Switched linear discrete time systems EECI Graduate School on - - PowerPoint PPT Presentation

Switched linear discrete time systems

EECI Graduate School on Control 2009 Jamal Daafouz March 2009

Outline

Switched Linear Discrete Time Systems

◮ Stability ◮ Structural Properties : Invertibility, Flatness

Applications

◮ Digital Control : Hot Strip Mill, Networked Control ◮ Encryption

Spectral radius

◮ Spectral radius of A : largest modulus of its eigenvalues

ρ(A) = max{|λ| : Av = λv}

◮ Spectral radius of a matrix power

ρ(Ak) = ρ(A)k Convergence condition : lim

k→∞ Ak = 0

⇐ ⇒ ρ(A) < 1

◮ Spectral radius as a limit of norms

ρ(A) = lim

k→∞ Ak1/k

Generalized spectral radius

Consider a (non necessarily bounded) set A of n × n matrices Ai A =

• Ai : i ∈ I
• ,

I =

• 1, ..., M
• The larget possible spectral radius of all products of the matrices

ρk(A) sup

• ρ(

k

• i=1

Ai) : Ai ∈ A for 1 ≤ i ≤ k

• ◮ GSR : The maximal asymptotic spectral radius of the product of

matrices choosen freely in A ρ(A) lim sup

k→∞

• ρk(A)

1/k

Joint spectral radius

The larget possible norm of all products of the matrices choosen in A ˆ ρk(A) sup

• k
• i=1

Ai : Ai ∈ A for 1 ≤ i ≤ k

• ◮ JSR :

ˆ ρ(A) lim sup

k→∞

• ˆ

ρk(A) 1/k

◮ GSR= JSR = spectral radius when A =

• A
• ρ(A) = ˆ

ρ(A) = ρ(A)

◮ GSR=JSR for any bounded set A

ρ(A) = ˆ ρ(A)

Stability of discrete switched systems

Consider a set A of matrices Ai, and the discrete linear inclusion xk+1 ∈

• Aσkxk

: Aσk ∈ A

• ,

x0 ∈ Rn arbitrary (1) The sequence (σk) is the switching signal depending on k and/or on xk. (x0, x1, . . . , xk, . . .) satisfying the inclusion (1) xk+1 = Aσkxk, for some sequence σk is a trajectory in the Rn space. The set of all possible switchings signals defines a whole set of possible trajectories.

GSR/JSR and UFS (J. Theys et al. 2005)

◮ Uniform Asymptotic Stability : UFS

Any trajectory (xk) of the discrete linear inclusion converges to the

• rigin

lim

k→∞ xk = 0,

∀ (σk) As this is supposed to hold for any x0, it is equivalent to saying that all matrix products converge to 0 lim

k→∞ AσkAσk−1 . . . Aσ1 = 0

◮ Theorem

For a bounded set A of matrices ˆ ρ(A) < 1 ⇐ ⇒ discrete linear inclusion is UFS ρ(A) < 1 ⇐ ⇒ discrete linear inclusion is UFS

Stability of discrete switched systems

• V. D. Blondel and J. N. Tsitsiklis. The boundedness of all products of a pair of

matrices is undecidable. Systems and Control Letters, 2000.

Theorem

Generalized/Joint spectral radius of a pair of matrices is not polynomial-time approximable. This is true even for a pair of matrices with {0, 1} entries. It is NP-hard to decide whether all products of two given real matrices A0 and A1 are bounded.

◮ GSR/JSR complex to compute ◮ not suitable for design purpose

Stability of discrete switched systems

◮ Lyapunov functions : In general, a sufficient stability condition.

A scalar function V such that V (xk) > 0, ∀xk = 0 and the derivative of V along the system trajectories must be decreasing as time evolves. Linear systems : xk+1 = Axk, x0 ∈ Rn Quadratic Lyapunov function V (xk) = xT

k Pxk,

P = PT > 0

Stability of discrete switched systems

V (xk+1) < V (xk), ∀xk = 0 xT

k+1Pxk+1 < xT k Pxk,

∀xk = 0 ATPA − P < 0 ⇐ ⇒ λ(A) < 1 Generalizing this to discrete time switched systems leads to the common quadratic Lyapunov function (CQLF) stability condition AT

i PAi − P < 0,

∀i which should hold simultaneously for all matrices Ai of the set A Existence of a CQLF is not a necessary condition.

Stability of discrete switched systems

◮ Example : (J. Theys et al. 2005)

xk+1 = Aσkxk, Aσk ∈ {A1, A2} A1 = −0.2 −0.4 0.4 −0.2

• ,

A2 = −0.2 −2.4 1/15 −0.2

• No CQLF but

0.9275 ≤ ˆ ρ ≤ 0.9510

Classical results

For more complex systems it is not clear :

◮ What form should the Lyapunov function have ? ◮ What form is necessary and sufficient for the stability ? ◮ How can we find it ? (analytical / numerical tool)

Solution :

◮ Use simple forms for which we can get LMI conditions

⇒ only sufficient stability conditions

◮ Conservatism :

V (x) does not exist but the system is stable

Stability Analysis of discrete time switched systems

Goal : Analyse global stability of a switched system xk+1 = Aσk xk where Aσk belongs to {Ai : i = 1, . . . , M} σk is the switching rule meaning that at each instant of time k Aσk = Ai, for some i ∈ {1, . . . , M} There is no complete solution for the stability problem even in the bidimensional case xk+1 = Aσk xk Aσk ∈ {A1, A2}, xk ∈ R2

Stability of Switched systems / Polytopic systems

Stability of a linear discrete time switched system xk+1 = Aσk xk, Aσk ∈ {Ai : i = 1, . . . , M} is equivalent to the stability of the linear discrete-time polytopic system xk+1 = Aξkxk, Aξk ∈ conv{A1, . . . , AM} that is Aξk = M

i=1 ξi(k)Ai

and M

i=1 ξi = 1, ξi ≥ 0

One has to prove that ρ(conv{A1, . . . , AM}) = ρ({A1 . . . AM})

(P. Mason, M. Sigalotti and J. Daafouz, CDC 2007)

Stability of linear switched discrete time systems

Different choices of parameter dependent quadratic Lyapunov functions for global stability analysis of a linear discrete-time polytopic system xk+1 = Aξkxk (k ∈ N) Aξ = M

i=1 ξiAi and M i=1 ξi = 1, ξi ≥ 0 (convex combination of Ai).

(1) Parameter dependent quadratic stability V (x, ξ) = xTP(ξ)x α1x2 ≤ V (x, ξ) ≤ α2x2 (2) Parameter and time dependent quadratic stability V (k, x, ξ) = xTP(k, ξ)x α1x2 ≤ V (k, x, ξ) ≤ α2x2 (3) Poly-quadratic stability V (x, ξ) = xTP(ξ)x with P(ξ) =

M

• i=1

ξiPi

Using parameter dependent Lyapunov functions

Proposition

The previous three types of quadratic stability are equivalent. Proof : (P. Mason, M. Sigalotti and J. Daafouz, CDC 2007) Clearly: (3) Polyquadratic stability (V (x, ξ) with P(ξ) = M

i=1 ξiPi)

⇓ (1) Parameter dependent quadratic stability (V (x, ξ) = xTP(ξ)x) ⇓ (2) Parameter and time dependent quadratic stability (V (k, x, ξ) = xTP(k, ξ)x) It remains to show that (2) ⇒ (3).

Scheme of the proof

(2) ⇒ (3) (i.e. ∃ a L.F. of the form xTP(k, ξ)x ⇒ ∃ a L.F. xT(M

i=1 ξiPi)x )

First step: Recall that xk+1 = Aξkxk where Aξ = M

i=1 ξiAi.

Define Pk,i = quadratic form associated with the vertex Ai at time k Then one proves that Πk,ξ(x) = xT(

M

• i=1

ξiPk,i)x is a (time and parameter dependent) quadratic LF. The key tool is the convexity of the function f (A) = ATPA for any positive definite matrix P.

Parameter dependent quadratic stability

Second step: By a compactness argument one can choose suitable k and h with h < k such that Π⋆

ξ(x) = k

• i=h

Πk,ξ(x) is a LF and has the desired form.

Remark

more general than the classical "static” notion of quadratic stability P(ξ) ≡ P Asymptotic stability does not imply parameter dependent quadratic stability

LMI stability condition (J. Daafouz et al 2001 and 2002)

◮ There exist a polyquadratic Lyapunov function

V (x, ξ) = xTP(ξ)x with P(ξ) =

M

• i=1

ξiPi for xk+1 = Aξkxk, Aξk ∈ conv{A1, . . . , AM}

◮ There exist a switched Lyapunov function

V (xk, σk) = xT

k Pσxk

for xk+1 = Aσk xk, Aσk ∈ {Ai : i = 1, . . . , M}

◮ There exist Pi, i = 1, . . . , M satisfying

AT

i PjAi − Pi < 0

∀i = 1, . . . , M, ∀j = 1, . . . , M (2)

Stability of Polytopic discrete time systems

Necessary and sufficient LMI condition for the existence of V = x⊤

k M

• i=1

ξiPixk

• r equivalently for the existence of

V = x⊤

k Pσxk

• Pi

A⊤

i Pj

PjAi Pj

• > 0

∀i = 1, . . . , M, ∀j = 1, . . . , M (3)

• Gi + Gi ⊤ − Si

Gi ⊤A⊤

i

AiGi Sj

• > 0,

∀i = 1, . . . , M, ∀j = 1, . . . , M (4) where Si = P−1

i

, ∀i = 1, . . . , M

Dwell Time (J.C Geromel et al 2006)

Consider σk = i ∈ {1, . . . , M}, k ∈ [lq, lq+1) where lq and lq+1 are succesive switching times satisfying lq+1 − lq ≥ ∆ ≥ 1, ∀q ∈ N

Theorem

Assume that, for some ∆ ≥ 1, these exists a collection of positive definite matrices {P1, . . . , M} such that AT

i PiAi − Pi < 0,

∀i = 1, . . . , M (5) (A∆

i )TPjA∆ i − Pi < 0,

∀i = j = 1, . . . , M (6) The switched system is globally asymptotically stable with a dwell time ∆.

Dwell Time

An upper bound for the minimum dwell time ∆∗ can be computed by taking the minimum value of ∆ satisfying the Theorem conditions. One has to solve the optimization problem min∆≥1,P1>0,...,PM>0

• ∆ : (5) − (6)

which, for ∆ ≥ 1 fixed, reduces to a convex programming feasibility problem with LMI constraints. ∆ = 1 : we recover the switched Lyapunov functions conditions AT

i PjAi − Pi < 0

∀i = 1, . . . , M, ∀j = 1, . . . , M

Dwell Time

◮ Example :

A1 = eB1T, A2 = eB2T with B1 =

• 1

−10 −1

• ,

B2 =

• 1

−0.1 −0.5

• ,

T = 0.5 Upper bound of the minimum dwell time ∆∗ ≤ 6 Necessary condition from linear periodic systems ∆per = 6 = ⇒ minimum dwell time ∆∗ = 6

Control design

Design a switched state feedback uk = Kσxk to stabilize xk+1 = Aσxk + Bσuk Find matrices Si = S⊤

i , matrices Gi and Ri, ∀i = 1, . . . , M such that

Gi + Gi ′ − Si (AiGi + BiRi)′ AiGi + BiRi Sj

• > 0

(7) The state feedback matrix gains are given by Ki = RiGi −1 ∀i = 1, . . . , M

Control design

Design a switched output feedback uk = Kσyk to stabilize xk+1 = Aσxk + Bσuk yk = Cσxk (8) Ci full rank ∀i = 1, . . . , M Find matrices Si = S⊤

i , matrices Gi, Ui and Vi, ∀i = 1, . . . , M such that

Gi + Gi ′ − Si (AiGi + BiUiCi)′ AiGi + BiUiCi Sj

• > 0

(9) ViCi = CiGi ∀i = 1, . . . , M The output feedback matrix gains are given by Ki = UiVi

−1

∀i ∈ E

Numerical Evaluation

Single input m = 1. System Success N = 2 N = 3 N = 4 N = 5 N = 6 n = 3, p = 1 SQ Without Gi With Gi 67 93 96 43 85 96 25 79 89 14 68 82 12 52 79 n = 3, p = 2 SQ Without Gi With Gi 78 98 100 46 87 95 42 93 98 16 80 94 17 70 88 n = 4, p = 2 SQ Without Gi With Gi 27 79 97 10 50 83 28 64 26 48 8 34 n = 5, p = 2 SQ Without Gi With Gi 6 48 81 1 11 49 4 25 1 7 1 4

Control design with pole placement

Design a switched state feedback such that :

◮ Each closed loop linear subsystem has its poles in desired location ◮ The closed loop switched system is stable under arbitrary switching

sequences Some motivations to study this problem :

◮ the dynamic of each subsystem is fixed so as to get a correct

behavior when a model keeps constant for any long time.

◮ When the switching is activated because of variation in the

parameters, or in the operating point, or by a supervisor which selects the best control strategy, the stability is ensured whatever the switching control can be.

Control design with pole placement

Consider the switched system defined by: xk+1 = Aσxk + Bσuk (10) Problem : Find a switched state feedback uk = Kσxk, (11) such that

◮ the closed loop switched system is globally asymptotically stable. ◮ the closed loop dynamic spectra satisfies

spec(Aσ + BσKσ) = {λσ1, . . . , λσn} (12) where the λσq , q = 1, ..., n, denote desired eigenvalues

Control design with pole placement

The problem of computing for a linear system a state feedback matrix gain K which ensures that (A + BK) has {λ1, . . . , λn} as its eigenvalues has been considered by Moore in 1976. The procedure leads to the computation of the matrix gains K in two steps :

• 1. For q = 1, . . . , n compute bases

» Mq Nq –

• f the null-space of

ˆ A − λqI B ˜ (13)

• 2. The matrix gain is then given by

K = “ ˆ N1 ... Nn ˜ diagn

q=1vq

| {z }

R

”“ ˆ M1 ... Mn ˜ diagn

q=1vq

| {z }

G

”−1 (14)

where vq are arbitrary vectors chosen so that invertibility is ensured.

Control design with pole placement

To consider complex eigenvalues, this procedure has to be modified. As complex eigenvalues occur in conjugate pairs (λr, λ(r+1)) with λ(r+1) = λ∗

r , the null-space computation must be replaced with

2 6 6 4 Mr Nr M(r+1) N(r+1) 3 7 7 5 null-space of » A − Re(λr)I B Im(λr)I −Im(λr)I A − Re(λr)I B – (15)

The matrix gain K is given by K = (NΨ)(MΨ)−1 (16) where M

• . . . , Miq, . . . , Mr, M(r+1), . . .
• N
• . . . , Nq, . . . , Nr, N(r+1), . . .
• Ψ
• diag
• . . . , vq, . . . , vr, v(r+1), . . .
• ,

vq ∈ Rp×1, vr = v(r+1) ∈ R2p×1 with the subscripts q for real eigenvalues and r, r + 1 for complex eigenvalues.

Control design with pole placement

Notice that in the framework of switched linear discrete systems, the stabilizing switched control structure Ki = RiG −1

i

is similar to the pole placement gain structure K = (NΨ

• R

)(MΨ

• G

)−1 (17) We can then select among the stabilizing switched controls the one ensuring that the closed loop matrices (Ai + BiKi), i ∈ E have {λi1, . . . , λin} as eigenvalues. Replace in the LMIs of switched control design Theorem the unknowns Ri and Gi by (NiΨi) and (MiΨi) respectively. The matrices Ψi, which gathers all degrees of freedom, are used to satisfy the Lyapunov constraints of the previous Theorem.

Control design with pole placement

This is summarized in the following Theorem.

Theorem

Assume that Mi and Ni have been computed as indicated previously. If there exist symmetric matrices Si and matrices Ψi solutions of:

» MiΨi + ΨT

i MT i

− Si ΨT

i MT i AT i + ΨT i NT i BT i

AiMiΨi + BiNiΨi Sj – > 0, (18)

∀(i, j) ∈ E × E, then a stabilizing switched state feedback exists and the resulting gains Ki are given by Ki = (NiΨi)(MiΨi)−1 Moreover, such gains ensure that the closed loop dynamic is characterized by spec(Ai + BiKi) = {λi1, . . . , λin}.

Control design with pole placement

The success of the proposed procedure is related to the availability of degrees of freedom in excess with respect to a pure pole placement. Hence, this approach is only valid for multi-input systems. Single input systems → Partial pole placement.

• 1. Compute for q = 1, . . . , l with l < n bases

Miq Niq

• f the null-space of [Ai − λiqI

Bi]

• 2. The matrix gains are given by

Ki RiG −1

i

, Ri NiΨNi , Gi MiΨMi with Mi

• Mi1, . . . , Mil, I, I, . . . , I
• n−l

times

• ,

Ni

• Ni1, . . . , Nil, I, I, . . . , I
• n−l

times

• ΨMi diag(vi1, . . . , vil, si1, . . . , si(n−l)),

ΨNi diag(vi1, . . . , vil, ti1, . . . , ti(n−l)) (19)

Control design with pole placement

Theorem

Let ΨMi , ΨNi matrices given by (19). If there exist symmetric matrices Si, matrices ΨMi , ΨNi solutions of:

• MiΨMi + (MiΨMi )T − Si

(•)T AiMiΨMi + BiNiΨNi Sj

• > 0,

(20) ∀(i, j) ∈ E × E, then a stabilizing switched state feedback exists and the resulting gains Ki are given by Ki = (NiΨNi )(MiΨMi )−1 Moreover, such gains ensure that the closed loop dynamic is characterized by {λi1, . . . , λil} ⊂ spec(Ai + BiKi).

Example

Consider a switched system given by xk+1 = Aσxk + Bσuk (21) where {Ai : i ∈ E} and {Bi : i ∈ E} are a family of matrices parameterized by an index set E = {1, 2} and A1 =

• 0.0094

0.3010 −3.0098 0.0094

• ,

A2 =

• 0.0094

3.0098 −0.3010 0.0094

• ,

B1 = B2 = 1

• Case (uk = 0) : The discrete time invariant subsystems characterized by

A1 and A2 are stable and the matrices A1 and A2 have the same eigenvalues: 0.0094 ± 0.9518i.

Example

If the switching signal is characterized by σ = 1 if x1

k x2 k ≥ 0,

with xk = [x1

k

x2

k ]′

2

• therwise

(22) then the switched system is unstable.

−0.5 0.5 1 1.5 2 2.5 3 x 10

48

−2 2 4 6 8 10 x 10

47

x1 x2 x1(0), x2(0)

Example

Design a stabilizing switched state feedback control with the property that each closed loop subsystem has λ = 0 as one of its eigenvalues. This example is single input and only a partial pole assignment may be achieved.

Example

Let i = 1, the first subsystem, compute for q = 1, ..., l with l < n the null-space of [Ai − λiqI Bi]. Here n = 2 and l = 1 since only one eigenvalue may be fixed for each subsystem. The result is :   0.0030 0.9576 −0.2883   = ⇒ M11 = 0.0030 0.9576

• ,

N11 = −0.2883 Build the matrices Mi

• Mi1, . . . , Mil, I, I, . . . , I
• n−l

times

• ,

Ni

• Ni1, . . . , Nil, I, I, . . . , I
• n−l

times

• that is

M1 =

• M11 I
• =
• 0.0030

1 0.9576 1

• ,

N1 =

• N11 I
• =
• − 0.2883 1

Example

One has to do similar computations for i = 2 the second subsystem and

• btain

M2 = 0.0098 1 0.3153 1

• ,

N2 =

• − 0.9490 1
• The next step is to check the feasibility of the Linear Matrix Inequalities
• f Theorem 4 where the unknowns are the Lyapunov matrices Si and the

matrices ΨMi =

• vi1

si1

• ,

ΨNi =

• vi1

ti1

• with

vi1 ∈ R, si1 ∈ R2 and ti1 ∈ R for i = 1 and i = 2;

Example

Such LMIs are found feasible and

S1 = 10−03 » 0.0100 0.0005 0.0005 0.1912 – , S2 = » 0.0026 0.0462 0.0462 1.4822 – ΨM1 = 2 4 0.0066 −0.0001 2.0158 3 5 , ΨN1 = » 0.0066 −0.6067 – ΨM2 = 2 4 0.2459 0.0230 2.1505 3 5 , ΨN1 = » 0.2459 −6.4726 –

This leads to the following control gains K1 = (N1ΨN1)(M1ΨM1)−1 = −0.01786 −0.30097 K2 = (N2ΨN2)(M2ΨM2)−1 =

• −0.0102

−3.0098

• Using the switched control characterized by K1 and K2

Example

◮ Each closed loop subsystem is characterized by

spec(A1 + B1K1) = {1.7 10−13, 0.00093} and spec(A2 + B2K2) = {−7 10−14, 0.0085}

◮ The closed loop switched system is globally asymptotically stable

2 4 6 8 10 12 14 16 18 20 −2 2 4 6 8 10 12 samples k x1(k) 2 4 6 8 10 12 14 16 18 20 −12 −10 −8 −6 −4 −2 samples k x2(k)