- Email: [email protected]
On tracking performance limits for tall systems
Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011
On tracking performance limits for tall systems ? Mat´ıas A. Garc´ıa ∗ Eduardo I. Silva ∗ Mario E. Salgado ∗ ∗
Departamento de Electr´ onica, Universidad T´ecnica Federico Santa Mar´ıa, Casilla 110-V, Valpara´ıso, Chile (e-mail: [email protected]).
Abstract: In this paper, a methodology to compute achievable performance bounds for stable, discrete-time tall MIMO systems is proposed. The methodology is based on a performance index, which cumulates the squared deviations of the system output with respect to the nearest vector to a constant reference direction. The results include expressions for the optimal value of this performance index and characterize the controllers that achieve such performance. For SIMO plants, closed form expressions depending on the dynamic features of the plant are also obtained. An example is presented to illustrate the results. Keywords: Linear multivariable systems, Optimal control theory, Analytic design 1. INTRODUCTION The study of performance bounds allows one to build benchmarks against which a given design strategy can be compared to. These bounds are calculated by considering a specific control architecture and a class of controllers and reference (or disturbance) signals. Most results consider a 2-norm based performance index that penalizes the tracking error (see, e.g., Chen et al. (2000); Toker et al. (2002); Chen et al. (2003); Silva and Salgado (2005)). This paper uses a quadratic index to measure performance. Performance bounds for MIMO continuous-time systems with more inputs than outputs, considering step references and one or two degree-of-freedom feedback loops, can be found in Chen et al. (2000). A conclusion of that work is that the best performance is limited not only by nonminimum phase (NMP) zeros, unstable poles and time delays, but also by their directions relative to the reference. The discrete-time counterpart of these results was investigated by Toker et al. (2002), where one degree-offreedom control loops with ramp and sinusoidal references were considered. Silva and Salgado (2005) extended Toker et al. (2002) to the case of MIMO systems with arbitrary delay structure, when step references are considered. Other extensions can be found in Chen et al. (2001); Su et al. (2003); Chen et al. (2003). The latter work shows that, when a quadratic term penalizing the control effort is added to the standard tracking error energy term, not only NMP zeros, unstable poles and delays limit the achievable performance, but also the plant frequency response. This paper focuses on tall systems, i.e., systems with more outputs than inputs. Thus, the conclusions of the above works, which apply to systems with fewer outputs than inputs, are not necessarily valid in our case. ? The authors acknowledge the support received from UTFSM and from CONICYT through grants ACT53, FONDECYT 1100692, and through the Advanced Human Capital Program.
978-3-902661-93-7/11/$20.00 © 2011 IFAC
Tall plants arise every time that fewer control signals than measurement signals are available. Some practical cases include the control of dams (Litrico (2002)), situations involving piezoactuators (Brinkerhoff and Devasia (2000)), magnetic bearing systems (Morse et al. (1998)), etc. One of the first works studying performance limitations for tall plants was carried out by Freudenberg and Middleton (1998). In that work, algebraic design trade-offs for singleinput two-output (SITO) plants are studied using the idea of plant-controller alignment. The same approach was then used by Freudenberg and Middleton (1999) to analyze almost singular two-input and two-output (TITO) systems. An integral (i.e., Bode-like) limitation for SITO systems was obtained by Woodyatt et al. (2001) using a special coordinate transformation that allowed one to focus on those references that the system can track without steady-state error. Such transformation was also used by Woodyatt et al. (2002) to study the cheap control of tall systems. In Chen et al. (2002), the best tracking performance for continuous-time single-input multiple-output (SIMO) plants is studied for reference vectors aligned with the plant DC-gain (i.e., with the matrix transfer function evaluated at zero frequency). In other words, the results by Chen et al. (2002) apply only to step references that can be tracked without steady-state error by the considered SIMO plants. Hara et al. (2007) study performance limits for both continuous and discrete-time SIMO plants, drawing connections between both cases by means of the delta transformation (Middleton and Goodwin (1990)). In that work, both the tracking error energy and the control energy is penalized assumingthe step reference vector is restricted to a limited set. The results by Hara et al. (2007) show, consistent with the work by Chen et al. (2003), that the plant frequency response, and the unstable poles and NMP zeros, impose performance constraints. All work on tall plants reported above considers reference
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signals that can be tracked with no error. However, in practice, it is usually the case that the references cannot be tracked without error and a compromise must be sought. In those situations, the standard 2-norm based performance index must be modified to account for the fact that the tracking error will not necessarily converge to zero. In this paper, we address this issue by penalizing the deviations of the plant output y with respect to its best opt steady-state value y∞ (in a sense to be defined later). More precisely, we consider the cost function ∞ X T opt opt Jυ , y∞ − y[k] y∞ − y[k] , (1) k=0
where the subscript υ indicates the dependence of the cost function respect to the constant reference vector υ . If the plant were right-invertible at zero frequency, or if the reference were aligned with the plant DC-gain in the opt tall plant case, then one could choose y∞ equal to the reference and the cost function given in (1) would reduce to the usual one (see, e.g., Toker et al. (2002)).
The main contribution of this paper is a closed-form expression for the optimal performance, as measured by an averaged version of the cost function given in (1). We also characterize the controllers that achieve such performance. When particularized to the SIMO case, our results show that the optimal performance depends explicitly on the dynamic features of the plant, including the DC-gain and the NMP zeros of the scalar components of it, and also on the plant frequency response. The results reported in this paper exploit techniques used by Chen et al. (2002) and Hara et al. (2007), but consider different assumptions on the reference direction. For SIMO plants, we show that, as intuition suggests, forcing perfect tracking on one output channel is more expensive than allowing for steady-state errors in all outputs. The remainder of the paper is organized as follows: Section 2 introduces notation. Section 3 defines the problem to be studied in the paper. Section 4 presents the main results of the paper. An example is included in Section 5, while in Section 6 conclusions are drawn. 2. NOTATION We use normalface for scalars and boldface for vector and matrices. If x is a complex scalar, then its magnitude is denoted by |x|; (·)T (resp. (·)H ) denotes the transpose (resp. conjugate transpose) of (·); tr {·} stands for trace, † and (·) for Moore-Penrose pseudoinverse. The identity matrix of dimensions n × n is denoted by In . Given a sequence r , {r[k], k = 0, 1, ...}, R[z] denotes its Z√ H transform. For a complex vector x, ||x|| , x x.
We denote by Rn×m (resp. Rn×m ) the set of all real p rational (resp. proper real rational) transfer matrices with n outputs and m inputs. When no confusion arises we omit the n × m superscript. If G[z] ∈ R, then G[z]∼ , T G z −1 . L2 is defined as the Hilbert space of all matrix functions measurable on the unit disc, with the usual inner product (Toker et al. (2002)). The norm induced by this product is known as 2-norm, and it is denoted by ||·||2 . H2 and H2⊥ are subspaces of L2 containing functions
e[k]
r[k]
u[k] C[z]
G[z]
y[k]
+
-
Fig. 1. Standard negative-feedback one degree-of-freedom control loop which are analytic for |z| > 1 and |z| < 1, respectively. Both subspaces are orthogonal and any G[z] ∈ L2 can be written as G[z] = (G[z])H2 + (G[z])H⊥ , where (G[z])H2 2 and (G[z])H⊥ denote the projections of G[z] over H2 and 2 H2⊥ respectively. RH∞ is the class of proper rational stable transfer functions. Given a matrix G[z] ∈ R, we define its poles and zeros as usual (Goodwin et al. (2001)). A zero at z = c is said to be a non-minimum phase (NMP) zero if and only if |c| > 1. (This implies that the zeros at infinity are NMP zeros too.) Any matrix E[z] ∈ R is said to be inner if and only if E[z] ∈ RH∞ and E[z]∼ E[z] = I. Thus, if E[z] is inner, then ||E[z]G[z]||22 = ||G[z]||22 for any G[z] ∈ L2 . A transfer matrix in R is said to be outer if it belongs to RH∞ and has a right inverse which is analytic for |z| > 1. Consider n×m G[z] ∈ RH∞ , n > m, with full normal rank. G[z] can be n×m factorized as G[z] = ΦG [z]ΘG [z], where ΦG [z] ∈ RH∞ m×m is inner and ΘG [z] ∈ RH∞ is outer. This factorization is known as inner-outer factorization (Francis (1987)). Clearly, G[z]∼ G[z] = ΘG [z]∼ ΘG [z]. If, in addition, G[z] has no zeros at z = 1, then G[1]† = ΘG [1]−1 ΦG [1]∼ . Given any transfer matrix G[z] ∈ Rp , controlled via a standard negative-feedback one degree-of-freedom control loop (see Fig. 1), where C[z] is the controller, it is well-known that all sensitivity functions associated with stabilizing controllers can be parameterized using the Youla parameterization (see, e.g., Goodwin et al. (2001)). 3. PROBLEM FORMULATION In this section, we state the control problem to be dealt with in this paper. We focus on the one degree-of-freedom feedback loop of Fig. 1, where G[z] is a tall LTI plant, C[z] is an LTI controller, r the reference signal, u is the control input, y is the plant output, and e is the tracking error. We will work under the following assumptions: Assumption 1. G[z] ∈ Rn×m , with n ≥ m, is strictly p proper, stable, has full normal rank, and has no zeros on the unit circle (i.e., for |z| = 1). 22 Our assumptions are mostly standard, except for the fact that we assume the plant to be stable. The unstable case is left out due to space constraints. In general, a tall plant cannot perfectly track an arbitrary step reference vector υ (unless υ belongs to the space spanned by G[1]). Hence, the standard performance measures which penalize the tracking error energy are not well-defined and have to be modified accordingly (cf. Toker et al. (2002), Silva and Salgado (2005) and Hara et al. (2007)). In this paper, we assume, without loss of generality, that prefect tracking has to be achieved in the first ` output channels (` ∈ {0, · · · , m}). We thus propose to use the cost function defined in (1), where
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opt y∞ ,
arg min y∞ =G[1]u∞ [I` 0](y∞ − υ )=0 u∞ ∈Rm
||y∞ − υ || .
opt The vector y∞ corresponds to the steady-state plant output value that is closest (in an Euclidean sense) to the reference υ , when zero steady-state tracking error is achieved in the first ` output channels. We thus see that the functional Jυ in (1) penalizes the deviations of the plant output with respect to the best possible steady-state opt output vector y∞ .
Partition G[z] as
G[z] =
G11 [z] G12 [z] G21 [z] G22 [z]
(3)
,
where G11 [z] ∈ R`×` , G12 [z] ∈ R`×(m−`) , G21 [z] ∈ opt R(n−`)×` , G22 [z] ∈ R(n−`)×(m−`) . To guarantee that y∞ is well-defined for an arbitrary reference direction υ , we introduce the following additional assumption: Assumption 2. If ` ≥ 1, then G11 [1] is non singular. 22 The problem in (2) is a least square problem with linear restrictions. By solving it, it follows that opt υ, y∞ = Nυ (4) where N , G[1]M, (5) the blocks of M are given by M11 , G11 [1]−1 I` +G12[1]H† G21 [1]G11 [1]−1 −1
M12 , −G11 [1]
G12 [1]H
4. OPTIMAL PERFORMANCE
(2)
This section presents the main results of this paper. We will begin by considering an arbitrary tall plant, and then we will particularize our results to the SIMO case. 4.1 General tall plants The next theorem provides a characterization of the optimal performance J opt and the optimal parameter Qopt [z], when the plant G[z] has arbitrary dimensions: Theorem 1. Consider a plant G[z] satisfying Assumptions 1 and 2. If ` ∈ {0, · · · , m} is given, then: (1) The optimal parameter Qopt [z] defined in (8) is
Qopt [z] = ΘG [z]−1 ΦG [1]∼ N. (9) opt (2) The optimal value of J, i.e., J in (8), is given by n o opt ∼ 0 J = − tr N ΦG [1]ΦG [1]∼ N , (10) 0
where ΦG [1] is the derivative of ΦG [z] respect to z, evaluated at z = 1.
Proof. (1) Pre-multiply the argument of the 2-norm of (7) by the unitary matrix (see, e.g., Chen et al. (2002)) ΦG [z]∼ Λ[z] = ∼ . In − ΦG [z]ΦG [z] Thus,
2 N ∼ J = (In − ΦG [z]ΦG [z] ) z − 1 2 ( ) (ΦG [z]∼ N−ΘG [z]Q[z]) 2 . + inf Q∈S z−1 2 (11) ∼ (ΦG [z]∼ −ΦG [1]∼ )N G [z]Q[z]) Since (ΦG [1] N−Θ ∈ H and 2 z−1 z−1 ∈ H2⊥ , we have that 2 N J opt = (In − ΦG [z]ΦG [z]∼ ) z − 1 2 (ΦG [z]∼ − ΦG [1]∼ ) N 2 (12) + z−1 2 (ΦG [1]∼ N−ΘG[z]Q[z]) 2 . + inf Q∈S z−1
†
opt
M21 , −H† G21 [1]G11 [1]−1
M22 , H† , and
H , G22 [1] − G21 [1]G11 [1]−1 G12 [1]. Using (4), Parseval’s Theorem and the Youla parameterization for stable plants (see, e.g., (Goodwin et al., 2001, Chapter 25)), (1) can be written as υ 2 zυ Jυ = (N − G[z]Q[z]) , (6) z − 1 2
where Q[z] is a parameter to be chosen in RH∞ . It is clear that the (optimal) value of Jυ depends on υ . In order to obtain an average performance measure which is independent of υ , we will assume that υ is an random υ } = 0 and E υ υ T = In . 1 The vector in Rn , with E {υ expected value of Jυ in (6) is thus given by 2 z J , E {Jυ } = (N − G[z]Q[z]) . (7) z − 1
2
Given Assumptions 1 and 2, we conclude from (12) that Qopt [z] is as in (9). (We note that Qopt [z] belongs to S. Indeed, the properties of the inner-outer factorization of G[z] and the definition of N given in (5) yields Q[1] = ΘG [1]−1 ΦG [1]∼ G[1]M = M.)
2
Define the set S , {Q ∈ RH∞ : Q[1] = M} and also J opt , inf J, Q[z]∈S
Qopt [z] , arg inf J.
(8)
Q[z]∈S
We note that Q[z] ∈ S is necessary and sufficient for J to be bounded.
opt
The remainder of this paper focuses in the optimization problem defined in (8). In particular, we will give a closedform expression for J opt and Qopt [z]. 1
E(·) denotes expectation.
(2) From (12) and (9), we conclude that J opt is given by (In − ΦG [z]ΦG [z]∼ ) N 2 opt J = z−1 2 (ΦG [z]∼ − ΦG [1]∼ ) N 2 + z−1 2 Z π ∼ 1 tr {N (In + ΦG [1]ΦG [1]∼ ) N} = dω 4π −π (1 − cos ω)
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− tr N∼ ΦG [1]ΦG [ejω ]∼ N dω (1 − cos ω) −π ∼ Z π − tr N ΦG [ejω ]ΦG [1]∼ N 1 + dω, 4π −π (1 − cos ω) (13) where we used the definition of the 2-norm. Since 2 Re ΦG [ejω ]ΦG [1]∼ = ΦG [1]ΦG [ejω ]∼ 1 + 4π
Z
π
+ ΦG [ejω ]ΦG [1]∼ ,
(13) can be written as Z −1 π − tr {N∼ (In + ΦG [1]ΦG [1]∼ ) N} J opt = dω 4π −π (1 − cos ω) Z π 2 · tr N∼ Re ΦG [ejω ]ΦG [1]∼ N 1 − dω. 4π −π (1 − cos ω) (14) The expression above is similar to the one considered in Lemma 1 in Hara et al. (2007). Define fR [z] , 2 · tr {N∼ (Re {ΦG [z]ΦG [1]∼ }) N} ,
fR1 , tr {N∼ (In + ΦG [1]ΦG [1]∼ ) N} . If fR [1] = fR1 , then we could use the Lemma 1 in Hara et al. (2007). Since fR [1] = 2 · tr {N∼ (Re {ΦG [1]ΦG [1]∼ }) N} (15) = 2 · tr {N∼ (ΦG [1]ΦG [1]∼ ) N} , fR [1] would be equal to fR1 if tr {N∼ N} = tr {N∼ (ΦG [1]ΦG [1]∼ ) N}. To show that this is indeed the case, we use the definition of N given in (5): tr {N∼ ΦG [1]ΦG [1]∼ N} = tr {M∼ G[1]∼ ΦG [1]ΦG [1]∼ G[1]M} = tr {M∼ ΘG [1]∼ ΘG [1]M} = tr {M∼ G[1]∼ G[1]M} = tr {N∼ N} . The result now follows from (14) by using Lemma 1 in Hara et al. (2007) with f [z] = 2 · tr {N∼ ΦG [z]ΦG [1]∼ N} .
Theorem 1 gives a closed-form characterization of the optimal performance, as measured by J, and of the Youla parameter that defines a controller achieving such performance. As expected, our results make explicit the fact that perfect tracking constraints play a relevant role in the best achievable performance (recall the definition of N in (5)). It is also worth noting that J opt depends on the inner factor of the plant, which has as zeros the NMP zeros of it. That is, the best achievable performance is related to the NMP plant zeros, which is consistent with previous results (Toker et al. (2002); Silva and Salgado (2005)). The role of plant inversion (see Goodwin et al. (2001)) is also apparent when one realizes that the optimal parameter Qopt [z] equals the “best possible” stable and proper plant inverse under the considered assumptions. This is more clearly seen if one analyzes two extreme cases: Corollary 1. Consider the setup and the assumptions of Theorem 1. Then:
(2) If ` = m, then Qopt [z] in (9) reduces to h i ¯ Qopt [z] = Q[z] 0 ,
m×m ¯ , ΘG [z]−1 ΘG [1]G11 [1]−1 ∈ RH∞ where Q .
Proof. (1) If ` = 0, then N = G[1]G[1]† = ΦG [1]ΦG [1]∼ . Replacing (18) in (9), we have (16). (2) If ` = m, then the matrix N is given by h i N = G[1] G11 [1]−1 0 .
(18)
Since G[1] = ΦG [1]ΘG [1], (17) follows from (9).
When ` = 0, no perfect tracking is enforced in any output channel. In this case, Qopt [z] equals the inverse of the plant outer factor, times ΦG [1]∼ . Since the plant outer factor corresponds to its right invertible factor, and the DC-gain of Qopt [z] is the Moore-Penrose pseudoinverse of the plant DC-gain, we see that the structure of Qopt [z] is that of the “best possible” stable and proper plant inverse foreshadowed above. On the other hand, when ` = m, perfect tracking is sought in the first m output channels. In this case, the outer plant factor is again inverted, but the additional factor corresponds to the only one that guarantees perfect tracking in the first m channels. 4.2 Single-input multiple-output plants In this section, we specialize our results to SIMO plants (hence m = 1 and ` ∈ {0, 1}). In particular, we provide closed-form expressions for the optimal performance J opt in terms of the dynamic features of the plant. Our main result is stated next: Theorem 2. Consider a plant G[z] satisfying Assumption 1 with m = 1, and partition it as T
G[z] = [G1 [z] G2 [z] . . . Gn [z]] . Denote by rd {Gi [z]} and by ηij the relative degree and the j th finite NMP zero of Gi [z], respectively, with i = 1, . . . , n and j = 1, . . . , nci . Then: (1) If ` = 0 and ||G[1]|| = 6 0, then opt opt opt (19) J = J1opt , J1z + J1p , where nci n 2 2 X X Gi [1] |ηij | −1 opt J1z , (20) 2 rd {Gi [z]}+ 2 ||G[1]|| i=1 j=1 |ηij −1| opt J1p ,
n X
G2i [1]
× 2π ||G[1]||2 # ! |Gi [1]| G ejω 1 log dω . (21) ||G[1]|| |Gi [ejω ]| 1−cos ω −π
"Zi=1 π
(2) If ` = 1 and Assumption 2 holds, then
opt
(1) If ` = 0, then Q [z] in (9) reduces to Qopt [z] = ΘG [z]−1 ΦG [1]∼ .
(17)
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J opt = J2opt , where J1opt is as in (19).
2
||G[1]|| opt J , G21 [1] 1
(22)
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
best achievable performance. It is worth noting that the NMP zeros of each element of G[z] (i.e., of each Gi [z]) affect J opt , and not only the NMP zeros of G[z].
Proof. (1) If ` = 0, then n o 0 J1opt = − tr N∼ ΦG [1]ΦG [1]∼ N n 0 o = − tr ΦG [1]ΦG [1]∼ ,
(23)
where we used the trace properties and the fact that NN∼ = ΦG [1]ΦG [1]∼ . Consider the partition T ΦG [z] , [φ1 [z] φ2 [z] . . . φn [z]] . Then, 0 n n X X 0 φi [1] opt 2 J1 = − φi [1]φi [1] = − φi [1] . (24) φi [1] i=1 i=1
Given that ΦG [1] = G[1]ΘG [1]−1 , and that ΘG [z] is scalar, we have G2 [1] G2i [1] φ2i [1] = 2i = (25) 2. ΘG [1] ||G[1]||
Using Lemma 2 in Hara et al. (2007) (see also Section 3.2 in Hara et al. (2007)), and since the NMP zeros of φi [z] are the same as those of Gi [z], we have 0 nci 2 X φi [1] |ηij | − 1 = − rd {Gi [z]} − 2 φi [1] j=1 |ηij − 1| Z φi [1] 1 π 1 − log dω, 2π −π φi [ejω ] 1 − cos ω |Gi [ejω ]| |Gi [ejω ]| where φi [ejω ] = |Θ[z]G | = ||G[ejω ]|| . Thus,
0 nci X φi [1] |ηij |2 − 1 = − rd {Gi [z]} − 2 φi [1] j=1 |ηij − 1| ! Z 1 π |Gi [1]| G ejω 1 − log dω. jω 2π −π ||G[1]|| |Gi [e ]| 1 − cos ω (26) Using (25) and (26) in (24), we get (19). (2) If ` = 1, then (recall that the plant G[z] is SIMO) i 1 h N= (27) G[1] 0 . G1 [1] Therefore, (" # ) h i 0 −1 G[1]∼ opt J2 = 2 tr ΦG [1]ΦG [1]∼ G[1] 0 G1 [1] 0 n o 0 −1 = 2 tr ΘG [1]∼ ΦG [1]∼ ΦG [1]ΘG [1] . G1 [1] (28) Since ΘG [z] is scalar, then o ΘG [1]∼ ΘG [1] n ∼ 0 J2opt = − tr Φ [1] Φ [1] G G G21 [1] (29) 2 n o 0 ||G[1]|| ∼ =− tr Φ [1]Φ [1] , G G G21 [1] where we used the fact that ΘG [1]∼ ΘG [1] = G[1]∼ G[1] = ||G[1]||2 . Given (23), (22) follows from (29).
Theorem 2 shows, in tune to previous work by Chen et al. (2002); Hara et al. (2007), that in both the constrained (` = 1) and unconstrained (` = 0) SIMO case, the plant DC-gain and frequency response, and the NMP zeros of each scalar component of G[z], play a relevant role on the
Since ||G[1]||2 = G21 [1] + . . . + G2n [1], it follows from Theorem 2 that J2opt > J1opt . We thus see that the optimal cost is smaller when no perfect tracking constraints are imposed, i.e., when ` = 0. This is natural since, when ` = 1, the optimization problem is equal to the one that arises when ` = 0, but with an additional constraint. It is up to the designer to decide whether or not it is convenient to trade off perfect tracking in one channel for smaller plant output deviations from the corresponding opt best possible steady-state value y∞ . 5. AN EXAMPLE To illustrate the results presented in this paper, we consider the actuator deficient multiple piezoactuator system studied in Brinkerhoff and Devasia (2000). That system considers, as an input, the voltage applied to the actuator and, as outputs, the displacement of the endpoint of the actuator ye , and the displacement of the midpoint of the actuator ym . Given that the system is described in continuous time, a discrete-time model must be obtained. In this case, a zero-order hold and a sample time of 30[µs] are used to obtain " the model # " # G11 [z] Ye [z] = U [z], (30) Ym [z] G21 [z] where Nle [z] Nlm [z] G11 [z] , , G21 [z] , , D[z] D[z] Nle [z] , −0.051835(z − 1.9024)(z + 0.7772) ×(z − 0.1357)(z 2 − 1.823z + 0.9482) ×(z 2 − 1.553z + 0.9341)(z 2 − 1.271z + 1.045),
and
Nlm [z] , −0.052039(z − 1.2077)(z − 0.7124) ×(z + 0.5237)(z 2 − 1.762z + 0.909) ×(z 2 − 1.424z + 1.027)(z 2 − 1.414z + 1.731), D[z] , (z 2 −1.975z +0.9925)(z 2 −1.788z +0.925) ×(z 2 + 0.3466z + 0.2753)(z 2 − 1.389z + 0.8923) ×(z 2 − 1.195z + 0.9435).
Since the system has one input and two outputs, Theorem 2 can be used. It can be seen that G11 [z] has three finite NMP zeros (at z = 1.9024 and z = 0.6355 ± 0.8008j), one zero at infinity and that G11 [1] = 1.8136. On the other hand, G21 [z] has five finite NMP zeros (at z = 1.2077, z = 0.7070 ± 1.1097j and z = 0.7122 ± 0.7208j), one zero at infinity and G21 [1] = 0.3789. We calculate the best achievable performance bound J opt in two cases: Case 1: ` = 0 (i.e., integration is not forced in any channel). In this case, 0.9582 0.2002 N= 0.2002 0.0418
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˜ opt [z] [0.9789 0.2045], where and Qopt [z] = Q 2 2 ˜ opt [z] , (z −1.975z +0.9925)(z +0.3466z +0.2753) Q 2 z(z +0.5941)(z −0.974z +0.2631) 6.06(z 2 −1.389z +0.8923)(z 2 −1.195z +0.9435) × . (z 2 −1.311z +0.7408)(z 2 −1.206z +0.8477)
A direct calculation shows that J opt = J1opt = 5.5384. This value can be broken down into two parts: the effect of the NMP zeros of the components Gi [z] of opt the system, J1z , and the contribution of the plant opt frequency response, J1p . The first term is given by opt opt J1z = 4.6878, and the second one by J1p = 0.8506. Case 2: ` = 1 (i.e., it is specified that ye [k] tends asymptotically to its set-point). In this case, 1 0 ˜ opt [z] [1.0216 0] , N= and Qopt [z] = Q 0.2089 0 ˜ opt [z] is as in the previous case. where Q In this case, J opt = J2opt = 5.7801. The contribution 2 of the NMP zeros of Gi [z] is given by ||G[1]|| J opt = G211 [1] 1z 4.8924 and that of the plant frequency response by ||G[1]||2 opt J = 0.8877, respectively. G2 [1] 1z 11
In both Case 1 and 2, the high value of J opt is mainly due to the NMP zeros, while the contribution of the plant frequency response has a minor impact. It is seen that, for the considered plant, the performance deterioration arising from the perfect tracking constraint is rather mild. This is 2 due to the fact that ||G[1]|| ≈ 1.0436 G211 [1]. 6. CONCLUSIONS In this paper, we have computed performance bounds for plants with more outputs than inputs, and arbitrary step references. The approach is based on a quadratic criterion, that cumulates the squared deviations of the system output with respect to the nearest vector to the reference direction. That nearest vector is obtained by minimizing the Euclidean norm of the difference between the steady-state plant output and the reference vector, subject to having zero steady-state errors in a subset of the outputs. For SIMO plants, closed-form expressions for the performance bounds are obtained. These bounds depend on the zeros and the DC-gain of the scalar transfer functions that form the plant, and also on the plant frequency response. These results extend existing results to the case when the direction of the reference does not lie in the space spanned by the columns of the plant DC gain. Consistent with intuition, our results show that forcing zero steady-state errors on one channel in a SIMO plant, has an additional performance cost. Future work should include the study if Theorem 1 is valid for systems with more inputs than outputs that are non-right-invertible, the extension of Theorem 2 to tall systems having more than one input, to the unstable plant case, and also to cases that include a penalty on the control effort in the cost function. REFERENCES Brinkerhoff, R. and Devasia, S. (2000). Output tracking for actuator deficient/redundant systems: Multiple
piezoactuator example. Journal of Guidance, Control, and Dynamics, 23(2), 370–373. Chen, G., Chen, J., and Middleton, R. (2002). Optimal tracking performance for SIMO systems. IEEE Transactions on Automatic Control, 47(10), 1770–1775. Chen, J., Hara, S., and Chen, G. (2003). Best tracking and regulation performance under control energy constraint. IEEE Transactions on Automatic Control, 48(8), 1320– 1336. Chen, J., Qiu, L., and Toker, O. (2000). Limitations on maximal tracking accuracy. IEEE Transactions on Automatic Control, 45(2), 326–331. Chen, J., Ren, Z., Hara, S., and Qiu, L. (2001). Optimal tracking performance: preview control and exponential signals. IEEE Transactions on Automatic Control, 46, 1647–1654. Francis, B.A. (1987). A course in H∞ Control Theory. Berlin New York : Springer-Verlag. Freudenberg, J. and Middleton, R. (1998). Properties of single input, two output feedback systems. Proceedings of the 1998 American Control Conference, 4, 2055–2060. Philadelphia, PA. Freudenberg, J. and Middleton, R. (1999). Feedback systems with an almost rank deficient plant. Proceedings of the 1999 American Control Conference, 1, 409–413. San Diego, CA. Goodwin, G., Graebe, S., and Salgado, M. (2001). Control System Design. Prentice Hall. Hara, S., Bakhtiar, T., and Kanno, M. (2007). The Best Achievable H2 Tracking Performance for SIMO Feedback Control Systems. Journal of Control Science and Engineering. Article ID 93904, 12 pages, 2007. Litrico, X. (2002). Robust IMC flow control of SIMO dam-river open-channel systems. IEEE Transactions on Control Systems Technology, 10(3), 432–437. Middleton, R. and Goodwin, G.C. (1990). Digital Control and Estimation. A Unified Approach. Prentice Hall, Englewood Cliffs, N.J. Morse, N., Smith, R., Paden, B., and Antaki, J. (1998). Position sensed and self-sensing magnetic bearing configurations and associated robustness limitations. Proceedings of the 37th IEEE Conference on Decision and Control, 3, 2599–2604. Tampa, FL. Silva, E. and Salgado, M. (2005). Performance bounds for feedback control of nonminimum-phase MIMO systems with arbitrary delay structure. IEE Proceedings Control Theory and Applications, 152(2), 211–219. Su, W., Qiu, L., and Chen, J. (2003). Fundamental performance limitations in tracking sinusoidal signals. IEEE Transactions on Automatic Control, 48(8), 1371– 1380. Toker, O., Chen, J., and Qiu, L. (2002). Tracking performance limitations in LTI multivariable discrete-time systems. IEEE Transactions on Circuits and Systems I, 49(5), 657–670. Woodyatt, A., Freudenberg, J., and Middleton, R. (2001). An integral constraint for single input two output feedback systems. Automatica, 37(11), 1717–1726. Woodyatt, A., Seron, M., Freudenberg, J., and Middleton, R. (2002). Cheap control tracking performance for non-right-invertible systems. International Journal of Robust and Nonlinear Control, 12(15), 1253–1273.
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On tracking performance limits for tall systems ? Mat´ıas A. Garc´ıa ∗ Eduardo I. Silva ∗ Mario E. Salgado ∗ ∗
Departamento de Electr´ onica, Universidad T´ecnica Federico Santa Mar´ıa, Casilla 110-V, Valpara´ıso, Chile (e-mail: [email protected]).
Abstract: In this paper, a methodology to compute achievable performance bounds for stable, discrete-time tall MIMO systems is proposed. The methodology is based on a performance index, which cumulates the squared deviations of the system output with respect to the nearest vector to a constant reference direction. The results include expressions for the optimal value of this performance index and characterize the controllers that achieve such performance. For SIMO plants, closed form expressions depending on the dynamic features of the plant are also obtained. An example is presented to illustrate the results. Keywords: Linear multivariable systems, Optimal control theory, Analytic design 1. INTRODUCTION The study of performance bounds allows one to build benchmarks against which a given design strategy can be compared to. These bounds are calculated by considering a specific control architecture and a class of controllers and reference (or disturbance) signals. Most results consider a 2-norm based performance index that penalizes the tracking error (see, e.g., Chen et al. (2000); Toker et al. (2002); Chen et al. (2003); Silva and Salgado (2005)). This paper uses a quadratic index to measure performance. Performance bounds for MIMO continuous-time systems with more inputs than outputs, considering step references and one or two degree-of-freedom feedback loops, can be found in Chen et al. (2000). A conclusion of that work is that the best performance is limited not only by nonminimum phase (NMP) zeros, unstable poles and time delays, but also by their directions relative to the reference. The discrete-time counterpart of these results was investigated by Toker et al. (2002), where one degree-offreedom control loops with ramp and sinusoidal references were considered. Silva and Salgado (2005) extended Toker et al. (2002) to the case of MIMO systems with arbitrary delay structure, when step references are considered. Other extensions can be found in Chen et al. (2001); Su et al. (2003); Chen et al. (2003). The latter work shows that, when a quadratic term penalizing the control effort is added to the standard tracking error energy term, not only NMP zeros, unstable poles and delays limit the achievable performance, but also the plant frequency response. This paper focuses on tall systems, i.e., systems with more outputs than inputs. Thus, the conclusions of the above works, which apply to systems with fewer outputs than inputs, are not necessarily valid in our case. ? The authors acknowledge the support received from UTFSM and from CONICYT through grants ACT53, FONDECYT 1100692, and through the Advanced Human Capital Program.
978-3-902661-93-7/11/$20.00 © 2011 IFAC
Tall plants arise every time that fewer control signals than measurement signals are available. Some practical cases include the control of dams (Litrico (2002)), situations involving piezoactuators (Brinkerhoff and Devasia (2000)), magnetic bearing systems (Morse et al. (1998)), etc. One of the first works studying performance limitations for tall plants was carried out by Freudenberg and Middleton (1998). In that work, algebraic design trade-offs for singleinput two-output (SITO) plants are studied using the idea of plant-controller alignment. The same approach was then used by Freudenberg and Middleton (1999) to analyze almost singular two-input and two-output (TITO) systems. An integral (i.e., Bode-like) limitation for SITO systems was obtained by Woodyatt et al. (2001) using a special coordinate transformation that allowed one to focus on those references that the system can track without steady-state error. Such transformation was also used by Woodyatt et al. (2002) to study the cheap control of tall systems. In Chen et al. (2002), the best tracking performance for continuous-time single-input multiple-output (SIMO) plants is studied for reference vectors aligned with the plant DC-gain (i.e., with the matrix transfer function evaluated at zero frequency). In other words, the results by Chen et al. (2002) apply only to step references that can be tracked without steady-state error by the considered SIMO plants. Hara et al. (2007) study performance limits for both continuous and discrete-time SIMO plants, drawing connections between both cases by means of the delta transformation (Middleton and Goodwin (1990)). In that work, both the tracking error energy and the control energy is penalized assumingthe step reference vector is restricted to a limited set. The results by Hara et al. (2007) show, consistent with the work by Chen et al. (2003), that the plant frequency response, and the unstable poles and NMP zeros, impose performance constraints. All work on tall plants reported above considers reference
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
signals that can be tracked with no error. However, in practice, it is usually the case that the references cannot be tracked without error and a compromise must be sought. In those situations, the standard 2-norm based performance index must be modified to account for the fact that the tracking error will not necessarily converge to zero. In this paper, we address this issue by penalizing the deviations of the plant output y with respect to its best opt steady-state value y∞ (in a sense to be defined later). More precisely, we consider the cost function ∞ X T opt opt Jυ , y∞ − y[k] y∞ − y[k] , (1) k=0
where the subscript υ indicates the dependence of the cost function respect to the constant reference vector υ . If the plant were right-invertible at zero frequency, or if the reference were aligned with the plant DC-gain in the opt tall plant case, then one could choose y∞ equal to the reference and the cost function given in (1) would reduce to the usual one (see, e.g., Toker et al. (2002)).
The main contribution of this paper is a closed-form expression for the optimal performance, as measured by an averaged version of the cost function given in (1). We also characterize the controllers that achieve such performance. When particularized to the SIMO case, our results show that the optimal performance depends explicitly on the dynamic features of the plant, including the DC-gain and the NMP zeros of the scalar components of it, and also on the plant frequency response. The results reported in this paper exploit techniques used by Chen et al. (2002) and Hara et al. (2007), but consider different assumptions on the reference direction. For SIMO plants, we show that, as intuition suggests, forcing perfect tracking on one output channel is more expensive than allowing for steady-state errors in all outputs. The remainder of the paper is organized as follows: Section 2 introduces notation. Section 3 defines the problem to be studied in the paper. Section 4 presents the main results of the paper. An example is included in Section 5, while in Section 6 conclusions are drawn. 2. NOTATION We use normalface for scalars and boldface for vector and matrices. If x is a complex scalar, then its magnitude is denoted by |x|; (·)T (resp. (·)H ) denotes the transpose (resp. conjugate transpose) of (·); tr {·} stands for trace, † and (·) for Moore-Penrose pseudoinverse. The identity matrix of dimensions n × n is denoted by In . Given a sequence r , {r[k], k = 0, 1, ...}, R[z] denotes its Z√ H transform. For a complex vector x, ||x|| , x x.
We denote by Rn×m (resp. Rn×m ) the set of all real p rational (resp. proper real rational) transfer matrices with n outputs and m inputs. When no confusion arises we omit the n × m superscript. If G[z] ∈ R, then G[z]∼ , T G z −1 . L2 is defined as the Hilbert space of all matrix functions measurable on the unit disc, with the usual inner product (Toker et al. (2002)). The norm induced by this product is known as 2-norm, and it is denoted by ||·||2 . H2 and H2⊥ are subspaces of L2 containing functions
e[k]
r[k]
u[k] C[z]
G[z]
y[k]
+
-
Fig. 1. Standard negative-feedback one degree-of-freedom control loop which are analytic for |z| > 1 and |z| < 1, respectively. Both subspaces are orthogonal and any G[z] ∈ L2 can be written as G[z] = (G[z])H2 + (G[z])H⊥ , where (G[z])H2 2 and (G[z])H⊥ denote the projections of G[z] over H2 and 2 H2⊥ respectively. RH∞ is the class of proper rational stable transfer functions. Given a matrix G[z] ∈ R, we define its poles and zeros as usual (Goodwin et al. (2001)). A zero at z = c is said to be a non-minimum phase (NMP) zero if and only if |c| > 1. (This implies that the zeros at infinity are NMP zeros too.) Any matrix E[z] ∈ R is said to be inner if and only if E[z] ∈ RH∞ and E[z]∼ E[z] = I. Thus, if E[z] is inner, then ||E[z]G[z]||22 = ||G[z]||22 for any G[z] ∈ L2 . A transfer matrix in R is said to be outer if it belongs to RH∞ and has a right inverse which is analytic for |z| > 1. Consider n×m G[z] ∈ RH∞ , n > m, with full normal rank. G[z] can be n×m factorized as G[z] = ΦG [z]ΘG [z], where ΦG [z] ∈ RH∞ m×m is inner and ΘG [z] ∈ RH∞ is outer. This factorization is known as inner-outer factorization (Francis (1987)). Clearly, G[z]∼ G[z] = ΘG [z]∼ ΘG [z]. If, in addition, G[z] has no zeros at z = 1, then G[1]† = ΘG [1]−1 ΦG [1]∼ . Given any transfer matrix G[z] ∈ Rp , controlled via a standard negative-feedback one degree-of-freedom control loop (see Fig. 1), where C[z] is the controller, it is well-known that all sensitivity functions associated with stabilizing controllers can be parameterized using the Youla parameterization (see, e.g., Goodwin et al. (2001)). 3. PROBLEM FORMULATION In this section, we state the control problem to be dealt with in this paper. We focus on the one degree-of-freedom feedback loop of Fig. 1, where G[z] is a tall LTI plant, C[z] is an LTI controller, r the reference signal, u is the control input, y is the plant output, and e is the tracking error. We will work under the following assumptions: Assumption 1. G[z] ∈ Rn×m , with n ≥ m, is strictly p proper, stable, has full normal rank, and has no zeros on the unit circle (i.e., for |z| = 1). 22 Our assumptions are mostly standard, except for the fact that we assume the plant to be stable. The unstable case is left out due to space constraints. In general, a tall plant cannot perfectly track an arbitrary step reference vector υ (unless υ belongs to the space spanned by G[1]). Hence, the standard performance measures which penalize the tracking error energy are not well-defined and have to be modified accordingly (cf. Toker et al. (2002), Silva and Salgado (2005) and Hara et al. (2007)). In this paper, we assume, without loss of generality, that prefect tracking has to be achieved in the first ` output channels (` ∈ {0, · · · , m}). We thus propose to use the cost function defined in (1), where
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opt y∞ ,
arg min y∞ =G[1]u∞ [I` 0](y∞ − υ )=0 u∞ ∈Rm
||y∞ − υ || .
opt The vector y∞ corresponds to the steady-state plant output value that is closest (in an Euclidean sense) to the reference υ , when zero steady-state tracking error is achieved in the first ` output channels. We thus see that the functional Jυ in (1) penalizes the deviations of the plant output with respect to the best possible steady-state opt output vector y∞ .
Partition G[z] as
G[z] =
G11 [z] G12 [z] G21 [z] G22 [z]
(3)
,
where G11 [z] ∈ R`×` , G12 [z] ∈ R`×(m−`) , G21 [z] ∈ opt R(n−`)×` , G22 [z] ∈ R(n−`)×(m−`) . To guarantee that y∞ is well-defined for an arbitrary reference direction υ , we introduce the following additional assumption: Assumption 2. If ` ≥ 1, then G11 [1] is non singular. 22 The problem in (2) is a least square problem with linear restrictions. By solving it, it follows that opt υ, y∞ = Nυ (4) where N , G[1]M, (5) the blocks of M are given by M11 , G11 [1]−1 I` +G12[1]H† G21 [1]G11 [1]−1 −1
M12 , −G11 [1]
G12 [1]H
4. OPTIMAL PERFORMANCE
(2)
This section presents the main results of this paper. We will begin by considering an arbitrary tall plant, and then we will particularize our results to the SIMO case. 4.1 General tall plants The next theorem provides a characterization of the optimal performance J opt and the optimal parameter Qopt [z], when the plant G[z] has arbitrary dimensions: Theorem 1. Consider a plant G[z] satisfying Assumptions 1 and 2. If ` ∈ {0, · · · , m} is given, then: (1) The optimal parameter Qopt [z] defined in (8) is
Qopt [z] = ΘG [z]−1 ΦG [1]∼ N. (9) opt (2) The optimal value of J, i.e., J in (8), is given by n o opt ∼ 0 J = − tr N ΦG [1]ΦG [1]∼ N , (10) 0
where ΦG [1] is the derivative of ΦG [z] respect to z, evaluated at z = 1.
Proof. (1) Pre-multiply the argument of the 2-norm of (7) by the unitary matrix (see, e.g., Chen et al. (2002)) ΦG [z]∼ Λ[z] = ∼ . In − ΦG [z]ΦG [z] Thus,
2 N ∼ J = (In − ΦG [z]ΦG [z] ) z − 1 2 ( ) (ΦG [z]∼ N−ΘG [z]Q[z]) 2 . + inf Q∈S z−1 2 (11) ∼ (ΦG [z]∼ −ΦG [1]∼ )N G [z]Q[z]) Since (ΦG [1] N−Θ ∈ H and 2 z−1 z−1 ∈ H2⊥ , we have that 2 N J opt = (In − ΦG [z]ΦG [z]∼ ) z − 1 2 (ΦG [z]∼ − ΦG [1]∼ ) N 2 (12) + z−1 2 (ΦG [1]∼ N−ΘG[z]Q[z]) 2 . + inf Q∈S z−1
†
opt
M21 , −H† G21 [1]G11 [1]−1
M22 , H† , and
H , G22 [1] − G21 [1]G11 [1]−1 G12 [1]. Using (4), Parseval’s Theorem and the Youla parameterization for stable plants (see, e.g., (Goodwin et al., 2001, Chapter 25)), (1) can be written as υ 2 zυ Jυ = (N − G[z]Q[z]) , (6) z − 1 2
where Q[z] is a parameter to be chosen in RH∞ . It is clear that the (optimal) value of Jυ depends on υ . In order to obtain an average performance measure which is independent of υ , we will assume that υ is an random υ } = 0 and E υ υ T = In . 1 The vector in Rn , with E {υ expected value of Jυ in (6) is thus given by 2 z J , E {Jυ } = (N − G[z]Q[z]) . (7) z − 1
2
Given Assumptions 1 and 2, we conclude from (12) that Qopt [z] is as in (9). (We note that Qopt [z] belongs to S. Indeed, the properties of the inner-outer factorization of G[z] and the definition of N given in (5) yields Q[1] = ΘG [1]−1 ΦG [1]∼ G[1]M = M.)
2
Define the set S , {Q ∈ RH∞ : Q[1] = M} and also J opt , inf J, Q[z]∈S
Qopt [z] , arg inf J.
(8)
Q[z]∈S
We note that Q[z] ∈ S is necessary and sufficient for J to be bounded.
opt
The remainder of this paper focuses in the optimization problem defined in (8). In particular, we will give a closedform expression for J opt and Qopt [z]. 1
E(·) denotes expectation.
(2) From (12) and (9), we conclude that J opt is given by (In − ΦG [z]ΦG [z]∼ ) N 2 opt J = z−1 2 (ΦG [z]∼ − ΦG [1]∼ ) N 2 + z−1 2 Z π ∼ 1 tr {N (In + ΦG [1]ΦG [1]∼ ) N} = dω 4π −π (1 − cos ω)
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
− tr N∼ ΦG [1]ΦG [ejω ]∼ N dω (1 − cos ω) −π ∼ Z π − tr N ΦG [ejω ]ΦG [1]∼ N 1 + dω, 4π −π (1 − cos ω) (13) where we used the definition of the 2-norm. Since 2 Re ΦG [ejω ]ΦG [1]∼ = ΦG [1]ΦG [ejω ]∼ 1 + 4π
Z
π
+ ΦG [ejω ]ΦG [1]∼ ,
(13) can be written as Z −1 π − tr {N∼ (In + ΦG [1]ΦG [1]∼ ) N} J opt = dω 4π −π (1 − cos ω) Z π 2 · tr N∼ Re ΦG [ejω ]ΦG [1]∼ N 1 − dω. 4π −π (1 − cos ω) (14) The expression above is similar to the one considered in Lemma 1 in Hara et al. (2007). Define fR [z] , 2 · tr {N∼ (Re {ΦG [z]ΦG [1]∼ }) N} ,
fR1 , tr {N∼ (In + ΦG [1]ΦG [1]∼ ) N} . If fR [1] = fR1 , then we could use the Lemma 1 in Hara et al. (2007). Since fR [1] = 2 · tr {N∼ (Re {ΦG [1]ΦG [1]∼ }) N} (15) = 2 · tr {N∼ (ΦG [1]ΦG [1]∼ ) N} , fR [1] would be equal to fR1 if tr {N∼ N} = tr {N∼ (ΦG [1]ΦG [1]∼ ) N}. To show that this is indeed the case, we use the definition of N given in (5): tr {N∼ ΦG [1]ΦG [1]∼ N} = tr {M∼ G[1]∼ ΦG [1]ΦG [1]∼ G[1]M} = tr {M∼ ΘG [1]∼ ΘG [1]M} = tr {M∼ G[1]∼ G[1]M} = tr {N∼ N} . The result now follows from (14) by using Lemma 1 in Hara et al. (2007) with f [z] = 2 · tr {N∼ ΦG [z]ΦG [1]∼ N} .
Theorem 1 gives a closed-form characterization of the optimal performance, as measured by J, and of the Youla parameter that defines a controller achieving such performance. As expected, our results make explicit the fact that perfect tracking constraints play a relevant role in the best achievable performance (recall the definition of N in (5)). It is also worth noting that J opt depends on the inner factor of the plant, which has as zeros the NMP zeros of it. That is, the best achievable performance is related to the NMP plant zeros, which is consistent with previous results (Toker et al. (2002); Silva and Salgado (2005)). The role of plant inversion (see Goodwin et al. (2001)) is also apparent when one realizes that the optimal parameter Qopt [z] equals the “best possible” stable and proper plant inverse under the considered assumptions. This is more clearly seen if one analyzes two extreme cases: Corollary 1. Consider the setup and the assumptions of Theorem 1. Then:
(2) If ` = m, then Qopt [z] in (9) reduces to h i ¯ Qopt [z] = Q[z] 0 ,
m×m ¯ , ΘG [z]−1 ΘG [1]G11 [1]−1 ∈ RH∞ where Q .
Proof. (1) If ` = 0, then N = G[1]G[1]† = ΦG [1]ΦG [1]∼ . Replacing (18) in (9), we have (16). (2) If ` = m, then the matrix N is given by h i N = G[1] G11 [1]−1 0 .
(18)
Since G[1] = ΦG [1]ΘG [1], (17) follows from (9).
When ` = 0, no perfect tracking is enforced in any output channel. In this case, Qopt [z] equals the inverse of the plant outer factor, times ΦG [1]∼ . Since the plant outer factor corresponds to its right invertible factor, and the DC-gain of Qopt [z] is the Moore-Penrose pseudoinverse of the plant DC-gain, we see that the structure of Qopt [z] is that of the “best possible” stable and proper plant inverse foreshadowed above. On the other hand, when ` = m, perfect tracking is sought in the first m output channels. In this case, the outer plant factor is again inverted, but the additional factor corresponds to the only one that guarantees perfect tracking in the first m channels. 4.2 Single-input multiple-output plants In this section, we specialize our results to SIMO plants (hence m = 1 and ` ∈ {0, 1}). In particular, we provide closed-form expressions for the optimal performance J opt in terms of the dynamic features of the plant. Our main result is stated next: Theorem 2. Consider a plant G[z] satisfying Assumption 1 with m = 1, and partition it as T
G[z] = [G1 [z] G2 [z] . . . Gn [z]] . Denote by rd {Gi [z]} and by ηij the relative degree and the j th finite NMP zero of Gi [z], respectively, with i = 1, . . . , n and j = 1, . . . , nci . Then: (1) If ` = 0 and ||G[1]|| = 6 0, then opt opt opt (19) J = J1opt , J1z + J1p , where nci n 2 2 X X Gi [1] |ηij | −1 opt J1z , (20) 2 rd {Gi [z]}+ 2 ||G[1]|| i=1 j=1 |ηij −1| opt J1p ,
n X
G2i [1]
× 2π ||G[1]||2 # ! |Gi [1]| G ejω 1 log dω . (21) ||G[1]|| |Gi [ejω ]| 1−cos ω −π
"Zi=1 π
(2) If ` = 1 and Assumption 2 holds, then
opt
(1) If ` = 0, then Q [z] in (9) reduces to Qopt [z] = ΘG [z]−1 ΦG [1]∼ .
(17)
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J opt = J2opt , where J1opt is as in (19).
2
||G[1]|| opt J , G21 [1] 1
(22)
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
best achievable performance. It is worth noting that the NMP zeros of each element of G[z] (i.e., of each Gi [z]) affect J opt , and not only the NMP zeros of G[z].
Proof. (1) If ` = 0, then n o 0 J1opt = − tr N∼ ΦG [1]ΦG [1]∼ N n 0 o = − tr ΦG [1]ΦG [1]∼ ,
(23)
where we used the trace properties and the fact that NN∼ = ΦG [1]ΦG [1]∼ . Consider the partition T ΦG [z] , [φ1 [z] φ2 [z] . . . φn [z]] . Then, 0 n n X X 0 φi [1] opt 2 J1 = − φi [1]φi [1] = − φi [1] . (24) φi [1] i=1 i=1
Given that ΦG [1] = G[1]ΘG [1]−1 , and that ΘG [z] is scalar, we have G2 [1] G2i [1] φ2i [1] = 2i = (25) 2. ΘG [1] ||G[1]||
Using Lemma 2 in Hara et al. (2007) (see also Section 3.2 in Hara et al. (2007)), and since the NMP zeros of φi [z] are the same as those of Gi [z], we have 0 nci 2 X φi [1] |ηij | − 1 = − rd {Gi [z]} − 2 φi [1] j=1 |ηij − 1| Z φi [1] 1 π 1 − log dω, 2π −π φi [ejω ] 1 − cos ω |Gi [ejω ]| |Gi [ejω ]| where φi [ejω ] = |Θ[z]G | = ||G[ejω ]|| . Thus,
0 nci X φi [1] |ηij |2 − 1 = − rd {Gi [z]} − 2 φi [1] j=1 |ηij − 1| ! Z 1 π |Gi [1]| G ejω 1 − log dω. jω 2π −π ||G[1]|| |Gi [e ]| 1 − cos ω (26) Using (25) and (26) in (24), we get (19). (2) If ` = 1, then (recall that the plant G[z] is SIMO) i 1 h N= (27) G[1] 0 . G1 [1] Therefore, (" # ) h i 0 −1 G[1]∼ opt J2 = 2 tr ΦG [1]ΦG [1]∼ G[1] 0 G1 [1] 0 n o 0 −1 = 2 tr ΘG [1]∼ ΦG [1]∼ ΦG [1]ΘG [1] . G1 [1] (28) Since ΘG [z] is scalar, then o ΘG [1]∼ ΘG [1] n ∼ 0 J2opt = − tr Φ [1] Φ [1] G G G21 [1] (29) 2 n o 0 ||G[1]|| ∼ =− tr Φ [1]Φ [1] , G G G21 [1] where we used the fact that ΘG [1]∼ ΘG [1] = G[1]∼ G[1] = ||G[1]||2 . Given (23), (22) follows from (29).
Theorem 2 shows, in tune to previous work by Chen et al. (2002); Hara et al. (2007), that in both the constrained (` = 1) and unconstrained (` = 0) SIMO case, the plant DC-gain and frequency response, and the NMP zeros of each scalar component of G[z], play a relevant role on the
Since ||G[1]||2 = G21 [1] + . . . + G2n [1], it follows from Theorem 2 that J2opt > J1opt . We thus see that the optimal cost is smaller when no perfect tracking constraints are imposed, i.e., when ` = 0. This is natural since, when ` = 1, the optimization problem is equal to the one that arises when ` = 0, but with an additional constraint. It is up to the designer to decide whether or not it is convenient to trade off perfect tracking in one channel for smaller plant output deviations from the corresponding opt best possible steady-state value y∞ . 5. AN EXAMPLE To illustrate the results presented in this paper, we consider the actuator deficient multiple piezoactuator system studied in Brinkerhoff and Devasia (2000). That system considers, as an input, the voltage applied to the actuator and, as outputs, the displacement of the endpoint of the actuator ye , and the displacement of the midpoint of the actuator ym . Given that the system is described in continuous time, a discrete-time model must be obtained. In this case, a zero-order hold and a sample time of 30[µs] are used to obtain " the model # " # G11 [z] Ye [z] = U [z], (30) Ym [z] G21 [z] where Nle [z] Nlm [z] G11 [z] , , G21 [z] , , D[z] D[z] Nle [z] , −0.051835(z − 1.9024)(z + 0.7772) ×(z − 0.1357)(z 2 − 1.823z + 0.9482) ×(z 2 − 1.553z + 0.9341)(z 2 − 1.271z + 1.045),
and
Nlm [z] , −0.052039(z − 1.2077)(z − 0.7124) ×(z + 0.5237)(z 2 − 1.762z + 0.909) ×(z 2 − 1.424z + 1.027)(z 2 − 1.414z + 1.731), D[z] , (z 2 −1.975z +0.9925)(z 2 −1.788z +0.925) ×(z 2 + 0.3466z + 0.2753)(z 2 − 1.389z + 0.8923) ×(z 2 − 1.195z + 0.9435).
Since the system has one input and two outputs, Theorem 2 can be used. It can be seen that G11 [z] has three finite NMP zeros (at z = 1.9024 and z = 0.6355 ± 0.8008j), one zero at infinity and that G11 [1] = 1.8136. On the other hand, G21 [z] has five finite NMP zeros (at z = 1.2077, z = 0.7070 ± 1.1097j and z = 0.7122 ± 0.7208j), one zero at infinity and G21 [1] = 0.3789. We calculate the best achievable performance bound J opt in two cases: Case 1: ` = 0 (i.e., integration is not forced in any channel). In this case, 0.9582 0.2002 N= 0.2002 0.0418
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
˜ opt [z] [0.9789 0.2045], where and Qopt [z] = Q 2 2 ˜ opt [z] , (z −1.975z +0.9925)(z +0.3466z +0.2753) Q 2 z(z +0.5941)(z −0.974z +0.2631) 6.06(z 2 −1.389z +0.8923)(z 2 −1.195z +0.9435) × . (z 2 −1.311z +0.7408)(z 2 −1.206z +0.8477)
A direct calculation shows that J opt = J1opt = 5.5384. This value can be broken down into two parts: the effect of the NMP zeros of the components Gi [z] of opt the system, J1z , and the contribution of the plant opt frequency response, J1p . The first term is given by opt opt J1z = 4.6878, and the second one by J1p = 0.8506. Case 2: ` = 1 (i.e., it is specified that ye [k] tends asymptotically to its set-point). In this case, 1 0 ˜ opt [z] [1.0216 0] , N= and Qopt [z] = Q 0.2089 0 ˜ opt [z] is as in the previous case. where Q In this case, J opt = J2opt = 5.7801. The contribution 2 of the NMP zeros of Gi [z] is given by ||G[1]|| J opt = G211 [1] 1z 4.8924 and that of the plant frequency response by ||G[1]||2 opt J = 0.8877, respectively. G2 [1] 1z 11
In both Case 1 and 2, the high value of J opt is mainly due to the NMP zeros, while the contribution of the plant frequency response has a minor impact. It is seen that, for the considered plant, the performance deterioration arising from the perfect tracking constraint is rather mild. This is 2 due to the fact that ||G[1]|| ≈ 1.0436 G211 [1]. 6. CONCLUSIONS In this paper, we have computed performance bounds for plants with more outputs than inputs, and arbitrary step references. The approach is based on a quadratic criterion, that cumulates the squared deviations of the system output with respect to the nearest vector to the reference direction. That nearest vector is obtained by minimizing the Euclidean norm of the difference between the steady-state plant output and the reference vector, subject to having zero steady-state errors in a subset of the outputs. For SIMO plants, closed-form expressions for the performance bounds are obtained. These bounds depend on the zeros and the DC-gain of the scalar transfer functions that form the plant, and also on the plant frequency response. These results extend existing results to the case when the direction of the reference does not lie in the space spanned by the columns of the plant DC gain. Consistent with intuition, our results show that forcing zero steady-state errors on one channel in a SIMO plant, has an additional performance cost. Future work should include the study if Theorem 1 is valid for systems with more inputs than outputs that are non-right-invertible, the extension of Theorem 2 to tall systems having more than one input, to the unstable plant case, and also to cases that include a penalty on the control effort in the cost function. REFERENCES Brinkerhoff, R. and Devasia, S. (2000). Output tracking for actuator deficient/redundant systems: Multiple
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