Modeling and Co-design of Control Tasks over Wireless Networking - - PowerPoint PPT Presentation
Department of information engineering, computer science and mathematics Center Of Excellence DEWS University of LAquila, Italy Modeling and Co-design of Control Tasks over Wireless Networking Protocols Alessandro DInnocenzo January
January 9, 2017 OptHySYS Workshop, Trento
“Removing cables undoubtedly saves cost, but often the real cost gains lie in the radically different design approach that wireless solutions permit. […] In order to fully benefit from wireless technologies, a rethink of existing automation concepts and the complete design and functionality of an application is required.” Jan-Erik Frey, R&D Manager ABB
Lower costs, easier installation
Broadens scope of sensing and control
Compositionality
Runtime adaptation and reconfiguration
Complexity
complexity from wireless channels and communication protocols Reliability
Security
Take into account communication protocol dynamics
Session Presentation Transport Network Data/Link Physical
Sensing/actuation
Session Presentation Transport Network Data/Link Physical
Wireless link
))) ((( S1
generic disturbances in the controller design
A1 A2
𝑣" 𝑙 𝑣$ 𝑙 y 𝑙 𝐯 𝐥 = 𝐠(𝐳 𝐥 )
Plant control law (e.g. PID, MPC)
A1 A2 S1
𝐯 𝐥 = 𝐠(𝐳 𝐥 )
Session Presentation Transport Network Data/Link Physical
Sensing/actuation
Session Presentation Transport Network Data/Link Physical
Wireless link
))) (((
Plant control law (e.g. PID, MPC)
generic disturbances in the controller design
TCP congestion control (e.g. Tahoe, Reno, Cubic) Routing control (e.g. RIP, Priority buff.) Medium access control (e.g. CSMA/CD) Power, coding & modulation control (e.g. UMTS inner loop)
Intra-layer control loops
Wireless network
Borderline between control over network and control of network disappears
Wireless network
Borderline between control over network and control of network disappears
M.C. Escher, Relativity Lithograph, 1953
A1 A2 S1
𝐯 𝐥 = 𝐠(𝐳 𝐥 )
Session Presentation Transport Network Data/Link Physical
Sensing/actuation
Session Presentation Transport Network Data/Link Physical
Wireless network
))) ((( Design network to meet control performance Control loop
[Park et al 2011], [Fischione et al 2009], …
Control specification
A1 A2 S1
𝐯 𝐥 = 𝐠(𝐳 𝐥 )
Session Presentation Transport Network Data/Link Physical
Sensing/actuation
Session Presentation Transport Network Data/Link Physical
Wireless network
))) ((( Network non-idealities Robust controller design Control specification
[Seiler&Sengupta 2001], [Jacobsson et al. 2004], [Sinopoli et al 2004], [Elia 2005], [Imer et al 2006], [Braslavsky et al 2007], [Gupta et al 2007], [Hespanha et al 2007], [Schenato et al 2007], [Heemels et al 2011, 2012], [Chiuso et al 2014], …
A1 A2 S1
𝐯 𝐥 = 𝐠(𝐳 𝐥 )
Session Presentation Transport Network Data/Link Physical
Sensing/actuation
Session Presentation Transport Network Data/Link Physical
Wireless network
))) ((( Controller & network co-design Control specification
[Park et al 2011], [Mesquita et al 2012], [Pajic et al 2012], [Antunes&Heemels 2013], [D’Innocenzo et al 2013], …
A1 A2 S1
𝐯 𝐥 = 𝐠(𝐳 𝐥 )
Session Presentation Transport Network Data/Link Physical
Sensing/actuation
Session Presentation Transport Network Data/Link Physical
Wireless network
))) (((
A1 A2 S1
𝐯 𝐥 = 𝐠(𝐳 𝐥 )
Session Presentation Transport Network Data/Link Physical
Sensing/actuation
Session Presentation Transport Network Data/Link Physical
Wireless network
))) (((
Example: exploit plant and network feedback to decide actuation signal and power [D’Innocenzo et al 2012], [Gatsis et al 2013], …
A1 A2 S1
𝐯 𝐥 = 𝐠(𝐳 𝐥 )
Session Presentation Transport Network Data/Link Physical
Sensing/actuation
Session Presentation Transport Network Data/Link Physical
Wireless network
))) (((
Example: exploit plant and network feedback to decide actuation signal and coding [Tatikonda&Mitter 2004], [Nair et al 2007], [Quevedo et al 2010], …
A1 A2 S1
𝐯 𝐥 = 𝐠(𝐳 𝐥 )
Session Presentation Transport Network Data/Link Physical
Sensing/actuation
Session Presentation Transport Network Data/Link Physical
Wireless network
))) (((
Example: exploit plant and network feedback to decide actuation signal and access to channel [Xu&Hespanha 2004], [Cogill et al 2007], [Li&Lemmon 2011], [Tabuada 2007], [Molin&Hirche 2009], [Rabi&Johansson 2009], [Anta&Tabuada 2010], [Donkers et al 2011],…
A1 A2 S1
𝐯 𝐥 = 𝐠(𝐳 𝐥 )
Session Presentation Transport Network Data/Link Physical
Sensing/actuation
Session Presentation Transport Network Data/Link Physical
Wireless network
))) (((
Example: exploit plant and network feedback to decide actuation signal and routing [Mesquita et al 2012], [Jungers et al. 2014], …
Lower costs, easier installation
Broadens scope of sensing and control
Compositionality
Runtime adaptation and reconfiguration
Complexity
complexity from wireless channels and communication protocols Reliability
Security
Take into account communication protocol dynamics
Security: a Systematic Mapping Study. Submitted for publication, preprint on arXiv. A systematic mapping study is a research methodology intended to provide an unbiased,
specific research area: cyber-physical systems security in our case.
studies in software engineering: An update,” Information and Software Technology, vol. 64,
Surprisingly, 100 out of 118 studies (i.e., 84,75%) do not explicitly consider any communication aspect or imperfection, while only 6 studies (i.e. 5,08%) address more than one aspect. Surprisingly, very few papers (attempt to) provide non-trivial mathematical models
explicitly considered in the CPS mathematical model.
6 6 3 3 5 3 2 1 1 2 3 4 5 6 7 Error control coding Transmission Scheduling Routing Time-varying sampling Variable Latency Packet loses and desorder Limited bandwidth Synchronisation errors
Application Session Presentation Transport Network Data/Link Physical
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25
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How to exploit redundancy optimally w.r.t. control performance metrics?
1. Separation of concerns 2. Co-design
§ …makes system tolerant to long- term link failures § …enables detection and isolation of failures and malicious attacks § …makes system robust to short-term link failures (e.g. packet losses)
𝐿(𝑢)
𝑢
𝐿(𝑢)
t+1
𝐿(𝑢)
t+…
𝐿(𝑢)
Different paths are associated with different delays.
t+…
𝐿(𝑢)
𝐵 = 𝐵; 𝐶; 𝐽 ⋮ ⋮ ⋮ ⋯ ⋯ ⋱ ⋮ ⋮ ⋯ 𝐽 ⋯ 𝐽 ⋯ 𝐶 𝜏 𝑢 = 𝐶𝜀E F ,H 𝐽𝜀E F ," ⋮ 𝐽𝜀E F ,IJ$ 𝐽𝜀E F ,IJ" 𝐽𝜀E F ,I Different paths are associated with different delays. Mathematical model: 𝑦 𝑢 + 1 = 𝐵𝑦 𝑢 + 𝐶 𝜏 𝑢 𝑤 𝑢 , 𝑢 ∈ ℕ, where 𝑦 𝑢 is the plant and network state, 𝜏 𝑢 ∈ Σ depends on routing/scheduling. The switching signal is considered as a disturbance.
t+…
Problem: Design a controller 𝐿(𝑢) s.t. the closed loop system is asymptotically stable. Given a state-feedback controller 𝐿(𝑢), the closed loop systems is asymptotically stable iff the Joint Spectral Radius of 𝐵 + 𝐶 𝜏 𝑢 𝐿 𝑢
E F ∈R is smaller than 1.
Insights: Switching systems analysis and design is a crowded research area:
to provide tailored results that
𝐿(𝑢)
Different paths are associated with different delays. Mathematical model: 𝑦 𝑢 + 1 = 𝐵𝑦 𝑢 + 𝐶 𝜏 𝑢 𝑤 𝑢 , 𝑢 ∈ ℕ, where 𝑦 𝑢 is the plant and network state, 𝜏 𝑢 ∈ Σ depends on routing/scheduling. The switching signal is considered as a disturbance.
𝑤 𝑢 is controllable
if for any initial state 𝑦H, any arbitrary final state 𝑦S and any switching signal 𝜏 𝑢 , there exists a finite time 𝑈 and a control sequence 𝑤 𝑢, 𝑦 𝑢 − 𝐸, … , 𝑢 , 𝜏 𝑢 − 𝐸, … , 𝑢 + 𝑂 , 𝑢 = 0, … , 𝑈 − 1, such that 𝑦 𝑈 = 𝑦S.
𝑤 𝑢 is stabilizable if
for any initial state 𝑦H and any switching signal 𝜏 𝑢 , there exists a control sequence 𝑤 𝑢, 𝑦 𝑢 − 𝐸, … , 𝑢 , 𝜏 𝑢 − 𝐸, … , 𝑢 + 𝑂 , 𝑢 ≥ 0, such that lim
F→\ 𝑦(𝑢) = 0.
𝐿(𝑢)
𝐿(𝑢)
Knowledge of the (routing) switching signal 𝜏(𝑢):
– Example: Existence of non-linear controllers that outperform any linear controller – Approximation methods for the design of 𝐿 [Cicone et al., CDC’15]
– Theorem: Characterize pathologic switching sequences that invalidate stabilizability – Configure network nodes to avoid such sequences
We can measure and keep memory of 𝝉 𝒖 and have a finite horizon knowledge of future 𝑶 switching signals: then 𝑳 𝒖 = 𝑳 𝝉 𝒖 − 𝒆 : 𝝉 𝒖 + 𝑶
verifying that for all switching signals of lenght ≤ 𝑜 + 2 𝐸 2 𝐸 a rank condition is satisfied, with 𝑜 plant state-space dimension, 𝐸 delays set cardinality. Moreover:
– If 𝑂 ≥ 𝑜 + 2 𝐸 2 𝐸 stabilizability is equivalent to controllability – If 𝑂 < 𝑜 + 2 𝐸 2 𝐸 controllability implies stabilizability but not vice-versa (by counter-example)
𝐿(𝑢)
systems with unmeasurable switching delays. 54th IEEE Conference on Decision and Control, Osaka, Japan, December 15-18, 2015.
with switching delays. 9th IFAC World Congress, Cape Town, South Africa, August 24-29, 2014.
13th European Control Conference (ECC14), Strasbourg, France, June 24-27, 2014. R.M. Jungers, A. D'Innocenzo, M.D. Di Benedetto. Feedback stabilization of dynamical systems with switched delays. 51st IEEE Conference on Decision and Control, Maui, Hawaii, December 10-13 2012. Jungers, Heemels et Al.: extension to deterministic packet losses
𝐿(𝑢)
1. Separation of concerns 2. Co-design
§ …makes system tolerant to long- term link failures § …enables detection and isolation of failures and malicious attacks § …makes system robust to short-term link failures (e.g. packet losses)
In the theoretical control literature, the most common model is
independently with a fixed probability pc,
In wireless channels, the errors typically come in bursts, due to the movement of the transmitter, receiver or small objects in the environment (Doppler spread) ⇒ time-selective fading. When the fading is slow (compared to a packet interval), the accurate model is given by finite-state Markov channel (FSMC).
extends the two-state channel (a.k.a Gilbert-Elliot) model by taking into account more channel states. Basically, FSMC is a Markov chain, having a different binary symmetric channel associated to each state: Such model has also been used
communication channels. IEEE transactions on vehicular technology, 44(1):163–171, 1995.
transactions on communications, 47(11):1688–1692, 1999.
fading channels - a survey of principles and applications. IEEE Signal Processing Magazine, 25(5):57–80, September 2008.
correlated Rayleigh fading. IEEE Journal on Selected Areas in Communications, 19(9), 1706–1717, 2001.
When the channel state information is available (through a channel estimation), FSMC can be portrayed by a Markov chain alone, but of double size (compared to a standard representation): Generally, the transition probability is defined as It is time-homogeneous when and is time-inhomogeneous otherwise.
pij(k) = pij ∀k
Network paths characteristics are often “at odds”
Very reliable but slow Very fast but unreliable Optimally exploit the best of each path by co designing controller and routing
𝑒", 𝜏"(𝑙) 𝑒$, 𝜏$(𝑙)
𝑒k, 𝜏k(𝑙) 𝑒l, 𝜏l(𝑙)
𝑣m 𝑙 = 𝐿m 𝑙 𝑦; 𝑙 , 𝐿 𝑙 = 𝐿"(𝑙) 𝐿$(𝑙) ⋮ 𝐿k(𝑙)
𝑒", 𝜏"(𝑙) 𝑒$, 𝜏$(𝑙)
𝑒k, 𝜏k(𝑙) 𝑒l, 𝜏l(𝑙)
… 𝜏k 𝑙 models occurrence of packet losses in the network
𝑒", 𝜏"(𝑙) 𝑒$, 𝜏$(𝑙)
𝑒k, 𝜏k(𝑙) 𝑒l, 𝜏l(𝑙)
¬𝑇𝐹𝑂𝐸 1 𝑇𝐹𝑂𝐸 , 𝑏 𝑙 = 𝑏" 𝑙 … 𝑏k 𝑙
𝑒", 𝜏"(𝑙) 𝑒$, 𝜏$(𝑙)
𝑒k, 𝜏k(𝑙) 𝑒l, 𝜏l(𝑙)
the actuator
𝑤 𝑙 = 𝑣m∗(𝑙 − 𝑒m∗), with 𝑗∗ the index of the path
associated to the smallest delay among received packets
𝑤 𝑙 = w 𝑣m(𝑙 − 𝑒m)
k mx"
𝑒", 𝜏"(𝑙) 𝑒$, 𝜏$(𝑙)
𝑒k, 𝜏k(𝑙) 𝑒l, 𝜏l(𝑙)
𝑒", 𝜏"(𝑙) 𝑒$, 𝜏$(𝑙)
𝑒k, 𝜏k(𝑙) 𝑒l, 𝜏l(𝑙)
Assume that the routing policy 𝑏 𝑙 is fixed- separation of concerns paradigm. We leverage discrete-time Markov-jump linear systems (MJLS). A MJLS is a switching linear system where the switching signal is a Markov chain. The transition probability matrix (TPM) 𝑄(𝑙) of the Markov chain 𝜏 𝑙 can be used to model the stochastic process that rules packet losses due to wireless communication. y 𝑦 𝑙 + 1 = 𝐵z { 𝑦 𝑙 + 𝐶z { 𝑣 𝑙 , 𝑦 0 = 𝑦H , 𝜄 0 = 𝜄H
𝑦 𝑙 ∈ ℂ~ is the state vector, u 𝑙 ∈ ℂ€ is the input vector, 𝑙 ∈ ℕH, 𝐵z { ∈ ℂ~×~, 𝐶z { ∈ ℂ~×€ are the state and input matrices associated with mode 𝜄 𝑙
⁄ , 𝑒" = 1; 𝑞$ = 0, 𝑒$ = 5
– LQR using only path 1 for any 𝑙 – LQR using only path 2 for any 𝑙 – LQR using both paths for any 𝑙
Plant state plot
Averaged cost Via path 1 ∼ 900 Via path 2 ∼ 250 Via paths 1 and 2 ∼ 100
Extension 1: In general wireless channels are time-varying and packet loss probability not easy to compute/measure: consider a MJLS where 𝑄(𝑙) time-varying in a polytope Extension 2: We want to control the routing policy 𝑏(𝑙): consider a MJLS where 𝜄(𝑙) is a Markov-decision process
Noiseless autonomous system ( x(k+1) = Aθ(k)x(k), x(0) = x0, θ(0) = θ0 A noiseless autonomous MJLS is mean square stable if for every initial state and for every initial probability distribution of In time-homogeneous case, the MJLS is MSS if and only if a spectral radius
moment of the state vector is where For example, x0 θ0, lim
k→∞ E[x(k)] = 0
and lim
k→∞ E[x(k)x∗(k)] = 0.
ρ A, x(k), < 1, A , (P T ⊗ In2)diag[ ¯ Ai ⊗ Ai]
Consider the MJLS with
A1 = 1 1.2
1.13 0.16 0.48
0.3 0.13 0.16 1.14
N =3 P = 0.35 0.65 0.6 0.4 0.4 0.6 =P1 A1 = (P T
1 ⊗ In2)diag[ ¯
Ai ⊗ Ai] ρ(A1)=0.901601
For the same values of state matrices let us consider another transition probability matrix A1 = 1 1.2
1.13 0.16 0.48
0.3 0.13 0.16 1.14
P2: P2 = 0.25 0.75 0.6 0.4 0.4 0.6 A2 = (P T
2 ⊗ In2)diag[ ¯
Ai ⊗ Ai] ρ(A2)=0.905686
What happens when the transition probability matrix is uncertain and is switching (randomly) between and P1 P2? P = 0.35 0.65 0.6 0.4 0.4 0.6 =P1 P2 = 0.25 0.75 0.6 0.4 0.4 0.6
What happens when the transition probability matrix is uncertain and is switching (randomly) between and P1 P2? P = 0.35 0.65 0.6 0.4 0.4 0.6 =P1 P2 = 0.25 0.75 0.6 0.4 0.4 0.6
works on convex sets and is widely used in robust control. A set is convex, when every point on the line segment connecting two elements of a convex set is in as well. Transition probability matrix is time-varying, with variations that are arbitrary within a bounded set of stochastic matrices. The current value of is unknown, but assumed to be a convex combination of the known vertices of a convex polytope: C C P P (k) P(k) =
L
X
l=1
λl(k)Pl, λl(k) ≥ 0,
L
X
l=1
λl(k) = 1.
Only sufficient stability conditions have been derived for MJLSs with such model of uncertainty:
nonhomogeneous Markovian jump linear systems": provides a sufficient condition for stochastic stability in terms
Markov jump linear systems using interval analysis" presents a sufficient condition for MSS of MJLS with interval transition probability matrix (TPM).
introduced by Rota and Strang in 1960, it characterizes
matrices taken in a set;
switching systems, approximation algorithms, etc. Formally, let be a family of complex square matrices Consider the set be a set of all possible products of length whose factors are elements of i.e. For any matrix norm
consider the supremum among the normalized norms of all products in i.e. The JSR of is defined as M Ml. Πk(M) k M, Πk(M) = {MlkMlk−1 · · · Ml1 | l1, . . . , lk ∈{1, ..., L}}. k·k Cn×n, Πk(M), ˆ ρk(M) , sup
Π∈Πk(M)
kΠk
1
/
k.
M ˆ ρ(M) , lim
k→∞ ˆ
ρk(M).
Joint spectral radius has two important properties: Convergence of matrix products: For any bounded set of matrices all matrix products converge to zero matrix as if and only if Convex hull: The convex hull of a set has the same joint spectral radius as the
M, Π ∈ Πk(M) k → ∞, ˆ ρ(M) < 1 ˆ ρ(conv M) = ˆ ρ(M).
linear systems. 55th IEEE Conference on Decision and Control, Las Vegas, US, December 12-14, 2016
A discrete-time noiseless MJLS with polytopic uncertainties on TPM is MSS if and only if JSR of is where is a set of all vertices of the polytope associated to the second moment of The proof is based on:
made up of all sequences of complex matrices is uniformly homeomorphic to
Euclidean space through the linear mapping which is based
has associativity property, and AL <1, L x(k). AL ,{Al}l∈L={1,...,L}, Al ,(P T
l ⊗ In2)diag[ ¯
Ai ⊗ Ai] Hm,n N m × n Nnm CNnm ˆ ϕ(·) ϕ(·). ⊗ ϕ(XY Z)=(ZT⊗X)ϕ(Y )
The computation and approximation of JSR in notoriously difficult. There exist tools for its computation, such as Vankeerberghen et al., "JSR: A Toolbox to Compute the Joint Spectral Radius”: The tool is implemented in Matlab and is freely downloadable. The JSR of the set of augmented matrices associated to the second moment
x(k) ˆ ρ(AL) = [ˆ ρmin(AL), ˆ ρmax(AL)] = [1.024442, 1.031096]
For MJLSs with polytopic uncertainties on TPMs, unless P = NP, there is no polynomial-time algorithm that decides whether it is mean square stable. The proof is obtained by:
which is well-known to be NP-hard.
linear systems. 55th IEEE Conference on Decision and Control, Las Vegas, US, December 12-14, 2016
The following assertions are equivalent for MJLSs with polytopic uncertainties
(S) : (S) (S) E[kx(k)k2] βζkkx0k2
2,
β 1, 0 < ζ < 1 (S)
∞
X
k=0
E[kx(k)k2] < 1
linear systems. 55th IEEE Conference on Decision and Control, Las Vegas, US, December 12-14, 2016
Autonomous MJLS with bounded noise: (with operational modes)
if and only if for every and . Also this proof is based on:
linear systems. Submitted.
( x(k + 1) = Aθ(k)x(k) + Gθ(k)w(k), x(0) = x0, θ(0) = θ0, (S) (S) ˆ ρ(AL) < 1 x = {x(k); k ∈ N0} ∈ Cn w = {w(k); k ∈ N0} ∈ Cr, x0 ∈ Cn θ0 ∈ Θ0 N
Controlled noiseless MJLS:
control sequence minimizing the quadratic cost
(S) : x(k + 1) = Aθ(k)x(k) + Bθ(k)u(k), y(k) = Cθ(k)x(k) + Dθ(k)u(k), x(0) = x0, θ(0) = θ0, x(k)∈Cn is a state vector, u(k)∈Cm is a control vector, θ(k) is an operational mode y(k)∈Cs is a (measured) output of the system u , [u(0), . . . , u(T −1)] J (θ0, x0, u) , max
p
J (θ0, x0, u, p) = max
p T −1
X
k=0
E[ky(k)k2]+E[x∗(T)Xθ(T )(T)x(T)], p , [pθ(0), . . . , pθ(T −1)], pθ(k) is a row of TPM at time step k X(T)=[X1(T), . . . , XN(T)]∈Hn+ is a vector of the terminal state weighs In a game-theoretic formulation, we assume that perturbation-player (environment and/or malicious adversary) has no information on the choice of the controller and vice-versa.
Denoting by the vertex of the convex polytope of TPM s.t. the is attained at the previous time step for an operational mode, we have that Multiple coupled Riccati difference equations (CRDEs)
v∈L={1, . . . , L} max
π
J (θ0, x0, u, π) u(k) = Kθ(k)|v(k)x(k), Ki|v(k) = −[D∗
i Di + B∗ i N
X
j=1
pij|vXj|l(j)(k+1)Bi]−1B∗
i N
X
j=1
pij|vXj|l(j)(k+1)Ai, Xi|v(k) = C∗
i Ci + A∗ i N
X
j=1
pij|vXj|l(j)(k+1)Ai + A∗
i N
X
j=1
pij|vXj|l(j)(k+1)BiKi|v(k).
Ω1(k) Ω2(k) !(#)
1. Parsimonious set of CRDEs includes only the members which ever achieve (2): if , then also if is a convex combination of other vertices, it is redundant. 2. If , then considering instead of introduces a numerical error of at most . 3. If the MJLS is stabilizable, i.e. s.t. , and is mean square stable, then after a transient, the finite-horizon LQR solution becomes unique.
Xl1
i|v(k) Xl2 i|v(k) ⌫ 0
x∗(k)Xl1
i|v(k)x(k) ≥ x∗(k)Xl2 i|v(k)x(k) ∀x(k);
Xl2
i|v(k)
Xl1
i|v(k)+✏In ⌫ Xl2 i|v(k)
Xl1
i|v(k)
Xl2
i|v(k)
✏kx(k)k2, 8x(k)2Cn ∃K =[K1, . . . , KN]∈Hn u(k)=Kθ(k)x(k) x(k+1)=(Aθ(k)+Bθ(k)u(k))x(k)
Inverted pendulum mounted to a motorized cart (discretized, sample time 0.02 s): The relative speed between controller and plant can vary between 1 and 1.5 km/h. The vertices of time-varying TPM of associated Markov chain are
0.4954 0.2667 0.2379 0.0681 0.7334 0.1658 0.0327 0.1282 0.7334 0.1384 0.0544 0.9456 0.4954 0.2667 0.2379 0.0681 0.7334 0.1658 0.0327 0.1282 0.7334 0.1384 0.0544 0.9456 , 0.6589 0.2000 0.1411 0.0709 0.8000 0.1140 0.0152 0.1151 0.8000 0.0849 0.0897 0.9103 0.6589 0.2000 0.1411 0.0709 0.8000 0.1140 0.0152 0.1151 0.8000 0.0849 0.0897 0.9103
A1 =A2 = 1.000000000000000 0.019961536921409 0.000304142666220 0.000002027774250 0.996155844513813 0.030413280758014 0.000304142666220 −0.000062069931882 1.003653767543144 0.020024353570176 −0.006206791991431 0.365567654428609 1.003653767543144 , B1 = 0.000384630785908 0.038441554861875 0.000620699318817 0.062067919914314 , B2 = , DT
1 D1 =DT 2 D2 =4 · 10−16, C1 =C2 =
1 1
After 1000 time steps, the MJLS has only few discrete states for some modes:
State space partitions for each mode, no approx. State space partitions for each mode, approx. with ✏ k 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 ✏ 999 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 998 2 1 2 2 2 1 1 1 2 1 2 2 2 1 1 1 997 5 9 8 8 5 6 8 4 2 2 1 1 2 2 1 1 10−8 996 16 16 4 2 16 16 4 2 3 4 2 1 8 8 2 1 10−5 995 139 307 12 2 176 336 12 2 2 3 1 1 4 8 1 1 10−4 994 48 48 5 1 48 48 5 1 3 3 1 1 10 7 2 1 10−4 993 129 179 5 1 180 180 5 1 5 2 1 1 14 3 1 1 10−4 992 148 191 3 2 252 280 3 1 4 2 1 1 11 3 1 1 10−4 991 85 59 3 2 159 160 3 1 5 2 1 1 9 3 1 1 10−4 990 63 49 4 1 161 178 3 1 6 3 1 1 6 3 1 1 10−4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 001 32 32 2 1 32 32 2 2 1 1 1 1 16 7 1 1 10−6 000 32 32 2 1 32 32 2 2 1 1 1 1 16 7 1 1 10−6
The time-homogeneous MJLS is stabilizable, with the following spectral radii: We observe that these values are much closer to 1. JSR can result to be > 1. We set dynamic approximation values and the approximation threshold, which triggers approximation when the number of CRDEs is > 12000. After 1000 time steps, the MJLS has only few discrete states for some modes.
ρ(Λ1)=0.904052453, ρ(Λ2)=0.904052372. ✏=[0, 10−6, 10−4]
Extension 1: In general wireless channels are time-varying and packet loss probability not easy to compute/measure: consider a MJLS where 𝑄(𝑙) time-varying in a polytope Extension 2: We want to control the routing policy 𝑏(𝑙): consider a MJLS where 𝜄(𝑙) is a Markov-decision process
Controlled noiseless Markov jump SWITCHED linear system:
switched linear systems. Submitted.
(S) : x(k + 1) = Aθ(k)x(k) + Bθ(k)u(k), y(k) = Cθ(k)x(k) + Dθ(k)u(k), x(0) = x0, θ(0) = θ0, x(k)∈Cn is a state vector, u(k)∈Cm is a control vector, θ(k) is an operational mode y(k)∈Cs is a (measured) output of the system σ(k) is a Markov-decision process,
pa
ij(k) , Pr{θ(k+1) = j | a, θ(k) = i} ≥ 0, N
X
j=1
pa
ij(k) = 1 if a 2 A(i),
pa
ij(k) = 0 if a 62 A(i)
switched linear systems. Submitted.
control policy minimizing the quadratic cost where π?
T ={u? k, a? k}T −1 k=0
J (θ0, x0, πT ) , max
p
J (θ0, x0, πT , p), p,{pa(k)
θ(k)(k)}T −1 k=0
X(T)=[X1(T), . . . , XN(T)]∈Hn+ vector of terminal continuous state weighs g=(g1, . . . , gN)∈RN
+ vector of terminal discrete state weights
J(θ0, x0, πT , p),E[x∗(T)Xθ(T )(T)x(T)+g(θ(T))]+
T− 1
X
k=0
(E[ky(k)k2+g(θ(k), a(θ(k), k))],
switched linear systems. Submitted.
linear systems. Submitted
a wireless network. 5th IFAC Workshop on Estimation and Control of Networked Systems (NecSys 2015), Philadelphia PA, September 10-11, 2015
multi-hop control network with packet losses. 14th European Control Conference (ECC 2015), Linz, Austria, July 15-17, 2015