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Control of disturbed LPV systems in a LMI setting
Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011
Control of disturbed LPV systems in a LMI setting Anamaria Luca ∗ Pedro Rodriguez-Ayerbe ∗ Didier Dumur ∗ ∗
SUPELEC Systems Sciences (E3S) - Automatic Control Department, 3, rue Joliot Curie, 91192, Gif-sur-Yvette Cedex, France (e-mail: [email protected], [email protected], [email protected]).
Abstract: This paper is concerned with the control of a discrete-time LPV system affected by bounded disturbances. First, an input-to-state stable observer-based controller is synthesized. In the sequel, we compute the invariant ellipsoid that assures constraints satisfaction and has the maximal projection on the initial state subspace. In order to increase the system robustness (quantified in terms of ellipsoidal set volume), a Youla parameter is designed. This parameter enlarges the invariant set projection but slows the closed-loop performances. A compromise between robustness and performance is obtained by imposing a new bound on the Lyapunov function decreasing rate. Keywords: LMI, S-procedure, ISS, Observer-based systems, Youla-Kuˇcera parameter, Invariant sets, LPV systems. 1. INTRODUCTION Linear parameter-varying (LPV) systems have gained a lot of attention in the control community due to their implication in treating nonlinear controller synthesis problems. The design of gain-scheduled controllers for non-linear systems is proposed in (Apkarian et al., 1994), (Becker et al., 1993), (Scorletti and Ghaoui, 1995). An observerbased controller is synthesized in (Heemels et al., 2009), the parameters are not exactly known and the parameter uncertainty is maximized while still guaranteeing closedloop stability. (Xie and Eisaka, 2004) designs a stable observer-based controller for an LPV system that does not suffer from disturbances. Invariant sets framework has significantly developed in control engineering over the last decades. The existence of an invariant set (ellipsoidal or polyhedral set) is equivalent with the existence of a Lyapunov function and hence with a stability test. In our case, the bounded disturbances presence and the LPV mode imposes the use of the “input-tostate stability” (ISS) notion in the case of norm bounded disturbances. ISS for this class of disturbances implies that the origin is an asymptotically stable point for the undisturbed LTI system and also that all state trajectories are bounded for all bounded disturbance sequences. More exactly, the state converges to the 0-reachable set (Blanchini and Miani, 2008). Furthermore, if the disturbance fades then the system asymptotically converges to the origin (Limon et al., 2008). In (Liberzon, 1999) the problem of achieving disturbance attenuation in the ISS and integralISS sense for nonlinear systems with bounded controls is considered. ISS for discrete-time nonlinear systems has been studied in (Jiang and Wang, 2001). To the best of the authors’ knowledge the results presented in this paper (the synthesis of an observer-based controller 978-3-902661-93-7/11/$20.00 © 2011 IFAC
and a Youla parameter for a LPV discrete-time system affected by disturbances) are novel. The observer-based controller assuring ISS for norm bounded disturbances is designed in Section 2. The inequalities giving this controller are obtained by considering an extension of the disturbances actuating in the system and by applying the S-procedure. In the sequel we search the invariant ellipsoid satisfying input constraints and having the maximal projection on the initial state subspace. In Section 3, a Youla parameter that enlarges the maximal ellipsoidal projection (assuring therefore a better robustness) is synthesized. The Youla-based system performances are improved by imposing a new bound on the Lyapunov function decreasing rate. These results are expressed in a LMI form (Boyd et al., 1994) and are validated in Section 4 with a numerical example. In Section 5 some concluding remarks are drawn. 2. OBSERVER-BASED DESIGN FOR A DISTURBED LPV SYSTEM 2.1 System description Consider the following discrete-time LPV system: x(k + 1) = A(θ(k))x(k) + Bu(k) + Bω ω(k), y(k) = Cx(k) + Bv v(k), together with an observer used to estimate the state: x ˆ(k + 1) = A(θ(k))ˆ x(k) + Bu(k) + L(θ(k))(y(k) − yˆ(k)), yˆ(k) = C x ˆ(k), where x(k) ∈ Rnx is the state, u(k) ∈ Rm the input, y(k) ∈ Rp the output, ω(k) ∈ R the state noise, v(k) ∈ R the output noise, θ(k) ∈ Θ the time varying parameter, x ˆ ∈ Rnx the estimated state, L ∈ Rnx xp the observer gain. The control law has the form: u(k) = −F (θ(k))ˆ x(k), where F (θ(k)) ∈ Rmxnx is the feedback gain.
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The varying parameter θ is fully known and available. This parameter lies in a polytope Θ of vertices θi , i = 1, N : N N X X µi (k) = 1}. (2) µi (k)θi ; µi (k) ≥ 0, θ(k) = { i=1
i=1
For the system with observer the following augmented state representation will be used: xo (k + 1) = Ao (θ(k))xo (k) + Bo (θ(k))n(k), (3) x(k) where: xo (k) = , ǫ(k) = x(k)− x ˆ(k) the estimation ǫ(k) A BF (θ(k)) ,AF =A(θ(k))−BF (θ(k)) error, Ao (θ(k))= F 0 AL Bω 0 AL =A(θ(k))−L(θ(k))C, Bo (θ(k)) = Bω −L(θ(k))Bv T
and n(k) = [ω(k) v(k)] is the disturbance vector.
The command can be written in the following form: u(k) = −Fo (θ(k)) · xo (k) with Fo (θ(k)) = [F (θ(k)) −F (θ(k))].
(4)
It is clear that the parameter-varying matrices range in a polytope of matrices whose vertices are the images of the vertices θ1 , . . . , θN : A(θ(k)), F (θ(k)), L(θ(k)), Ao (θ(k)), Bo (θ(k)), F( θ(k)) = N N X X µi (k) [A(θi ), µi (k) [Ai , Fi , Li , Aoi , Boi , Foi ] = { i=1
i=1
F (θi ),L(θi ),Ao (θi ),Bo (θi ),Fo (θi )] ,µi (k)≥ 0,
N X
µi (k)=1}.
i=1
The disturbance vector is assumed to be bounded: nT n ≤ 1.
(5)
2.2 Design of a stabilizing observer-based LPV control In this section we state the theorem giving an ISS observerbased LPV controller for the LPV system affected by bounded disturbances (5). Lemma 1. (ISS-Lyapunov function) Goulart et al. (2005) The system (3) is ISS for the norm bounded disturbances (5) if there exist K∞ functions α1 (·), α2 (·), α3 (·), a K function δ(·) and a continuous function V : Rn → R, V (0) = (0) such that: α1 (|| x(k) ||) ≤ V (k) ≤ α2 (|| x(k) ||), (6) V (k + 1) − V (k) ≤ −α3 (|| x ||) + δ(|| n ||). (7) Theorem 1. Let the discrete-time LPV system (3) affected by bounded disturbances (5). If there exist Gi = GTi ≻ 0, Pi = PiT ≻ 0, QG , QP , Yi , Zi , i = 1, N , α > 0, β ≥ 0 and γ ≥ 0 such that the following inequalities are satisfied: T QG + QG −Gi 0 αQTG QTG ATi −YiT B T T 0 βI 0 [Bω 0] ≻ 0, (8) αQG 0 αGi 0 Ai QG −BYi [Bω 0] 0 Gj Pi 0 αPi ATi QTP −C T ZiT 0 γI 0 [QP Bω −Zi Bv ]T ≻0, αPi 0 αPi 0 QP Ai −Zi C [QP Bω −Zi Bv ] 0 QTP + QP −Pj (9)
α − β ≥ 0, α − γ ≥ 0, i, j = 1, N (10) then the system is input-to-state stable in the case of bounded norm disturbances (5). The feedback gains and the observer gains that stabilize the system are given by: −1 Fi = Yi Q−1 (11) G , Li = QP Zi , i = 1, N . We can calculate F (θ) and L(θ) as: F (θ(k)) =
N X
µi (k)Fi and L(θ(k)) =
N X
µi (k)Li . (12)
i=1
i=1
Proof. As in (Daafouz and Bernussou, 2001) a parameterdependent Lyapunov function V (k)=xo (k)T Po (θ(k))xo (k), N X T µi (k)Poi , Poi =Poi =G−1 with Po (θ(k))= oi ≻ 0 is considi=1
ered. Let us define A = Ao (θ(k)), B = Bo (θ(k)), P = Po (θ(k)) and P + = Po (θ(k + 1)). Then ∆V (k) = V (k + 1)−V (k)=xo (k+1)T P + xo (k+1)−xo (k)T Pxo (k)= T T + xo (k) A P A−P AT P + B xo (k) = . n(k) n(k) BT P + A BT P + B Condition (6) is satisfied for α1 =ψ || x || (ψ a sufficiently N X λmax (Poi ) || x ||. small positive scalar) and α2 = i=1
Further we prove that if: ∆V (k) < 0, (13) for any xo and n satisfying: xo (k)T Pxo (k) ≥ 1 and n(k)T n(k) ≤ 1 (14) the condition (7) is obtained for the case of norm bounded disturbances (5). For the sequel of the demonstration the following extensions will be made: Bx 0 Bω 0 0 0 , = B →B= v 0 Bǫ 0 0 Bω −L(θ(k))B (15) ω ω n n → n = 1 , n1 = 1 , n2 = 2 , v2 v1 n2 nT1 n1 ≤ 1, nT2 n2 ≤ 1. With these extensions we assume that the system and the observer can be affected by different disturbances. This augmentation does not change our problem and it is a necessary tool for manipulating the inequalities in order to obtain (8) and (9). From the S-procedure (Boyd (2009), Boyd et al. (1994) pp.82-84) we have that (14) implies (13) if there exist α > 0, β ≥ 0 and γ ≥ 0 such that: ∆V (k) − α(1 − xTo Pxo ) − β(nT1 n1 − 1) − γ(nT2 n2 − 1) < 0. (16) For α ≥ (β + γ) (Boyd et al. (1994) pp.82-84), (16) can be written as in (7) with α3 = αϑ || x || (ϑ a sufficiently small positive scalar) and δ = max{β, γ}. By applying Schur complement (16) becomes: P h 0 i αP AT P + βI 0 0 0 BT P + 0 γI ≻ 0, αP 0 αP 0 P +A P +B 0 P+ α − β − γ ≥ 0.
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
”⇒” Following the steps in (Boyd et al. (1994) pp. 117118) we prove first that the inequality (17) has a solution if and only if the inequalities (8), (9) and (10) do. Suppose that the inequalities (17) and (18) have a solution P, α > 0 and β ≥ 0, where: P11 P12 ≻ 0. (19) P= T P12 P22 For P in the form above, we pre- and post-multiply (17) with , respectively, Γ and ΓT , I 0 0 0 0 I 0 0 0 0 0
0 0 0 Γ = 0 0
0 I 0 0 0 0 0 0
0 0 0 0 0 I 0 0
0 0 I 0 0 0 0 0
0 0 0 0 0 0 I 0
0 0 0 I 0 0 0 0
0 0 0 0 . 0 0 0 I
(20)
Now, based on Schur complement it can be easily seen that the inequality (17) implies: + P11 0 αP11 AF T P11 + 0 βI 0 BxT P11 ≻0, (21) αP11 0 αP11 0 + + + P11 P11 AF P11 Bx 0 α − β ≥ 0.
(22)
−1 We pre- and postmultiply (21) with diag(QTG , I, P11 , + −1 −1 + −1 T (P11 ) ) and diag(QG , I, P11 , (P11 ) ). Since (QG − −1 −1 P11 )P11 (QG − P11 ) 0 the following relaxation can −1 T be made: QG P11 QG QTG + QG − P11 . By making the −1 + −1 notations G(θ(k)) = P11 , G(θ(k + 1)) = (P11 ) and Y (θ(k)) = F (θ(k))QG and because the new inequality is
affine in all the variables, condition (8) results. The inequality (17) also implies: + P22 0 αP22 AL T P22 + 0 γI 0 BǫT P22 ≻ 0, αP22 0 αP22 0 + + + P22 P22 AL P22 Bǫ 0
0 0 (8)(θ(k)) − λ 0 BF (θ(k))
|
{z
0 0 0 0
S(θ(k))
0 0 0 0
0 0 ·(9)−1 (θ(k)) · S T (θ(k)) ≻ 0, 0 0
}
α − λβ − γ ≥ 0. (24) These inequalities are satisfied for any positive λ such that: ν > λη and α − γ ≥ λβ where: ν = ς, (ς a sufficiently small positive scalar) N N X X −1 λmax (Si (9) SiT ). Since (8)(θ(k)), (9)(θ(k)) η = i=1 j=1
and (10) are feasible, such λ will always exist.
There exist LPV systems that cannot be stabilized using a fixed observer-based controller independent of the parameter θ, for this reason we have chosen to design a LPV feedback law and a LPV observer gain. S-procedure introduces new variables. Because one of this variable α multiplies Pi , Gi and QG , the inequalities (8) and (9) are now BMIs (bilinear matrix inequalities). Since α is a scalar, an αoptim can be found by executing a simple loop. In order to avoid the loop the PENBMI solver (Koevara and Stingl, 2006) can be used. The results can be extended to parameter dependent input and output matrices, B(θ(k)) and C(θ(k)), by using the relaxations stated in (Sala and Ari˜ no, 2007). 2.3 Invariant ellipsoid calculus An invariant set is a subset of the state space with the property that, if it contains the system state at some time, then it will contain it also in the future (Blanchini and Miani, 2008). For the observer-based controller (11), we search the maximal invariant set (the biggest state region) satisfying input constraints despite the disturbance presence and the LPV mode. Input constraints, disturbances presence and the LPV mode influence the invariant set volume (larger the possible disturbance signal is, the smaller the maximal ellipsoid gets).
Because of their association with powerful tools such as the Lyapunov equation or LMIs, we approximate the real + −1 We pre and postmultiply (23) with diag(I, I, I, QP (P22 ) ). invariant set by an ellipsoidal set: + + −1 + Since (QP − P22 )(P22 ) (QTP − P22 ) 0 the following reE = {xo | xTo Q−1 xo ≤ 1} (25) + −1 T + laxation can be made: QP (P22 ) QP QP +QTP −P22 . By where Q ∈ R(2nx )x(2nx ) is a symmetric positive-definite + , P (θ(k)) = P22 matrix (not necessarily block diagonal). Since we are intermaking the notations P (θ(k + 1)) = P22 and Z(θ(k)) = QP L(θ(k)) and because the new inequality ested more in the initial system state, x(k), we maximize is affine in all the variables, inequality (9) results. the invariant set projection onto x-subspace instead of maximizing the entire set volume. The projection of E ”⇐” Suppose that P (θ(k)), G(θ(k)), QP , QG , Y (θ(k)), (25) onto x is given by: Z(θ(k)), α, β and γ satisfy the inequalities (8), (9) (written −1 Ex = {x | xT (T QT T )−1 x ≤ 1, x = T xo }. (26) in θ(k)) and (10) and define F (θ(k)) = Y (θ(k))QG and −1 L(θ(k)) = QP Z(θ(k)). The objective is to prove that The projection of an ellipsoid on a subspace is an ellipsoid (Blanchini and Miani, 2008). λG(θ(k))−1 0 there exists a λ > 0 such that P = 0 P (θ(k)) The constraints on the input are: satisfies (17) and (18), assuring therefore ISS for (5). || u ||2 ≤ umax . (27) Because β is a variable, we can suppose β → λβ. By con- Theorem 2. Consider an ISS observer-based LPV system gruence of (17) with Γ, and after with diag(QTG , I, G(θ(k)), (3) face of bounded disturbances (5). The maximization of G(θ(k + 1)), I, I, I, QP G(θ(k + 1))) we obtain via Schur Ex subject to input constraint (27) is achieved by solving: complement that the observer-based system is ISS for the min − log det(T QT T ) (28) bounded disturbances (5) if λ > 0 satisfies: α − γ ≥ 0.
(23)
Goi ,Q,Qo ,α,β
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subject to: Qo + QTo − Goi 0 αQTo QTo AToi T 0 βI 0 Boi ≻ 0, αQo 0 αGoi 0 Aoi Qo Boi 0 Goj Goi Q, α − β ≥ 0, α > 0, β ≥ 0, T Qo + QTo − Goi QTo Foi 0, i = 1, N . Foi Qo u2max I
where:
Ao (θ(k))−Be Dq (θ(k))Ce −Be Cq (θ(k)) , Bq (θ(k))Ce Aq (θ(k)) B B −Le (θ(k))Bv −Be Dq (θ(k))Bv ,Be = By (θ(k))= ωe 0 0 Bq (θ(k))Bv xo 0 Bω . ,xy = ,Le (θ(k))= Ce = [0 C] ,Bωe = L(θ(k)) xq Bω Ay (θ(k))=
(29) (30) (31)
Proof. The conditions (17) and (18) guarantee the state invariance for bounded disturbances. By congruence of (17) with diag(QTo , I, P −1 , (P + )−1 ), considering G = P −1 , G + = (P + )−1 and Qo G −1 QTo ≥ Qo + QTo − G an affine inequality is obtained leading to (29). In order to reduce the calculation load one can consider non extended noise signal and α = β > 0 (Boyd et al. (1994) pp.82-84). For F = Foi (θ(k)) we have: || u ||22 = || Fxo ||22 ≤ || FP −1/2 ||22 || P 1/2 xo ||22 = = λmax (FP −1 F T )(xTo Pxo )≤λmax (FP −1 F T ), where λmax is the maximal eigenvalue. Now, by applying Schur complement and by congruence with diag(QTo , I) we obtain that || u ||2 ≤ umax if LMI (31) is feasible. The ellipsoid volume is inversely proportional with the eigenvalues product (that is the determinant). For rendering the problem concave, the “logarithm” operator is used. Because the most MatLab tools can only determine the minimum of a convex problem our optimization criterion becomes min − log det(T QT T ). 3. A YOULA PARAMETER DESIGN FOR A DISTURBED LPV SYSTEM 3.1 System description In order to improve the robustness of the observer basedsystem a Youla parameter is introduced in the closed-loop: xq (k + 1) = Aq (θ(k))xq (k) + Bq (θ(k))˜ y (k), u ˜(k) = Cq (θ(k))xq (k) + Dq (θ(k))˜ y (k), (32) y˜(k) = y(k) − yˆ(k), ˜ ∈ Rm where xq ∈ Rnq the state of Youla parameter, u is its output of Q and y˜ ∈ Rp the input. Fig. 1 gives a block-diagram overview of the structure.
The command is written in the following form: u(k) = −Fx (θ(k)) · xy (k) − Fn (θ(k)) · n(k) (35) with: Fx (θ(k)) = [Fo (θ(k)) + Dq (θ(k))Ce Cq (θ(k))] and Fn (θ(k)) = [0 Dq (θ(k))Bv ]. The augmented system has a polytopic description: N X µi (k) [Ayi , [Ay (θ(k)), By (θ(k), Fx (θ(k)), Fn (θ(k)))] = { i=1
Byi , Fxi , Fni ] , µi (k) ≥ 0,
N X
µi (k) = 1}.
i=1
The disturbance has the same bounds as in (5). 3.2 Youla synthesis for robustness requirements For the Youla LPV system, consider the invariant ellipsoid: Ey = {xy | xTy Q−1 y xy ≤ 1}. The goal here is to synthesize a Youla parameter that maximizes the projection of Ey on the initial state subspace: Exy = {x | xT (ZQy Z T )−1 x ≤ 1, x = Zxy }. The gain in robustness will be given by the difference of volume between the observer-based system ellipsoidal projection and the Youla-based system ellipsoidal projection. The same input constraints as in (27) are considered. Theorem 3. Consider the discrete-time LPV system (34) subject to input constraints (27) and affected by bounded disturbances (5). The design of the Youla parameter that maximizes the projection Exy is achieved by solving: − log det(T QT T )
min
(36)
X, Y, Mi , Ni , Hi , Dqi , α, β, T Q, Gi,11 , Gi,21 = Gi,12 , Gi,22
subject to: LM Ii,j ≻ 0,
(37)
Gi,11 ≻ Q,
System(θ) Observer(θ)
T
(38)
X + X − Gi,11 ⋆ ⋆ I + R − Gi,21 Y + Y T − Gi,22 2 umax I [Foi X + Mi Foi + Dqi Ce ] T 0 Fni i, j = 1, N
⋆ 0, ⋆ I
(39)
and
F (θ) Q(θ)
Fig. 1. State-space controller with Youla parametrization. Now the control law has the form: u(k) = −F (θ(k))ˆ x(k) − u ˜(k).
(33)
For the Youla-based LPV system the following augmented state representation will be used: xy (k + 1) = Ay (θ(k))xy (k) + By (θ(k))n(k) (34)
α − β ≥ 0, α > 0, β ≥ 0, where R = Y T X + V T U, Mi = Dqi Ce X + Cqi U, Ni = −Y T Be Dqi + V T Bqi , Hi = Y T Aoi X − Y T Be Dqi Ce X + V T Bqi Ce X− −Y T Be Cqi U + V T Aqi U, i = 1, N
(40)
(41)
and ⋆ is referring to the symmetrical value. Assuming that the maximization problem has a solution one gets:
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LM Ii,j
X + X T − Gi,11 ⋆ I + R − Gi,21 Y + Y T − Gi,22 0 X + XT − G ⋆ = i,11 α I + R − Gi,21 Y + Y T − Gi,22
hA
oi X
− Be Mi Aoi − Be Dqi Ce Hi Y Aoi + Ni Ce
i
⋆ βI 0 Bωe −Lei Bv − Be Dqi Bv Y T Bωe −Y T Lei Bv + Ni Bv
i=1
i=1
Cq (θ(k))=
i=1
µi (k)Cqi , Dq (θ(k))=
N X
α
U, V obtained by factorization V T U = R − Y T X, Bqi = V −T (Mi + Y T Be Dqi ), Cqi = (Ni − Dqi Ce X)U −1 , Aqi = V −T (Hi − Y T Aoi X + Y T Be Dqi Ce X− − V T Bqi Ce X + Y T Be Cqi U )U −1 , i = 1, N (42) N N X X µi (k)Bqi , µi (k)Aqi , Bq (θ(k))= and Aq (θ(k))= N X
⋆
µi (k)Dqi .
i=1
Proof. The inequality that verifies the stability of a Youlabased system is obtained by analogy with (29) substituting Aoi by Ayi , Boi by Byi , Goi by Gyi and Qo by Qy with: X U1 Y V1 −1 Qy = , Qy = (43) U • V • Q−1 y , nq xnq
uniquely determined where • are blocks of Qy and . The transformation by X, Y ∈ R2nx x2nx , U , V ∈ R (43) is similar to that used in (Scherer et al., 1997; Luca et al., 2010). The size of the Youla parameter is nq = 2nx .
Pre and post-multiplying this new inequality by diag(ΠT , T T I, Π , Π ) and diag(Π, I, Π, Π) respectively, where Π = I Y and considering the decision variables as in (41), 0 V inequality (37) is obtained. It can be considered α = β > 0. For Fx = Fx (θ(k)), Fn = Fn (θ(k)), Py = (Gy )−1 = Py (θ(k)) = Py−1 (θ(k)) we have: || u ||22 = || Fx xy + Fn n ||22 ≤|| Fx xy ||22 + || Fn n ||22 ≤ ≤ λmax (Fx Py−1 FxT ) + λmax (Fn FnT ). Now, using Schur complement in two steps and after pre and post-multiplying with, respectively, diag(ΠT QTy , I, I) and diag(ΠQy , I, I), LMI (39) is derived. The optimization criterion is obtained by analogy with (28) since x = Zxy = T xo . 3.3 Compromise between robustness and performance The synthesis of a Youla parameter in terms of finding the maximal invariant ellipsoid projection may lead to slow closed-loop performances. An effective way to improve the response transients is to consider Lyapunov functions V (k) = xTy Py xy with a decrease rate bigger than the output norm scaled by γ1 : 1 1 V (k)−V (k + 1)≥ (|| Cx(k) ||2 + || Bv v(k) ||2 )≥ || y(k) ||2 . (44) γ γ
Theorem 4. Consider the discrete-time LPV system (34) with input constraints (27) and affected by bounded disturbances (5). For a given scalar γ ≥ 0, the design of
⋆
⋆ X + X T − Gi,11 ⋆ I + R − Gi,21 Y + Y T − Gi,22
0
⋆ ⋆
h
Gj,11 ⋆ Gj,21 Gj,22
i
the Youla parameter that maximizes the projection Exy guaranteeing (44) is achieved by solving: − log det(T QT T )
min
(45)
X, Y, Mi , Ni , Hi , Dqi , α, β, T Q, Gi,11 , Gi,21 = Gi,12 , Gi,22
subject to:
⋆ ⋆ ⋆ ⋆
⋆ ⋆ ⋆ ⋆
LM Ii,j 0, i, j = 1, N (46) [Cf X Cf ] 0 0 0 γI ⋆ 0 [0 Bv ] 0 0 0 γI (38),(39) and (40), with the notations made in (41) and Cf = [C 0]. Assuming that the maximization problem has a solution, the Youla parameter is obtained as in (42). Proof. Making the notation Cz = [Cf 0] and Bz = [0 Bv ], one has: || Cx(k) ||2 + || Bv v(t) ||2 = T T xy xy Cz 0 Cz 0 . n n 0 Bz 0 BzT For Ay = Ay (θ(k)), By = By (θ(k)) we get that (44) is satisfied if: T + Py 0 αPy AT 1 Cz 0 y Py I 0 0 βI 0 ByT Py+ 0 BzT γ ⋆ 0. − 0 0 αP 1 0 αP 0 y
y
Py+ Ay Py+ By
0
Py+
0
0
0
γ
I
After applying Schur theorem, we pre- and postmultiply this inequality with diag(ΠT QTy , I, ΠT Gy , ΠT Gy+ , I, I) and diag(Qy Π, I, Gy Π, Gy+ Π, I, I). After relaxation and considering the same notations as is in (41), the inequality (46) is obtained. 4. NUMERICAL EXEMPLE Consider" a discrete-time # " with: # disturbed " # LPV system 0 0 0.8 0.5 0 A(θ) = −0.4 1.1θ 0.7 , B = 0.7 , Bω = 0.01 0.7 , 0 1 0 0 1 C= [0 1 0] , Bv =0.01 and θ ∈ [−1 1]. An integral action was considered in order to cancel the steady-state errors. By applying Theorem 1 with α = 0.5 and β = γ = 0.427, we obtain the following ISS observer-based controller: F1 = [−0.011 − 0.8925 0.997], L1 = [1.5813 − 0.1711 2.0989]T , F2 = [−0.0147 1.4035 1.00074] and L2 = [1.5813 2.0283 2.099]T . For this controller and considering the input constraint || u ||2 ≤ 1 we search (via Theorem 2) the invariant ellipsoid having the biggest projection on the initial state subspace (x(k)). This projection is represented by the dark ellipsoid in Fig. 2 and was found for an optimal α = β = 0.0042.
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
By introducing the Youla parameter a new maximal projection is computed. This projection corresponds to the blue ellipsoid in Fig. 2 and was obtained for an optimal α = β = 0.0084. For lack of space we skip the values obtained for the Youla parameter.
Fig. 2. Maximal projection. Considering the performance constraints (44) with γ = 1, the maximal projection is now given by the magenta ellipsoid in Fig. 2. This projection and the associated Youla parameter were obtained for an optimal α = β = 0.0213. In Fig. 3 we have plotted the time-domain representation of the output for a step disturbance (with the amplitude 0.1) in the output signal at time 0.05s and a switch between the two vertices at time 0.1s. By adding the
Fig. 3. Output time-domain simulation. Youla parameter (and so, some degrees of freedom) we obtain an improvement in terms of volume (Fig. 2). The gain is more significant when performance constraints are not considered. On the other hand, the Youla parameter leads to slow closed-loop performances (Fig. 3). These performances can be improved by imposing new bounds on the Lyapunov function decreasing rate. These results were obtained using the software Yalmip (Lofberg, 2004) with the Lmilab solver in MatLab environment (the operator used for “log det” is “geomean”). 5. CONCLUSION Considering a discrete-time LPV system affected by bounded disturbances, this paper proposes the design of an ISS observer-based controller. For this controller we search the maximal projection of the invariant ellipsoid that satisfies input constraints despite the disturbance presence. In the sequel we synthesize a Youla parameter that enlarges the invariant ellipsoidal projection. Because
large closed-loop transients may result, a compromise between performance and robustness is reached by imposing a new bound on the Lyapunov function decreasing rate. REFERENCES Apkarian, P., Gahinet, P., and Becker, G. (1994). Selfscheduled H∞ control of linear parameter-varying systems. American Control Conference, pp. 856-860. Becker, G., Packard, A., Philbrick, D., and Balas, G. (1993). Control of parametrically-dependent linear systems: A single quadratic Lyapunov approach. American Control Conference, pp. 2795-2799. Blanchini, F. and Miani, S. (2008). Set-theoretic methods in control. Systems & Control: Foundations & Applications, Birkhauser, Boston–Basel–Berlin. Boyd, S. (2009). Linear matrix inequalities and the Sprocedure. Course EE363, Lecture 15, Standford. Boyd, S., Ghaoui, L.E., Feron, E., and Balakrishnan, V. (1994). Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia. Daafouz, J. and Bernussou, J. (2001). Parameter dependent Lyapunov functions for discrete time systems with time varying parameters uncertainties. Systems & Control Letters, Vol. 43 (5), pp.355-359. Goulart, P., Kerrigan, E., and Maciejowski, J. (2005). Optimization over state feedback policies for robust control with constraints. Automatica, Vol.42, pp.523–533. Heemels, W., Daafouz, J., and Millerioux, G. (2009). Design of observer-based controllers for LPV systems with unknown parameters. 38th IEEE Conference on Decision and Control. Jiang, Z.P. and Wang, Y. (2001). Input-to-state stability for discrete-time nonlinear systems. Automatica, Vol. 37 (6), pp.857-869. Koevara, M. and Stingl, S. (2006). PENBMI user’s guide. Penopt GbR, version 2.1, available on www.penopt.com. Liberzon, D. (1999). ISS and integral-ISS disturbance attenuation with bounded controls. 38th IEEE conference on Decision and Control, pp.2501-2506. Limon, D., Alamo, T., Raimondo, D., de la Pena, D.M., Bravo, J., and Camacho, E. (2008). Input-to-state stability: an unifying framework for robust model predictive control. Int. Workshop on Assessment and Future Directions of NMPC, Pavia, Italy. Lofberg, J. (2004). YALMIP: A toolbox for modeling and optimization in MATLAB. CACSD Conference, Taiwan. Luca, A., Rodriguez-Ayerbe, P., and Dumur, D. (2010). Youla-Kucera parameter synthesis using invariant sets techniques. American Control Conference, pp.56265631. Sala, A. and Ari˜ no, C. (2007). Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: Applications of Polya’s theorem. Fuzzy Sets and Systems, Vol. 158, pp.2671-2686. Scherer, C., Gahinet, P., and Chilali, M. (1997). Multiobjective output-feedback control via LMI optimization. IEEE Trans. on Auto. Control, Vol.42 (7), pp.896-911. Scorletti, G. and Ghaoui, L.E. (1995). Improved linear matrix inequality conditions for gain scheduling. 34th IEEE Conf. on Decision and Control, pp. 3626-3631. Xie, W. and Eisaka, T. (2004). Design of LPV control systems based on Youla parametrization. IEE Proc. Control Theory Appl., vol. 151 (4), pp.465-472.
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Control of disturbed LPV systems in a LMI setting Anamaria Luca ∗ Pedro Rodriguez-Ayerbe ∗ Didier Dumur ∗ ∗
SUPELEC Systems Sciences (E3S) - Automatic Control Department, 3, rue Joliot Curie, 91192, Gif-sur-Yvette Cedex, France (e-mail: [email protected], [email protected], [email protected]).
Abstract: This paper is concerned with the control of a discrete-time LPV system affected by bounded disturbances. First, an input-to-state stable observer-based controller is synthesized. In the sequel, we compute the invariant ellipsoid that assures constraints satisfaction and has the maximal projection on the initial state subspace. In order to increase the system robustness (quantified in terms of ellipsoidal set volume), a Youla parameter is designed. This parameter enlarges the invariant set projection but slows the closed-loop performances. A compromise between robustness and performance is obtained by imposing a new bound on the Lyapunov function decreasing rate. Keywords: LMI, S-procedure, ISS, Observer-based systems, Youla-Kuˇcera parameter, Invariant sets, LPV systems. 1. INTRODUCTION Linear parameter-varying (LPV) systems have gained a lot of attention in the control community due to their implication in treating nonlinear controller synthesis problems. The design of gain-scheduled controllers for non-linear systems is proposed in (Apkarian et al., 1994), (Becker et al., 1993), (Scorletti and Ghaoui, 1995). An observerbased controller is synthesized in (Heemels et al., 2009), the parameters are not exactly known and the parameter uncertainty is maximized while still guaranteeing closedloop stability. (Xie and Eisaka, 2004) designs a stable observer-based controller for an LPV system that does not suffer from disturbances. Invariant sets framework has significantly developed in control engineering over the last decades. The existence of an invariant set (ellipsoidal or polyhedral set) is equivalent with the existence of a Lyapunov function and hence with a stability test. In our case, the bounded disturbances presence and the LPV mode imposes the use of the “input-tostate stability” (ISS) notion in the case of norm bounded disturbances. ISS for this class of disturbances implies that the origin is an asymptotically stable point for the undisturbed LTI system and also that all state trajectories are bounded for all bounded disturbance sequences. More exactly, the state converges to the 0-reachable set (Blanchini and Miani, 2008). Furthermore, if the disturbance fades then the system asymptotically converges to the origin (Limon et al., 2008). In (Liberzon, 1999) the problem of achieving disturbance attenuation in the ISS and integralISS sense for nonlinear systems with bounded controls is considered. ISS for discrete-time nonlinear systems has been studied in (Jiang and Wang, 2001). To the best of the authors’ knowledge the results presented in this paper (the synthesis of an observer-based controller 978-3-902661-93-7/11/$20.00 © 2011 IFAC
and a Youla parameter for a LPV discrete-time system affected by disturbances) are novel. The observer-based controller assuring ISS for norm bounded disturbances is designed in Section 2. The inequalities giving this controller are obtained by considering an extension of the disturbances actuating in the system and by applying the S-procedure. In the sequel we search the invariant ellipsoid satisfying input constraints and having the maximal projection on the initial state subspace. In Section 3, a Youla parameter that enlarges the maximal ellipsoidal projection (assuring therefore a better robustness) is synthesized. The Youla-based system performances are improved by imposing a new bound on the Lyapunov function decreasing rate. These results are expressed in a LMI form (Boyd et al., 1994) and are validated in Section 4 with a numerical example. In Section 5 some concluding remarks are drawn. 2. OBSERVER-BASED DESIGN FOR A DISTURBED LPV SYSTEM 2.1 System description Consider the following discrete-time LPV system: x(k + 1) = A(θ(k))x(k) + Bu(k) + Bω ω(k), y(k) = Cx(k) + Bv v(k), together with an observer used to estimate the state: x ˆ(k + 1) = A(θ(k))ˆ x(k) + Bu(k) + L(θ(k))(y(k) − yˆ(k)), yˆ(k) = C x ˆ(k), where x(k) ∈ Rnx is the state, u(k) ∈ Rm the input, y(k) ∈ Rp the output, ω(k) ∈ R the state noise, v(k) ∈ R the output noise, θ(k) ∈ Θ the time varying parameter, x ˆ ∈ Rnx the estimated state, L ∈ Rnx xp the observer gain. The control law has the form: u(k) = −F (θ(k))ˆ x(k), where F (θ(k)) ∈ Rmxnx is the feedback gain.
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
The varying parameter θ is fully known and available. This parameter lies in a polytope Θ of vertices θi , i = 1, N : N N X X µi (k) = 1}. (2) µi (k)θi ; µi (k) ≥ 0, θ(k) = { i=1
i=1
For the system with observer the following augmented state representation will be used: xo (k + 1) = Ao (θ(k))xo (k) + Bo (θ(k))n(k), (3) x(k) where: xo (k) = , ǫ(k) = x(k)− x ˆ(k) the estimation ǫ(k) A BF (θ(k)) ,AF =A(θ(k))−BF (θ(k)) error, Ao (θ(k))= F 0 AL Bω 0 AL =A(θ(k))−L(θ(k))C, Bo (θ(k)) = Bω −L(θ(k))Bv T
and n(k) = [ω(k) v(k)] is the disturbance vector.
The command can be written in the following form: u(k) = −Fo (θ(k)) · xo (k) with Fo (θ(k)) = [F (θ(k)) −F (θ(k))].
(4)
It is clear that the parameter-varying matrices range in a polytope of matrices whose vertices are the images of the vertices θ1 , . . . , θN : A(θ(k)), F (θ(k)), L(θ(k)), Ao (θ(k)), Bo (θ(k)), F( θ(k)) = N N X X µi (k) [A(θi ), µi (k) [Ai , Fi , Li , Aoi , Boi , Foi ] = { i=1
i=1
F (θi ),L(θi ),Ao (θi ),Bo (θi ),Fo (θi )] ,µi (k)≥ 0,
N X
µi (k)=1}.
i=1
The disturbance vector is assumed to be bounded: nT n ≤ 1.
(5)
2.2 Design of a stabilizing observer-based LPV control In this section we state the theorem giving an ISS observerbased LPV controller for the LPV system affected by bounded disturbances (5). Lemma 1. (ISS-Lyapunov function) Goulart et al. (2005) The system (3) is ISS for the norm bounded disturbances (5) if there exist K∞ functions α1 (·), α2 (·), α3 (·), a K function δ(·) and a continuous function V : Rn → R, V (0) = (0) such that: α1 (|| x(k) ||) ≤ V (k) ≤ α2 (|| x(k) ||), (6) V (k + 1) − V (k) ≤ −α3 (|| x ||) + δ(|| n ||). (7) Theorem 1. Let the discrete-time LPV system (3) affected by bounded disturbances (5). If there exist Gi = GTi ≻ 0, Pi = PiT ≻ 0, QG , QP , Yi , Zi , i = 1, N , α > 0, β ≥ 0 and γ ≥ 0 such that the following inequalities are satisfied: T QG + QG −Gi 0 αQTG QTG ATi −YiT B T T 0 βI 0 [Bω 0] ≻ 0, (8) αQG 0 αGi 0 Ai QG −BYi [Bω 0] 0 Gj Pi 0 αPi ATi QTP −C T ZiT 0 γI 0 [QP Bω −Zi Bv ]T ≻0, αPi 0 αPi 0 QP Ai −Zi C [QP Bω −Zi Bv ] 0 QTP + QP −Pj (9)
α − β ≥ 0, α − γ ≥ 0, i, j = 1, N (10) then the system is input-to-state stable in the case of bounded norm disturbances (5). The feedback gains and the observer gains that stabilize the system are given by: −1 Fi = Yi Q−1 (11) G , Li = QP Zi , i = 1, N . We can calculate F (θ) and L(θ) as: F (θ(k)) =
N X
µi (k)Fi and L(θ(k)) =
N X
µi (k)Li . (12)
i=1
i=1
Proof. As in (Daafouz and Bernussou, 2001) a parameterdependent Lyapunov function V (k)=xo (k)T Po (θ(k))xo (k), N X T µi (k)Poi , Poi =Poi =G−1 with Po (θ(k))= oi ≻ 0 is considi=1
ered. Let us define A = Ao (θ(k)), B = Bo (θ(k)), P = Po (θ(k)) and P + = Po (θ(k + 1)). Then ∆V (k) = V (k + 1)−V (k)=xo (k+1)T P + xo (k+1)−xo (k)T Pxo (k)= T T + xo (k) A P A−P AT P + B xo (k) = . n(k) n(k) BT P + A BT P + B Condition (6) is satisfied for α1 =ψ || x || (ψ a sufficiently N X λmax (Poi ) || x ||. small positive scalar) and α2 = i=1
Further we prove that if: ∆V (k) < 0, (13) for any xo and n satisfying: xo (k)T Pxo (k) ≥ 1 and n(k)T n(k) ≤ 1 (14) the condition (7) is obtained for the case of norm bounded disturbances (5). For the sequel of the demonstration the following extensions will be made: Bx 0 Bω 0 0 0 , = B →B= v 0 Bǫ 0 0 Bω −L(θ(k))B (15) ω ω n n → n = 1 , n1 = 1 , n2 = 2 , v2 v1 n2 nT1 n1 ≤ 1, nT2 n2 ≤ 1. With these extensions we assume that the system and the observer can be affected by different disturbances. This augmentation does not change our problem and it is a necessary tool for manipulating the inequalities in order to obtain (8) and (9). From the S-procedure (Boyd (2009), Boyd et al. (1994) pp.82-84) we have that (14) implies (13) if there exist α > 0, β ≥ 0 and γ ≥ 0 such that: ∆V (k) − α(1 − xTo Pxo ) − β(nT1 n1 − 1) − γ(nT2 n2 − 1) < 0. (16) For α ≥ (β + γ) (Boyd et al. (1994) pp.82-84), (16) can be written as in (7) with α3 = αϑ || x || (ϑ a sufficiently small positive scalar) and δ = max{β, γ}. By applying Schur complement (16) becomes: P h 0 i αP AT P + βI 0 0 0 BT P + 0 γI ≻ 0, αP 0 αP 0 P +A P +B 0 P+ α − β − γ ≥ 0.
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
”⇒” Following the steps in (Boyd et al. (1994) pp. 117118) we prove first that the inequality (17) has a solution if and only if the inequalities (8), (9) and (10) do. Suppose that the inequalities (17) and (18) have a solution P, α > 0 and β ≥ 0, where: P11 P12 ≻ 0. (19) P= T P12 P22 For P in the form above, we pre- and post-multiply (17) with , respectively, Γ and ΓT , I 0 0 0 0 I 0 0 0 0 0
0 0 0 Γ = 0 0
0 I 0 0 0 0 0 0
0 0 0 0 0 I 0 0
0 0 I 0 0 0 0 0
0 0 0 0 0 0 I 0
0 0 0 I 0 0 0 0
0 0 0 0 . 0 0 0 I
(20)
Now, based on Schur complement it can be easily seen that the inequality (17) implies: + P11 0 αP11 AF T P11 + 0 βI 0 BxT P11 ≻0, (21) αP11 0 αP11 0 + + + P11 P11 AF P11 Bx 0 α − β ≥ 0.
(22)
−1 We pre- and postmultiply (21) with diag(QTG , I, P11 , + −1 −1 + −1 T (P11 ) ) and diag(QG , I, P11 , (P11 ) ). Since (QG − −1 −1 P11 )P11 (QG − P11 ) 0 the following relaxation can −1 T be made: QG P11 QG QTG + QG − P11 . By making the −1 + −1 notations G(θ(k)) = P11 , G(θ(k + 1)) = (P11 ) and Y (θ(k)) = F (θ(k))QG and because the new inequality is
affine in all the variables, condition (8) results. The inequality (17) also implies: + P22 0 αP22 AL T P22 + 0 γI 0 BǫT P22 ≻ 0, αP22 0 αP22 0 + + + P22 P22 AL P22 Bǫ 0
0 0 (8)(θ(k)) − λ 0 BF (θ(k))
|
{z
0 0 0 0
S(θ(k))
0 0 0 0
0 0 ·(9)−1 (θ(k)) · S T (θ(k)) ≻ 0, 0 0
}
α − λβ − γ ≥ 0. (24) These inequalities are satisfied for any positive λ such that: ν > λη and α − γ ≥ λβ where: ν = ς, (ς a sufficiently small positive scalar) N N X X −1 λmax (Si (9) SiT ). Since (8)(θ(k)), (9)(θ(k)) η = i=1 j=1
and (10) are feasible, such λ will always exist.
There exist LPV systems that cannot be stabilized using a fixed observer-based controller independent of the parameter θ, for this reason we have chosen to design a LPV feedback law and a LPV observer gain. S-procedure introduces new variables. Because one of this variable α multiplies Pi , Gi and QG , the inequalities (8) and (9) are now BMIs (bilinear matrix inequalities). Since α is a scalar, an αoptim can be found by executing a simple loop. In order to avoid the loop the PENBMI solver (Koevara and Stingl, 2006) can be used. The results can be extended to parameter dependent input and output matrices, B(θ(k)) and C(θ(k)), by using the relaxations stated in (Sala and Ari˜ no, 2007). 2.3 Invariant ellipsoid calculus An invariant set is a subset of the state space with the property that, if it contains the system state at some time, then it will contain it also in the future (Blanchini and Miani, 2008). For the observer-based controller (11), we search the maximal invariant set (the biggest state region) satisfying input constraints despite the disturbance presence and the LPV mode. Input constraints, disturbances presence and the LPV mode influence the invariant set volume (larger the possible disturbance signal is, the smaller the maximal ellipsoid gets).
Because of their association with powerful tools such as the Lyapunov equation or LMIs, we approximate the real + −1 We pre and postmultiply (23) with diag(I, I, I, QP (P22 ) ). invariant set by an ellipsoidal set: + + −1 + Since (QP − P22 )(P22 ) (QTP − P22 ) 0 the following reE = {xo | xTo Q−1 xo ≤ 1} (25) + −1 T + laxation can be made: QP (P22 ) QP QP +QTP −P22 . By where Q ∈ R(2nx )x(2nx ) is a symmetric positive-definite + , P (θ(k)) = P22 matrix (not necessarily block diagonal). Since we are intermaking the notations P (θ(k + 1)) = P22 and Z(θ(k)) = QP L(θ(k)) and because the new inequality ested more in the initial system state, x(k), we maximize is affine in all the variables, inequality (9) results. the invariant set projection onto x-subspace instead of maximizing the entire set volume. The projection of E ”⇐” Suppose that P (θ(k)), G(θ(k)), QP , QG , Y (θ(k)), (25) onto x is given by: Z(θ(k)), α, β and γ satisfy the inequalities (8), (9) (written −1 Ex = {x | xT (T QT T )−1 x ≤ 1, x = T xo }. (26) in θ(k)) and (10) and define F (θ(k)) = Y (θ(k))QG and −1 L(θ(k)) = QP Z(θ(k)). The objective is to prove that The projection of an ellipsoid on a subspace is an ellipsoid (Blanchini and Miani, 2008). λG(θ(k))−1 0 there exists a λ > 0 such that P = 0 P (θ(k)) The constraints on the input are: satisfies (17) and (18), assuring therefore ISS for (5). || u ||2 ≤ umax . (27) Because β is a variable, we can suppose β → λβ. By con- Theorem 2. Consider an ISS observer-based LPV system gruence of (17) with Γ, and after with diag(QTG , I, G(θ(k)), (3) face of bounded disturbances (5). The maximization of G(θ(k + 1)), I, I, I, QP G(θ(k + 1))) we obtain via Schur Ex subject to input constraint (27) is achieved by solving: complement that the observer-based system is ISS for the min − log det(T QT T ) (28) bounded disturbances (5) if λ > 0 satisfies: α − γ ≥ 0.
(23)
Goi ,Q,Qo ,α,β
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subject to: Qo + QTo − Goi 0 αQTo QTo AToi T 0 βI 0 Boi ≻ 0, αQo 0 αGoi 0 Aoi Qo Boi 0 Goj Goi Q, α − β ≥ 0, α > 0, β ≥ 0, T Qo + QTo − Goi QTo Foi 0, i = 1, N . Foi Qo u2max I
where:
Ao (θ(k))−Be Dq (θ(k))Ce −Be Cq (θ(k)) , Bq (θ(k))Ce Aq (θ(k)) B B −Le (θ(k))Bv −Be Dq (θ(k))Bv ,Be = By (θ(k))= ωe 0 0 Bq (θ(k))Bv xo 0 Bω . ,xy = ,Le (θ(k))= Ce = [0 C] ,Bωe = L(θ(k)) xq Bω Ay (θ(k))=
(29) (30) (31)
Proof. The conditions (17) and (18) guarantee the state invariance for bounded disturbances. By congruence of (17) with diag(QTo , I, P −1 , (P + )−1 ), considering G = P −1 , G + = (P + )−1 and Qo G −1 QTo ≥ Qo + QTo − G an affine inequality is obtained leading to (29). In order to reduce the calculation load one can consider non extended noise signal and α = β > 0 (Boyd et al. (1994) pp.82-84). For F = Foi (θ(k)) we have: || u ||22 = || Fxo ||22 ≤ || FP −1/2 ||22 || P 1/2 xo ||22 = = λmax (FP −1 F T )(xTo Pxo )≤λmax (FP −1 F T ), where λmax is the maximal eigenvalue. Now, by applying Schur complement and by congruence with diag(QTo , I) we obtain that || u ||2 ≤ umax if LMI (31) is feasible. The ellipsoid volume is inversely proportional with the eigenvalues product (that is the determinant). For rendering the problem concave, the “logarithm” operator is used. Because the most MatLab tools can only determine the minimum of a convex problem our optimization criterion becomes min − log det(T QT T ). 3. A YOULA PARAMETER DESIGN FOR A DISTURBED LPV SYSTEM 3.1 System description In order to improve the robustness of the observer basedsystem a Youla parameter is introduced in the closed-loop: xq (k + 1) = Aq (θ(k))xq (k) + Bq (θ(k))˜ y (k), u ˜(k) = Cq (θ(k))xq (k) + Dq (θ(k))˜ y (k), (32) y˜(k) = y(k) − yˆ(k), ˜ ∈ Rm where xq ∈ Rnq the state of Youla parameter, u is its output of Q and y˜ ∈ Rp the input. Fig. 1 gives a block-diagram overview of the structure.
The command is written in the following form: u(k) = −Fx (θ(k)) · xy (k) − Fn (θ(k)) · n(k) (35) with: Fx (θ(k)) = [Fo (θ(k)) + Dq (θ(k))Ce Cq (θ(k))] and Fn (θ(k)) = [0 Dq (θ(k))Bv ]. The augmented system has a polytopic description: N X µi (k) [Ayi , [Ay (θ(k)), By (θ(k), Fx (θ(k)), Fn (θ(k)))] = { i=1
Byi , Fxi , Fni ] , µi (k) ≥ 0,
N X
µi (k) = 1}.
i=1
The disturbance has the same bounds as in (5). 3.2 Youla synthesis for robustness requirements For the Youla LPV system, consider the invariant ellipsoid: Ey = {xy | xTy Q−1 y xy ≤ 1}. The goal here is to synthesize a Youla parameter that maximizes the projection of Ey on the initial state subspace: Exy = {x | xT (ZQy Z T )−1 x ≤ 1, x = Zxy }. The gain in robustness will be given by the difference of volume between the observer-based system ellipsoidal projection and the Youla-based system ellipsoidal projection. The same input constraints as in (27) are considered. Theorem 3. Consider the discrete-time LPV system (34) subject to input constraints (27) and affected by bounded disturbances (5). The design of the Youla parameter that maximizes the projection Exy is achieved by solving: − log det(T QT T )
min
(36)
X, Y, Mi , Ni , Hi , Dqi , α, β, T Q, Gi,11 , Gi,21 = Gi,12 , Gi,22
subject to: LM Ii,j ≻ 0,
(37)
Gi,11 ≻ Q,
System(θ) Observer(θ)
T
(38)
X + X − Gi,11 ⋆ ⋆ I + R − Gi,21 Y + Y T − Gi,22 2 umax I [Foi X + Mi Foi + Dqi Ce ] T 0 Fni i, j = 1, N
⋆ 0, ⋆ I
(39)
and
F (θ) Q(θ)
Fig. 1. State-space controller with Youla parametrization. Now the control law has the form: u(k) = −F (θ(k))ˆ x(k) − u ˜(k).
(33)
For the Youla-based LPV system the following augmented state representation will be used: xy (k + 1) = Ay (θ(k))xy (k) + By (θ(k))n(k) (34)
α − β ≥ 0, α > 0, β ≥ 0, where R = Y T X + V T U, Mi = Dqi Ce X + Cqi U, Ni = −Y T Be Dqi + V T Bqi , Hi = Y T Aoi X − Y T Be Dqi Ce X + V T Bqi Ce X− −Y T Be Cqi U + V T Aqi U, i = 1, N
(40)
(41)
and ⋆ is referring to the symmetrical value. Assuming that the maximization problem has a solution one gets:
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
LM Ii,j
X + X T − Gi,11 ⋆ I + R − Gi,21 Y + Y T − Gi,22 0 X + XT − G ⋆ = i,11 α I + R − Gi,21 Y + Y T − Gi,22
hA
oi X
− Be Mi Aoi − Be Dqi Ce Hi Y Aoi + Ni Ce
i
⋆ βI 0 Bωe −Lei Bv − Be Dqi Bv Y T Bωe −Y T Lei Bv + Ni Bv
i=1
i=1
Cq (θ(k))=
i=1
µi (k)Cqi , Dq (θ(k))=
N X
α
U, V obtained by factorization V T U = R − Y T X, Bqi = V −T (Mi + Y T Be Dqi ), Cqi = (Ni − Dqi Ce X)U −1 , Aqi = V −T (Hi − Y T Aoi X + Y T Be Dqi Ce X− − V T Bqi Ce X + Y T Be Cqi U )U −1 , i = 1, N (42) N N X X µi (k)Bqi , µi (k)Aqi , Bq (θ(k))= and Aq (θ(k))= N X
⋆
µi (k)Dqi .
i=1
Proof. The inequality that verifies the stability of a Youlabased system is obtained by analogy with (29) substituting Aoi by Ayi , Boi by Byi , Goi by Gyi and Qo by Qy with: X U1 Y V1 −1 Qy = , Qy = (43) U • V • Q−1 y , nq xnq
uniquely determined where • are blocks of Qy and . The transformation by X, Y ∈ R2nx x2nx , U , V ∈ R (43) is similar to that used in (Scherer et al., 1997; Luca et al., 2010). The size of the Youla parameter is nq = 2nx .
Pre and post-multiplying this new inequality by diag(ΠT , T T I, Π , Π ) and diag(Π, I, Π, Π) respectively, where Π = I Y and considering the decision variables as in (41), 0 V inequality (37) is obtained. It can be considered α = β > 0. For Fx = Fx (θ(k)), Fn = Fn (θ(k)), Py = (Gy )−1 = Py (θ(k)) = Py−1 (θ(k)) we have: || u ||22 = || Fx xy + Fn n ||22 ≤|| Fx xy ||22 + || Fn n ||22 ≤ ≤ λmax (Fx Py−1 FxT ) + λmax (Fn FnT ). Now, using Schur complement in two steps and after pre and post-multiplying with, respectively, diag(ΠT QTy , I, I) and diag(ΠQy , I, I), LMI (39) is derived. The optimization criterion is obtained by analogy with (28) since x = Zxy = T xo . 3.3 Compromise between robustness and performance The synthesis of a Youla parameter in terms of finding the maximal invariant ellipsoid projection may lead to slow closed-loop performances. An effective way to improve the response transients is to consider Lyapunov functions V (k) = xTy Py xy with a decrease rate bigger than the output norm scaled by γ1 : 1 1 V (k)−V (k + 1)≥ (|| Cx(k) ||2 + || Bv v(k) ||2 )≥ || y(k) ||2 . (44) γ γ
Theorem 4. Consider the discrete-time LPV system (34) with input constraints (27) and affected by bounded disturbances (5). For a given scalar γ ≥ 0, the design of
⋆
⋆ X + X T − Gi,11 ⋆ I + R − Gi,21 Y + Y T − Gi,22
0
⋆ ⋆
h
Gj,11 ⋆ Gj,21 Gj,22
i
the Youla parameter that maximizes the projection Exy guaranteeing (44) is achieved by solving: − log det(T QT T )
min
(45)
X, Y, Mi , Ni , Hi , Dqi , α, β, T Q, Gi,11 , Gi,21 = Gi,12 , Gi,22
subject to:
⋆ ⋆ ⋆ ⋆
⋆ ⋆ ⋆ ⋆
LM Ii,j 0, i, j = 1, N (46) [Cf X Cf ] 0 0 0 γI ⋆ 0 [0 Bv ] 0 0 0 γI (38),(39) and (40), with the notations made in (41) and Cf = [C 0]. Assuming that the maximization problem has a solution, the Youla parameter is obtained as in (42). Proof. Making the notation Cz = [Cf 0] and Bz = [0 Bv ], one has: || Cx(k) ||2 + || Bv v(t) ||2 = T T xy xy Cz 0 Cz 0 . n n 0 Bz 0 BzT For Ay = Ay (θ(k)), By = By (θ(k)) we get that (44) is satisfied if: T + Py 0 αPy AT 1 Cz 0 y Py I 0 0 βI 0 ByT Py+ 0 BzT γ ⋆ 0. − 0 0 αP 1 0 αP 0 y
y
Py+ Ay Py+ By
0
Py+
0
0
0
γ
I
After applying Schur theorem, we pre- and postmultiply this inequality with diag(ΠT QTy , I, ΠT Gy , ΠT Gy+ , I, I) and diag(Qy Π, I, Gy Π, Gy+ Π, I, I). After relaxation and considering the same notations as is in (41), the inequality (46) is obtained. 4. NUMERICAL EXEMPLE Consider" a discrete-time # " with: # disturbed " # LPV system 0 0 0.8 0.5 0 A(θ) = −0.4 1.1θ 0.7 , B = 0.7 , Bω = 0.01 0.7 , 0 1 0 0 1 C= [0 1 0] , Bv =0.01 and θ ∈ [−1 1]. An integral action was considered in order to cancel the steady-state errors. By applying Theorem 1 with α = 0.5 and β = γ = 0.427, we obtain the following ISS observer-based controller: F1 = [−0.011 − 0.8925 0.997], L1 = [1.5813 − 0.1711 2.0989]T , F2 = [−0.0147 1.4035 1.00074] and L2 = [1.5813 2.0283 2.099]T . For this controller and considering the input constraint || u ||2 ≤ 1 we search (via Theorem 2) the invariant ellipsoid having the biggest projection on the initial state subspace (x(k)). This projection is represented by the dark ellipsoid in Fig. 2 and was found for an optimal α = β = 0.0042.
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
By introducing the Youla parameter a new maximal projection is computed. This projection corresponds to the blue ellipsoid in Fig. 2 and was obtained for an optimal α = β = 0.0084. For lack of space we skip the values obtained for the Youla parameter.
Fig. 2. Maximal projection. Considering the performance constraints (44) with γ = 1, the maximal projection is now given by the magenta ellipsoid in Fig. 2. This projection and the associated Youla parameter were obtained for an optimal α = β = 0.0213. In Fig. 3 we have plotted the time-domain representation of the output for a step disturbance (with the amplitude 0.1) in the output signal at time 0.05s and a switch between the two vertices at time 0.1s. By adding the
Fig. 3. Output time-domain simulation. Youla parameter (and so, some degrees of freedom) we obtain an improvement in terms of volume (Fig. 2). The gain is more significant when performance constraints are not considered. On the other hand, the Youla parameter leads to slow closed-loop performances (Fig. 3). These performances can be improved by imposing new bounds on the Lyapunov function decreasing rate. These results were obtained using the software Yalmip (Lofberg, 2004) with the Lmilab solver in MatLab environment (the operator used for “log det” is “geomean”). 5. CONCLUSION Considering a discrete-time LPV system affected by bounded disturbances, this paper proposes the design of an ISS observer-based controller. For this controller we search the maximal projection of the invariant ellipsoid that satisfies input constraints despite the disturbance presence. In the sequel we synthesize a Youla parameter that enlarges the invariant ellipsoidal projection. Because
large closed-loop transients may result, a compromise between performance and robustness is reached by imposing a new bound on the Lyapunov function decreasing rate. REFERENCES Apkarian, P., Gahinet, P., and Becker, G. (1994). Selfscheduled H∞ control of linear parameter-varying systems. American Control Conference, pp. 856-860. Becker, G., Packard, A., Philbrick, D., and Balas, G. (1993). Control of parametrically-dependent linear systems: A single quadratic Lyapunov approach. American Control Conference, pp. 2795-2799. Blanchini, F. and Miani, S. (2008). Set-theoretic methods in control. Systems & Control: Foundations & Applications, Birkhauser, Boston–Basel–Berlin. Boyd, S. (2009). Linear matrix inequalities and the Sprocedure. Course EE363, Lecture 15, Standford. Boyd, S., Ghaoui, L.E., Feron, E., and Balakrishnan, V. (1994). Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia. Daafouz, J. and Bernussou, J. (2001). Parameter dependent Lyapunov functions for discrete time systems with time varying parameters uncertainties. Systems & Control Letters, Vol. 43 (5), pp.355-359. Goulart, P., Kerrigan, E., and Maciejowski, J. (2005). Optimization over state feedback policies for robust control with constraints. Automatica, Vol.42, pp.523–533. Heemels, W., Daafouz, J., and Millerioux, G. (2009). Design of observer-based controllers for LPV systems with unknown parameters. 38th IEEE Conference on Decision and Control. Jiang, Z.P. and Wang, Y. (2001). Input-to-state stability for discrete-time nonlinear systems. Automatica, Vol. 37 (6), pp.857-869. Koevara, M. and Stingl, S. (2006). PENBMI user’s guide. Penopt GbR, version 2.1, available on www.penopt.com. Liberzon, D. (1999). ISS and integral-ISS disturbance attenuation with bounded controls. 38th IEEE conference on Decision and Control, pp.2501-2506. Limon, D., Alamo, T., Raimondo, D., de la Pena, D.M., Bravo, J., and Camacho, E. (2008). Input-to-state stability: an unifying framework for robust model predictive control. Int. Workshop on Assessment and Future Directions of NMPC, Pavia, Italy. Lofberg, J. (2004). YALMIP: A toolbox for modeling and optimization in MATLAB. CACSD Conference, Taiwan. Luca, A., Rodriguez-Ayerbe, P., and Dumur, D. (2010). Youla-Kucera parameter synthesis using invariant sets techniques. American Control Conference, pp.56265631. Sala, A. and Ari˜ no, C. (2007). Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: Applications of Polya’s theorem. Fuzzy Sets and Systems, Vol. 158, pp.2671-2686. Scherer, C., Gahinet, P., and Chilali, M. (1997). Multiobjective output-feedback control via LMI optimization. IEEE Trans. on Auto. Control, Vol.42 (7), pp.896-911. Scorletti, G. and Ghaoui, L.E. (1995). Improved linear matrix inequality conditions for gain scheduling. 34th IEEE Conf. on Decision and Control, pp. 3626-3631. Xie, W. and Eisaka, T. (2004). Design of LPV control systems based on Youla parametrization. IEE Proc. Control Theory Appl., vol. 151 (4), pp.465-472.
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