tutorial ecole macs: (discrete-time) switched...
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Tutorial Ecole MACS:(Discrete-time) Switched systems
BourgesJune 16th 2015
Marc JUNGERS
Outline of the tutorial
What are switched systems ?
About stability
Stability results for discrete-time switched systems (solving P1)
Stability results with constrained switching law (solving P2)
Stabilization results for discrete-time switched systems (solving P3)
Tutorial switched systems 2 / 58 M. Jungers
Aims of the tutorial
Goals :
• Have an overview about switched systems.• Consider discrete-time linear autonomous switched systems.• Understand the main properties of switched systems.• Be familiar with stability and stabilization of switched systems.
Tutorial switched systems 3 / 58 M. Jungers
Outline of the tutorial
What are switched systems ?Definition and link with hybrid systemsIllustrations and motivation
About stability
Stability results for discrete-time switched systems (solving P1)
Stability results with constrained switching law (solving P2)
Stabilization results for discrete-time switched systems (solving P3)
Tutorial switched systems 4 / 58 M. Jungers
Definition of switched systems
Definition :Switched systems are the association of a finite set of dynamical systems(modes) and a switching law σ(·) that indicates at each time which mode isactive.
Let I = 1; · · · ; N, where N ∈ N is the number of modes.
Continuous-time
x(t) = fσ(t)(x(t), u(t), t), ∀t ∈ R+,
where• x(t) ∈ Rn is the state,• u(t) the input.
• σ the switching law
σ : R→ I.
Discrete-time
xk+1 = fσ(k)(xk , uk , k), ∀k ∈ N, (1)
where• xk ∈ Rn is the state,• uk the input.
• σ the switching law
σ : N→ I.
Tutorial switched systems 5 / 58 M. Jungers
Assumptions for the switching law
Several assumptions :
• σ(·) is arbitrary.σ(·) is seen as a perturbation. The results should be true for all the switchinglaws. The generation of the signal k 7→ σ(k) could be very difficult to take intoaccount.
• σ(·) is state dependent.Here we have σ(k) = g(xk ).
• σ(·) is time dependent or has time constraints.This is for instance the case when σ(·) is periodic, or has a time constraintsuch a dwell time.
• σ(·) is a control input.The issue here is to design the switching law σ(·).
Tutorial switched systems 6 / 58 M. Jungers
Particular case of hybrid systems
Hybrid system :Heterogenous interaction between continuous and discrete dynamics :
If z(t) ∈ C, z(t) ∈ F (z(t), u(t)), (flow map)
If z(t) ∈ D, z(t+) ∈ G(z(t), u(t)), (jump map).(2)
For continuous-time switched systems, we have :
C = D = Rn × I, z(t) =
(x(t)σ(t)
)∈ Rn+1, (3)
F (z(t)) =
(fi (x(t), u(t))i∈I
0
); G(z(t)) =
(x(t)I
). (4)
Tutorial switched systems 7 / 58 M. Jungers
IllustrationsSaturated systems :Let x(t) ∈ R2, with
x(t) =
[−1 10 −5
]x(t) +
[0.21
]sat([
1 −1]
x(t)).
with
sat(u) =
−1 if u < −1,+1 if u > +1,u if − 1 ≤ u ≤ +1.
0 1 2 3 4 5 6 7 8 9 10−2
0
2
4
6
8
10
12
Tutorial switched systems 8 / 58 M. Jungers
Illustrations
Boost converter :
Cdvo
dt= (2− σ)iL −
1R
vo, σ(t) ∈ 1; 2
LdiLdt
= vin − (2− σ)vo.
Tutorial switched systems 9 / 58 M. Jungers
Illustrations
Multiagent systems :The new position of each agent i is a mean of the position of agents, who are inthe current neighborhood (depending on time k ). Existence of a consensuslimk→+∞ x (i)
k = x∗ ?
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
xk+1 = A1xk
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
xk+1 = A2xk
Tutorial switched systems 10 / 58 M. Jungers
Illustrations
Switching controllers :
System
Controller 1
Controller N
...
Switched law
Tutorial switched systems 11 / 58 M. Jungers
IllustrationsSliding modes :
Let x(t) ∈ R, with
x(t) = −sign(x(t)) =
−1 if x(t) > 0,+1 if x(t) < 0,undefined if x(t)=0.
0 1 2 3 4 5 6 7 8 9 10−1
0
1
2
3
4
5
5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10−0.03
−0.02
−0.01
0
0.01
0.02
0.03
Difficulties : Well-posed solution ? Possible presence of Zeno phenomenon.
Tutorial switched systems 12 / 58 M. Jungers
Typical examples of embedded systems
Tutorial switched systems 13 / 58 M. Jungers
Framework of the talk
Consider discrete-time switched systems :• Avoid well-posedness of solutions (different kinds of solutions : Filipov
solution etc),• Avoid Zeno phenomenon,• Simplicity and richness of this class of systems.
Assume also for this talk :• The modes are time invariant,• The modes are autonomous (or already in their closed-loop form).
To sum up, we consider in the following (with distinct assumptions on σ(·)) :
xk+1 = Aσ(k)xk . (5)
Tutorial switched systems 14 / 58 M. Jungers
Outline of the tutorial
What are switched systems ?
About stabilityDefinitionsStability of time invariant discrete-time linear systemsProperties/Complexity of switched systemsMain problems
Stability results for discrete-time switched systems (solving P1)
Stability results with constrained switching law (solving P2)
Stabilization results for discrete-time switched systems (solving P3)
Tutorial switched systems 15 / 58 M. Jungers
Definitions relative to stability
The definitions are relative to an equilibrium point. Here we assume that theequilibrium point is the origin x∗ = 0. In addition, the following definitions arevalid for linear switched systems, for which there does not exist finite timeescape.
Global asymptotic stability (GAS) : ensure that
limk 7→+∞
xk = 0, ∀(x0, σ(0)) ∈ Rn × I. (6)
Global uniform asymptotic stability (GUAS) : ensure that
limk 7→+∞
xk = 0, ∀(x0, σ(0)) ∈ Rn × I, ∀σ : N 7→ I. (7)
The term uniform means uniformly in σ(·).
Tutorial switched systems 16 / 58 M. Jungers
Geometric approach
We recall stability results for the time invariant discrete-time linear system :
xk+1 = Axk , ∀k ∈ N. (8)
The solution is given byxk = Ak x0, ∀k ∈ N.
Theorem : The system (8) is GAS if and only if
ρ(A) = maxi∈|λi (A)| < 1. (9)
Tutorial switched systems 17 / 58 M. Jungers
Lyapunov function approach
Theorem : Consider the system xk+1 = Axk and V : Rn → R, such that• V (x)→ +∞ as ‖x‖ → +∞. (radially unbounded).• V (0) = 0 and V (x) > 0 if x 6= 0. (positive definite).• V (Ax)− V (x) < 0, ∀x 6= 0. (decreasing)
Then the origin x∗ = 0 is GAS.
The function V is called a Lyapunov function and is an extended energy of thesystem, which should decrease to zero along all trajectories.
Tutorial switched systems 18 / 58 M. Jungers
Converse theorem
Theorem : If the origin x∗ = 0 is GAS for the system xk+1 = Axk , then thereexists a Lyapunov function V (·).
In such a case, the difficulty is to obtain the expression of the Lyapunov functionV (·).
Tutorial switched systems 19 / 58 M. Jungers
Stability for linear systems with Lyapunov functions
Theorem : the following statements are equivalent :
1. The linear system xk+1 = Axk is GAS.
2. There is a quadratic Lyapunov function
V (x) = xT Px , (10)
where P is a positive definite matrix P > 0n such that the following Lyapunovinequality (Linear Matrix Inequality LMI) is satisfied :
AT PA− P < 0. (11)
3. There is a quadratic Lyapunov function
V (x) = xT Px , (12)
where P is the positive definite matrix P > 0n associated with any Q > 0such that the following Lyapunov equation is satisfied.
AT PA− P = −Q. (13)
Tutorial switched systems 20 / 58 M. Jungers
Sketch of proof3)⇒ 2) . Trivial
AT PA− P = −Q < 0.2)⇒ 1) If the inequality AT PA− P < 0 has a positive definite solution
P > 0n, then there exists sufficient small 1 > ε > 0 such that
AT PA− P < −εP < 0.
Then, by considering V (x) = xT Px , and xk 6= 0,
V (xk+1)− V (xk ) = xTk (AT PA− P)xk < −εxT
k Pxk < 0,
which implies, with λmin(P)‖x‖2 ≤ xT Px ≤ λmax(P)‖x‖2, that
xTk Pxk ≤ (1− ε)k V (x0); ‖xk‖2 ≤ λmax(P)
λmin(P)‖x0‖2(1− ε)k .
1)⇒ 3) If the system xk+1 = Axk is GAS, then the Grammian associatedwith the pair (Q,A), with any Q > 0 is well-defined (the sumconverges). ∑
k∈N
(AT)k
QAk ,
and is a solution of the Lyapunov equation. To end the proof, wehave only to prove that P > 0.
Tutorial switched systems 21 / 58 M. Jungers
Main difficulty concerning the stability
The stability of a switched system is not intuitive
xk+1 = Aσ(k)xk , x(0) = x0; xk ∈ R2 (14)
where σ : N→ 1, 2 is the switching rule, which imposes the active mode.
A1 =
[0.9960 −0.01000.0100 0.9960
]; A2 =
[0.9960 −0.19920.0005 0.9960
]; (15)
A1 and A2 have the same eigenvalues λ± = 0.9960± 0.0100i and are stable(Schur : ‖λ±‖ < 1) .
Tutorial switched systems 22 / 58 M. Jungers
Stability of mode 1
60 50 40 30 20 10 0 10 204
2
0
2
4
6
8
σ(k) = 1, ∀k ∈ N.
Tutorial switched systems 23 / 58 M. Jungers
Stability of mode 2
60 50 40 30 20 10 0 10 204
2
0
2
4
6
8
σ(k) = 2, ∀k ∈ N.
Tutorial switched systems 24 / 58 M. Jungers
Comparing modal trajectories
60 50 40 30 20 10 0 10 204
2
0
2
4
6
8
σ(k) = 1,
σ(k) = 2, ∀k ∈ N.
Tutorial switched systems 25 / 58 M. Jungers
Is the switched system stable for all switching laws ? (I)
60 50 40 30 20 10 0 10 204
2
0
2
4
6
8
σ(k) = 1 if xk,(1)xk,(2) ≤ 0σ(k) = 2 if xk,(1)xk,(2) > 0
Stable
Tutorial switched systems 26 / 58 M. Jungers
Is the switched system stable for all switching laws ? (II)
3000 2500 2000 1500 1000 500 0 50050
0
50
100
150
200
250
σ(k) = 1 if xk,(1)xk,(2) > 0σ(k) = 2 if xk,(1)xk,(2) ≤ 0
Unstable
Tutorial switched systems 27 / 58 M. Jungers
Main problems
See [LM99].
P1 Find stability conditions such that the switched system isasymptotically stable for any switching law.
P2 Given a switching law, determine if the switched system isasymptotically stable.
P3 Give the switching signal which makes the system asymptoticallystable. P3 is called the stabilization problem.
Tutorial switched systems 28 / 58 M. Jungers
Outline of the tutorial
What are switched systems ?
About stability
Stability results for discrete-time switched systems (solving P1)The joint spectral radiusThe common Lyapunov function approach
Stability results with constrained switching law (solving P2)
Stabilization results for discrete-time switched systems (solving P3)
Tutorial switched systems 29 / 58 M. Jungers
Geometric approach : the joint spectral radius
The joint spectral radius of a set of matrices A = A1, · · · ,AN, denoted ρ(A) isan extension of the radius of a matrix A (i.e. ρ(A)) and gives a necessary andsufficient condition for the stability of the system (5) and solves P1. See [The05].
Remark : the joint spectral radius is the maximal growing rate which may beobtained by using long products of matrices from a given set.
We defineρ(A) = lim supp→+∞ρp(A), (16)
whereρp(A) = sup
Ai1,Ai2
,··· ,Aip∈A
∥∥Ai1 Ai2 × · · · × Aip
∥∥ 1p .
Theorem : The switched system (5) is GAS if and only if
ρ(A) < 1. (17)
Main difficulty : this is difficult in the generic case to practically compute the jointspectral radius. Several approximations are provided in the literature.
Tutorial switched systems 30 / 58 M. Jungers
The common Lyapunov function approach
Theorem If all the modes share a common Lyapunov function, then the switchedsystem is GUAS.
Theorem If the switched system is GUAS, then all the modes share a commonLyapunov function.
Remark : be careful, there is no assumption concerning the class of theLyapunov function. Especially, this Lyapunov function is not necessary on theform V (x) = xT Px as it will be seen in the following. This existence result doesnot help roughly speaking about how to find this Lyapunov function. In addition,there exists a common Lyapunov function on the form V (x) = xT P(x)x , whereP(λx) = P(x), ∀λ 6= 0 (homogeneous of degree zero).
Tutorial switched systems 31 / 58 M. Jungers
The common Lyapunov function approach : sufficient conditions
The previous theorem suggests to look for a common quadratic Lyapunovfunction in the class V (x) = xT Px .
Theorem : Consider the discrete-time linear switched system (5). If there exists amatrix P ∈ Rn×n such that
P > 0n (18)
andAT
i PAi − P < 0, ∀i ∈ I, (19)
then the system (5) admits the common quadratic Lyapunov function V (x) and isGUAS.
Remark : the system (5) may be GUAS without feasible LMI (19).
Tutorial switched systems 32 / 58 M. Jungers
The common Lyapunov function approach : unfeasibility test
To complete the previous remark, we have the following theorem.
Theorem : If there exist positive definite matrices Ri ∈ Rn×n, Ri > 0n such that∑i∈I
AiRiATi − Ri > 0n, (20)
then there does not exist P > 0n such that
ATi PAi − P < 0, ∀i ∈ I, (21)
Proof : If there exist Ri (∈ I) such that Inequalities (20) hold, then for everyP > 00,
0 < Tr
[P
(∑i∈I
AiRiATi − Ri
)]= Tr
[Ri
(AT
i PAi − P)],
then there exists i0 ∈ I such that ATi0 PAi0 − P > 0.
Tutorial switched systems 33 / 58 M. Jungers
Multiple Lyapunov functions
Definition : We consider functions of the form
V (σ(k), xk ) = Vσ(k)(xk ) = xTk P(σ(k), xk )xk . (22)
Theorem : If there exist Pi , i ∈ I such that Pi > 0 and
ATi PjAi − Pi < 0, ∀(i, j) ∈ I2, (23)
then the discrete-time switched system (5) is GUAS.
Sketch of proof : By chosing i = σ(k) and j = σ(k + 1), we haveVσ(k+1)(xk+1)− Vσ(k)(xk ) < 0, ∀xk 6= 0. A common Lyapunov function is
Vmax(x) = maxi∈I
xT Pix . (24)
Tutorial switched systems 34 / 58 M. Jungers
Outline of the tutorial
What are switched systems ?
About stability
Stability results for discrete-time switched systems (solving P1)
Stability results with constrained switching law (solving P2)A periodic switching lawDwell time constraint
Stabilization results for discrete-time switched systems (solving P3)
Tutorial switched systems 35 / 58 M. Jungers
Periodic switching lawA K -periodic switching law is defined by σ : N→ I such that
σ(k + K ) = σ(k), k ∈ N. (25)
We define the monodromy matrix as
Φk = Aσ(k+K−1)Aσ(k+K−2) × · · · × Aσ(k). (26)
Theorem : the eigenvalues of the monodromy matrix Φk are called characteristicmultipliers and are independent of i . The system (5) is GUAS if its characteristicmultipliers belong strictly to the unit circle.
Then there exists W > 0n such that ΦTk W Φk −W < 0. Moreover there exists a
K -periodical Lyapunov function V (xk , k) = xTk P(k)xk ,l with P(k + K ) = P(k),
such that 0 ≤ ∀k ≤ K − 2 :
ATσ(k)Pk+1Aσ(k) − Pk = 0n; (27)
ATσ(K−1)P0Aσ(K−1) − PK−1 < 0n. (28)
Sketch of proof : choose PK−1 = W , and because P0 = PK , then
PK−2 = ATσ(K−2)WAσ(K−2); PK−3 = AT
σ(K−3)ATσ(K−2)WAσ(K−2)Aσ(K−3); · · · (29)
Tutorial switched systems 36 / 58 M. Jungers
Dwell time constraint
Definition : For an integer ∆ ∈ N∗, the set of the switching laws satisfying a dwelltime at least equal to ∆ is defined by
D∆ =σ : N→ I; ∃`qq∈N, `q+1 − `q ≥ ∆;
σ(k) = σ(`q), ∀`q ≤ k < `q+1;σ(`q) 6= σ(`q+1).
Theorem : (See [GC06]) If there exist Pi (i ∈ I) such that
ATi PiAi − Pi < 0n, ∀i ∈ I (30)
(ATi )∆PjA∆
i − Pi < 0n, ∀(i, j) ∈ I2, i 6= j, (31)
then the system (5) is GAS for any switching law σ ∈ D∆.
Tutorial switched systems 37 / 58 M. Jungers
Outline of the tutorial
What are switched systems ?
About stability
Stability results for discrete-time switched systems (solving P1)
Stability results with constrained switching law (solving P2)
Stabilization results for discrete-time switched systems (solving P3)Lyapunov–Metzler inequalitiesGeometric approachLMI sufficient conditionPeriodic stabilizability
Tutorial switched systems 38 / 58 M. Jungers
Stabilization of linear discrete-time switched systems
The problem P3 is to design a switching law that stabilizes the system (5).
Assumption : Ai (∀i ∈ I) are not Schur.
This assumption is to avoid a trivial solution : if there exists i0 such that Ai0 isSchur, then σ(k) = i0 globally asymptotically stabilizes the system.
Tutorial switched systems 39 / 58 M. Jungers
Lyapunov-Metzler BMI conditions : sufficient conditions
Let consider theMd the set of the Metzler matrices in discrete-time, that is thematrices whose elements are nonnegative and
∑j∈I πji = 1.
Theorem (see [GC06]. If there exist Pi > 0 (i ∈ I) and π ∈Md such that
ATi
∑j∈I
πjiPj
Ai − Pi < 0, ∀i ∈ I (32)
holds, then the switched system is globally asymptotically stabilizable with themin-switching strategy
σ(k) ∈ arg mini∈I
xTk Pixk . (33)
The inequality (32) is a Bilinear Matrix Inequality (BMI). The condition impliesthat the homogeneous function induced by
⋃i∈I E(Pi ) (where
E(P) = x ∈ Rn, xT Px ≤ 1) is a control Lyapunov function.
Tutorial switched systems 40 / 58 M. Jungers
Sketch of proofLyapunov function considered
Vmin :
Rn → R,xk 7→ min
i∈IxT
k Pixk ,(34)
Notation : (P)p,i =∑∈Iπ`iP`.
Elements of proof• By post-multiplying by xk 6= 0 and pre-multiplying by x ′k ,
x ′k+1(P)p,ixk+1 − x ′k Pixk < 0 (35)
• the minimum scalar value of convex polytopes is reached on one of thevertices
Vmin(xk+1) = minj∈I
x ′k+1Pjxk+1 = min∑j∈I λj =1λj∈R+;
∑j∈I
λjx ′k+1Pjxk+1. (36)
Each column of the Metzler matrix Π ∈M is in the unit simplex, then
Vmin(xk+1) ≤ x ′k+1(P)p,ixk+1. (37)
⇒ global asymptotic stability holds with
Vmin(xk+1)− Vmin(xk ) < 0, ∀xk 6= 0. (38)
Tutorial switched systems 41 / 58 M. Jungers
Example of state-partition
With
A1 =
[−1.1 0
1 0.4
], A2 =
[0.2 00 1.3
], x0 =
(−0.50.5
)we have
Π =
[0.3 0.70.7 0.3
],P1 =
[1.7097 0.37340.3734 0.4786
], P2 =
[1.1978 0.63980.6398 1.3173
].
0 5 10 15 20 25
0.05
0.1
0.15
0.2
0.25
0.3
0.35
k
γ1=0.3 et γ2=0.3
x1
x 2
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Tutorial switched systems 42 / 58 M. Jungers
Geometric tools
A C-set is a compact, convex set containing the origin in its interior.Definition A set Ω ⊆ Rn is a C∗-set if it is compact, star-convex with respect tothe origin and 0 ∈ int(Ω).
Notice a set is• convex if ∀x0 ∈ Ω and ∀x ∈ Ω, then αx0 + (1− α)x ∈ Ω, ∀α ∈ [0, 1].• star-convex if ∃x0 ∈ Ω, such that ∀x ∈ Ω, thenαx0 + (1− α)x ∈ Ω, ∀α ∈ [0, 1].
Minkowski function of a C∗-set Ω : ΨΩ(x) = minαα ∈ R : x ∈ αΩ.
• Any C-set is a C∗-set.• Given a C∗-set Ω, we have that αΩ is a C∗-set and αΩ ⊆ Ω for all α ∈ [0, 1].• ΨΩ(·) is : defined on Rn ; homogenous of degree one ; positive definite and
radially unbounded. But nonconvex in general !
Tutorial switched systems 43 / 58 M. Jungers
Geometric approach
Algorithm 1 Control λ-contractive C-set for the switchedsystem (5).• Initialization : given the C∗-set Ω ⊆ Rn, define
Ω0 = Ω and k = 0 ;• Iteration for k ≥ 0 :
Ωik+1 = A−1
i Ωk , ∀i ∈ I,Ωk+1 =
⋃i∈I
Ωik+1;
• Stop if Ω ⊆ int
⋃j∈Nk+1
Ωj
; denote N = k + 1 and
Ω =⋃
j∈1;···;NΩj .
3 2 1 0 1 2 32.5
2
1.5
1
0.5
0
0.5
1
1.5
2
2.5
x1
x 2
6 4 2 0 2 4 6
5
4
3
2
1
0
1
2
3
4
5
x1
x 2
!6 !4 !2 0 2 4 6
!4
!3
!2
!1
0
1
2
3
4
x1
x 2
Tutorial switched systems 44 / 58 M. Jungers
Geometric approach
Geometrical interpretation :• the set Ωi
k is the set of x that can be stirred in Ω in k steps by a switchingsequence beginning with i ∈ I ;
• then Ωk is the set of points that can be driven in Ω in k steps ;• and hence Ω the set of those which can reach Ω in N or less steps, by an
adequate switching law.
Necessary and sufficient condition for stabilizability.
Theorem There exists a control Lyapunov function for the switched system if andonly if the Algorithm 1 ends with finite N.
Tutorial switched systems 45 / 58 M. Jungers
Example 1
Non-Schur switched system with q = n = 2.
A1 =
[1.2 0−1 0.8
], A2 =
[−0.6 −2
0 −1.2
],
6 4 2 0 2 4 6
4
3
2
1
0
1
2
3
4
x1
x 2
0 5 10 15 20 25 30 35 400
1
2
3
Timex kP ix k
0 5 10 15 20 25 30 35 400.5
1
1.5
2
2.5
Time k
mk
3 2 1 0 1 2 32.5
2
1.5
1
0.5
0
0.5
1
1.5
2
2.5
x1
x 2
6 4 2 0 2 4 6
5
4
3
2
1
0
1
2
3
4
5
x1
x 2
−6 −4 −2 0 2 4 6−5
−4
−3
−2
−1
0
1
2
3
4
5
x1
x 2
Tutorial switched systems 46 / 58 M. Jungers
Example 2
System with q = 4, n = 2 and
A1 =
[1.5 00 −0.8
], A2 = 1.1 R( 2π
5 )
A3 = 1.05 R( 2π5 − 1), A4 =
[−1.2 0
1 1.3
].
The matrices Ai , with i ∈ N4, are not Schur. Notice : only one stable eigenvalue !
!4 !3 !2 !1 0 1 2 3 4
!3
!2
!1
0
1
2
3
4
x1
x 2
0 5 10 15 20 25 30 35 400
2
4
6
Time
x kP ix k
0 5 10 15 20 25 30 35 40
1
2
3
4
Time k
mk
Tutorial switched systems 47 / 58 M. Jungers
Example 3Switched system with
A1 =
[0 −1.011 −1
], A2 =
[0 −1.011 −0.5
].
The technique based on Lyapunov-Metzler inequalities [GC06] has beennumerically checked (gridding) and it results not feasible.
Nevertheless...
−4 −3 −2 −1 0 1 2 3 4 5
−4
−3
−2
−1
0
1
2
3
x1
x2
0 5 10 15 20 25 30 35 400
5
10
15
Time
xkP
ixk
0 5 10 15 20 25 30 35 400.5
1
1.5
2
2.5
Time k
σk
Tutorial switched systems 48 / 58 M. Jungers
Example 4
Switched system with
A1 =
[1.3 00 0.9
] [cos(θ) − sin(θ)sin(θ) cos(θ)
], A2 =
[1.4 00 0.8
],
for θ = 0 (left) and θ = π5 (right).
−3 −2 −1 0 1 2 3
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
x1
x2
−6 −4 −2 0 2 4 6
−4
−3
−2
−1
0
1
2
3
4
x1
x2
Tutorial switched systems 49 / 58 M. Jungers
LMI sufficient condition
Our next aim is to formulate an alternative condition that could be checkedefficiently, a convex one.Theorem The switched system is stabilizable if there exist N ∈ N and η ∈ RN
such that η ≥ 0,∑i∈I
ηi = 1 and ∑i∈I
ηiATi Ai < I.
with Ai =k∏
j=1
Aij = Aik · · ·Ai1 .
The condition is just sufficient (except for particular cases), is it also necessary ?No !
Tutorial switched systems 50 / 58 M. Jungers
CounterexampleConsider the three modes given by the matrices
A1 = AR(0), A2 = AR(
2π3
), A3 = AR
(−2π
3
), with A =
[a 00 a−1
]and a = 0.6. The geometric condition holds with N = 1.
−2 −1 0 1 2−2
−1
0
1
2
For every N and every Bi with i ∈ I, the related Ai is such that det(ATi Ai ) = 1
and Tr(ATi Ai ) ≥ 2.
Notice that, for all the matrices Q > 0 in R2×2 such that det(Q) = 1, thenTr(Q) ≥ 2 and Tr(Q) = 2 if and only if Q = I.
Thus, for every subset K ⊆ I, we have that∑i∈K
ηiATi Ai < I, cannot hold, since
either Tr(ATi Ai ) > 2 or AT
i Ai = I.
Tutorial switched systems 51 / 58 M. Jungers
Periodic stabilizability
A periodic switching law is given by σ(k + K ) = σ(k).
The stabilizability through periodic switching law, i.e. periodic stabilizability, isformalized below.definition The switched system is periodic stabilizable if there exist a periodicswitching law σ : N→ I, such that the system is stabilizable for all x ∈ Rn.
Notice that for stabilizability the switching function might be state-dependent,hence a state feedback, whereas for having periodic stabilizability the switchinglaw must be independent on the state.
Is there an equivalence relation between periodic stabilizability and the LMIcondition ? The answer is below.
Theorem : A stabilizing periodic switching law for the switched system exists ifand only if the LMI condition holds.
Tutorial switched systems 52 / 58 M. Jungers
Sum up on the stabilizability
Lyapunov-MetzlerCS
ConditionLMI
Geometriccondition
Stabilisabilityissue
PeriodicStabilisability
Tutorial switched systems 53 / 58 M. Jungers
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Perspective and results on the stability and stabilization of hybrid systems.Proceedings of the IEEE, 88 :1069–1082, 2000.
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Hybrid Systems with Constraints.Wiley-ISTE, July 2013.
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Necessary and sufficient condition for stabilizability of discrete-time linear switched systems : aset-theory approach.Automatica, regular paper, 50(1) :75–83, January 2014.
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Stability of discrete linear inclusion.Linear Algebra and its Applications, 231 :47–85, 1995.
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Min-switching local stabilization for discrete-time switching systems with nonlinear modes.Nonlinear Analysis : Hybrid Systems, 9 :18–26, August 2013.
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Handbook of Hybrid Systems Control : Theory, Tools, Applications.Cambridge, 2009.
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Basic problems in stability and design of switched systems.IEEE Control Systems Magazine, 19 :59–70, 1999.
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Joint Spectral Radius : theory and approximations.PhD thesis, UCL Belgium, 2005.
Tutorial switched systems 57 / 58 M. Jungers
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Tutorial switched systems 58 / 58 M. Jungers