49th IEEE Conference on Decision and Control December 15-17, 2010 Hilton Atlanta Hotel, Atlanta, GA, USA
Average passivity for discrete-time and sampled-data linear systems Fernando Tiefensee, Salvatore Monaco and Doroth´ee Normand-Cyrot
Abstract— Average passivity concepts are proposed for linear time-invariant discrete-time as well as sampled-data dynamics to remove the direct input-output link obstacle. Necessary and sufficient conditions for average passivity are given in terms of LMI’s. Direct discrete-time stabilizing and damping control strategies are proposed for average passive systems or Lyapunov stable dynamics. Linear port-controlled Hamiltonian - PCH systems are treated as an example to illustrate the stabilizing and damping performances through a simulated example.
conservativeness is preserved under sampling and a direct sampled-data damping controller is described. Simulation results are discussed. II. D ISCRETE - TIME SYSTEMS A. Preliminaries and problem settlement Consider a discrete-time LTI system Σd
I. I NTRODUCTION The notion of passivity inspired by electrical energy dissipation through resistors in circuits theory, is nowadays a recognized concept for systems stabilization at large (see [21], [6], [2], [3], [15]). While passivity is well defined for continuous-time systems, it is not so in the discretetime case, where usual passivity definition requires a direct input/output link (D 6= 0). It results that any continuoustime passive system without direct link loses such a property under sampling. This problem motivates several approaches aiming to ensure passivity in discrete-time or sampled-data contexts [1], [9], [14]. More recently in [4], it was shown that passivity can be preserved under sampling with respect to some suitably defined sampled-data output; the same concept was generalized to nonlinear systems in [11] and the definition of average passivity has been introduced for discrete-time and sampled-data nonlinear systems. Using this new definition, in [12] it was shown that passivity and Hamiltonian conservation can be preserved by nonlinear Port Controlled Hamiltonian (PCH) systems under sampling. In [18], [19], [13], feedback controllers which stabilize nonlinear systems by damping improvement are inspired by the nonlinear average passivity definition. In this paper we specialize those concepts to discretetime linear time-invariant (LTI) systems. Average passivity conditions in the usual format of linear matrix inequalities are set in Section II where stabilizing and damping controllers are also proposed. Section III deals with passivity under sampling; we show that the sampled system is average passive with respect to a modified output and possibly with respect to the output itself. The resulting stabilizing and damping controllers are given. The example of a PCH system is treated in Section IV; it is shown that Hamiltonian F. Tiefensee and D. Normand-Cyrot are with Laboratoire des Signaux et Syst`emes, CNRS-Supelec, Plateau de Moulon, 91190 Gif-sur-Yvette, France.
[email protected],
[email protected] S. Monaco is with Dipartimento di Informatica e Sistemistica ’Antonio Ruberti’, Universit`a La Sapienza, via Ariosto 25, 00185, Roma, Italy.
[email protected]
978-1-4244-7744-9/10/$26.00 ©2010 IEEE
xk+1
= Ad xk + Bd uk
(1)
yk
= Cd xk + Dd uk
(2)
with Ad , Bd , Cd and Dd matrices of appropriate dimensions, T xk ∈ IRn , yk and uk = uk1 . . . ukm . Definition 2.1: Σd is passive with quadratic storage function V = 12 xT Px, P = PT > 0 if for all xk and uk the stored energy does not exceed the supply rate, s(y, u), defined as the inner product uTk yk ; i.e. V (xk+1 ) −V (xk ) ≤ uTk yk where
T
(3)
indicates the matrix transposition.
Equivalently, Σd is passive if and only there exists a matrix P = PT > 0 verifying the linear matrix inequality (LMI) below Ad T PAd − P Ad T PBd −CdT ≤ 0. Bd T PAd −Cd Bd T PBd − Dd + DTd LMI’s solvability conditions are discussed in [7], and can be interpreted as the linear discrete-time version of KalmanYakubovich-Popov (KYP) properties.
Proposition 2.1: [7] Σd is passive with quadratic storage V (x) = 21 xT Px if and only if there exist real matrices P = PT > 0, L and W s.t. (i) ATd PAd − P = −LT L (ii) (iii)
ATd PBd −CdT = −LW BTd PBd − DTd + Dd =
(4) T
−W W.
(5) (6)
From (4), one deduces that Σd is passive only if there are no eigenvalues outside the unitary disk of the complex plane and those on the boundary are simple ones (stability). From (6), one concludes that Dd cannot be equal to zero so that any discrete-time LTI system without direct input/output link cannot be passive according to Definition 2.1. Remark 2.1: To avoid the necessity of a direct inputoutput link, dissipativity can be related to different supply rates set as s(y, u) = yT Qy + 2uT Sy + uT Ru, with Q = QT , R = RT and S, matrices of appropriate dimensions [14].
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B. Average passivity Following [11] where a new concept of average passivity is proposed for nonlinear discrete-time systems adopting the Differential/Difference Representation - DDR [10], one sets equivalently to (1) +
x := Ad x;
dx+ (u) := Bd du
+
+
with x (0) = x .
Z uk 0
Bd dv = Ad xk + Bd uk .
(7)
(8)
In such a framework, it is natural to associate with dynamics (7), the output y+ (xk , u) = Cd xk+ (u) and to define the u-average output as follows y+ av (xk , uk ) := :=
Z m 1 uki + y (xk , v)dvi ∑
i=1 uki m
∑
0
Z 1 uki
i=1 uki
0
Cd x+ (v)dvi
:= Cd Ad xk +Cd Bd
(9)
uk . 2
Accordingly, one defines average dissipativity. Definition 2.2: The discrete-time LTI system Σd without direct input/output link (Dd = 0), is said average passive with quadratic storage V (x) = 21 xT Px (P = PT > 0) if for all xk and uk , one verifies the inequality V (xk+1 ) −V (xk ) ≤ uTk y+ av (xk , uk )
(10)
where the u-average output y+ av (xk , uk ) is defined in (9). Σd is said average lossless when V (xk+1 ) −V (xk ) = uTk y+ av (xk , uk ). Rewritting (10) in terms of the LMI ATd PAd − P ATd (PBd −CdT ) ≤0 (BTd P −Cd )Ad (BTd P −Cd )Bd necessary and sufficient conditions are easily deduced. Proposition 2.2: -Average passivity without direct inputoutput link- The discrete-time LTI system Σd , with (Dd = 0) is average passive with quadratic storage V (x) = 12 xT Px if and only if there exist real matrices P = PT > 0, L and W verifying (i) ATd PAd − P = −LT L (ii) ATd PBd −CdT = −LW (iii) BTd P −Cd Bd = −W T W.
y+D av (xk , uk ) := =
More precisely, x+ (u) represents a curve in IRn parameterized by u and (1) is described as two differential and difference equations modeling respectively the free and controlled evolutions. From xk , one computes xk+ (0) = Ad xk and one integrates the differential equation between 0 and uk to recover (1); i.e. xk+ (uk ) := xk+1 = Ad xk +
Assuming Dd 6= 0 in Σd , the u-average output is computed according to
(11) (12) (13)
1 ∑ uki i=1
Z uki 0
(Cd (x+ (vi )) + Dd v)dvi
1 Cd Ad xk + (Cd Bd + Dd ) uk . 2
Proposition 2.3: -Average passivity with direct inputoutput link- The discrete-time LTI system Σd , with a direct input/output link (Dd 6= 0) is average passive with quadratic storage function V (x) = 12 xT Px if and only if there exists a real matrix P = PT > 0 verifying the LMI: ATd PAd −P ATd PBd −CdT ≤ 0. (14) BTd P −Cd Ad BTd PBd − Dd −Cd Bd
The solvability of such a LMI is conditioned to the existence of real matrices P = PT > 0, L and W verifying (11), (12) and
(iiiD ) BTd PBd − Dd −Cd Bd = −W T W. An analogy between the sampled-data time-average output proposed in [4] (which preserve passivity under sampling) and the discrete-time u-average output y+ av , described in (9), can be set in the sens that the first one is calculated by a temporal averaging of the continuous-time output during one sampled period, while the second one is computed by an output average with respect to the control input through one step (in discrete-time the output evolution stay constant during one step and thus a temporal average makes no sense). C. Average-passivity based control The design of stabilizing control laws based on passivity criteria is well known. Roughly speaking, if Σd is detectable (i.e. the non-observable subsystem is asymptotically stable) and uk → yk passive according to the usual definition, then the negative output control law uk = −Kyk with K > 0 ensures V (xk+1 ) − V (xk ) ≤ −Ky2k < 0 for all xk 6= 0 and asymptotic stabilization is guaranteed by the Lasalle’s principle (see [6] and [2]). The goal of this section is to describe stabilizing controllers for systems without direct input-output link assumed average passive. Theorem 2.1: Given the discrete-time LTI system Σd without direct input/output link, suppose it is average passive with quadratic storage 12 xT Px (P = PT > 0 and the output y+ (x, 0) = Cd Ad x detectable, then the discrete-time feedback −1 K uk = −K I + Cd Bd Cd Ad xk , (15) 2 with K > 0, asymptotically stabilizes Σd . Proof: It is a matter of computations to show that uk in (15), computed according to uk = −Ky+ av (xk , uk ) satisfies
Σd is average lossless iff (11)-(13) hold true with L = W = 0. Remark 2.2: Comparing Proposition 2.1 and Proposition 2.2, we note that average passivity is made possible provided Cd Bd is different from zero.
m
T
V (k + 1) −V (xk ) ≤ −Ky+ av (xk , uk )y+ av (xk , uk ) ≤ 0. Asymptotic stability follows from detectability and the Lasalle’s principle.
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For a system Σd with direct input-output link, the controller (15) generalizes as follows −1 K uk = −K I + (Cd Bd + Dd ) Cd Ad xk . (16) 2 We note that even though the so defined controllers can be formally seen as negative average output feedbacks, their computations require state measurements. Work is progressing to complete the design with observers schemes. D. Lyapunov stability and average passivity The next result shows how to define a suitable output link with respect to which a given Lyapunov stable dynamics is made average passive.Then, a control strategy improving damping performances is described.
A. Passivity properties under sampling Given a continuous-time LTI system Σc x(t) ˙ = Ax(t) + Bu(t) y(t) = Cx(t)
with A, B and C matrices with appropriate dimensions, Σc is said passive with quadratic storage V = 12 xT Px, if there exists a real matrix P = PT > 0 s.t. V˙ (x(t)) ≤ uT (t)y(t)
Assuming the output ymod (x, 0) = BTd Px+ (0) = BTd PAd x) detectable, then the static-state feedback −1 K uk = −K I + BTd PBd BTd PAd xk (17) 2 achieves asymptotic stabilization. Proof: As (1) is Lyapunov stable, one has ATd PAd − P ≤ 0, and thus 1 V (xk+1 )−V (xk ) ≤ uTk BTd PAd xk + uTk BTd PBd uk = uTk ydav (xk , uk ) 2
where by definition
ydav (x, u) =
Z 1 u
u
0
(20)
for all t ≥ 0, u ∈ IRm , or equivalently if there exists a real matrix P = PT > 0 verifying the continuous-time KYPproperties: (i) PA + AT P ≤ 0 (ii) C = BT P.
(21) (22)
The equivalent sampled-data system is given by Σδ
Proposition 2.4: Given the LTI dynamics (1), suppose it is stable with quadratic Lyapunov V = 12 xT Px, P = PT > 0, then it is average passive with respect to the output ymod (x, v) = BTd Px+ (v).
(18) (19)
xk+1 yk
= Aδ xk + Bδ uk = Cxk
(23) (24)
with Aδ := eδ A , Bδ := 0δ eτ A Bdτ , δ ∈]0, T ∗ ] a finite time interval so that the passivity criteria (3) is transformed into the LMI below " # T T Aδ PAδ − P Aδ PBδ −CT ≤ 0. (25) T T Bδ PBδ Bδ PAδ −C R
Even though Σc satisfies (20), Σδ cannot verify (25) T because the term Bδ PBδ is strictly positive. Consequently, passivity is lost under sampling for any system without direct input-output link. B. Sampled-data δ u-average passivity We refer in this section to the DDR of the sampled-data dynamics, equivalent to a given continuous-time dynamics (18); i.e. from (23)
BTd Px+ (v)dv
is the u-average output associated with yd (x, v) = BTd Px+ (v) and average passivity holds true. Asymptotic stabilization under the assumption of detectability is a direct consequence of Theorem 2.1. Such a controller can be interpreted as a discrete-time version of the usual continuous-time stabilizing and damping controller through energy dissipation.
x+
= Aδ x;
dx+ (δ v) Bδ = d(δ v) δ
with x+ (0) = x+(26)
Given a passive system Σc , is its sampled-data equivalent average passive ? Unfortunately such property does not hold in general. However, as shown in Theorem 3.1 below, it turns out that average passivity holds with reference to the output T yδ (x+ (u)) := Bδ Px+ (u). Theorem 3.1: Let the continuous-time dynamics (18) assumed stable with Lyapunov function V = 21 xT Px, P = PT > 0, then its sampled-data equivalent dynamics with output
III. S AMPLED - DATA SYSTEMS As well known, passivity is lost under sampling for systems without direct input-output link (see [3], [5], [11]). The solution proposed in [4] consists in looking forward in time to define a new output mapping with respect to which usual passivity can be preserved. Hereafter, we specialize the average passivity concepts introduced for purely discretetime dynamics to sampled-data dynamics which inherit some properties of the continuous-time dynamics.
T
yδ (x, u) = Bδ Px+ (u)
(27)
is average passive with quadratic storage V . Proof: Specializing (3) on (26) one has
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V (xk+1 ) −V (xk )
=
1 T δT δ xk A PA − P xk 2 T T 1 +uTk Bδ PAδ xk + uTk Bδ PBδ uk . (28) 2
The stability condition of the continuous-time dynamics (AT P + PA ≤ 0) ensures negativity of the first term on the right hand side of (28) (A stable implies Aδ stable); i.e. Z 1 T δT δ 1 (k+1)δ xk A PA − P xk = x(τ )T (AT P + PA)x(τ )dτ ≤ 0 2 2 kδ
Under sampling Hc can be rewritten in the DDR form (26) R with P = Q, Aδ = eδ (J−R)Q and Bδ = 0δ eτ (J−R)Q Bdτ .
and thus
V (xk+1 ) −V (xk )
≤
T T 1 uTk Bδ PAδ xk + uTk Bδ PBδ uk 2
T
so verifying average passivity of the link u → Bδ Px+ (u); i.e. V (xk+1 ) −V (xk )
≤
uTk yδav (xk , uk )
1 δ uk
T
Z δ uk 0
δ
y
δ xk + Dδav uk (x, v)dv := Cav
(29)
(30)
T
δ := Bδ PAδ , Dδ := 1 Bδ PBδ . and Cav av 2
On these bases the following result can be stated for sampled-data stabilization with damping improvement. Theorem 3.2: Given a continuous-time LTI dynamics, suppose it is stable with Lyapunov function V = 21 xT Px, T P = PT > 0, let the output yδ (x, 0) = Bδ PAδ x supposed detectable for its sampled-data equivalent dynamics, then the sampled-data control law δ xk uk = uδk = −K(I + KDδav )−1Cav
with K > 0
T
The δ u-average output is computed as follows: T 1 T yδavH (x, u) = Bδ QAδ x + Bδ QBδ u. 2 V. E XAMPLE - THE MASS SPRING SYSTEM The performances of the proposed sampled-data stabilizing controller by damping improvement is illustrated by the mass-spring system. Simulations are worked out comparing uk in (31) with a sampled-data control law designed in [4]. The studied system consists in three masses connected through two springs modeled as a 6-dimensional PCH system (see [16]):
q˙ p˙
(31)
ensures asymptotic stabilization and damping improvement at the sampling instants. Proof: By construction, the controller (31) makes V (xk+1 )−V (xk ) not positive because of (29). Asymptotic staT bilization follows from detectability of yδ (x, 0) = Bδ PAδ x. IV. P ORT-C ONTROLLED H AMILTONIAN AS AN E XAMPLE From average passivity concepts we illustrate in this section how Hamiltonian conservativeness and Port Controlled Hamiltonian - PCH passivity (see [20]) are preserved under sampling. The δ u-average output is characterized in both cases. A. LTI-PCH systems A continuous-time PCH dynamics Hc , with a quadratic Hamiltonian H(x) = 12 xT Qx, Q = QT > 0 is defined by x(t) ˙ = (J − R)Qx(t) + Bu(t)
Proposition 4.1: Given a continuous time LTI-PCH system with Q = QT > 0 and R = RT > 0 (resp. R = 0), then for all δ ∈]0, T ∗ ] its sampled-data equivalent is average passive (resp. average lossless) with respect to yδHmod = Bδ QAδ xk .
with yδav (xk , uk ) :=
T of Hc , with output Q = QT > 0, J = −J T = yh = B Qx, 1 −1 −1 T T − Q A and R = R − 21 AQ−1 + Q−1 AT semi2 AQ positive (resp. positive).
=
y
=
# " ∂H I3 0 ∂ q (q, p) + u (33) ∂ H (q, p) −R p Bp ∂p " # ∂∂Hq (q, p) BTp (34) ∂ H (q, p) ∂p
−Rq −I3
0
where q = (q1 , q2 , q3 )T and p = (p1 , p2 , p3 )T denote the generalized coordinates vector and the generalized momenta vector with respect to m1 , m2 and m3 , respectively. I3 is the 3-dimensional identity matrix. Supposing that the system is actuated by an horizontal force u applied on m1 and that the output y is the displacement velocity of m1 on the horizontal plan, one has B p = [1 0 0]T . The system losses are represented by the dissipation matrices Rq = RTq > 0 and R p = RTp > 0. The Hamiltonian function is defined by the sum of kinetic and potential energies =
H
# " i p23 p22 1h 1 p21 + k1 (q1 − q2 )2 + k2 (q2 − q3 )2 + + 2 m1 m2 m3 2
with k1 , k2 > 0 the spring elastic constants, then H = 12 xT Qx with Q=
(32)
κ 0
0 M −1
k1 , κ = −k1 0
−k1 k1 + k2 −k2
0 −k2 , M = k2
m1 0 0
0 m2 0
0 0 . m3
with x ∈ IR , u and y ∈ IR. The interconnection matrix J ∈ IRn×n is constant and skew-symmetric, i.e. J = −J T and xT Jx = 0; the dissipativity matrix R ∈ IRn×n , is also constant, positive definite and symmetric, i.e. R = RT ≥ 0. B ∈ IRn×m is constant. Adopting the Hamiltonian as system storage function and specializing condition (21) on Hc , one verifies that its free evolution is stable (J − R)Q2 + Q2 (J − R)T = −2RQ2 ≤ 0.
It is easy to see that the system is passive since its free evolution is stable, i.e. ∂H T ∂H ∂H T ∂H ˙ H(t) = − Rq Rp − < 0. ∂q ∂q ∂p ∂p
Remark 4.1: In [16], it is shown that any LTI stable (resp. asymptotically stable) system can be rewritten in the form
injects additional damping to the system and its convergence speed is controlled by the gain K.
n
A static-state feedback uc (t) = −Ky(t) = −
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K p1 m1
5000
200
4000
H(t)
3500 3000 2500
H(t)
150
H(t)| δ u k H(t)|u⋆ k H(t)|u d0
100 50
y(t)
4500
0 −50
y(t)
2000 −100
y(t)| δ u k y(t)|u⋆ k y(t)|u d0
1500 −150
1000
−200
500 0 0
0.2
0.4
0.6
0.8
1 t
1.2
1.4
1.6
1.8
−250 0
2
0.2
0.4
(a) Evolution of H
0.6
0.8
1 t
1.2
1.4
1.6
1.8
2
(b) Output y evolution 500
5
q(t) 0
u(t)
400
q(t)|u d0
−5
−15 −20
u(t)
y(t)| δ u k q(t)|u⋆ k
−10
200
uδ k u⋆k
100
ud0
300
0 −25 5
−100 0 −5 −10 −15 −20 −25
−200 −300
−25
−20
−15
−5
−10
0
5
10
15
20
−400 0
(c) q1 × q2 × q3 evolution Fig. 1.
is written in the form The sampled-data model of (23)-(24), with Rq κ M −1 A= , BT = [0 BTp ] and C = [0 BTp M −1 ]. −κ R p M − 1 Since the free-evolution of the system is stable, the sampled-data stabilizing control uδk in (31) asymptotically stabilizes the origin of (33)-(34). B. Alternate sampled-data strategy Another approach is presented in [4]. It is shown that the system (23)-(24) is passive under sampling with respect to the output Z (k+1)δ kδ
y(τ )dτ =
with C⋆ = C 0δ eτ A dτ and D⋆ = C controller is given by R
C⋆ x + D⋆ u δ
(35)
Rδ τ 0 B dτ . Then, a stabilizing
δ u⋆ = −K(I + KD⋆ )−1C⋆ xk
0.6
0.8
1 t
1.2
1.4
1.6
1.8
2
Control performances for δ = 75ms.
(Aδ , Bδ ,C)
1 δ
0.4
(d) Control input amplitudes
A. Sampled-data design
y⋆ (x, u) =
0.2
with K > 0.
oscillations appear on the output y and in q trajectory (figures 1(b) and 1(c)); as a consequence, ud0 assumes high amplitudes to ensure suitable damping assignment (figure 1(d)). Even if the alternate solution u⋆ slightly improves the closed-loop system response, softly reducing control-input amplitudes and oscillations on y and q, these improvements are not so convincing.In the other hand uδk ensures a good damping assignment under sampling reducing control amplitudes and oscillations on y and q. uδk robustness with respect of δ is put in light in figures 2(a)-2(d). In fact the damping performances are maintained for a δ = 150ms. One concludes that in practical applications, when a sampled-data passivity based controller can be performed using small δ , the emulated solution can be implemented, since it consists in an output feedback and ensures acceptable performances. However, when larger sampling periods and higher control performances are required, the proposed sampled-data damping assignment (which consists in a state feedback) should be priorized.
(36)
C. Simulation Results Figure 1(a)-1(b) compare the storage function and the output evolution of the closed-loop continuous-time system with the sampled-data ones under emulated ud0 = −K pmk11 , uδk and u⋆ controllers for δ = 150ms. Even if the emulated control consists in an output feedback, its performances are degraded for increasing δ , since uk → yk passivity of (33)(34) is lost under sampling and ud0 ensures strict negativity of H(xk+1 ) − H(xk ) only to approximations in the order δ (with an error in O(δ 2 )). It results that the inter-sampling H evolution (figure 1(a)) presents increasing regions, some
VI. C ONCLUSION Average passivity concepts are specified for discrete-time and sampled-data LTI systems. It is shown that using such a concept, the direct input/output link obstacle is removed and necessary and sufficient conditions of average passivity are set. Feedback control laws constructed from the uaverage output and ensuring asymptotic stability through damping improvement are described. Their performances are illustrated and compared with a previous solution proposed in [4]. A port-controlled Hamiltonian example illustrates the results putting in light the controller robustness to larger sampling lengths.
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5000
50
4500
H(t) 0
4000
H(t)| δ u k
−50
3000
y(t)
H(t)
3500
2500
−100
2000 −150 1500
y(t)
1000 −200
y(t)| δ u k
500 0 0
0.2
0.4
0.6
0.8
1 t
1.2
1.4
1.6
1.8
−250 0
2
0.2
0.4
0.6
0.8
1 t
1.2
1.4
1.6
1.8
2
(b) Output y evolution
(a) Evolution of H 5 500
q(t)
0 −5
u(t) 400
y(t)| δ u k
−10
uδ k
300
u(t)
−15 −20
200
−25 100 0 0
−10 −20 −25
−20
−15
−5
−10
0
5
−100 0
(c) q1 × q2 × q3 evolution Fig. 2.
0.2
0.4
0.6
0.8
1 t
1.2
1.4
1.6
1.8
2
(d) Control input amplitudes Control performances for δ = 150ms.
Working is progressing and a interesting theoretical extension should be the interpretation of average passivity concepts in a frequency domain. In a practical way, these concepts can be suitable applied to design feedback controllers to linearized models of VTOLS [8] and for the energetic management of a hybrid fuel cell/supercapacitor supply system [17]. VII. ACKNOWLEDGMENTS Work partially supported by a research mobility grant VINCI- of the UFI/UFI - French/Italian - Italian/French University.
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