We have remarked earlier (see the footnote on page 122) that a time-in- variant system has a diagonal modulated transfer matrix. Hence, there is a clear connection between having a time-invariant system as a design objective and achieving diagonal decoupling for a MIMO time-invariant system. The latter problem was studied in §4.3.4 of Chapter 4, where it was shown that there is usually a sensitivity cost associated with diago- nal decoupling. We will see next that the same is true for the problem of having a time-invariant closed loop for periodic systems.
Consider the periodic feedback system of Figure 5.1, which, for simplic- ity, is assumed to be a SISO system. Assume that, inter-alia, the control objective is to achieve a time-invariant map from the reference to the out- put. This implies that, in the frequency domain raised system, ¯T(ejθ)and S(e¯ jθ)are required to be block diagonal. We then have the following re- sult.
Theorem 5.4.1.Letq=rqejθq,rq> 1, be a zero of the plant ¯G, and letΨ∈
Nn,Ψ6=0, be its output direction. Assume that the modulated sensitivity function, ¯S, is proper and stable, and such that ¯S(ejθ)is diagonal. Then, for each indexkinIΨ,
1 2π
Zπ
−π
log|S¯kk(ejθ)| r2q−1
1−2rqcos(θ−θq) +r2qdθ≥0 , (5.22) or equivalently
1 2π
Zπ
−π
log|S¯11(ejθ)| r2q−1
1−2rqcos(θ−θq+ (k−1)θN) +r2qdθ≥0 , (5.23) for each indexkinIΨ.
Proof. The constraint (5.22) is immediate from (5.19), on noting that, if S(e¯ jθ)is diagonal, then ¯Sik(ejθ) =0fori6=k.
For the constraint (5.23), note that ¯Skk(ejθ) =S¯11(ej(θ−(k−1)θN)), which follows from Proposition 5.1.1. We then obtain, using (5.22), that
1 2π
Zπ
−π
log|S¯11(ej(θ−(k−1)θN))| r2q−1
1−2rqcos(θ−θq) +r2qdθ≥0 . Making the change of variablesφ=θ−(k−1)θN, and invoking periodicity of the integrand, we finally arrive at the constraint (5.23).
Note that ¯S11(ejθ)is the frequency response of the error to the refer- ence signal, i.e.,E(ejθ) =S¯11(ejθ)R(ejθ), and there are no terms involving shifted functions ofR(ejθ)because it is assumed that the closed loop has a time-invariant input-output map.
128 5. Extensions to Periodic Systems The inequality (5.23) implies that ¯S11 satisfies as many integral con- straints as there are nonzero elements of the zero directionΨ(i.e., for each kinIΨ). Moreover, in each of this integrals, the weighting function
Wq(θ+ (k−1)θN), r2q−1
1−2rqcos(θ−θq+ (k−1)θN) +r2q (5.24) is shifted in angle for different values ofk, and thus, their corresponding maximum values are located at different angles.
The impact of this constraint is more evident if we assume also that
|S¯11(ejθ)|is intended to be reduced in some setΘ1 ⊂(−π, π), i.e.,
|S¯11(ejθ)|≤α < 1 , ∀θ∈Θ1 ,[−θ1, θ1], θ1< π . (5.25) Let the weighted length of the intervalΘ1be defined, for eachkinIΨ, as (cf. (3.50) in Chapter 3)
Θkq(θ1), Zθ1
−θ1
Wq(θ+ (k−1)θN)dθ . Then, from (5.23), we obtain
[2π−Θkq(θ1)]logkS¯11k +Θkq(θ1)logα≥0 , ∀k∈IΨ, which is equivalent to
kS¯11k ≥ 1
α
Θkq(θ1) 2π−Θkq(θ1)
, ∀k∈IΨ. (5.26) Note that the value ofΘkq(θ1)(for somek∈IΨ) will, in general, be large if the frequency rangeΘ1where sensitivity reduction is required includes the maximum value of the corresponding weighting functionWqin (5.24).
Therefore, in view of (5.26), peaks in|S¯11(ejθ)|are more likely to be large when sensitivity reduction is required over ranges that include the maxi- mum value of some of the different weighting functions.
We then note that this situation is, in general, worse for periodic systems than it is for time-invariant systems. Indeed, in the latter case, we need consider only one weighting function for each nonminimum phase zero rather than as many as there are nonzero elements in the zero direction.
We also note that the situation becomes potentially worse as the number of elements inIΨincreases or as the dimension of the null space associated with the zero grows (see §4.3.4 in Chapter 4).
We conclude from the previous analysis that, for periodic systems, the problem of sensitivity reduction with the additional requirement that the closed-loop map from reference to output be time-invariant is very likely to result, in general, in large peaks for|S(e¯ jθ)|.
5.4 Design Interpretations 129 As an illustration of the above results we study a simple situation where it will turn out that there is no extra sensitivity cost associated with having a time-invariant closed-loop map.
Example 5.4.1. Consider a SISO plant described by xk+1 =Axk+Bkuk,
yk=Cxk+Dkuk , (5.27)
whereAand Care constant matrices, and{Bk}and{Dk}areN-periodic sequences. Assume further that the very special condition holds that there exists a periodic scalar gainKk6=0,∀k, such that
BkKk=B , and DkKk =D ,
whereB andD are constant. It is then clear that one can achieve time- invariance by simply redefining the input. We will use our previous anal- ysis to show that, in this particular case, there is no penalty in achieving time-invariance for the closed-loop system.
Let ¯Gbe as in (5.7), i.e., ¯Gis the modulated transfer matrix associated with the plant (5.27). Similarly, let ¯Kbe the modulated transfer matrix as- sociated with the periodic gainKk. Using the relationship (5.9) between time and frequency domain raising, it is easy to see that ¯Kis given by5
K¯ =Ws−1
K0 0 . . . 0 0 K1 . . . ...
... ... ... ...
0 . . . KN−1
Ws,
and we see that ¯Kis nonsingular sinceKk6=0for allk.
LetNGD−1G be a right coprime factorization of ¯G. We will show that the output zero directions ofNGare canonical, i.e., for each zeroqofNG, there is a basis for the left null space ofNG(q)constructed of vectors having only one nonzero element.
We first note that there are no unstable pole-zero cancelations in ¯GK,¯ and, clearly NG(K¯−1DG)−1 is a right coprime factor of ¯GK. Moreover,¯ sinceKkwas chosen so thatBkKkandDkKkare constant, it is clear that G¯K¯ is diagonal. Because of this, it is possible to build a right coprime frac- tion ¯ND¯−1=G¯K¯ where ¯Nand ¯Dare diagonal. We then conclude that6
NG=NR ,¯ (5.28)
5The time domain raised transfer matrix corresponding to the scalar gain is a diago- nal matrix having the elements in its diagonal (see Feuer and Goodwin (1996)).
6This is because any two coprime factorizations are related by units of the ring of proper and stable transfer matrices, i.e., biproper and bistable matrices (Vidyasagar, 1985, Theo- rem 4.1.43, p.75).
130 5. Extensions to Periodic Systems whereRis biproper and bistable.
Since ¯Nis diagonal, it is straightforward to build a canonical basis for the left null space of ¯N(q)and, because of (5.28), this basis will also be a canonical basis for the left null space ofNG(q).
LetΨ be a vector in the left null space ofNG(q); it follows thatΨ is canonical, i.e.,IΨhas only one nonzero element. Hence, there is only one weighting function of the form (5.24), which implies that there is only one integral constraint (5.23) associated with the zero with directionΨ. This situation is no worse than that corresponding to a time-invariant system.
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