Y ET A NOTHER H∞ D ISCRETIZATION by
Leonid Mirkin and Gilead Tadmor
T ECHNICAL R EPORT ETR-2002-03 October 2002
."$"%- ./0$'( '&("12' , ()*+#( #,"("$%& !"%- !"#$%&' TECHNION — Israel Institute of Technology, Faculty of Mechanical Engineering
Yet Another H∞ Discretization∗ Leonid Mirkin†
Faculty of Mechanical Eng. Technion — IIT Haifa 32000, Israel
Gilead Tadmor‡
Depart. of Electrical & Computer Eng. Northeastern University Boston, MA 02115, U.S.A.
Abstract A new approach to the conversion of the sampled-data H∞ problem to an equivalent pure discrete-time, lumped-variables counterpart, is presented. The approach is independent of controller causality constraints. In particular, it is applicable to problems, such as preview tracking and fixed-lag smoothing, where existing reduction methods fail. We demonstrate this by solving Open Problem no. 51 from (Blondel et al., 1999).
1 Introduction & Problem Outline This note concerns H∞ control of the sampled-data (SD) system in Fig. 1(a). Direct motivations for the present developments include design problems with relaxed compensator causality constraints, such as preview tracking and finite lag smoothing. As in other SD settings, the system consists of a continuous-time generalized 1 plant and a discrete-time controller connected by A/D (sampler) and D/A (hold) devises. We assume that the plant P is time invariant with the transfer matrix A B 1 B2 P(s) = C1 D11 D12 (1) C2 0 0 (the partitioning is consistent with that in Fig. 1(a)), S h is the ideal sampler, and Hh is the zeroorder hold. The fact that both D21 and D22 in (1) are set to zero reflects the inclusion of an anti-aliasing filter in the generalized plant and guarantees the boundedness of the sampling ¯ so that the H∞ (L2 -induced) operation. The problem is to design the discrete-time controller K norm of the closed-loop mapping from w to z is smaller than a given γ > 0. The book (Chen and Francis, 1995) and references therein provide a detailed discussion of the SD H ∞ control problem. The analysis of SD systems is complicated by the facts that their continuous-time behavior is periodically time-varying and the dynamics are hybrid (continuous/discrete). However, the system can be transformed into an equivalent (in the sense of preserving the induced inputoutput norm) pure discrete-time time-invariant counterpart by use of the so-called lifting transformation, which was adopted to SD systems by (Yamamoto, 1994; Toivonen, 1992; Tadmor, This research was supported by the U.S.-Israel Binational Science Foundation (grant no. 2000167). E-mail:
[email protected] ‡ E-mail:
[email protected] 1 That is, including weighting functions, the antialiasing filter, etc.
∗
†
2 w
-
P
u ¯ Hh u¯ K
z -
y Sh y¯
(a) In the time domain
-
w ˘
u¯
˘
P
¯ K
z˘ y¯
(b) In the lifted domain
Figure 1: General sampled-data setup. 1992; Bamieh and Pearson, 1992). The idea is to regard continuous-time signals as discrete-time sequences taking values in a functional space rather than in R r , for instance in L2 [0, h]. Lifting transforms the system in Fig. 1(a) to the equivalent representation in Fig. 1(b), where w ˘ and z˘ ˘ are lifted versions of w and z, respectively, and P is the (pure discrete-time and time-invariant) lifted plant. The lifted plant realization is of the form ¯ B´ 1 B ¯2 A ˘ ˘ 11 D ` 12 . (2) P(z) = C` 1 D C2 0 0 ¯ indicates an operator whose where we use the following operator nomenclature: a bar — O, ` indicates a finite diinput and output spaces are both finite dimensional; a grave accent — O, 2 ´ indicates a mensional input space and a distributed, L [0, h] output space; an acute accent — O, 2 finite dimensional output space and a distributed, L [0, h] input space; finally, a breve accent — ˘ indicates that both the input and output take distributed, L2 [0, h] values. O, ˘ The detailed expressions for the parameters of P(z) can be found in (Chen and Francis, 1995) and are not essential to follow the discussion. We highlight here some key properties. The “A” component in (2) remains a matrix, reflecting the fact that lifting does not change ` 12 are all state dimension. Additionally, the ranks of the distributed coefficients B´ 1 , C` 1 , and D ˘ finite. In contrast, D11 is an infinite rank operator, a fact that constitutes the main challenge in the analysis of the system in Fig. 1(b). In H ∞ control the problems caused by the presence ˘ 11 can be circumvented by the use of the so-called loop shifting technique of the infinite-rank D of (Safonov et al., 1989). The idea is to transform P˘ by a norm preserving linear fractional transformation so that the resulting generalized plant, say P˘ α , has a zero (1, 1) feed through subblock. Applying to SD systems, loop shifting was first exploited by Bamieh and Pearson ˘ BP ? P˘ with the unitary (1992), who proposed to use the transformation 2 Φ 2I − D ˘ 11 ˘ 11 D ˘ ∗ )1/2 1 − D (γ . 11 ˘ BP = , Φ ˘ 1/2 ˘∗ ˘∗ D D γ (γ2 I − D 11 11 11 ) ˘ 11 . The resulting P˘ α can then be transformed into an equivalent plant, say which eliminates D P¯ e (z), with only finite-dimensional matrix parameters. The entire procedure, which reduces the SD H∞ problem to an equivalent purely discrete-time, finite-dimensional one, is referred to as H∞ discretization. Some modifications of the original transformation were proposed. For example, Hayakawa et al. (1994) proposed to apply an additional loop shifting to guarantee ˘ ˘ BP ), which may that the (2, 2) subblock of P¯ e (z) is equal to that of P(z) (modified by the use of Φ be of value in some applications. 2
Here “?” stands for the star product (Zhou et al., 1995).
3 Available H∞ discretizations require that . ˘ γh = k D 11 kL2 [0,h] < γ.
(3)
¯ is constrained to be causal, since then γ h is a lower bound for This is not a limitation when K ∞ achievable H performance. Yet (3) is conservative, and does become an obstacle when the ¯ is relaxed. Preview tracking control (Kojima and Ishijima, 1999) and causality constraint on K finite-lag smoothing (Mirkin, 2001) are examples, and indeed, outstanding challenges in the SD case (Yamamoto and Hara, 1999). In this paper we propose a new H∞ discretization procedure, based on a norm-preserving ˘ 11 completely, the purpose of this loop shifting transformation. Instead of aiming to remove D ˘ 11 by a finite-rank operator, in a procedure that requires new transformation is to replaces D ˘ 11 is not completely removed is not quite new: an appropriate relaxation of (3). The fact that D ∞ the H discretization of (Hayakawa et al., 1994) also results in a nonzero “D 11 ” block, yet it includes an intermediate step for which (3) is required. Unlike this, our direct transformation ˘ 11 to a finite-rank operator enables us to relax condition (3) to the milder condition of D . ˘ ˘ γˆ h = kD 11 ΠB kL2 [0,h] < γ,
(4)
˘ B is the orthogonal projection of L2 [0, h] on the null space N(B´ 1 ). Moreover, we show where Π that γˆ h is a lower bound on achievable H∞ performance, regardless of causality constraints on ¯ so that (4) causes no loss of generality. Consequently, the proposed H ∞ discretization is apK, ¯ are relaxed or even altogether removed, plicable to problems where causality constraints on K such as the two examples mentioned above: preview tracking and finite-lag smoothing. The paper is organized as follows. In Section 2 the main result is formulated (Theorem 1) and proved. Computational formulae for the proposed H ∞ discretization are provided in §2.1. To illustrate the advantages of the method, Section 3 outlines its use in SD problems arising in digital signal processing. In particular, the first approximation-free solution of Open Problem no. 51 from (Blondel et al., 1999) is given.
2 H∞ Discretization ˘B = Let Π˘ B denote the orthogonal projection onto the null space of B´ 1 , as above. Explicitly, Π ∗ ∗ # ´ ´ ´ ´ I − B1 (B1 B1 ) B1 , where “#” indicates the pseudoinverse. Let ¯ A B¯ 1 B¯ 2 ¯ 11 D ¯ 12 , P¯ e (z) = C¯ 1 D C2 0 0
(5)
be a discrete-time transfer matrix, where B¯ 1 is any matrix satisfying ´ 1 B´ ∗1 , B¯ 1 B¯ 10 = B ¯ 11 = D ¯ aB ¯ 1 , and C¯ 1 , D ¯ a and D ¯ 12 are any matrices satisfying D 0 C¯ 1 ` ∗z I − 12 D ¯a D ¯ 12 = O ` z, ¯ a0 C¯ 1 D ˘ 11 Π˘ B D ˘ ∗11 −1 O D γ 0 ¯ D12
(6a)
(6b)
4 where . ` ˘ ´∗ ´ ´∗ # ` `z = O C1 D11 B1 (B1 B1 ) D12 .
In these terms, the following is the main result of this paper: ˘ K¯ k∞ ≥ γˆ h for every K, ¯ where γˆ h is defined by (4). Furthermore, ∀γ > γˆ h the Theorem 1. kF` P, following statements are equivalent: ˘ K¯ k∞ < γ; i) K¯ internally stabilizes P˘ and kF` P, ii) K¯ internally stabilizes P¯ e and kF` P¯ e , K¯ k∞ < γ.
˘ Proof. The proof goes through several intermediate transformations of P(z), each of which pre∞ serves both closed-loop internal stability and the H norm. First, define ¯ A 0 B´ 1 B¯ 2 ˘ 11 Π˘ B D ˘ 11 (I − Π ˘ B) D ` 12 , P˘ 1 (z) = C` 1 D C2 0 0 0
which is obtained from P˘ through multiplication of its first input by the co-inner operator ˘B I−Π ˘ B . The purpose of this step is to formally partition the exogenous input into a comΠ ponent in Im(I − Π˘ B ), which affects the dynamics of the lifted (discrete-time) state x k = x(kh), and the component in Im(Π˘ B ), which appears only in the feedthrough expression. Obviously, this transformation affects neither stability nor the closed-loop gain. The feedthrough compo˘ 11 Π ˘ B is isolated in P˘ 1 , and is not affected by K, ¯ whereby γˆ h is indeed a lower bound for nent D ∞ achievable H performance. To simplify notations we normalize the system throughout the rest of the proof, so that γ = 1 and γˆ h < 1. Define the operator ˘ 11 Π˘ B ˘ 11 Π ˘ BD ˘ ∗ )1/2 −D 0 (I − D 11 . ˘ 11 Π ˘ B )1/2 0 ˘ BD ˘∗ ˘∗ D Ψ˘ = (I − Π˘ B D Π 11 11 0 I 0
One easily verifies that Ψ˘ is inner. Thus, by the same arguments as in (Bamieh and Pearson, . 1992, Lemma 5), the system P˘ 2 = Ψ˘ ? P˘ 1 is such that ¯ k∞ < 1 kF` P˘ 2 , K kF` P˘ 1 , K¯ k∞ < 1 . ⇐⇒ in stable closed loop in stable closed loop Applying the formulae from (Zhou et al., 1995, §10.4) to Ψ˘ ? P˘ 1 , it can be shown by tedious yet straightforward algebra that ¯ 0 A B´ 1 B¯ 2 ˘D ˘ 11 (I − Π ˘ B ) ∆˘ D ` 12 , P˘ 2 (z) = ∆˘ C` 1 0 ∆ C2 0 0 0 . ˘ 11 Π˘ B D ˘ ∗ )−1/2 . Moreover, since the first component of the external signal (i.e., where ∆˘ = (I − D 11 the component in N(B´ 1 )) has no effect on the outputs of P˘ 2 , it can be removed, along with the zero column in the realization of P˘ 2 , resulting in ¯ ´1 A B B¯ 2 ˘ 11 (I − Π˘ B ) ∆˘ D ` 12 . P˘ 3 (z) = ∆˘ C` 1 ∆˘ D C2 0 0
5 Unlike the procedures in (Bamieh and Pearson, 1992; Hayakawa et al., 1994), the loop shifting by Ψ˘ does not remove the (1, 1) subblock of the feedthrough part of P˘ 3 . Yet the result˘ 11 (I − Π ˘ B ), has finite rank. This follows from the fact that I − Π ˘ B is the oring subblock, ∆˘ D ⊥ ∗ ´ ´ thogonal projection onto N(B1 ) = Im(B1 ), which is obviously finite dimensional. Moreover, ´ ∗ )# B´ 1 , so that ˘ B = B´ ∗ (B´ 1 B I−Π 1 1
˘ ` ˘P3 (z) = ∆Oz 0
¯ A I 0 0 I 0 C2
I 0 I 0 0
B¯ 2 0 0 I 0
B´ 1 0 0 I .
The equalities (6a) and (6b) then lead to the equality of the induced input-output norm of P˘ 3 with that of P¯ e . Remark. It is worth mentioning that the (2, 2) part of the H ∞ discretized plant P¯ e , as defined in ˘ in (2). As noted earlier, this fact may be of (5), coincides with that of the original lifted plant P, value in some applications. For example, in multi-rate control problems it may be important to preserve the block triangular structure of the (2, 2) part of the lifted plant. Whereas the H ∞ discretization of (Bamieh and Pearson, 1992) destroys this structure, the discretization of Theorem 1 does not. In this sense the discretization of Theorem 1 is similar to that in (Hayakawa et al., 1994).
2.1 Computational formulae The result of Theorem 1 is not readily usable and explicit expressions for the parameters of P¯ e , based on (6), are needed. Whereas (6a) is rather standard in the SD literature (see, e.g., (Chen and Francis, 1995)), the right-hand side of (6b) is not, and may appear cumbersome at first look, owing to the presence of the projection operator Π˘ B . The computation of γˆ h , as defined in (4), ˘ B. also relies upon the ability to handle Π As it turns out, however, the operators in (4) and (6b) are identical to those arising in the computation of the frequency response gains of SD systems, in (Mirkin and Palmor, 2002). The formulae of Mirkin and Palmor (2002) can therefore be used to compute γˆ h and the parameters in (5). We include these formulae here, for completeness. To this end we assume that D 11 = 0. A nonzero D11 can be removed by substituting 0 0 −1 A B2 + B1 D11 (γ2 I − D11 D11 ) C1 D12 → A B2 , 0 −1/2 C1 0 D12 → C1 D11 D12 , ) (I − γ12 D11 D11 B1 (I −
1 D 0 D )−1/2 γ2 11 11
→ B1
(the inversions above exist since γˆ h ≥ kD11 k). The result in (Mirkin and Palmor, 2002) also requires that (A, B1 ) be controllable. Here we refer to its extension to the general case. To formulate the result, define the following Hamiltonian matrix: 1 0 C 0 D 0 γ12 D12 B20 D 1 12 2 12 γ A B1 B10 B2 . 0 Hγ = 0 − γ12 C10 C1 −A 0 − γ12 C10 D12 0 0 0 0
6
z
? i
-
6 P yˆ a
Dδh
Hh K ¯ u¯ e
Sh y¯
y F w
Figure 2: Delayed signal reconstruction setup and define the (symplectic) matrix exponential function I Γ12 (t) Γ13 (t) Γ14 (t) 0 Γ22 (t) Γ23 (t) Γ24 (t) . Γ (t) = eHγ t = 0 Γ32 (t) Γ33 (t) Γ34 (t) . 0 0 0 I . To simplify notations, we write hereafter Γ ij instead of Γij (h) and denote Λ = limγ→∞ Γ (note 0 that Λ12 , Λ14 , Λ32 , and Λ34 are all zero matrices and Λ−1 33 = Λ22 ). Lemma 1. Let n1 denote the dimension of the controllable subspace of (A, B 1 ). Then γ > γˆ h iff rank(Γ23 (t)) = n1 , ∀t ∈ (0, h]. Moreover, for every γ > γˆ h M11 M12 M13 ˘ 11 Π ˘ BD ˘ ∗11 −1 O ` z = M 0 M22 M 0 , ` ∗z I − 12 D O 12 32 γ 0 M13 M32 M33 where 0 # 0 M11 = (I − Λ22 Γ33 )Γ23 (Λ22 − Γ22 ) − Λ22 Γ32 , 0 # M12 = (I − Λ22 Γ33 )Γ23 , 0 0 # Γ34 , M13 = (I − Λ22 Γ33 )Γ23 (Λ24 − Γ24 ) − Λ22 # , M22 = Λ33 Λ#23 − Γ33 Γ23 0 # M32 = (Γ13 − Λ13 Λ22 Γ33 )Γ23 , 0 # 0 M33 = (Γ13 − Λ13 Λ22 Γ33 )Γ23 (Λ24 − Γ24 ) + Γ14 − Λ13 Λ22 Γ34 .
¯ = Λ22 , B ¯ 2 = Λ24 , and B¯ 1 = (Λ23 Λ 0 )1/2 . Additionally, A 22 Proof. The proof is essentially the same as for the counterpart result in (Mirkin and Palmor, 2002). The extension to the case of uncontrollable (A, B 1 ) is by an appeal to Kalman’s canonical decomposition of (A, B1 ). The reader is referred to (Fujioka and Mirkin, 2002) for an alternative expression of M ij and for the numerical algorithm for the computation of γˆ h .
3 A Quick’n’Dirty Solution of SD Problems Arising in Digital Signal Processing ¯ arise H∞ problems for the lifted setup, in Fig. 1(b), with relaxed causality constraints on K, naturally in digital signal processing applications, like sampled signal reconstruction (Khargonekar and Yamamoto, 1996) and sampling rate conversion (Ishii et al., 1999). Some delay or
7 latency between measurement and estimation can be tolerated in these applications. They can therefore be thought of as problems with a “controller” of the form z δ K¯ e (z), for some proper ¯ e and a natural δ, called the smoothing lag. K Currently available treatments of such problems are based either on imposing the restrictive assumption (3) or on approximations. Khargonekar and Yamamoto (1996) suggested to use a rational approximation of the continuous-time delay, whereas Ishii et al. (1999) suggested a fast ˘ 11 . These approximations, however, are likely to considerably sampling approximation of D increase the dimension – and computational complexity – of the problem. Theorem 1 yields an attractive alternative. Consider for instance the sampled signal reconstruction problem (Khargonekar and Yamamoto, 1996; Yamamoto and Hara, 1999). The problem is to reconstruct a continuous-time signal y using its sampled values y. ¯ The problem setup is depicted in Fig. 2, where F is a given LTI continuous-time filter, whose frequency ¯ e is the discrete-time estimator to be designed, P a shape reflects a-priori information about y, K is a given LTI continuous-time amplifier, and D δh is the delay (by δh time units) operator in continuous time. The goal is to minimize the H ∞ norm of the operator from w to z = Dδh y − y, ˆ which is the error between the delayed version of y and the estimate y. ˆ The tolerated delay between the measurement and estimation, D δh , is introduced to improve the reconstruction performance. Its presence reflects the availability of previewed information to the estimator. As shown in (Khargonekar and Yamamoto, 1996), the setup in Fig. 2 is equivalent to that in Fig. 1(a) with F −Pa (7) P= F 0 ¯ e (z). We can then calculate γˆ h (which depends only upon F in this case) for this ¯ and K(z) = zδ K system. Furthermore, for every γ > γˆ h and every δ > 0 the verification of whether or not γ is an upper bound for the achievable performance can be done via the following steps: 1. H∞ discretize P given by (7) using Theorem 1 and Lemma 1. 2. Since the (2, 2) part of the discretized discrete-time plant, P¯ e,22 , is kept zero, we have: ¯ e P¯ e,21 ¯ e = P¯ e,11 + P¯ e,12 zδ K F` P¯ e , zδ K ¯ e P¯ e,21 ) = zδ (z−δ P¯ e,11 + P¯ e,12 K = zδ F` P¯ aug , K¯ e ,
where the augmented plant . z−δ P¯ e,11 (z) P¯ e,12 (z) ¯Paug (z) = P¯ e,21 (z) 0
is finite dimensional for every finite smoothing lag δ. 3. Since zδ does not affect the H∞ norm, the problem is reduced to the standard discrete ¯ e such that kF` P¯ aug , K¯ e k∞ < γ. time H∞ problem of finding a causal stabilizing K
Embedded into the standard γ-iteration procedure, the algorithm above yields the complete solution to the sampled-data signal reconstruction problem. Thus, we actually solve Open Problem no. 51 from the book (Blondel et al., 1999), see (Yamamoto and Hara, 1999). Our solution, however, has two severe drawbacks. First, the computational burden (e.g., the dimension of the Riccati equations to be solved) grows as δ gets larger. Second, the structure of the original
8 problem gets lost in the augmented plant, so that the properties of the resulting system are not readily (if at all) seen. We are convinced that these drawbacks are not intrinsic for this kind of problems, taking into account the recent progress in the continuous-time preview tracking and smoothing (Kojima and Ishijima, 1999; Mirkin, 2001). Discrete-time solutions that are not based on the plant augmentation and potentially more elegant direct solutions of SD H ∞ problems with preview are currently under development.
4 Concluding Remarks A new approach to the conversion of sampled-data H ∞ problems to equivalent pure discretetime, lumped-variables counterparts (H ∞ discretization), has been presented. Unlike existing H∞ discretization methods, the proposed procedure is not limited to proper controller design and can therefore be used to treat preview control and estimation problems. We have demonstrated this by solving Open Problem no. 51 from (Blondel et al., 1999).
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