Frequency-Weighted Model Reduction with Applications to Structured Models

Transcription

1 27 American Control Conference Frequency-Weighted Model Reduction ith Applications to Structured Models Henrik Sandberg and Richard M. Murray Abstract In this paper, e generalize a recently proposed method for model reduction of linear systems to the frequencyeighted case. The method uses convex optimization and can be used both ith sample data and exact models. We also derive simple a priori bounds on the frequency-eighted error. We combine the method ith a rank-imization heuristic, to approximate multi-input multi-output systems. We also present to applications environment compensation and simplification of interconnected models here e argue the proposed methods are useful. I. INTRODUCTION The frequency-eighted model reduction problem is arguably a more important problem than the uneighted one; at least in the context of closed-loop control systems. For process models it is usually mainly around the cross-over frequency a good model match is needed, for example. For controller reduction problems, frequency eights are also essential, see []. It may seem that eights should not make the problem much more complicated, and that simple extensions of balanced truncation [2] or optimal Hankelnorm approximation [3] should solve the problem. This is only partly true. For example, in [4], an extension of balanced truncation to the frequency-eighted case is proposed. Even though the proposed method often orks ell, it is hard to prove hen it ill ork and one can even find examples here unstable approximations of stable models result. Many more extensions and improvements have been proposed. The books [], [5] contain many of these methods. In [6], [7], a ne approach is taken to solve the model reduction problem. Instead of solving high-dimensional Lyapunov equations, as in balanced truncation and Hankel-norm approximation, a relaxation that makes the problem convex is introduced. The problem can be solved using convex optimization ith frequency-domain data. In [7], a priori error bounds on the approximation error are obtained. The method is flexible and alays delivers stable approximations. In this paper, e sho ho frequency-eights can be added to the procedure, and ho the a priori error bounds are changed. We also combine the method ith a rank-imization heuristic, introduced in [8], to approximate multi-input multi-output MIMO) systems. Furthermore, e present to applications here e argue that frequency-eighted model reduction is useful. The first application deals ith finding lo-complexity updates to This ork as supported by the Hans Werthén foundation and a postdoctoral grant from the Sedish Research Council. H. Sandberg and R.M. Murray are ith California Institute of Technology, Control and Dynamical Systems, M/C 7-8, Pasadena, CA 925, USA. {henriks,murray}@cds.caltech.edu existing feedback controllers. The updates are introduced to compensate for complex environments that disturb the controlled system. The idea of including a model of the environment and to compensate for it has been used in distributed control of vehicle formations, see [9]. Here e deal ith a simple linear frameork. In [9], a more complicated nonlinear problem as addressed. The other application is the problem of reducing the complexity of an interconnected linear system, hilst taking its structure into account. Model reduction of interconnected systems is a problem that has received some attention recently, see, for example [] [2]. This problem is motivated by the many uses of netorked control systems [3]. These systems are often very large, and ho to systematically reduce their size and complexity is often a difficult task. Further evaluation of the proposed applications is a topic for our future ork. The organization of the paper is as follos: In Section II, the frequency-eighted model reduction technique is described along ith an example. In Section III, the application to environment compensation is presented, along ith an example. In Section IV, the application to interconnected linear systems is described. In Section V, some conclusions and suggested future ork are given. Notation H and H denote the sets of stable and anti-stable transfer-function matrices TFMs), respectively. RH are the stable rational TFMs, and R n H the stable rational TFMs of McMillan degree less or equal to n. Similar definitions hold for H. The TFM Gs) belongs to L if the norm G sup σgjω)), ω is finite, here σ is the largest singular value. H denotes the Hankel norm, see [4]. We define G s) G s) T, and Z Z ) are the non-negative) integers, R the real numbers, and C the complex numbers ith j being the imaginary unit and the complex conjugate. II. FREQUENCY-WEIGHTED MODEL REDUCTION The problems e end up solving in this paper are frequency-eighted model reduction problems. There are many methods available for solving such problems, see, for example, [], [4]. Typically these methods use state-space techniques and it is hard to bound their approximation error a priori. Error bounds are important since they can be used to guarantee good approximations. The methods e suggest here do alays preserve stability, comes ith error bounds,

2 and can be used ith frequency data samples and ith exact models. A. Frequency-eighted approximation problem The problem e ould like to solve can be formulated as W o G Ĝ)W i subject to Ĝ R r H, ) Ĝ here G RH is a given TFM together ith frequencydependent eights W i, W o RH and r Z. To the best knoledge of the authors, no polynomial time algorithm is available to solve ), and the suboptimal methods mentioned above are frequently used instead. In this paper, e relax the problem ) and thereby obtain a problem that can be solved ith convex optimization. The first method, presented in Section II-B, only deals ith single-input single-output SISO) models. The second method, presented in Section II-C, can be applied to multiinput multi-output MIMO) models as ell. Ho the methods can be combined is also discussed in Section II-C. B. SISO frequency-eighted approximation It is not knon ho to solve the desired approximation problem ) using convex optimization. In [6], a relaxation technique that makes the uneighted discrete-time problem convex is introduced. We here use an analogous relaxation for the eighted continuous-time problem. Instead of the desired problem ), e suggest to solve the problem a,b,c γ subject to G b a c a ) < γ, here a,, are Huritz polynomials, G RH and SISO, and as) = s r a r s r... a s a bs) = b r s r b r s r... b s b cs) = c r s r c r 2 s r 2... c s c b a R rh,, R d H, c a R rh. Just as in [6], e can re-parameterize the problem. Define here Bs) As) a s)bs) as)cs) a s)as) As) = ) r s 2r A 2r 2 s 2r 2... A 2 s 2 A Bs) = B 2r s 2r B 2r s 2r... B s B. There is a one-to-one correspondence beteen the set of polynomials {As), Bs)} and {as), bs), cs)}. The direction {as), bs), cs)} {As), Bs)} is obvious. The other direction follos if e enforce the condition Ajω) > for all ω. Since Ajω) >, and As) = A s) by construction, e can compute a spectral factor as) of As). It follos that 2) e can choose as) as a Huritz polynomial of degree r. Once as) is detered, e can solve for bs), cs) as the unique solution to the polynomial equation a s)bs) as)cs) = Bs), for instance by constructing a Sylvester matrix from as) and a s) hich are coprime). Hence, instead of solving 2), e can equivalently solve the the quasi-convex optimization problem γ subject to A,B jω) Gjω)Ajω) Bjω)) jω) < γajω), 3) Ajω) > for all ω. 4) The above problem is quasi-convex because for each fixed γ, the constraints 3) 4) are convex in the unknon polynomial coefficients {A k }, {B k }. The approximation accuracy γ can be imized using a bisection algorithm. The constraint 4) can be enforced for all ω ith a linear matrix inequality LMI) using the Kalman-Yakubovich- Popov KYP) lemma [5], or sum-of-squares techniques, see, for example, [6], [7]. Whereas 4) should alays be enforced for all ω to guarantee existence of a stable spectral factor as), 3) can be enforced on a grid {ω k }. This is of interest if Gjω) only is knon on this grid. Enforcing 3) for a high-order Gs) for all ω using the KYP lemma leads to an LMI of high dimension, and may not be possible to solve. It can then be effective to sample Gjω) on a grid. This of course requires that Gjω) does not vary much beteen the samples. In the problem 2), an unstable term c/a is introduced to make the problem convex. This may seem like an odd thing to do, but a similar idea is also used in optimal Hankel-norm approximation, see [3]. The folloing theorem shos that the unstable term c/a can be bounded, and ho stable approximations Ĝ can be chosen. A discrete-time uneighted counterpart is given in [7]. Theorem : Assume that G b a c a ) < γ 5) here G RH and,, a, b, c satisfy the assumptions in 2). i) Define Ĝ = b a R rh. Then Ĝ) G < γ 2r ii) Define Ĝ2 = b a c 2 R rd H here s)cs) s)a s) = c s) a s) c 2s) s) is a stable/anti-stable decomposition. Then Ĝ2) G < γ 2r). ).

3 iii) Assume that γ is the smallest γ such that 5) holds. Then Ĝ) G γ. Ĝ R rh Proof: i) We first give a bound on c/a. This can be done by using an argument from [7] hich is based on Nehari s theorem [4]. We have that G b a c < γ. a Since G b/a) RH and c/a R r H, e have that c H a < γ, and using a standard bound from [4] e have c a < 2rγ. Using this and the triangle inequality in 5) gives the stated bound. ii) Using the stable/anti-stable decomposition, e have that G b a c 2 ) c a < γ. No, use a similar Nehari theorem argument as in i) to obtain c /a < 2rγ notice that the eight does not appear no). The bound in the statement again follos from the triangle inequality. iii) The approximation b/a c/a belongs to a set that is larger than and contains) R r H. Hence, γ a,b,c G b a c ) a Ĝ) G. Ĝ R rh It should be remembered that the bounds derived in Theorem mainly should be seen as a theoretical justification for the method and to help guide in the choice of approximations and eights. We expect the bound in i) to be conservative in general. This is because e used the submultiplicative property of the L -norm to derive it. If the eight attains large and small values, the bound is alays large. Hoever, numerical experiments sho that Ĝ often are good approximations. It may be possible to construct examples here the orst-case bound really is attained. The bound in ii) is more attractive since it only depends on γ and the approximation order r. The price is that Ĝ2 has d more states than Ĝ. It is then important to choose locomplexity eights. A typical lo-complexity eight can be in the form s) s) = kω2 s/α ) 2 s 2 2ζω s ω 2. Here ω is typically chosen to be close to the cross-over frequency of G here good approximation is desired), ith a damping parameter < ζ <. The constant k > deteres ho important it is to have good model matching at lo frequencies. The factor s/α) 2 in the numerator is introduced to make the eight biproper. A biproper eight is a common assumption in eighted model reduction, see [], and is also assumed in Theorem. Often it is reasonable to choose α ω. Then Ĝ2 has to poles in α hose break points are far aay from the cross-over frequency. In this case, it may be reasonable to discard these poles and simply use Ĝ 2 = b a c 2) ) R rh, as approximation if the break points in c 2 s) also are large). The bound iii) is interesting since it shos that the problem e solve actually gives a loer bound on hat can be achieved at all ith any stable model of McMillan degree r. The upper and loer bounds together give us an a priori estimate on ho far aay our approximations Ĝ and Ĝ2 at orst are from an optimal solution. C. MIMO frequency-eighted approximation The MIMO method e suggest here requires that the stable poles {p i } r of Ĝs) are fixed from the start. The poles could be detered by first running the SISO approximation technique in Section II-B on each entry G ij s) of the p m)-dimensional TFM Gs), for example. Once the poles are fixed, the problem is to find the zeros of Ĝ such that the McMillan degree of Ĝ is as small as possible, hile the eighted error W o G Ĝ)W i is also small. To solve this problem, e use a heuristic similar to the one suggested in [8]. Assug that p i are distinct, use the parametrization r Ĝs) = Ĝ Ĝ i, 6) s p i and e shall fix Ĝi C p m, here Ĝ i = Ĝj hen p i = p j. The McMillan degree of 6) is given by r rank Ĝi. Minimization of the rank of a matrix subject to LMIs is knon as a difficult and nonconvex problem. Hoever, there exist simple and effective heuristics, such as the one in [8]. There the trace-class or nuclear) norm of Ĝi, Ĝi = {p,m} k= σ k Ĝi), here σ i are the singular values, is imized instead of the rank. The imization problem e solve is the folloing: Fix a desired approximation accuracy γ >. Then solve Ĝ i r Ĝi subject to W o G Ĝ)W i < γ, 7) Generically p i are distinct. If not, e have to modify the parametrization in 6) slightly.

4 Magnitude db) Frequency rad/sec) Magnitude db) Frequency rad/sec) PSfrag replacements y ref K Ĝ l G P y n Magnitude db) Magnitude db) 2 4 Fig. 2. The environment compensator using the feedback ). 8 2 Frequency rad/sec) Frequency rad/sec) Fig.. The 28th-order model G solid) and the 3th-order approximation Ĝ dashed) from Example. The relative approximation criteria 8) has been used. here Ĝ is given by 6). This is a convex optimization problem. Ho to solve this problem by means of LMIs is shon in [8] for the case hen 7) is enforced on a frequency grid {ω k } and ithout eights. To add eights only require or changes. To enforce 7) for all ω, one can again use the KYP lemma. A problem is that the resulting LMI is often of high dimension. Ho to enforce similar conditions more effectively for all ω is shon in [8]. As γ is decreased to obtain a better approximation, the McMillan degree of Ĝ increases until there no longer is a feasible solution to 7). Hence, there is a trade off beteen approximation accuracy and complexity. An upper bound on the McMillan degree of Ĝ is r {p, m}. If G is SISO, the McMillan degree of Ĝ is alays equal to r. Then e do not need to imize the sum of trace norms. Instead e can simply imize γ as in Section II-B. D. Implementation and an example We implements and solve the above methods using the LMI solver SeDuMi [9] ith YALMIP [2]. We use the method in Section II-B to fix the poles for the method in Section II-C, hich delivers the final approximation. Example : Here e seek to imize the relative error G Ĝ)G. 8) This is a very common criteria in model reduction, see []. We have a model Gs) of McMillan degree 28, shon in Fig., and W o = I and W i = G. We assume knoledge of Gjω) on a 75-point frequency grid in the interval [., 3] rad/s. In the first step of the approximation procedure, described in Section II-B, e approximate each entry of Gs) sepa- rately, Ĝ ij G ij Ĝij)/G ij, i, j =, 2, using r = 2 or r = 4 depending on the entry. This gives us a set of stable poles {p i } to be used in the second step. In the method in Section II-C, e have a trade off beteen the accuracy γ, and the degree of Ĝ. In Table I, this trade off is shon. We choose γ =.34, hich gives a 3th-order approximation. Ĝ is plotted together ith G in Fig., and there is seen to be a good fit over the interval [., 3] rad/s. TABLE I ACCURACY γ AND CORRESPONDING MCMILLAN DEGREE OF Ĝ. γ Degree III. APPLICATION : ENVIRONMENT COMPENSATOR In this section, e study a TFM P s) the plant ) that is interacting ith an environment modeled by Gs): y = P u ) = Gy l, 9) here the output yt) R m is available for feedback control of P and also influences the environment. The control signal is ut) R o. The signal t) R p represents the influence from the environment on the plant, and lt) R p is an additional external disturbance. The environment is assumed to be stable and is modeled by the p m)-dimensional TFM G RH. It influences P through the feedback 9). We assume throughout that the feedback connection of P and G is internally stable, and hence I P G) RH. The environment G may be a TFM of high McMillan degree. The problem e consider here is to find a lo-complexity feedback controller for P s) that compensates for the disturbances that are generated by the environment Gs). The controller ill consist of to parts: One part depends on P s), and is assumed to be fixed. The other part depends on the environment Gs).

5 This problem should be of interest hen a plant is orking in a possibly changing and complex environment. Applications e have in d for this set up include a vehicle P driving in a formation of other vehicles G, compare ith [9]; a generator or subnetork P acting in a larger poer system G; and a cell P producing proteins in an hostile environment G. We ill not discuss these applications further here. Instead, e focus on ho the problem can be formulated and solved using eighted model reduction. In particular, e sho ho the methods proposed in Section II are useful. We use the feedback u = Ky ref y) Ĝy, ) here the TFM K is a ell-tuned controller for P, designed ithout taking the environment G into account. y ref is a reference signal, and Ĝ an approximate model of the environment. The controlled system is shon in Fig. 2. The closed-loop transfer function is y = I P K )) P Ky ref l Ĝ K)n), here = G Ĝ is the environment model error. If =, the response to references, y ref, and to disturbances, l, is the same as hen P is not connected to the environment G. Notice, hoever, that the response to measurement noise, n, depends on Ĝ. One rationale for choosing Ĝ hen eliation of load disturbances l is of interest, is to match the closed-loop transfer functions from l to y. If K has been chosen to fulfill requirements from the unconnected system, e then choose Ĝ so that the error I P K) P I P K P ) P I P K) P I P K) P, is small. Here e have used a first-order Taylor expansion, hich is valid for small P. This leads to the eights W o = W i = I P K) P in the approximation problem ). Of course, the above technique can be generalized to other cases, such as hen reference folloing is the main concern. Example 2: In this example, e assume that the plant is P s) = /s ) 4 and e choose K as a PID-controller Ks) = 2 2.5s s.5s ). A Bode diagram of the environment G is shon in Fig. 3. It is seen to be a highly resonant system ith many poles and zeros close to the imaginary axis. Such systems are generally hard to approximate ith lo-order systems. We use W i = W o = P K) P, hich is also shon in Fig. 3. Furthermore, e assume knoledge of frequency samples on an uniform grid {ω k } = {.2,.2,...,.2}. To obtain Ĝ, e first use the method in Section II-B ith r = 4 and r = 6 to fix the poles of Ĝ. When r = 4 e obtain γ =.24 and hen r = 6 e obtain γ =.4. Magnitude db) Magnitude db) Frequency rad/sec) 2 Frequency rad/sec) Fig. 3. The upper plot shos magnitude data from the environment G solid), a fourth-order approximation Ĝ dashed), and a sixth-order approximation Ĝ dash-dotted) from Example 2. The loer plot shos the eight P 2 /P K) 2 used in the approximation. There is good agreement beteen the models for the relevant frequencies Time Fig. 4. The load step response test in Example 2. In the upper plot, P is just controlled by K. In the noal case solid) it is not connected to the environment G, and in the other case dashed) it is connected to G. The resonant environment G introduces oscillations in the system. In the loer plot, fourth- and sixth-order models Ĝ dashed and solid, respectively) are added to the feedback ), and are seen to almost bring the performance back to noal. Instead of using Ĝ or Ĝ2 as approximations, e extract the poles {p i }, and use them to get improved approximations ith the method in Section II-C. Using the method in Section II-C, e obtain the imum γ.4 ith both r = 4 and r = 6. The resulting Ĝ are shon in Fig. 3. Notice, that since this example is SISO, the McMillan degree of Ĝ is alays equal to r. A load step response test is shon in Fig. 4, ith and ithout the environment model Ĝ in ). As can be seen, adding just a lo-order model Ĝ almost brings the behavior back to noal, even though the environment G is very complex.

6 IV. APPLICATION 2: APPROXIMATION OF INTERCONNECTED LINEAR SYSTEMS In this section, e ill formulate to eighted model reduction problems that are relevant for simplification of interconnected linear systems. We use the model setup from [2]. Consider a collection of n TFMs G i s) that models the subsystems in the interconnected structure. The interconnected linear system is given by bs) = b s). b n s) Gs)as) = as) = Kbs) Hus) ys) = F bs) G s) a s).... G n s) a n s) here u is the external input, y the output, and a, b are interconnection signals. K, H, and F are real constant matrices of appropriate dimensions. K is the connectivity matrix and contains the interconnection structure of G i s). We assume that the interconnected system is internally stable so that I KG) RH. The TFM of the interconnected system is given by ys) = F I Gs)K) Gs)Hus). A. Simplification of subsystem dynamics G i s) We are looking for an approximation Ĝs) ith the same block-diagonal structure as Gs). Using the method in Section II-C, e can seek approximations in the form r Ĝs) = Ĝ Ĝ i, s p i here Ĝi has the same block-diagonal structure as Gs). Such a structure is easily enforced in LMI solvers. To simplify the subsystem dynamics, one can for example solve Ĝ i r Ĝi subject to I KG) KĜ G) < γ, block structuregs)) = block structureĝi), here the eight comes from the small-gain theorem. If γ <, it guarantees that the interconnected system is stable using Ĝ instead of G. Other eights result if e try to match closed-loop transfer functions, as as done in Section III. B. Simplification of interconnection structure K Another interesting problem is to simplify the interconnection structure. One complexity measure of the interconnection structure is the rank of K. If the rank of K is equal to l, then there are only l independent signals that connects the subsystems. If e simplify the system ith the respect to the rank of K, e gain insight about hat signals are most important in the structure. We can use the rank imization heuristic in Section II-C to simplify the interconnection structure hilst maintaining the stability of the interconnected system. For example, using the small-gain theorem, e obtain ˆK subject to ˆK I GK) G ˆK K) < γ. If γ <, e have guaranteed stability. As the approximation error tolerance γ is increased, e obtain an approximation ˆK of loer rank. V. CONCLUSION AND FUTURE WORK In this paper, e have shon ho a recently proposed model-reduction technique [6] can be used together ith frequency eights. Incorporation of eights is often very important hen closed-loop systems are dealt ith. Yet it is not a straightforard thing to do using the traditional methods, see []. We also derived upper and loer error bounds on the eighted error, and they turned out to be simple. The technique can also be combined ith a rankimization heuristic to deal ith MIMO systems. We presented to applications here the suggested techniques are useful. First, there as an application here the problem as to update an existing feedback controller ith a simple compensator for complex environments. We called this environment compensation, and gave a simple example. Second, e shoed ho the methods can be used to simplify linear models hile maintaining their interconnection structure. Future ork ill include more ork on the applications in Sections III and IV. Acknoledgment The authors ould like to thank Maryam Fazel for useful discussions. REFERENCES [] G. Obinata and B. Anderson, Model Reduction for Control System Design. London, UK: Springer-Verlag, 2. [2] B. Moore, Principal component analysis in linear systems: controllability, observability, and model reduction, IEEE Transactions on Automatic Control, vol. 26, no., pp. 7 32, February 98. [3] K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and their L -error bounds, International Journal of Control, vol. 39, pp. 5 93, 984. [4] D. Enns, Model reduction ith balanced realizations: an error bound and a frequency eighted generalization, in Proceedings of the IEEE Conference on Decision and Control, Las Vegas, Nevada, 984. [5] A. C. Antoulas, Approximation of Large-Scale Dynamical Systems. Philadelphia: SIAM, 25. [6] K. Sou, A. Megretski, and L. Daniel, A quasi-convex optimization approach to parametrized model order reduction, in Proceedings of Design Automation Conference, Anaheim, CA, USA, June 25. [7] A. Megretski, H-infinity model reduction ith guaranteed suboptimality bound, in Proceedings of the American Control Conference, Minneapolis, MN, June 26, pp [8] M. Fazel, H. Hindi, and S. Boyd, A rank imization heuristic ith application to imum order system approximation, in Proceedings of the American Control Conference, vol. 6, June 2, pp [9] W. B. Dunbar and R. M. Murray, Receding horizon control of multivehicle formations: A distributed implementation, Proceedings of the 43nd IEEE Conference on Decision and Control, 24.

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