On the H 2 Two-Sided Model Matching Problem with Preview

Transcription

1 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 1, JAN. 1 (TO APPEAR; SUBMITTE AUG. 1, REVISE FEB. 11, MAY 11) 1 On the H Two-Sided Model Matching Problem with Preview Maxim Kristalny and Leonid Mirkin Abstract The H optimization problem with preview and asymptotic behavior constraints is considered in a general two-sided model matching setting. The solution is obtained in terms of two constrained Sylvester equations, associated with asymptotic behavior, and stabilizing solutions of two algebraic Riccati equations. The Riccati equations do not depend on the preview length, yet are affected by asymptotic behavior constraints and are thus different from the standard H Riccati equations arising in problems with no steady-state requirements or in one-sided problems. Index Terms Model matching, preview control, H optimization. I. INTROUCTION AN PROBLEM FORMULATION This note studies the following H optimization problem: OP: Given proper rational transfer matrices G 1, G, G and a constant h, find K H 1 guaranteeing that T e sh G 1 G KG H \ H 1 (1) and minimizing the H -norm of T, i.e., kt k. The constant h represents the length of preview available to K. The systems G i may not be stable due to the use of weights having j!-axis poles. In this setting the input-output stability requirement T H 1 accounts for asymptotic behavior of the system subject to non-decaying input signals. The minimization of the H norm of T accounts then for the transient behavior. One-sided particular cases of OP, corresponding to either G I or G I, are known as the preview tracking and fixed-lag smoothing problems, respectively. These problems are currently well studied in both H [1] [6] and H 1 [7] [9] settings, see also discretetime results in [1] [1]. Many problems of interest, however, have an intrinsic two-sided structure, like in the example addressed in Section IV. This motivates our study of the general two-sided problem with preview, which, to the best of our knowledge, has not been addressed in the literature yet. The main challenge in the required extension stems from the fact that the considered problem formulation combines preview and asymptotic behavior constraints, captured by the stability requirement. Indeed, on the one hand, the two-sided problem with stable data can be solved by a straightforward extension of one-side solutions, like those in [1], [6]. On the other hand, a solution of the twosided problem with asymptotic behavior constraints can be inferred from [1], [14], but is not readily extendible to problems with preview. In this paper we resolve this problem via the recent result on input-output model matching stabilization from [15] (see also [16, Chapter ]). It is shown there that under mild simplifying assumptions all stabilizing solutions of the two-sided model matching problem can be characterized by an affine parameterization in terms of a single stable but otherwise arbitrary parameter. This serves as a starting point for the current study. We show that a nontrivial change of variables in the parameterization of [15] facilitates the required optimization procedure. This enables us to derive an explicit and numerically efficient solution of the considered problem, given in This research was supported by THE ISRAEL SCIENCE FOUNATION (grant No. 18/8). M. Kristalny is with the ept. of Automatic Control, Lund University, Box 118, SE-1 Lund, Sweden. maxim.kristalny@control.lth.se. L. Mirkin is with the Faculty of Mechanical Eng., Technion IIT, Haifa, Israel. mirkin@technion.ac.il. terms of two matrix Sylvester equations and two algebraic Riccati equations. The latter equations differ from the standard H Riccati equations by shifts in their A matrices, which can be considered as an alternative to the concept of semi-stabilizing solutions of Riccati equations from [1]. We consider OP under the following set of assumptions: A 1 : G 1.1/, A : G.1/ and G.1/ have full row and column ranks, respectively, A : G.s/ and G.s/ have no j!-axis zeros, A 4 : G.s/ and G.s/ have no poles in the open right-hand plane. The first assumption is technical and imposes no loss of the generality. Indeed, OP is solvable only if there exists K that renders T H and, in particular, guarantees that T.1/. This, in turn, is possible only if there exists a matrix k such that G 1.1/ G.1/ k G.1/. In this case, the design parameter can be shifted as K KQ C e sh k to yield a problem of designing KQ in which A 1 is satisfied. Assumptions A, are standard and rule out problem redundancy and singularity. The fourth assumption is practically not restrictive, since, typically, unstable poles present in OP originate from unstable weights with imaginary axis instabilities. This assumption rules out the possibility of the coincidence between unstable poles and zeros of G = and facilitates the stabilization procedure, see [16] for more details. The paper is organized as follows. In Section II the rationale behind the proposed solution is described in the frequency domain. In Section III an explicit solution of OP are derived using the statespace machinery. In Section IV an illustrative example is discussed. Finally, some concluding remarks are available in Section V. The main result of this paper is formulated in Theorem in on p. 4. Notation: For any left-invertible A R nm, the matrices A C R mn and A? R.n m/n denote a pseudo inverse of A and its complement satisfying A C A A I and det A C? A.? Likewise, if A is right invertible, A C R mn and A? m.m n/ R satisfy A A C A? I and det A C A?. Given a transfer matrix G.s/, its pseudo inverse is denoted by G #.s/ and its conjugate by G.s/ ŒG. s/. If G L,.G/ C refers to the projection of G onto H. For any rational strictly proper transfer function given by its state-space realization G.s/ C.sI A/ 1 B, the completion operator is defined [8] as h G.s/ C e Ah.sI A/ 1 B e sh C.sI A/ 1 B and is an FIR (finite impulse response) linear system. oubly coprime factorizations for each of the transfer matrices involved in OP are denoted by Xi G i N i Mi 1 QM i 1 QN i ; Y i Mi QY i QN i QM i Q N i X i I I (a) ; (b) for i f1; ; g. The notion of bilateral iophantine equation (BE) over RH 1 refers to an equation of the form MX CYN P, where M; N; P RH 1 are given and X; Y RH 1 are to be found.

2 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 1, JAN. 1 (TO APPEAR; SUBMITTE AUG. 1, REVISE FEB. 11, MAY 11) II. FREQUENCY OMAIN SOLUTION In this section a frequency domain solution of OP is derived. The results presented below are not readily computable and the aim of this section is to spell out the main steps of the solution, which will be implemented by the state-space techniques later on to get the final result. Our first step will be to resolve stability constraints imposed on T and to reduce the problem to an unconstrained L optimization problem. The following result is essentially from [15], [16]: Lemma 1: Let A 1 4 hold. Then there exists K H 1 stabilizing (1) iff QM G 1 M H 1 and there exist U, V, U, and V in H 1 satisfying the following BEs: N U C V QM Q X QM G 1 M Y ; U QN C M V QY QM G 1 M X : If these conditions hold, all K H 1 stabilizing (1) and all corresponding stabilized T s can be characterized as K e sh N K 1 C N K Q N K and T e sh 1 Q ; () where K N1 G N 1 Y Q QM G 1 M Y C M U C U QM M QM X Q QM G 1 M X C V QN C N V N QN and Q H 1 but otherwise arbitrary. Following the ideas from [9], we may assume without loss of the generality that 1, and involved in () satisfy I; N G I; and N G 1 H? (4) (the last expression also implies that 1 is strictly proper). It will be shown in the next section that coprime factorizations of G and G with these properties can alway be constructed. Properties (4) facilitate the derivation of compact formulae for the frequency domain solution and greatly simplify further state-space derivations. Thus, OP can be reformulated as the problem of finding Q H 1, which minimizes the L norm of T given by (). As H 1 is not a Hilbert space, an additional step is required. Note that under assumptions A, there are (rational) pseudo-inverses # ; # L1. Post- and pre-multiplying () by these transfer functions results in Q e sh # N G 1 # # T #. Taking into account that 1.1/, we have that T L iff Q L. Hence, the domain of the optimization parameter can be effectively replaced with Q H. Thus, OP reduces to the following L optimization problem: ; ; Q opt arg min QH ke sh 1 Q k ; (5) which can be solved using standard Hilbert space arguments. By the Projection Theorem [17] the optimal solution can be characterized by the orthogonality of the optimal error to the subspace generated by Q, i.e., by he sh 1 Q opt ; Q i, 8Q H. Because the transfer function of the adjoint operator is the conjugate transfer function and because the inner product over L [18] involves the trace operator, satisfying tr.ab/ tr.ba/, the condition above reads h.e sh 1 Q opt / ; Qi. Using (4), we then obtain that the optimal Q must satisfy e sh G N 1 Q opt H?, which, in turn, yields Q opt e sh G N 1 C : (6) The expression for the minimal norm of the error is then given by kt opt k k 1 k k Q opt k k 1 k kq opt k ; where we made use of the facts that and are co-inner and inner, respectively (cf. (4)). At this point, going back to the original problem, i.e., substituting (6) into the parameterizations from Lemma 1, yields the frequency domain solution of OP. Theorem 1: Let the conditions of Lemma 1 hold and assume that parameterizations () are constructed to satisfy (4). Then OP is solvable and the optimal K with the corresponding minimal kt k are given by K opt e sh N K 1 C N K.e sh N G 1 / C N K ; (7) kt opt k k 1 k k.e sh N G 1 / Ck : (8) Note that, according to (4), the system G N 1 is anti-causal. As a result, we have that.e sh G N 1 / C h f G N 1 g is an FIR system. III. STATE-SPACE SOLUTION Having described the main steps of the solution in the frequency domain, we aim at implementing them using state-space techniques in order to derive computable formulae for the optimal solution and for the achievable performance. To this end, consider the following composite system given by its minimal state-space realization: G1 G G G 4 A B 1 B C 1 C 5 : (9) To simplify the exposition, assume hereafter that A 5 : I and I, which is a matter of scaling and can be assumed without loss of generality, provided that A holds true. Bringing in the canonical decomposition of the A matrix of (9) with respect to the second input and the second output leads to the following form: A A 1 A 1 B 11 B 1 A 1i A B 1 G 6 A B 1 4 C 11 C 1 C 1 C 7 5 ; (1) where the pair.a ; B 1 / is controllable and the pair.a ; C / is observable. Without loss of generality, we may assume that the realization above has an additional structure: Au A A 1 A 1 (11a) A s A A 4 A s A u 5 ; (11b) where A s and A s are Hurwitz, A u and A u are anti-stable, and stand for irrelevant blocks. efine matrices E and E through the equalities A E E A u and E A A u E, i.e., E I and E I with the dimensions and partitioning compatible with those of A and A, respectively. Pick any F s and L s such that A C B 1 F s E and A C E L s C are Hurwitz. To derive the statespace form of Lemma 1, introduce the following pair of constrained Sylvester equations: and A u Z Z.A B 1 C / E.A 1 B 11 C /; (1a) E B 11 C Z B 1? (1b) Z A u.a B 1 C 11/Z.A 1 B 1 C 1/E ; (1a)?.C 1E C C 11 Z / : (1b)

3 ON THE H TWO-SIE MOEL MATCHI PROBLEM WITH PREVIEW, BY KRISTALNY AN MIRKIN K N1 4 A FL L K B F K I C I 5 and G N A G B.F F G / B 1 C L G B.L L G /C A FL.L L G / B 7 C 1 C F G.F F G / 5 (19) C C efine also J E Z and J Z, E where the partitioning corresponds to that of the A matrix in (1). Straightforward algebra yields then that J and J satisfy the following equalities J.A B 1 C / A u J ; J B 1? ; (14).A B C 1/J J A u ;? C 1J : (15) efine L G J L s, F G F s J and finally E.J B 1 C Z E L s / L K 4 5 ; (16a) E L s F K F s E. C 1J C F s E Z /E (16b) (the partitioning corresponds to that in (1)), which satisfy J.L K C B 1 / and.f K C C 1/J : (17) The following result, which is a state-space counterpart of Lemma 1, can now be formulated: Lemma ([15], [16]): Let G be given by its minimal state-space realization (9), having the structure specified by (1) and (11). Then there exists K H 1 that stabilize (1) iff A 1i is Hurwitz and there exist Z and Z satisfying (1) and (1), respectively. If these conditions hold, then all stabilizing K H 1 and all the corresponding stabilized T s can be characterized by (), where Q H 1 but otherwise arbitrary and G N 1 K N1 4 4 A K L K B F K I C I A G B 1 C L G B C 1 C F G C 5 ; (18a) 5 ; (18b) with A G A C B F G C L G C, where L G, F G are as defined in the discussion preceding the lemma, and A K A C B F K C L K C, where L K, F K are as defined in (16). This lemma can be considered as an extension of the state-space solution of the one-sided stabilization problem, see [8, Lemma 5]. One- and two-sided solutions, however, are qualitatively different due to the fact that in the one-sided problem the gain matrix, L, is unconstrained and the freedom in its choice can be exploited to construct a parameterization satisfying (4). In the two-sided case, the gain matrices F K and L K defined by (16) are constrained, which impedes the extension of the one-sided solution method to the twosided case. To overcome this difficulty, our next step is to show that the constraints on the gain matrices in (18) can be partially removed at the expense of increasing the order of the resulting stabilized setup. The following result shows that plugging any stabilizing gain matrices satisfying (17) into (18) will still render K in () a complete parameterization of stabilizing solutions. Lemma : Given that the conditions of Lemma hold, all stabilizing K H 1 and all corresponding stabilized T s can be characterized by (), where Q H 1 but otherwise arbitrary and i and i, i f1; ; g, are given by (19) on top of this page with A G as in Lemma and A FL A C B F C LC, where L, F are any matrices of an appropriate dimensions satisfying J.L C B 1 / and.f C C 1/J () and guarantee that A FL is Hurwitz. Proof: To prove this result, one should consider the parameterizations of Lemma and shift its free parameter as Q! Q C Q p, where A C B F C LC Q p L L G : F F G The rest follows by a straightforward albeit tedious algebra, see [16, Lemma 4.11] for details. Although the choice of the gain matrices F and L in (19) is still restricted by (), it will be shown below that the constraints imposed by these equalities are sufficiently mild to allow the construction of parameterizations satisfying (4). To this end, introduce the following AREs: and.a C L G C / X C X.A C L G C /.XB C C 1 /.B X C C 1/ C C 1 C 1.A C B F G /Y C Y.A C B F G /.YC C B 1 /.C Y C B 1 / C B 1B 1 ; (1a) (1b) which are defined only if Z and Z involved in the definitions of F G and L G are well defined. These Riccati equations are different from the standard H AREs [18, 14.5] associated with problems with internal stability requirements. We, however, do not impose any internal stability requirement, so in our case the pairs.a; C / and.a; B / are not necessarily detectable and stabilizable, respectively. Hence, the standard AREs might not have stabilizing solutions in our case. At the same time, the AREs (1) always have: Claim 1: If OP is stabilizable, AREs (1) have unique stabilizing solutions X and Y. Proof: The proof follows by standard arguments of [18, Corollary 1.1] and using A. The details can be found in [16, Claim 4.1]. We need another result: Claim : If OP is stabilizable, the stabilizing solutions of (1) satisfy XJ and J Y. Proof: We will prove only the X part of the Claim, the proof for Y follows by similar arguments. Pre- and post-multiplying (1a) by J and J, respectively, and using (15), we have: A u J XJ C J XJ A u J XB B XJ C J C L G XJ C J XL GC J : Substituting the definitions of L G, F G and reorganizing the equation above, we get.a u C L s C E / J XJ C J XJ.A u C L s C E / J XB B XJ : By the construction of L s, the matrix A u C L s C E is Hurwitz. Considering the equality above as a Lyapunov equation in J XJ, which is constrained to be positive semi-definite, and taking into account that the right-hand side of the equality is also positive semidefinite, implies that J XJ and, as a result, XJ. Claim implies that the stabilizing solutions of (1) satisfy XL G C XJ L s C and B F G Y B F s J Y, which,

4 4 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 1, JAN. 1 (TO APPEAR; SUBMITTE AUG. 1, REVISE FEB. 11, MAY 11) in turn, implies that they satisfy also the standard H Riccati equations. Note, however, that the stabilizing solutions of the shifted equations are not necessarily stabilizing for the standard AREs. The relation between the two-sided H optimization problem with stability constraints and non-stabilizing solutions of standard AREs was perceived earlier in [14], were a problem without preview was studied and the notion of the semi-stabilizing solution of AREs was introduced. In the current work, the use of shifted Riccati equations is proposed as an alternative to the definition of semi-stabilizing solutions. At this point we can choose the stabilizing gain matrices for (19) as F B X C 1 and L YC B 1 ; () where X and Y are the stabilizing solutions of (1). It can then be shown by direct algebra that with this choice of L and F the matrix A FL is Hurwitz. Note also that the result of Claim together with () imply that.f C C 1/J B XJ and J.L C B 1 / J YC, showing that the choice of F and L in () does satisfy the conditions of Lemma. Finally, it can be shown by straightforward, yet tedious, calculations that for this choice of the gain matrices the conditions in (4) hold and.a C B F /.C 1 C F / C 1 Y XL G N 1 4.A C LC / C 5 : B C 1Y Substituting this realization, as well as (18), into (7) and (8), we obtain the following solution of OP, which is the main result of the paper: Theorem : Let G be given by its minimal state-space realization (9), having the structure specified by (1) and (11). Then OP is solvable iff A 1i is Hurwitz and there exist Z and Z satisfying (1) and (1) respectively. If these conditions hold, then the optimal solution and the corresponding minimal kt k are given by K opt AFL B F I h G N 1 A FL L C I e sh AFL L () F and kt opt k A G B F B 1 C L G 4 L C A FL L 5 C 1 C F G F h G N 1 ; (4) respectively, where A G is as defined in Lemma, X and Y are the stabilizing solutions of AREs (1), A FL A C B F C LC with L and F as defined in (), F F F G, and L L L G. Remark 1: If only one of the factors of K in T is unstable, e.g., if G is stable, the solvability condition associated with the constrained Sylvester equation (1) is not relevant. As a result, Z, J, and F G vanish and (1b) reduces to the standard H Riccati equation. In this case () does not change and (4) is valid with F G NULL. Theorem concludes the work, providing an explicit and numerically feasible solution of OP. An important difference between the one- and two-sided solution is that the latter relies on the modified AREs, with A matrices shifted by terms obtained from the solution of the constrained Sylvester equations (1) and (1). The computation of these terms can be complicated by the fact that (1a) and (1a) may have multiple solutions. It may be shown, however, that, taking into account the constraints (1b) and (1b), Z and Z are unique once they exist and there exists an efficient method for their calculation, see [16] for more details. Fig. 1. W a W w W v W u a w v u y y w y r P 11 s P 1 P 1 1 s P P 11 P 1 s P 1 P 1 Active suspension control 1111 (a) Quartercar model u K (b) Control scheme IV. ILLUSTRATIVE EXAMPLE Consider the active suspension control setup depicted in Fig. 1. A conventional quarter-car model is presented in Fig. 1(a), where the signals y w, y and y r refer to the elevations of the wheel, the vehicle chassis and the road, respectively. The suspension is assumed to contain a controlled actuator that applies force u between the chassis and the wheel. This system is described by the following motion equations: M Ry k.y w y/ C c. Py w y/ C u; e sh m Ry m k.y y w / C c. Py Py w / u C k w.y r y w /: where M 1 kg is a quarter of the chassis mass, m 4 kg is the mass of the wheel with its semi-axis, k w kn m corresponds to a tire spring coefficient and k 18 kn m, c 1 knsec m represent spring and damping coefficients of a passive part of the suspension (these values are taken from [19]). The transfer matrix from.y r ; u/ to.y; y w /, our plant, is then denoted by P. To quantify the suspension performance, introduce a Ry, w y w y r and v y y w. The acceleration of the chassis, a, is associated with the ride comfort and should be kept small. The suspension stroke and tire deflection, w and v, should not be too large in order to meet geometric limitations and to provide safety and road handling. Let us consider a scenario, in which the car is traveling with a constant speed and has to pass a bump. We aim at reducing the maximal chassis acceleration by a factor of two with respect to the performance of a passive system, assuming that the active force is limited by [kn]. To this end, we adopt an open-loop control scheme, depicted in Fig. 1(b), were d refers to the second derivative of the road profile. We assume that the measurement of d, corrupted by a high-frequency noise n, is available to the controller K with a preview of h seconds. We assume also that n contains a harmonic component with the known frequency w n rad/sec, which should be asymptotically rejected. In the simulations below we will assume that the bump, whose form is depicted in Fig. 1(a), has the height of :1 m and the length of :5 m and that the car speed is 1 m/sec. We will assume also that the the amplitude of the harmonic component in the measurement noise corresponds to the measurement error of 4 m/sec. We consider static weights W a, W w, W v and W d as our tuning parameters and choose W n s, which reflects our assumptions s Cwn about n. To penalize the control signal at high frequencies, we choose W u scwu sc1w u, where w u is another tuning parameter, which can be interpreted as a delimiter between the high and the low frequencies. d n W d W n

5 ON THE H TWO-SIE MOEL MATCHI PROBLEM WITH PREVIEW, BY KRISTALNY AN MIRKIN a [m/sec ] 8 u [kn] a [m/sec ] a [m/sec ] (a) Chassis acceleration. 1 (b) Active force. 1 (a) Without noise. 1 (b) With noise Fig.. Passive suspension (dotted line) vs. active suspension without preview (dashed lines) vs. active suspension with : sec preview (solid lines), no measurement noise With these choices, our problem casts as an OP with G1 G G 6 4 W a P 11 W d W a s P 1 1 W w.p s 1 1/W d W w P.P s 11 P 1 /W d W v.p 1 P / 7 W u 5 W v 1 W d The presence of the measurement noise renders the resulting model matching setup not only two-sided but an inherently four-block problem. Choosing the H norm as the performance criteria renders this problem an OP, which can be solved using the result of Theorem. Let us start with considering the controller design that does not use previewed information. The behavior of active suspension obtained by solving OP with h, w u 16, W d 1, W a :4, W w 1 5 and W v 1 5 is presented in Fig. by dashed lines (the behavior of passive system is presented by the dotted line in Fig. (a)). We see that the use of the active element improves the ride quality by damping low frequency oscillation of the chassis. The first peak of acceleration is also reduced, yet, its level still does not meet the design requirement. It is intuitively clear that preview of the road profile can help to reduce the first acceleration peak. Indeed, in this case the design objective can be achieved by introducing a preview of : sec. The behavior of the system obtained with h :, w u 17, W d 1, W a :6, W w 1:71 5, and W v 1:71 5 is presented in Fig. by solid lines. To understand the strategy of the controller, let us focus on the plot of the active force depicted in Fig. (b). It indicates that when the controller receives information about the upcoming bump it tries to retrieve the wheel in order to gain elevation of the chassis in advance. Then, at the moment when the wheel reaches the bump, the controller retracts the wheel in order to reduce the impact. This demonstrates the use of preview to meet the design objective, while keeping the active force within the limits. It might be of interest to compare the results of the two-sided design described above with the results of the one-sided optimization that does not account for the measurement noise. This can be cast as a one-sided setup, in which G 1 keeps only its first column, G is replaced with G W d, and G 1. Consider then the controller obtained by solving this one-sided problem with h :, w u 18, W a :9, W w 1:4 1 5, and W v 1: Fig. (a) indicates that if a noise-free measurement of d is available to the controller, the behaviors of the one- and two-sided designs are similar. This, however, is not the case once the measurements are corrupted with the noise. Fig. (b) shows that the system resulted from the one-sided design is severely affected by the noise while the twosided design is rather resilient. W n Fig.. Two-sided (solid lines) vs. one-sided (dashed lines) design V. CONCLUI REMARKS In this paper we have studied the H two-sided model matching problem, which can be considered as a unified setting for control and estimation problems with preview and asymptotic behavior constraints. First, recent result from [15] have been used to sort out asymptotic behavior constraints and reduce the problem to an unconstrained L optimization. Then, the optimization problem has been solved using standard Hilbert space arguments. The resulting frequency domain solution has been exploited to derive explicit state-space formulae for the optimal solution and for the optimal achievable performance. The final result relies upon two constrained algebraic Sylvester equations, associated with asymptotic behavior, and two algebraic Riccati equations (AREs), which have the same dimension as in the filtering case and do not depend on preview length. Unlike the one-sided case, in the two-sided setting the standard H AREs might have no stabilizing solutions even for solvable problems. We circumvented this problem by considering modified AREs with A matrices shifted by terms obtained from the solutions of the Sylvester equations. This can be considered as an alternative to the concept of a semi-stabilizing solution of ARE proposed in [14]. This alternative is convenient in the context of numerical implementation, since shifted equations can be solved using the standard routines for computing the stabilizing solutions of AREs. An illustrative example demonstrating practical merits of the proposed solution has also been presented. REFERENCES [1] M. 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