Discrete Time

Frequency-Domain Performance Analysis for Multi-Rate Sampled-Data Sys- tern

2.3.2 Discrete Time

In an analgous manner to the continuous-time case, one can specify performance re- quirements based on the H2- and H,-norm of the discrete-time closed-loop operator.

Discrete-time performance analysis should be adequate for continuous-time systems if sampling time is chosen to be insignificant relative to the closed-loop bandwidth

and appropriate &anti-aliasingn of the measurements are performed [3,50].

H2

Performance Measure for SR Discrete-Tie Systems

Suppose that M A is a discrete-time closed-loop operator for a single-rate (SR) discrete-time system where all measurements are available a t every time unit. Con- sider a hypothetical experiment where a unit impulse is injected to each input channel one by one. The H2-norm of MA measures the sum of the squared .t2-norm of the output vectors el(k). In other words,

where 6; is the discrete-time unit pulse in the ith channel. Again, by appropriate normalization of el, we can define the following two performance objectives:

1. The nominal performance is achieved if

where

Mmm = Wp(G22

+

^G23K(I

^-

G33K)-lG32)%

2. The robust performance is achieved if

As for the continuous-time case, there is no method to test the condition (2.59) at present.

Hz

Performance Measure for MR Discrete-Time Systems

When the measurements are sampled a t multiple rates, the resulting MA is a periodi- cally shift-varying (PSV) system with the period of 78. Hence, 6; entering the system at t = (k, j ) and t = (m,n) lead to different outputs MA&; if j

#

n. However, the periodicity of the system ensures that the outputs are the same for k

#

^rn as long as j = n. The natural way of extending the definition of Hz-norm to MR systems is as follows:

1 ^{N-1 dim{wl) w}

( M ~ [ 6 i l e ( t ) ) ~ ~ A [ 6 & ( t ) (2.60) e=o ^{i=1 t=(O,O)}

where [Silt denotes unit impulse entering at ith channel at t = (0, l). Note that the above definition reduces to the standard Hz-norm definition if the system is shift- invariant. In addition, the resulting H2-norm for a PTV system is independent of the choice of time zero. The nominal performance and robust performance can be defined in exactly the same way as for SR systems.

H ,

Performance Measure for SR Discrete-Time Systems

Consider all input signals d' such that llBllLz

<

1. AS a performance objective, we may want to minimize the "worst-possible" 12-norm of the output e'(k). This "worst possible"

e2

norm of e'(k) is the H,-norm of MA. As before, we can define the following two objectives:

1. The nominal performance is achieved if

2. The robust performance is achieved if

The condition (2.62) can be tested through the following condition on the Structured Singular Value:

max I I M A ~ ~ ~

<

1 (2.63)

P A E P ~ .

if and only if

1. Nominal Stability:

M,, is stable.

2. Structured Singular Value Condition:

hl,

Performance Measure for MW Discrete-The Systems

For systems where one or more measurements are available only at every integer- multiple rs, the resulting MA is a shift-varying operator. Let us define the H,-norm for a PSV operator as its induced &-norm. Clearly, the induced norm does not depend on the choice of time zero even for PSV operators. The main difficulty with the H, performance analysis of a MR system, however, lies in that the system is shift-varying and the pulse transfer function representation of the closed-loop system does not exist (in terms of time unit T ~ ) . Hence, the frequency-domain techniques just described cannot be applied to MR systems straightforwardly. As for the sampled-data systems in the continuous-time domain, the samplers can either be approximated as shift- invariant operators or be bounded using conic sectors.

Frequency-Domain Performance Analysis for MR Discrete-Tie Systems

Suppose the system is represented as a discrete-time system of TS, which is a common divider of all sampling times. Consider a continuous-time signal y&. The fourier transform of the signal sampled at every 7s is

If the sampling time of y& is N1rS, then the fourier transform of the sampled-signal

Now let ym(k) be a general discrete-time signal where each time unit represents the time interval of 7s and denote its sampled signal sampled a t every N time unit as (ym(k))bl. From the above discussion, it is apparent that the fourier transform of (ym(k))kl (expressed as impulse trains) can be expressed as

where 2{ym (k) } represents

Cr=o

Ym (k)$. Assuming that the inputs to the samplers

f l 7r

are band-limited ( i . e . , 2{ym(k)}

l,=&,.,

_{0 for}

fi <

^w

<

^;),

1 7r

'{(~m)b

}lz=eju,

⁼

^-

_NI ^Z{ym(k)) l z = , p T s for

o 5

w

5 -

_{Ni 7s (2.69)}

Typical discrete-time multi-rate sampled-data systems are represented schemat- cally in Figure 2.8. Under the assumption of band-limitedness of the signals, the above discussion implies that each sampler may be replaced with l / N i where N; is its respective sampling time expressed in terms of the discrete time unit, 7s. Similar arguments can be made to the samplers within the controller as well. The samplers

within the controller may be replaced by N I / N o where NI and No are the sampling intervals (in terms of T ~for the input and output signals of the sampler respectively )

(see Figure 2.8). A more conservative approach is to represent these samplers as LSI operator plus the norm-bounded LSV block as in the conic sector approach for the continuous time system. Details of this method will not be discussed in this thesis.