) Hypercubes are well suited for quantization For best results should somehow combine the two methodologies Tailored to switched systems (time-dependent switching) but can be adopted / refined for hybrid systems

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**SLOW - SWITCHING and DATA - RATE ASSUMPTIONS**

dwell time (lower bound on time between switches) average dwell time (ADT) s.t. number of switches on (sampling period) Implies: switch on each sampling interval We’ll see how large should be for stability \le 1 switch on each sampling interval – just for simplicity, could generalize to any known upper bound on the number of switches Gap between 4) and necessary condition: hypercube vs. using different N’s for different state dimensions based on eigenvalues Define (usual data-rate bound for individual modes)

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**ENCODING and CONTROL STRATEGY**

Goal: generate, on the decoder / controller side, a sequence of points and numbers s.t. (always -norm) Let Pick s.t is Hurwitz Define state estimate on by We’ll say on next slide how such bounds will be generated, but here suppose we have them Repeat that what’s sent over the channel is the number identifying the yellow box Define control on by

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**GENERATING STATE BOUNDS**

Choosing a sequence that grows faster than system dynamics, for some we will have Inductively, assuming we show how to find s.t. Case 1 (easy): sampling interval with no switch Let This bound is independent of what control is being applied

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**GENERATING STATE BOUNDS**

Case 2 (harder): sampling interval with a switch – unknown to the controller Before the switch: as on previous slide, but this is unknown known Instead, pick some and use as center (triangle inequality) In the picture the green point is before the black one, but this need not be the case I’m skipping the formula for D_{k+1} but it’s not very difficult to compute it It depends on \bar t – eventually we will take max over \bar t – but the center does not Intermediate bound:

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**GENERATING STATE BOUNDS**

After the switch: on , closed-loop dynamics are , or Auxiliary system in : lift project onto Here the estimate and the control are generated using the “wrong” dynamics Can draw a picture (use the board if available) and explain two points: e dynamics now depend individually on x and \hat x, and x and \hat x deviate from each other so \hat x is no longer a good approximation to use for x Then take maximum over to obtain final bound

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**STABILITY ANALYSIS: OUTLINE**

sampling interval with no switch : on This is exp. stable DT system w. input data-rate assumption exp and Thus, the overall is exp. stable “cascade” system Lyapunov function : satisfies if from to , then contains a switch P_p is a positive definite matrix, \rho_p is a positive number If satisfies then ADT as same true for exp Intersample bound, Lyapunov stability – see [L, Automatica, Feb’14]

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**SIMULATION EXAMPLE (data-rate assumption holds)**

switch Observe the initial ``zooming-out" phase and the nonsmooth behavior of $x$ when $\hat x$ experiences a jump (causing a jump in $u$) After the first switch, we can see that \hat x continues smoothly until the next sampling time, and then jumps higher; after subsequent switches this effect is less noticeable In the paper the theoretical ADT bound is around 85 (can be decreased to around 75 by adjusting control gains) but as Oliver pointed out it to me it can be reduced to about 50 by doing the calculation a bit more carefully Theoretical lower bound on is about 50

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HYBRID SYSTEMS Switching triggered by switching surfaces (guards) in state space Previous result applies if we can use relative location of switching surfaces to verify slow-switching hypotheses Can just run the algorithm and verify convergence on-line Can use the extra info to improve reachable set bounds For example: keep discard Knowledge of switching surfaces can also be used to obtain estimates of switching times, which in turn can be used to refine our bounds State jumps – easy to incorporate

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**CONCLUSIONS and FUTURE WORK**

Contributions: Stabilization of switched / hybrid systems with quantization Main step: computing over-approximations of reachable sets Data-rate bound is the usual one, maximized over modes Extensions: Refining reachable set bounds (set shapes, choice of ) Relaxing slow-switching assumptions ( ) Less frequent transmissions of discrete mode value Challenges: Output feedback (Wakaiki and Yamamoto, MTNS’14) External disturbances (ongoing work with Yang) Modeling uncertainty Nonlinear dynamics Zonotopes are affine transforms of hypercubes – can consider pre-processing the state Optimal t’ and t’’ – ? Wakaiki and Yamamoto assume switching signal is known, and use a different encoding strategy (quantizer center not moving) Once we have the result with disturbances in the ISS framework, we’ll be able to say something about modeling uncertainty as well

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**CONCLUSIONS and FUTURE WORK**

Contributions: Stabilization of switched / hybrid systems with quantization Main step: computing over-approximations of reachable sets Data-rate bound is the usual one, maximized over modes Extensions: Refining reachable set bounds (set shapes, choice of ) Relaxing slow-switching assumptions ( ) Less frequent transmissions of discrete mode value Challenges: Output feedback (Wakaiki and Yamamoto, MTNS’14) External disturbances (ongoing work with Yang) Modeling uncertainty Nonlinear dynamics Talk about knowing vs. not knowing disturbance bound, mention work with Dragan

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