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ROBUST SAMPLED-DATA H∞ CONTROL AND FILTERING
Preprints of the 5th IFAC Symposium on Robust Control Design
ROCOND'06, Toulouse, France, July 5-7, 2006
ROBUST SAMPLED-DATA H∞ CONTROL AND FILTERING V. Suplin, E. Fridman and U. Shaked ∗
∗
School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel, e-mail:[email protected]
Abstract: Sampled-data H∞ control of linear systems is considered. The measured output is sampled and the only restriction on the sampling is that the distance between sequel sampling times does not exceed a given bound. A novel performance index is introduced which takes into account the sampling rates of the measurement and it is thus related to the energy of the measurement noise. A novel structure is adopted for the proposed controllers where the dynamics of the controller is affected by the continuous-time state vector and the sampled value of this vector. A new approach, which was recently introduced to sampled-data stabilization is developed: the system is modeled as a continuous-time one, where the measurement output has a piecewise-continuous delay. A simple solution to the H∞ control problem is derived in terms of Linear Matrix Inequalities (LMIs). Keywords: Sampled-data control, H∞ filtering, LMI, delay model.
1. INTRODUCTION Sampled-data H∞ control of systems has been studied in a number of papers (see e.g. (Bamieh & Pearson, 1992; Sivashankar & Khargonekar, 1994; Basar& Bernard, 1995; Chen & Francis, 1995; Sagfors & Toivonen, 1997; Lall & Dullerud, 2001) and the references therein). Two main approaches have been used. The first one is based on the lifting technique (Bamieh & Pearson, 1992; Yamamoto 1990) in which the problem is transformed to equivalent finite-dimensional discrete H∞ control problem. The second, a more direct approach, is based on the representation of the system in the form of hybrid discrete/continuous model and the solution is obtained in terms of differential Riccati equations with jumps (Sivashankar & Khargonekar, 1994;
Basar& Bernard, 1995). These approaches provide necessary and sufficient conditions and lead to equivalent solutions. The LMI solution to sampled-data output-feedback H∞ control was derived by (Lall & Dullerud, 2001) for the lifted discrete system when the sampling and the hold operators are periodic and their rates are commensurable. This solution is computationally complicated because it includes the evaluation of the matrices of the lifted system. The hybrid system approach has been applied recently to robust H∞ filtering under sampleddata measurements (Xu & Chen, 2003). Sampling interval-independent LMI conditions have been derived there which are quite restrictive.
Preprints of the 5th IFAC Symposium on Robust Control Design
Modelling of continuous-time systems with digital control as continuous systems with delayed control input was introduced by (Mikheev et. al. 1988). The digital control law may be represented as delayed control as follows. u(t) = ud (tk ) = ud (t−(t−tk )) = ud (t−τ (t)), (1) tk ≤ t < tk+1 , τ (t) = t − tk , where ud is a discrete-time control signal and the time-varying delay τ (t) = t − tk is piecewise linear with derivative τ˙ (t) = 1 for t 6= tk . Moreover, τ < tk+1 − tk . Recently, this approach was applied to robust sampled-data stabilization (Fridman et. al., 2004) and to H∞ control (Fridman et. al., 2005). In (Fridman et. al., 2005) a conventional performance index (Sagfors & Toivonen, 1997;Xu & Chen, 2003) is considered, which has no physical meaning in the case of non-uniform sampling. The solution of the output-feedback control problem in (Fridman et. al. 2005) is based on introducing some special filters that precede the sampling of the measurement and the control input and that recover the filtering property of the sample and hold which filters out the high frequency part of the sampled signal. In the present paper we introduce a novel performance index, taking into consideration the energy of the measurement noise. We construct directly the filter by deriving a new Bounded Real Lemma(BRL) for systems with time varying state delay and piecewise constant disturbances. Moreover, an improved stability and BRL conditions for systems with time-varying delay τ , where τ˙ ≤ 1 (for almost all t) are derived. This is achieved by developing the input-output approach (see e.g. Gu et. al., 2003), where stability of systems with constant or slow-varying delays with τ˙ ≤ q < 1 was analyzed, to the BRL and to the fast-varying delay with τ˙ ≤ 1. Notation: Throughout the paper the superscript ‘T ’ stands for matrix transposition, Rn denotes the n dimensional Euclidean space with vector norm | · |, Rn×m is the set of all n × m real matrices, and the notation P > 0, for P ∈ Rn×n means that P is symmetric and positive definite. Let L2 [0, ∞) be the space of the square integrable functions with the norm k · k2 . We also denote xt (θ) = x(t + θ) (θ ∈ [−h, 0]), the vector [aT bT ]T is denoted by col{a, b} and diag{A, B} is a block diagonal matrix with A and B on the diagonal.
ROCOND'06, Toulouse, France, July 5-7, 2006
2. MAIN RESULTS 2.1 Problem Formulation Consider the system x(t) ˙ = A0 x(t) + B1 w(t) + B2 u(t), z(t) = C1 x(t) + D12 u(t),
x(0) = 0
(2)
where x(t) ∈ Rn is the state vector, w(t) ∈ Rq is the disturbance, u(t) ∈ Rp is the control input and z(t) ∈ Rr is the signal to be estimated. The measurement output yk ∈ Rm is assumed to be available at discrete sampling instants: 0 = σ0 < σ1 < ... < σk < ..., lim σk = ∞ k→∞
and it may be corrupted by vk = v(σk ), where v(t) is a measurement noise signal: yk = C2 x(σk ) + D21 vk , k = 0, 1, 2, ... We assume that: A1 σk+1 − σk ≤ h,
(3)
∀k ≥ 0.
We consider the following two problems: i. The output-feedback control problem: We define the following performance index for a prescribed scalar γ > 0: Z∞ Jc (w, v) = [z T (s)z(s) − γ 2 wT (s)w(s)]ds (4) ∞ X 0 −γ 2 (σk+1 − σk )v T (σk )v(σk ) k=0
and we seek a controller of the form x˙ c (t) = Ac0 xc (t)+Ac1 xck +Bc yk , xck = xc (σk ), u(t) = Cc xc (t), σk ≤ t < σk+1 ,
(5)
where xc ∈ Rn , which, for all sampling times satisfying A1 and for a prescribed value of γ, stabilizes the system and leads to Jc < 0, for all nonzero w ∈ L2 [0, ∞) and v(σk ) ∈ l2 [0, ∞). ii. The filtering problem: We define the cost function Z∞
Jf (w, v) = [˜ z T (s)˜ z (s) − γ 2 wT (s)w(s)]ds (6) ∞ X 0 −γ 2 (σk+1 −σk )v T (σk )v(σk ), z˜(t)=z(t)−zf (t), k=0
where we assume that u(t) ≡ 0 in (2), and where we consider the following filter: x˙ f (t) = Af 0 xf (t) + Af 1 xf k + Bf yk , xf k = xf (σk ), zf (t) = Cf xf (t), σk ≤ t < σk+1 .
(7)
This filter should guarantee an estimation error level of γ, namely, Jf < 0 for all non-zero w ∈ L2 [0, ∞), and v(σk ) ∈ l2 [0, ∞).
Preprints of the 5th IFAC Symposium on Robust Control Design
ROCOND'06, Toulouse, France, July 5-7, 2006
Remark 1. In the definition of Jc and Jf the energies of the signals w(t) and v(t) are properly considered. The last summation in this definition is a rectangular approximation of the energy entailed in the measurement noise. In the past, an attempt has been made to solve the sampling problem by taking the sum of the squared L2 norm of w(t) and the squared l2 -norm of {v(σk )} (Sagfors & Toivonen, 1997;Xu & Chen, 2003). Unfortunately, the latter summation does not make physical sense since it does not take the sampling rate into account and, in fact, it puts an increasing weight on the measurement noise the shorter the sampling interval becomes. Remark 2. The above controller (filter) is a generalization of the standard continuous-time n-th order controller (filter).Its dynamics is affected not only by the continuous control state (estimation signal) xc (t) (xf (t)) but also by the sampled values of this signal at the sampling instants σk , k = 1, 2, .... The additional degree of freedom introduced by Ac1 (Af 1 ) should clearly lead to a lower value of γ for which a solution can be found. This fact is demonstrated by the results of the examples below. 2.2 The output delay model
τ (t) = t − σk ,
σk ≤ t < σk+1 .
(8)
¿From A1 it follows that 0 ≤ τ (t) < h and it also d found from (8) that dt τ = 1 over [σk , σk+1 ), for all k ≥ 0. Defining ξ = col{x, xc } we obtain the following augmented model for the output-feedback controller. ˙ ¯ ¯1 v(t−τ (t)), ξ(t)= A¯0 ξ(t)+ A¯1 ξ(t−τ (t))+ Bw(t)+ B ¯ ξ(t) = 0, t ∈ [−h 0], z(t) = Lξ(t), where A B Cc 0 0 ¯ B1 A¯0= 0 2 , A¯1= , B= 0 Ac0 Bc C2 Ac1 0 0 ¯1= ¯ = [C1 D12 Cc ]. B and L Bc D21
In this section we introduce a ’comparison’ system the stability of which implies that system (9) is asymptotically stable (see e.g. Gu et. al., 2003). In order to treat a non-uniform sampled data case, the system (9) is transformed into the following. ˙ = A¯0 ξ(t)+ A¯1 ξ(t−τ (t))+ Bw(t)+ ¯ ¯1 v(t−τ (t)) ξ(t) B t−h/2 Z ˙ ¯ ˆ ¯ ˆ = (A0 + F )ξ(t)+(A1−F )ξ(t−h/2)+ Fˆ ξ(s)ds
(9)
t
t−h/2−η(t) Z
˙ ¯ ¯1 v(t−τ (t)), ξ0 (θ)= 0 +A¯1 ξ(s)ds+ Bw(t)+ B t−h/2
where τ (t) = h/2 + η(t), −h/2 ≤ η(t) < h/2 and Fˆ is an arbitrary matrix. The following inequalities, which derived in (Fridman & Shaked, 2005), will be useful in the sequel: t−h/2−η(t) Z h˙ ˙ ||ξ(t−h/2)||2 ≤ ||ξ(t)||2 , || ξ(s)ds|| 2 ≤ || ξ(t)||2 2 t−h/2 t−h/2 Z
h˙ ˙ || ξ(s)ds|| 2 ≤ || ξ(t)||2 2
We define the new signals ω1 (t), ω2 (t) and ω3 (t) that satisfy the following: Zt Zt T ω1 (s)ω1 (s)ds ≤ ξ T (s)ξ(s)ds
Z0 t 0
0
ω2T (s)ω2 (s)ds ω3T (s)ω3 (s)ds
2
Zt
≤ h /4 2
Z0 t
≤ h /4
y1 (s)T y1 (s)ds
(12)
y2 (s)T y2 (s)ds
0
where ˆ T )ξ(t)+(A¯1 − H ˆ T )ω1 (t) y1 (t) = (A¯0 + H 1 1 ˆ 1T ω2 (t)+ A¯1 ω3 (t)+ Bw(t)+ ¯ ¯1 v(t−τ (t)) +H B ˆ 2T )ξ(t)+(A¯1 − H ˆ 2T )ω1 (t) y2 (t) = (A¯0 + H
(13)
ˆ 2T ω2 (t)+ A¯1 ω3 (t)+ Bw(t)+ ¯ ¯1 v(t−τ (t)), +H B ˆ 1, H ˆ 2 are arbitrary matrices. Defining y0 (t) = and H ξ(t) we obtain the following ’closed-loop’ comparison system where ω1 , ω2 and ω3 are the feedback signals that are the outputs of the feedback operators which act on the measurements y0 , y1 and y2 , respectively. ˙¯ = (A¯ + Fˆ )ξ(t)+( ¯ ξ(t) A¯1 − Fˆ )ω1 (t)+ Fˆ ω2 (t) 0 ¯ ¯ ¯1 v(t−τ (t)), z˜(t) = L ¯ ξ(t). +A¯1 ω3 (t)+ Bw(t)+ B
(10)
(11)
t
Z0 t
We consider the following piecewise-constant measurement y(t−τ (t)) = C2 x(t−τ (t))+D21 v(t−τ (t)),
2.3 New BRL for systems with state and disturbance delays
(14)
The idea behind the latter system is that this system does not incorporate any delay on one hand, but its stability guarantees, by the small
Preprints of the 5th IFAC Symposium on Robust Control Design
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gain theorem, the stability of (9) on the other hand. It is also found that applying to (14) the standard BRL for systems with norm-bounded uncertainty (Boyd et. al. 1994), the upper-bound on the H∞ -norm of (14) will also be an upper bound on the H∞ -norm of the system (9). Here we consider the operator that operate on yi−1 (t) to produce ωi , i = 1, 2, 3 as the norm-bounded uncertainty. We seek conditions for the following inequality to hold for all nonzero ω ∈ L2 [0, ∞) and {vk } ∈ l2 [0, ∞). d ¯ + z˜T (t)˜ V (ξ) z (t) dt 2 T −γ (w (t)w(t) + v T (t − τ (t))v(t − τ (t))) < 0 where
¯ = ξ¯T P ξ, ¯ V (ξ)
2.4 Controller design Consider (16). Define P −1 = Q, denote the partitions: Y NT X MT and P = (17) Q= M
0 N
¯ I, I, I} from the right Multiplying (16) by diag{J, T and by diag{J¯ , I, I, I} from the left we obtain: ¯ T ¯ ¯ ¯ ¯ Φ=
(15)
0 < P ∈ Rn×n .
If the latter inequality holds, then integrating (15) in t from 0 to ∞ and taking into account that v(t−τ (t)) = v(σk ), t ∈ [σk , σk+1 ) the requirement (6) will be satisfied.
Φ11 P A1 −F F P A1 ∗ −S 0 0 ∗ ∗ −R1 0 ∗ ∗ ∗ −R2 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
¯T R2 +H2 A 0 ¯ AT 1 R2 −H2 H2 AT 1 R2 0 −4R2 /h2 ∗ ∗ ∗
¯ PB ¯1 L ¯T PB 0 0 0 0 0 0 0 0 0 ¯ R1 B ¯1 0 R1 B ¯ R2 B ¯1 0 R2 B −γ 2 I 0 0 ∗ −γ 2 I 0 ∗ ∗ −I
A0 R1 +H1 ¯T R1 −H1 A 1 H1 AT 1 R1 −4R1 /h2 ∗ ∗ ∗ ∗
AT 1 0 4 P − ε2 h2
B 0 0 0 B
B1 0 0 0 B1
LT 0 0 0 0
B
B1
0
−γ 2 I 0 0 ∗ −γ 2 I 0 ∗ ∗ −I
< 0,
(18)
where ¯ 11 = A0 + AT0 + F¯ + F¯ T + S, ¯ Φ A0 X +B2 Z A0 0 0 A0 = , A1 = ˆ , Aˆ0 Y A0 A1 U C2 ¯ X ¯ B1 0 X B= , B1 = , P= ¯ , Y B1 U D21 X Y and L = C1 +D12 Z C1 , and where we take R1 = ε1 P and R2 = ε2 P , for positive tuning scalars ε1 and ε2 ,
<0
Φ11 A1 − F F A1 A0 + H1 ¯ ∗ −S¯ 0 0 AT 1 − H1 ¯1 ∗ ∗ −ε1 P 0 H ∗ ∗ ∗ −ε2 P AT 1 4 P ∗ ∗ ∗ ∗ − ε1 h2 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
¯ AT 0 + H2 T ¯2 A1 − H ¯2 H
The condition for (15) to hold is obtained by the following LMI version of the BRL for systems with norm-bounded uncertainties. ¯ ¯T ¯ Φ=
N W
T
and define: I Y J= , J¯ = diag{QJ, QJ, QJ, QJ, QJ, QJ}.
(16)
Φ11 = P A¯0 + A¯T0 P + F + F T + S, ˆ i Ri , i = 1, 2. and where we define F = P Fˆ , Hi = H We thus proved the following. Lemma 3. Consider (9). For a prescribed γ > 0, the cost function (6) achieves Jc (w, v) < 0 for all nonzero w ∈ L2 [0, ∞), v(σk ) ∈ l2 [0, ∞) and for τ (t) = t − σk < h, t ∈ [σk , σk+1 ), if there exist matrices P > 0, S > 0,F, Hi and Ri = RiT , i = 1, 2 that satisfy (16).
Aˆ0 = Y A0 X +Y B2 Z +N T Ac0 M, U = N T Bc , Aˆ1 = U C2 X +N T Ac1 M, F¯ = J T QF QJ ¯ i = J T QHi QJ, i = 1, 2. Z = Cc M, S¯ =J T QSQJ, H For given values of 1 and 2 the latter inequality is ¯ Z, U , F¯i , linear in the decision variables: X, Y , S, ¯ i and Aˆi , i = 1, 2. This leads to the following. H Theorem 4. For a prescribed γ > 0, there exists a controller of the form (5) that achieves Jc (w, v) < 0 for all nonzero w ∈ L2 [0, ∞), v(σk ) ∈ l2 [0, ∞) and for τ (t) = t − σk < h, t ∈ [σk , σk+1 ), if there ¯ i, exist matrices X > 0, Y > 0, S¯ > 0, U, Z, F¯ , H Aˆi , and tuning scalars i > 0, i = 1, 2 that satisfy (18).
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The controller can be calculated using the following equations: Ac0= N −T(Aˆ0 −Y(A0 X+B2 ))ZM −1, Bc=N −TU Ac1=N −T (Aˆ1 −U C2 X))M −1 , Cc=ZM −1 .
(19)
The matrices M and N in (19) are not decision variables in Theorem 4 and (18). The question arise how these matrices can be found and whether they are nonsingular. The definition of M and N is given in (17). It follows from this definition that M T N = I − XY. (20) It also follows from the definition of P, and the fact that this matrix appears on the diagonal of ¯ in (18), that Y − X is nonsingular (see also Φ Gahinet & Apkarian, 1994) and thus that M T N is nonsingular. It also follows from (20) that N can be arbitrarily chosen to be any nonsingular n × n matrix and the corresponding matrix M is then determined. For N = In the corresponding M is In − Y X which leads to the following. Corollary 5. If the LMI condition of Theorem 4 possesses a solution, then the filter of (5) that achieves the estimation performance dictated by γ is given by: Ac0 = (Aˆ0 − Y (A0 X + B2 Z))(I − Y X)−1 Ac1 = (Aˆ1 − U C2 X))(I − Y X)−1 Bc = U, Cc = Z(I − Y X)−1 . In the case where Aˆc1 is chosen a priori to be zero, namely the case where the filter (5) is a standard continuous-time controller, the transfer function matrix of the controller is given by: G(s) = Z(s(I − Y X) − Aˆ0 + Y (A0 X + B2 Z))−1 U. Remark 6. In Theorem 4, two scalar parameters 1 and 2 are sought for which the resulting LMI of (18) possesses a solution. These parameters cannot be obtained via the LMI solver due to the product of these parameters by P. A simple choice for these parameters is 1 = 2 = ˆ, where ˆ is found by a line search. It is shown in the example below that taking an arbitrary value of, say, ˆ = 1 may lead to a significant overdesign. Performing a line search on ˆ leads to a solution that is very close to the result that is obtained by applying fminsearch, the nonlinear optimization routine of Matlab , which performs a 2-dimensional search for the optimal pair of 1 and 2 , where the
resulting minimum achievable γ is used as the objective function of the optimization method. 2.5 Robust filter design The problem of finding an estimator (7) that satisfies Jf < 0, for all non-zero w ∈ L2 [0, ∞), and v(σk ) ∈ l2 [0, ∞), is readily obtained by applying Lemma 3 for B2 = 0 and D12 = −Ir . The matrices Af 0 , Af 1 , Bf and Cf are then the resulting Ac0 , Ac1 , Bc and Cc , respectively, and the estimate of C1 x is given by Cf xf . 3. EXAMPLE We consider here an example of (Sagfors & Toivonen, 1997). Given the system: 0 0 1 w(t), x(t) + x(t) ˙ = 16 −16 −4.8 z(t) = 1 0 x(t), yk = [1 0]x(σk )+0.1vk , k = 0, 1, ... with h = 0.1. Applying Theorem 4, we obtain γmin = 0.9916 for 1 = 2 = ˆ = 1. Performing a line search for ˆ a minimum value of γmin = 0.486 is obtained for ˆ = 23.4992 . The corresponding filter matrices are: −0.9689 −0.6754 22.0689 , Bf = Af 0 = 0.0762 −0.5974 −4.2364 −7.9520 2.5868 Af 1 = , Cf = −8.424 1.992 . 0.6257 −0.2033 Solving this problem for Af 1 = 0, this by manipulating (8)), a minimum value of γ = 0.6552 is obtained. Clearly, here, the existence of the sampling and hold part of the system plays a major role in the estimation performance. The question arises next whether taking 1 6= 2 and applying the fminsearch program will significantly improve the result. Applying the latter optimization routine, it is found that the optimization on the two different scalar parameters hardly improves the estimation performance. An alternative method to the one presented in the present paper is one that still applies the output delay model of (3) but the delay τ (t) is considered to be a fast time-varying delay with unbounded rate of change that lies in the interval [0, h]. The solution in this case which is denoted in (Fridman & Shaked, 2003) as ’case B’ is obtained ¯ 1 = A1 and S¯ = 0 in (18) by taking F¯ = H and deleting the sixth matrix row and column there. Applying this method a minimum value of γ = 0.6864 is obtained. The superiority of the method of Theorem 4 is clearly evident.
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4. CONCLUSIONS A new approach, which was recently introduced to sampled-data H∞ control, is improved and simplified in the case of sampled-data H∞ filtering and output-feedback control. The system is modeled as a continuous-time one, where the measurement output has piecewise-continuous delays. It is assumed that the maximum holding interval is not greater than a given value h > 0. hdependent sufficient conditions are derived for the H∞ estimation of such systems by applying the comparison system approach to time-delay systems. The comparison system is free of the delay but it entails time-varying feedback operators that are norm-bounded. Standard BRL result for such system provide the condition that guarantees the stability of the estimation process and its performance level. In the present paper, a non conventional filter/controller has been suggested that possesses an additional degree of freedom compared to the standard continuous-time filters. A sampled value of the filter states is used in this filter dynamics and the resulting estimate is shown, in the example, to be much improved compared to the application of the standard filters.
REFERENCES Bamieh, B. and J. Pearson (1992), A general framework for linear periodic systems with applications to H∞ sampled-data control. IEEE Trans. on Automat. Contr., 37, 418-435. Basar, T. and P. Bernard (1995). H∞ Optimal Control and Related Minimax Design Problems. A Dynamic Game Approach. Systems and Control: Foundation and Applications. Birkhauser. Boyd, S., L. El Ghaoui, E. Feron and V. Balakrishnan (1994). Linear Matrix Inequality in Systems and Control Theory SIAM. Chen, T. and B. Francis (1995). Optimal SampledData Control Systems. New York: SpringerVerlag. Fridman, E. and U. Shaked (2003). Delaydependent stability and H∞ control: constant and time-varying delays. Int. J of Robust and Nonlinear Control, 1, 48-60. Fridman, E., A. Seuret and J.-P. Richard (2004). Robust sampled-data stabilization of linear sys-
ROCOND'06, Toulouse, France, July 5-7, 2006
tems: an input delay approach. Automatica 40, 1441-1446. Fridman, E. and U. Shaked (2005). Input-output approach to stability and L2 -gain analysis of systems with time-varying delays. Proc. of the 44th Conf. on Decision and Control, Sevilia, Spain. Fridman, E., U. Shaked and V. Suplin (2005). Input/output delay approach to robust sampleddata H∞ control. Systems & Control Letters, 54, pp 271-282. Gahinet, P. and P. Apkarian (1994). A linear matrix inequality approach to H∞ control. Int. J of Robust and Nonlinear Control, 4, 421-448. Gu, K., V. Kharitonov and J. Chen (2003). Stability of time-delay systems. Birkhauser: Boston. Lall, S. and G. Dullerud (2001). An LMI solution to the robust synthesis problem for multi-rate sampled-data systems. Automatica, 37, 19091920. Matlab, The MathWorks Inc., Natick, MA, USA Mikheev, Yu., V. Sobolev and E. Fridman (1988). Asymptotic analysis of digital control systems, (English, Russian original). Autom. Remote Control, 49, 1175-1180. Sagfors, M. S. and H. T. Toivonen (1997). , H∞ and LQG control of asynchronous sampleddata systems. Automatica, 33, 1663-1668. Sivashankar, N. and P. Khargonekar (1994). Characterization of the L2 -induced norm for linear systems with jumps with applications to sampled-data systems. SIAM J. Control and Optimization, 32, 1128-1150. Xu, S. and T. Chen (2003). Robust H∞ filtering for uncertain impulsive stochastic systems under sampled measurements. Automatica, 39, 509-516. Yamamoto, Y. (1990). New approach to sampleddata control systems - a function space method. Proc. 29th Conf. on Decision and Control, Honolulu, HW. 1882-1887.
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ROBUST SAMPLED-DATA H∞ CONTROL AND FILTERING V. Suplin, E. Fridman and U. Shaked ∗
∗
School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel, e-mail:[email protected]
Abstract: Sampled-data H∞ control of linear systems is considered. The measured output is sampled and the only restriction on the sampling is that the distance between sequel sampling times does not exceed a given bound. A novel performance index is introduced which takes into account the sampling rates of the measurement and it is thus related to the energy of the measurement noise. A novel structure is adopted for the proposed controllers where the dynamics of the controller is affected by the continuous-time state vector and the sampled value of this vector. A new approach, which was recently introduced to sampled-data stabilization is developed: the system is modeled as a continuous-time one, where the measurement output has a piecewise-continuous delay. A simple solution to the H∞ control problem is derived in terms of Linear Matrix Inequalities (LMIs). Keywords: Sampled-data control, H∞ filtering, LMI, delay model.
1. INTRODUCTION Sampled-data H∞ control of systems has been studied in a number of papers (see e.g. (Bamieh & Pearson, 1992; Sivashankar & Khargonekar, 1994; Basar& Bernard, 1995; Chen & Francis, 1995; Sagfors & Toivonen, 1997; Lall & Dullerud, 2001) and the references therein). Two main approaches have been used. The first one is based on the lifting technique (Bamieh & Pearson, 1992; Yamamoto 1990) in which the problem is transformed to equivalent finite-dimensional discrete H∞ control problem. The second, a more direct approach, is based on the representation of the system in the form of hybrid discrete/continuous model and the solution is obtained in terms of differential Riccati equations with jumps (Sivashankar & Khargonekar, 1994;
Basar& Bernard, 1995). These approaches provide necessary and sufficient conditions and lead to equivalent solutions. The LMI solution to sampled-data output-feedback H∞ control was derived by (Lall & Dullerud, 2001) for the lifted discrete system when the sampling and the hold operators are periodic and their rates are commensurable. This solution is computationally complicated because it includes the evaluation of the matrices of the lifted system. The hybrid system approach has been applied recently to robust H∞ filtering under sampleddata measurements (Xu & Chen, 2003). Sampling interval-independent LMI conditions have been derived there which are quite restrictive.
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Modelling of continuous-time systems with digital control as continuous systems with delayed control input was introduced by (Mikheev et. al. 1988). The digital control law may be represented as delayed control as follows. u(t) = ud (tk ) = ud (t−(t−tk )) = ud (t−τ (t)), (1) tk ≤ t < tk+1 , τ (t) = t − tk , where ud is a discrete-time control signal and the time-varying delay τ (t) = t − tk is piecewise linear with derivative τ˙ (t) = 1 for t 6= tk . Moreover, τ < tk+1 − tk . Recently, this approach was applied to robust sampled-data stabilization (Fridman et. al., 2004) and to H∞ control (Fridman et. al., 2005). In (Fridman et. al., 2005) a conventional performance index (Sagfors & Toivonen, 1997;Xu & Chen, 2003) is considered, which has no physical meaning in the case of non-uniform sampling. The solution of the output-feedback control problem in (Fridman et. al. 2005) is based on introducing some special filters that precede the sampling of the measurement and the control input and that recover the filtering property of the sample and hold which filters out the high frequency part of the sampled signal. In the present paper we introduce a novel performance index, taking into consideration the energy of the measurement noise. We construct directly the filter by deriving a new Bounded Real Lemma(BRL) for systems with time varying state delay and piecewise constant disturbances. Moreover, an improved stability and BRL conditions for systems with time-varying delay τ , where τ˙ ≤ 1 (for almost all t) are derived. This is achieved by developing the input-output approach (see e.g. Gu et. al., 2003), where stability of systems with constant or slow-varying delays with τ˙ ≤ q < 1 was analyzed, to the BRL and to the fast-varying delay with τ˙ ≤ 1. Notation: Throughout the paper the superscript ‘T ’ stands for matrix transposition, Rn denotes the n dimensional Euclidean space with vector norm | · |, Rn×m is the set of all n × m real matrices, and the notation P > 0, for P ∈ Rn×n means that P is symmetric and positive definite. Let L2 [0, ∞) be the space of the square integrable functions with the norm k · k2 . We also denote xt (θ) = x(t + θ) (θ ∈ [−h, 0]), the vector [aT bT ]T is denoted by col{a, b} and diag{A, B} is a block diagonal matrix with A and B on the diagonal.
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2. MAIN RESULTS 2.1 Problem Formulation Consider the system x(t) ˙ = A0 x(t) + B1 w(t) + B2 u(t), z(t) = C1 x(t) + D12 u(t),
x(0) = 0
(2)
where x(t) ∈ Rn is the state vector, w(t) ∈ Rq is the disturbance, u(t) ∈ Rp is the control input and z(t) ∈ Rr is the signal to be estimated. The measurement output yk ∈ Rm is assumed to be available at discrete sampling instants: 0 = σ0 < σ1 < ... < σk < ..., lim σk = ∞ k→∞
and it may be corrupted by vk = v(σk ), where v(t) is a measurement noise signal: yk = C2 x(σk ) + D21 vk , k = 0, 1, 2, ... We assume that: A1 σk+1 − σk ≤ h,
(3)
∀k ≥ 0.
We consider the following two problems: i. The output-feedback control problem: We define the following performance index for a prescribed scalar γ > 0: Z∞ Jc (w, v) = [z T (s)z(s) − γ 2 wT (s)w(s)]ds (4) ∞ X 0 −γ 2 (σk+1 − σk )v T (σk )v(σk ) k=0
and we seek a controller of the form x˙ c (t) = Ac0 xc (t)+Ac1 xck +Bc yk , xck = xc (σk ), u(t) = Cc xc (t), σk ≤ t < σk+1 ,
(5)
where xc ∈ Rn , which, for all sampling times satisfying A1 and for a prescribed value of γ, stabilizes the system and leads to Jc < 0, for all nonzero w ∈ L2 [0, ∞) and v(σk ) ∈ l2 [0, ∞). ii. The filtering problem: We define the cost function Z∞
Jf (w, v) = [˜ z T (s)˜ z (s) − γ 2 wT (s)w(s)]ds (6) ∞ X 0 −γ 2 (σk+1 −σk )v T (σk )v(σk ), z˜(t)=z(t)−zf (t), k=0
where we assume that u(t) ≡ 0 in (2), and where we consider the following filter: x˙ f (t) = Af 0 xf (t) + Af 1 xf k + Bf yk , xf k = xf (σk ), zf (t) = Cf xf (t), σk ≤ t < σk+1 .
(7)
This filter should guarantee an estimation error level of γ, namely, Jf < 0 for all non-zero w ∈ L2 [0, ∞), and v(σk ) ∈ l2 [0, ∞).
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Remark 1. In the definition of Jc and Jf the energies of the signals w(t) and v(t) are properly considered. The last summation in this definition is a rectangular approximation of the energy entailed in the measurement noise. In the past, an attempt has been made to solve the sampling problem by taking the sum of the squared L2 norm of w(t) and the squared l2 -norm of {v(σk )} (Sagfors & Toivonen, 1997;Xu & Chen, 2003). Unfortunately, the latter summation does not make physical sense since it does not take the sampling rate into account and, in fact, it puts an increasing weight on the measurement noise the shorter the sampling interval becomes. Remark 2. The above controller (filter) is a generalization of the standard continuous-time n-th order controller (filter).Its dynamics is affected not only by the continuous control state (estimation signal) xc (t) (xf (t)) but also by the sampled values of this signal at the sampling instants σk , k = 1, 2, .... The additional degree of freedom introduced by Ac1 (Af 1 ) should clearly lead to a lower value of γ for which a solution can be found. This fact is demonstrated by the results of the examples below. 2.2 The output delay model
τ (t) = t − σk ,
σk ≤ t < σk+1 .
(8)
¿From A1 it follows that 0 ≤ τ (t) < h and it also d found from (8) that dt τ = 1 over [σk , σk+1 ), for all k ≥ 0. Defining ξ = col{x, xc } we obtain the following augmented model for the output-feedback controller. ˙ ¯ ¯1 v(t−τ (t)), ξ(t)= A¯0 ξ(t)+ A¯1 ξ(t−τ (t))+ Bw(t)+ B ¯ ξ(t) = 0, t ∈ [−h 0], z(t) = Lξ(t), where A B Cc 0 0 ¯ B1 A¯0= 0 2 , A¯1= , B= 0 Ac0 Bc C2 Ac1 0 0 ¯1= ¯ = [C1 D12 Cc ]. B and L Bc D21
In this section we introduce a ’comparison’ system the stability of which implies that system (9) is asymptotically stable (see e.g. Gu et. al., 2003). In order to treat a non-uniform sampled data case, the system (9) is transformed into the following. ˙ = A¯0 ξ(t)+ A¯1 ξ(t−τ (t))+ Bw(t)+ ¯ ¯1 v(t−τ (t)) ξ(t) B t−h/2 Z ˙ ¯ ˆ ¯ ˆ = (A0 + F )ξ(t)+(A1−F )ξ(t−h/2)+ Fˆ ξ(s)ds
(9)
t
t−h/2−η(t) Z
˙ ¯ ¯1 v(t−τ (t)), ξ0 (θ)= 0 +A¯1 ξ(s)ds+ Bw(t)+ B t−h/2
where τ (t) = h/2 + η(t), −h/2 ≤ η(t) < h/2 and Fˆ is an arbitrary matrix. The following inequalities, which derived in (Fridman & Shaked, 2005), will be useful in the sequel: t−h/2−η(t) Z h˙ ˙ ||ξ(t−h/2)||2 ≤ ||ξ(t)||2 , || ξ(s)ds|| 2 ≤ || ξ(t)||2 2 t−h/2 t−h/2 Z
h˙ ˙ || ξ(s)ds|| 2 ≤ || ξ(t)||2 2
We define the new signals ω1 (t), ω2 (t) and ω3 (t) that satisfy the following: Zt Zt T ω1 (s)ω1 (s)ds ≤ ξ T (s)ξ(s)ds
Z0 t 0
0
ω2T (s)ω2 (s)ds ω3T (s)ω3 (s)ds
2
Zt
≤ h /4 2
Z0 t
≤ h /4
y1 (s)T y1 (s)ds
(12)
y2 (s)T y2 (s)ds
0
where ˆ T )ξ(t)+(A¯1 − H ˆ T )ω1 (t) y1 (t) = (A¯0 + H 1 1 ˆ 1T ω2 (t)+ A¯1 ω3 (t)+ Bw(t)+ ¯ ¯1 v(t−τ (t)) +H B ˆ 2T )ξ(t)+(A¯1 − H ˆ 2T )ω1 (t) y2 (t) = (A¯0 + H
(13)
ˆ 2T ω2 (t)+ A¯1 ω3 (t)+ Bw(t)+ ¯ ¯1 v(t−τ (t)), +H B ˆ 1, H ˆ 2 are arbitrary matrices. Defining y0 (t) = and H ξ(t) we obtain the following ’closed-loop’ comparison system where ω1 , ω2 and ω3 are the feedback signals that are the outputs of the feedback operators which act on the measurements y0 , y1 and y2 , respectively. ˙¯ = (A¯ + Fˆ )ξ(t)+( ¯ ξ(t) A¯1 − Fˆ )ω1 (t)+ Fˆ ω2 (t) 0 ¯ ¯ ¯1 v(t−τ (t)), z˜(t) = L ¯ ξ(t). +A¯1 ω3 (t)+ Bw(t)+ B
(10)
(11)
t
Z0 t
We consider the following piecewise-constant measurement y(t−τ (t)) = C2 x(t−τ (t))+D21 v(t−τ (t)),
2.3 New BRL for systems with state and disturbance delays
(14)
The idea behind the latter system is that this system does not incorporate any delay on one hand, but its stability guarantees, by the small
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gain theorem, the stability of (9) on the other hand. It is also found that applying to (14) the standard BRL for systems with norm-bounded uncertainty (Boyd et. al. 1994), the upper-bound on the H∞ -norm of (14) will also be an upper bound on the H∞ -norm of the system (9). Here we consider the operator that operate on yi−1 (t) to produce ωi , i = 1, 2, 3 as the norm-bounded uncertainty. We seek conditions for the following inequality to hold for all nonzero ω ∈ L2 [0, ∞) and {vk } ∈ l2 [0, ∞). d ¯ + z˜T (t)˜ V (ξ) z (t) dt 2 T −γ (w (t)w(t) + v T (t − τ (t))v(t − τ (t))) < 0 where
¯ = ξ¯T P ξ, ¯ V (ξ)
2.4 Controller design Consider (16). Define P −1 = Q, denote the partitions: Y NT X MT and P = (17) Q= M
0 N
¯ I, I, I} from the right Multiplying (16) by diag{J, T and by diag{J¯ , I, I, I} from the left we obtain: ¯ T ¯ ¯ ¯ ¯ Φ=
(15)
0 < P ∈ Rn×n .
If the latter inequality holds, then integrating (15) in t from 0 to ∞ and taking into account that v(t−τ (t)) = v(σk ), t ∈ [σk , σk+1 ) the requirement (6) will be satisfied.
Φ11 P A1 −F F P A1 ∗ −S 0 0 ∗ ∗ −R1 0 ∗ ∗ ∗ −R2 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
¯T R2 +H2 A 0 ¯ AT 1 R2 −H2 H2 AT 1 R2 0 −4R2 /h2 ∗ ∗ ∗
¯ PB ¯1 L ¯T PB 0 0 0 0 0 0 0 0 0 ¯ R1 B ¯1 0 R1 B ¯ R2 B ¯1 0 R2 B −γ 2 I 0 0 ∗ −γ 2 I 0 ∗ ∗ −I
A0 R1 +H1 ¯T R1 −H1 A 1 H1 AT 1 R1 −4R1 /h2 ∗ ∗ ∗ ∗
AT 1 0 4 P − ε2 h2
B 0 0 0 B
B1 0 0 0 B1
LT 0 0 0 0
B
B1
0
−γ 2 I 0 0 ∗ −γ 2 I 0 ∗ ∗ −I
< 0,
(18)
where ¯ 11 = A0 + AT0 + F¯ + F¯ T + S, ¯ Φ A0 X +B2 Z A0 0 0 A0 = , A1 = ˆ , Aˆ0 Y A0 A1 U C2 ¯ X ¯ B1 0 X B= , B1 = , P= ¯ , Y B1 U D21 X Y and L = C1 +D12 Z C1 , and where we take R1 = ε1 P and R2 = ε2 P , for positive tuning scalars ε1 and ε2 ,
<0
Φ11 A1 − F F A1 A0 + H1 ¯ ∗ −S¯ 0 0 AT 1 − H1 ¯1 ∗ ∗ −ε1 P 0 H ∗ ∗ ∗ −ε2 P AT 1 4 P ∗ ∗ ∗ ∗ − ε1 h2 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
¯ AT 0 + H2 T ¯2 A1 − H ¯2 H
The condition for (15) to hold is obtained by the following LMI version of the BRL for systems with norm-bounded uncertainties. ¯ ¯T ¯ Φ=
N W
T
and define: I Y J= , J¯ = diag{QJ, QJ, QJ, QJ, QJ, QJ}.
(16)
Φ11 = P A¯0 + A¯T0 P + F + F T + S, ˆ i Ri , i = 1, 2. and where we define F = P Fˆ , Hi = H We thus proved the following. Lemma 3. Consider (9). For a prescribed γ > 0, the cost function (6) achieves Jc (w, v) < 0 for all nonzero w ∈ L2 [0, ∞), v(σk ) ∈ l2 [0, ∞) and for τ (t) = t − σk < h, t ∈ [σk , σk+1 ), if there exist matrices P > 0, S > 0,F, Hi and Ri = RiT , i = 1, 2 that satisfy (16).
Aˆ0 = Y A0 X +Y B2 Z +N T Ac0 M, U = N T Bc , Aˆ1 = U C2 X +N T Ac1 M, F¯ = J T QF QJ ¯ i = J T QHi QJ, i = 1, 2. Z = Cc M, S¯ =J T QSQJ, H For given values of 1 and 2 the latter inequality is ¯ Z, U , F¯i , linear in the decision variables: X, Y , S, ¯ i and Aˆi , i = 1, 2. This leads to the following. H Theorem 4. For a prescribed γ > 0, there exists a controller of the form (5) that achieves Jc (w, v) < 0 for all nonzero w ∈ L2 [0, ∞), v(σk ) ∈ l2 [0, ∞) and for τ (t) = t − σk < h, t ∈ [σk , σk+1 ), if there ¯ i, exist matrices X > 0, Y > 0, S¯ > 0, U, Z, F¯ , H Aˆi , and tuning scalars i > 0, i = 1, 2 that satisfy (18).
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The controller can be calculated using the following equations: Ac0= N −T(Aˆ0 −Y(A0 X+B2 ))ZM −1, Bc=N −TU Ac1=N −T (Aˆ1 −U C2 X))M −1 , Cc=ZM −1 .
(19)
The matrices M and N in (19) are not decision variables in Theorem 4 and (18). The question arise how these matrices can be found and whether they are nonsingular. The definition of M and N is given in (17). It follows from this definition that M T N = I − XY. (20) It also follows from the definition of P, and the fact that this matrix appears on the diagonal of ¯ in (18), that Y − X is nonsingular (see also Φ Gahinet & Apkarian, 1994) and thus that M T N is nonsingular. It also follows from (20) that N can be arbitrarily chosen to be any nonsingular n × n matrix and the corresponding matrix M is then determined. For N = In the corresponding M is In − Y X which leads to the following. Corollary 5. If the LMI condition of Theorem 4 possesses a solution, then the filter of (5) that achieves the estimation performance dictated by γ is given by: Ac0 = (Aˆ0 − Y (A0 X + B2 Z))(I − Y X)−1 Ac1 = (Aˆ1 − U C2 X))(I − Y X)−1 Bc = U, Cc = Z(I − Y X)−1 . In the case where Aˆc1 is chosen a priori to be zero, namely the case where the filter (5) is a standard continuous-time controller, the transfer function matrix of the controller is given by: G(s) = Z(s(I − Y X) − Aˆ0 + Y (A0 X + B2 Z))−1 U. Remark 6. In Theorem 4, two scalar parameters 1 and 2 are sought for which the resulting LMI of (18) possesses a solution. These parameters cannot be obtained via the LMI solver due to the product of these parameters by P. A simple choice for these parameters is 1 = 2 = ˆ, where ˆ is found by a line search. It is shown in the example below that taking an arbitrary value of, say, ˆ = 1 may lead to a significant overdesign. Performing a line search on ˆ leads to a solution that is very close to the result that is obtained by applying fminsearch, the nonlinear optimization routine of Matlab , which performs a 2-dimensional search for the optimal pair of 1 and 2 , where the
resulting minimum achievable γ is used as the objective function of the optimization method. 2.5 Robust filter design The problem of finding an estimator (7) that satisfies Jf < 0, for all non-zero w ∈ L2 [0, ∞), and v(σk ) ∈ l2 [0, ∞), is readily obtained by applying Lemma 3 for B2 = 0 and D12 = −Ir . The matrices Af 0 , Af 1 , Bf and Cf are then the resulting Ac0 , Ac1 , Bc and Cc , respectively, and the estimate of C1 x is given by Cf xf . 3. EXAMPLE We consider here an example of (Sagfors & Toivonen, 1997). Given the system: 0 0 1 w(t), x(t) + x(t) ˙ = 16 −16 −4.8 z(t) = 1 0 x(t), yk = [1 0]x(σk )+0.1vk , k = 0, 1, ... with h = 0.1. Applying Theorem 4, we obtain γmin = 0.9916 for 1 = 2 = ˆ = 1. Performing a line search for ˆ a minimum value of γmin = 0.486 is obtained for ˆ = 23.4992 . The corresponding filter matrices are: −0.9689 −0.6754 22.0689 , Bf = Af 0 = 0.0762 −0.5974 −4.2364 −7.9520 2.5868 Af 1 = , Cf = −8.424 1.992 . 0.6257 −0.2033 Solving this problem for Af 1 = 0, this by manipulating (8)), a minimum value of γ = 0.6552 is obtained. Clearly, here, the existence of the sampling and hold part of the system plays a major role in the estimation performance. The question arises next whether taking 1 6= 2 and applying the fminsearch program will significantly improve the result. Applying the latter optimization routine, it is found that the optimization on the two different scalar parameters hardly improves the estimation performance. An alternative method to the one presented in the present paper is one that still applies the output delay model of (3) but the delay τ (t) is considered to be a fast time-varying delay with unbounded rate of change that lies in the interval [0, h]. The solution in this case which is denoted in (Fridman & Shaked, 2003) as ’case B’ is obtained ¯ 1 = A1 and S¯ = 0 in (18) by taking F¯ = H and deleting the sixth matrix row and column there. Applying this method a minimum value of γ = 0.6864 is obtained. The superiority of the method of Theorem 4 is clearly evident.
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4. CONCLUSIONS A new approach, which was recently introduced to sampled-data H∞ control, is improved and simplified in the case of sampled-data H∞ filtering and output-feedback control. The system is modeled as a continuous-time one, where the measurement output has piecewise-continuous delays. It is assumed that the maximum holding interval is not greater than a given value h > 0. hdependent sufficient conditions are derived for the H∞ estimation of such systems by applying the comparison system approach to time-delay systems. The comparison system is free of the delay but it entails time-varying feedback operators that are norm-bounded. Standard BRL result for such system provide the condition that guarantees the stability of the estimation process and its performance level. In the present paper, a non conventional filter/controller has been suggested that possesses an additional degree of freedom compared to the standard continuous-time filters. A sampled value of the filter states is used in this filter dynamics and the resulting estimate is shown, in the example, to be much improved compared to the application of the standard filters.
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