1. 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

2. 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

3. 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])

4. 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?

5. 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

6. 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

7. 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)

8. 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

9. 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

10. 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:

11. 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

12. 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]

13. 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

14. 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

15. 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

16. 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